theory of machines_static and dynamic force analysis
TRANSCRIPT
Static and Dynamic Force Analysis
Prepared by -Prof K N WakchaureSRESCollege of Engineering KopargaonSavitribai Phule PUNE UNIVERSITY
Theory of Machines I
2Force
In physics a force is any interaction which tends to change the motion of an object In other words a force can cause an object with mass to change its velocity (which includes
to begin moving from a state of rest) ie to accelerate Force can also be described by intuitive concepts such as a push or a pull A force has both magnitude and direction making it a vector quantity It is measured in the
SI unit of newtons and represented by the symbol F The original form of Newtons second law states that the net force acting upon an object is
equal to the rate at which its momentum changes with time If the mass of the object is constant this law implies that the acceleration of an object is
directly proportional to the net force acting on the object is in the direction of the net force and is inversely proportional to the mass of the object
As a formula this is expressed as Related concepts to force include thrust which increases the velocity of an object drag
which decreases the velocity of an object and torque which produces changes in rotational speed of an object In an extended body each part usually applies forces on the adjacent parts the distribution of such forces through the body is the so-called mechanical stress
Pressure is a simple type of stress Stress usually causes deformation of solid materials or flow in fluids
Aristotle famously described a force as anything that causes an object to undergo unnatural motion
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
3Types of Forces
There are different types of forces that act in different ways on structures such as bridges chairs buildings in fact any structure
The main examples of forces are shown below Study the diagram and text and then draw a diagrampictogram to represent each of these forces
A Static Load A good example of this is a person seen on the left He is holding a stack of books on his back but he is not moving The force downwards is STATIC
A Dynamic Load A good example of a dynamic load is the person on the right He is carrying a weight of books but walking The force is moving or DYNAMIC
Internal Resistance The person in the diagram is sat on the mono-bicycle and the air filled tyre is under great pressure The air pressure inside it pushes back against hisher weight
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
2Force
In physics a force is any interaction which tends to change the motion of an object In other words a force can cause an object with mass to change its velocity (which includes
to begin moving from a state of rest) ie to accelerate Force can also be described by intuitive concepts such as a push or a pull A force has both magnitude and direction making it a vector quantity It is measured in the
SI unit of newtons and represented by the symbol F The original form of Newtons second law states that the net force acting upon an object is
equal to the rate at which its momentum changes with time If the mass of the object is constant this law implies that the acceleration of an object is
directly proportional to the net force acting on the object is in the direction of the net force and is inversely proportional to the mass of the object
As a formula this is expressed as Related concepts to force include thrust which increases the velocity of an object drag
which decreases the velocity of an object and torque which produces changes in rotational speed of an object In an extended body each part usually applies forces on the adjacent parts the distribution of such forces through the body is the so-called mechanical stress
Pressure is a simple type of stress Stress usually causes deformation of solid materials or flow in fluids
Aristotle famously described a force as anything that causes an object to undergo unnatural motion
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
3Types of Forces
There are different types of forces that act in different ways on structures such as bridges chairs buildings in fact any structure
The main examples of forces are shown below Study the diagram and text and then draw a diagrampictogram to represent each of these forces
A Static Load A good example of this is a person seen on the left He is holding a stack of books on his back but he is not moving The force downwards is STATIC
A Dynamic Load A good example of a dynamic load is the person on the right He is carrying a weight of books but walking The force is moving or DYNAMIC
Internal Resistance The person in the diagram is sat on the mono-bicycle and the air filled tyre is under great pressure The air pressure inside it pushes back against hisher weight
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
3Types of Forces
There are different types of forces that act in different ways on structures such as bridges chairs buildings in fact any structure
The main examples of forces are shown below Study the diagram and text and then draw a diagrampictogram to represent each of these forces
A Static Load A good example of this is a person seen on the left He is holding a stack of books on his back but he is not moving The force downwards is STATIC
A Dynamic Load A good example of a dynamic load is the person on the right He is carrying a weight of books but walking The force is moving or DYNAMIC
Internal Resistance The person in the diagram is sat on the mono-bicycle and the air filled tyre is under great pressure The air pressure inside it pushes back against hisher weight
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
4Types of Forces
Tension The rope is in ldquotensionrdquo as the two people pull on it This stretching puts the rope in tension
Compression The weight lifter finds that his body is compressed by the weights he is holding above his head
Shear Force A good example of shear force is seen with a simple scissors The
two handles put force in different directions on the pin that holds the two parts together The force applied to the pin is called shear force
Torsion The plastic ruler is twisted between both hands The ruler is said to be in a state of torsion
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
5Laws of Motion
Newtons laws of motion are three physical laws that together laid the foundation for classical mechanics They describe the relationship between a body and the forces acting upon it and its motion in response to said forces They have been expressed in several different ways over nearly three centuries and can be summarised as follows
The three laws of motion were first compiled by Isaac Newton in his Philosophiaelig Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) first published in 1687
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
First law When viewed in an inertial reference frame an object either remains at rest or continues to move at a constant velocity unless acted upon by an external force
Second law The vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration vector a of the object F = ma
Third law When one body exerts a force on a second body the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body
6Moment of Inertia
Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation
Moment of inertia depends on the shape of the body and may be different around different axes of rotation A larger moment of inertia around a given axis requires more torque to increase the rotation or to stop the rotation of a body about that axis
Moment of inertia depends on the amount and distribution of its mass and can be found through the sum of moments of inertia of the masses making up the whole object under the same conditions
For example if ma + mb = mc then Ia + Ib = Ic In classical mechanics moment of inertia may also be called mass moment of inertia
rotational inertia polar moment of inertia or the angular mass When a body is rotating around an axis a torque must be applied to change its
angular momentum The amount of torque needed for any given change in angular momentum is proportional to the size of that change
Moment of inertia may be expressed in terms of kilogram-square metres (kgmiddotm2) in SI units and pound-square feet (lbmmiddotft2) in imperial or US units
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
7Moment of Inertia
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Four objects racing down a plane while rolling without slipping From back to front spherical shell (red) solid sphere (orange) cylindrical ring (green) and solid cylinder (blue) The time for each object to reach the finishing line depends on their moment of inertia
8Moment of Inertia
Suppose a body of mass m is made to rotate about an axis z passing through the bodys center of mass
The body has a moment of inertia Icm with respect to this axis The parallel axis theorem states that if the body is made to rotate instead about a new axis zprime which is parallel to the first axis and displaced from it by a distance d then the moment of inertia I with respect to the new axis is related to Icm by
Explicitly d is the perpendicular distance between the axes z and zprime The parallel axis theorem can be applied with the stretch rule and
perpendicular axis theorem to find moments of inertia for a variety of shapes
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
9Moment of Inertia
The second moment of area also known as moment of inertia of plane area area moment of inertia or second area moment is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis
The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane Its unit of dimension is length to fourth power L4
In the field of structural engineering the second moment of area of the cross-section of a beam is an important property used in the calculation of the beams deflection and the calculation of stress caused by a moment applied to the beam
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
10
6Moment of Inertia
Moment of inertia is the mass property of a rigid body that determines the torque needed for a desired angular acceleration about an axis of rotation
Moment of inertia depends on the shape of the body and may be different around different axes of rotation A larger moment of inertia around a given axis requires more torque to increase the rotation or to stop the rotation of a body about that axis
Moment of inertia depends on the amount and distribution of its mass and can be found through the sum of moments of inertia of the masses making up the whole object under the same conditions
For example if ma + mb = mc then Ia + Ib = Ic In classical mechanics moment of inertia may also be called mass moment of inertia
rotational inertia polar moment of inertia or the angular mass When a body is rotating around an axis a torque must be applied to change its
angular momentum The amount of torque needed for any given change in angular momentum is proportional to the size of that change
Moment of inertia may be expressed in terms of kilogram-square metres (kgmiddotm2) in SI units and pound-square feet (lbmmiddotft2) in imperial or US units
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
7Moment of Inertia
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Four objects racing down a plane while rolling without slipping From back to front spherical shell (red) solid sphere (orange) cylindrical ring (green) and solid cylinder (blue) The time for each object to reach the finishing line depends on their moment of inertia
8Moment of Inertia
Suppose a body of mass m is made to rotate about an axis z passing through the bodys center of mass
The body has a moment of inertia Icm with respect to this axis The parallel axis theorem states that if the body is made to rotate instead about a new axis zprime which is parallel to the first axis and displaced from it by a distance d then the moment of inertia I with respect to the new axis is related to Icm by
Explicitly d is the perpendicular distance between the axes z and zprime The parallel axis theorem can be applied with the stretch rule and
perpendicular axis theorem to find moments of inertia for a variety of shapes
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
9Moment of Inertia
The second moment of area also known as moment of inertia of plane area area moment of inertia or second area moment is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis
The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane Its unit of dimension is length to fourth power L4
In the field of structural engineering the second moment of area of the cross-section of a beam is an important property used in the calculation of the beams deflection and the calculation of stress caused by a moment applied to the beam
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
10
7Moment of Inertia
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Four objects racing down a plane while rolling without slipping From back to front spherical shell (red) solid sphere (orange) cylindrical ring (green) and solid cylinder (blue) The time for each object to reach the finishing line depends on their moment of inertia
8Moment of Inertia
Suppose a body of mass m is made to rotate about an axis z passing through the bodys center of mass
The body has a moment of inertia Icm with respect to this axis The parallel axis theorem states that if the body is made to rotate instead about a new axis zprime which is parallel to the first axis and displaced from it by a distance d then the moment of inertia I with respect to the new axis is related to Icm by
Explicitly d is the perpendicular distance between the axes z and zprime The parallel axis theorem can be applied with the stretch rule and
perpendicular axis theorem to find moments of inertia for a variety of shapes
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
9Moment of Inertia
The second moment of area also known as moment of inertia of plane area area moment of inertia or second area moment is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis
The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane Its unit of dimension is length to fourth power L4
In the field of structural engineering the second moment of area of the cross-section of a beam is an important property used in the calculation of the beams deflection and the calculation of stress caused by a moment applied to the beam
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
10
8Moment of Inertia
Suppose a body of mass m is made to rotate about an axis z passing through the bodys center of mass
The body has a moment of inertia Icm with respect to this axis The parallel axis theorem states that if the body is made to rotate instead about a new axis zprime which is parallel to the first axis and displaced from it by a distance d then the moment of inertia I with respect to the new axis is related to Icm by
Explicitly d is the perpendicular distance between the axes z and zprime The parallel axis theorem can be applied with the stretch rule and
perpendicular axis theorem to find moments of inertia for a variety of shapes
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
9Moment of Inertia
The second moment of area also known as moment of inertia of plane area area moment of inertia or second area moment is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis
The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane Its unit of dimension is length to fourth power L4
In the field of structural engineering the second moment of area of the cross-section of a beam is an important property used in the calculation of the beams deflection and the calculation of stress caused by a moment applied to the beam
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
10
9Moment of Inertia
The second moment of area also known as moment of inertia of plane area area moment of inertia or second area moment is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis
The second moment of area is typically denoted with either an for an axis that lies in the plane or with a for an axis perpendicular to the plane Its unit of dimension is length to fourth power L4
In the field of structural engineering the second moment of area of the cross-section of a beam is an important property used in the calculation of the beams deflection and the calculation of stress caused by a moment applied to the beam
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
10
05012023PROF K N WAKCHAURE
10
11Simple Pendulum
A simple pendulum in its simplest form consists of heavy bob suspended at the end of a light inextensible and flexible string The other end of the string is fixed at O as shown in Fig
Let L = Length of the stringm = Mass of the bob in kgW = Weight of the bob in newtons = mg andθ = Angle through which the stringis displaced
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
Periodic time
Frequency of oscillation
05012023PROF K N WAKCHAURE
12Compound Pendulum
When a rigid body is suspended vertically and it oscillates with a small amplitude under the action of the force of gravity the body is known as compound pendulum as shown in Fig
Let m = Mass of the pendulum in kg W = Weight of the pendulum in newtons = mg k = Radius of gyration about an axis through the centre of gravity G and perpendicular to the plane of motion h = Distance of point of suspension O from the centre of gravity G of the body
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
13Compound Pendulum
If the pendulum is given a small angular displacement θ then the couple tending to restore the pendulum to the equilibrium position OAbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since θ is very small
mass moment of inertia about the axis of suspension O
there4 Angular acceleration of the pendulum
angular acceleration is directly proportional to angular displacement therefore the pendulum executes simple harmonic motion
We know that the periodic time
14Compound Pendulum
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that the periodic time
Frequency of oscillation
15Compound Pendulum
A small flywheel of mass 85 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 100 mm and the flywheel makes 100 oscillations in 145 seconds Find the moment of inertia of the flywheel through the centre of gravity
Given m = 85 kg h = 100 mm = 01 m Since the flywheel makes 100 oscillations in 145 seconds therefore frequency of
oscillation n = 100145 = 069 Hz
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 02061 mI =36 kg-m2
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
12Compound Pendulum
When a rigid body is suspended vertically and it oscillates with a small amplitude under the action of the force of gravity the body is known as compound pendulum as shown in Fig
Let m = Mass of the pendulum in kg W = Weight of the pendulum in newtons = mg k = Radius of gyration about an axis through the centre of gravity G and perpendicular to the plane of motion h = Distance of point of suspension O from the centre of gravity G of the body
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
13Compound Pendulum
If the pendulum is given a small angular displacement θ then the couple tending to restore the pendulum to the equilibrium position OAbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since θ is very small
mass moment of inertia about the axis of suspension O
there4 Angular acceleration of the pendulum
angular acceleration is directly proportional to angular displacement therefore the pendulum executes simple harmonic motion
We know that the periodic time
14Compound Pendulum
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that the periodic time
Frequency of oscillation
15Compound Pendulum
A small flywheel of mass 85 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 100 mm and the flywheel makes 100 oscillations in 145 seconds Find the moment of inertia of the flywheel through the centre of gravity
Given m = 85 kg h = 100 mm = 01 m Since the flywheel makes 100 oscillations in 145 seconds therefore frequency of
oscillation n = 100145 = 069 Hz
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 02061 mI =36 kg-m2
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
13Compound Pendulum
If the pendulum is given a small angular displacement θ then the couple tending to restore the pendulum to the equilibrium position OAbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since θ is very small
mass moment of inertia about the axis of suspension O
there4 Angular acceleration of the pendulum
angular acceleration is directly proportional to angular displacement therefore the pendulum executes simple harmonic motion
We know that the periodic time
14Compound Pendulum
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that the periodic time
Frequency of oscillation
15Compound Pendulum
A small flywheel of mass 85 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 100 mm and the flywheel makes 100 oscillations in 145 seconds Find the moment of inertia of the flywheel through the centre of gravity
Given m = 85 kg h = 100 mm = 01 m Since the flywheel makes 100 oscillations in 145 seconds therefore frequency of
oscillation n = 100145 = 069 Hz
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 02061 mI =36 kg-m2
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
14Compound Pendulum
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that the periodic time
Frequency of oscillation
15Compound Pendulum
A small flywheel of mass 85 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 100 mm and the flywheel makes 100 oscillations in 145 seconds Find the moment of inertia of the flywheel through the centre of gravity
Given m = 85 kg h = 100 mm = 01 m Since the flywheel makes 100 oscillations in 145 seconds therefore frequency of
oscillation n = 100145 = 069 Hz
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 02061 mI =36 kg-m2
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
15Compound Pendulum
A small flywheel of mass 85 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 100 mm and the flywheel makes 100 oscillations in 145 seconds Find the moment of inertia of the flywheel through the centre of gravity
Given m = 85 kg h = 100 mm = 01 m Since the flywheel makes 100 oscillations in 145 seconds therefore frequency of
oscillation n = 100145 = 069 Hz
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 02061 mI =36 kg-m2
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
16Compound Pendulum
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
Given h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
kG= 0377 m I =532 kg-m2
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
17
A connecting rod is suspended from a point 25 mm above the centre of small end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscillate the time period is found to be 187 seconds Find the mass moment of inertia when pendulum is located at the small end
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
18Compound Pendulum
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Frequency of oscillation
When the axis of oscillation coincides with the small end centre then frequency of oscillationn1 = 100190 = 0526 HzWhen the axis of oscillation coincides with the big end centre the frequency of oscillationn2 = 100165 = 0606 Hz
h1 = 0767 mkG=0316mI=60 times 01 = 6 kg-m2
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
19
The connecting rod of an oil engine has a mass of 60 kg the distance between the bearing centres is 1 metre When suspended vertically with a knife-edge through the small end it makes 100 oscillations in 190 seconds and with knife-edge through the big end it makes 100 oscillations in 165seconds Find the moment of inertia of the rod in kg-m2 and the distance of CG from the small end centre
Solution Given m = 60 kg h1 + h2 = 1 m
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
20Equivalent length of simple pendulum
Equations of periodic time of simple pendulum and compound pendulum are given belowbull Simple pendulum
bull Theory and analysis of Compound Pendulum
bull Concept of equivalent length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Periodic time of simple pedulum
Periodic time of compound pendulum
By comparing above equation we see that the equivalent length of a simple pendulum which gives the same frequency as compound pendulum is
equivalent length of simple pendulum (L) depends upon the distance between the point ofsuspension and the centre of gravity (G) therefore L can be changed by changing the position of point of suspension This will obviously change the periodic time of a compound pendulum The periodic time will be minimum if L is minimum
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
21 Bifilar Suspension
The moment of inertia of a body may be determined experimentally by an apparatus called bifilar suspension
The body whose moment of inertia is to be determined (say AB) is suspended bytwo long parallel flexible strings as shown in Fig
When the body is twisted through a small angle θ about a vertical axis through the centre of gravity G it will vibrate with simple harmonic motion in a horizontal plane
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
bull Let m=Mass of the bodybull W=Weight of the body in newtons = mgbull kG=Radius of gyration about an axis through the centre
of gravitybull I=Mass moment of inertia of the body about a vertical
axis through 2Gbull l=Length of each stringbull x=Distance of A from G (ie AG)bull y=Distance of B from G (ie BG)θ=Small angular
displacement of the body from the equilibrium position in the horizontal plane
bull φA and φB=Corresponding angular displacements of the strings
bull andα=Angular acceleration towards the equilibrium position
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
22 Bifilar Suspension
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
23 Bifilar Suspension
When the body is stationary the tension in the strings are given by
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
24 Bifilar Suspension
When the body is stationary the tension in the strings are given bybull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When the body is displaced from its equilibrium position in a horizontal plane through a small angle θ then the angular displacements of the strings are given by
Component of tension TA and TB in the horizontal plane
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
25 Bifilar Suspension
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
These components of tensions TA and TB are equal and opposite in direction which gives rise to a couple The couple or torque applied to each string to restore the body to its initial equilibrium position ie restoring torque
and accelerating (or disturbing) torque
for equilibrium condition Restoring torque = accelerating torque Hence
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
26 Bifilar Suspension bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravityGiven m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of oscillation n = 2040 = 05 Hz
kG=0107 mI= 0017 kg-m2
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
27
A small connecting rod of mass 15 kg is suspended in a horizontal plane by two wires 125 m long The wires are attached to the rod at points 120 mm on either side of the centre of gravity If the rod makes 20 oscillations in 40 seconds find the radius of gyration and the mass moment of inertia of the rod about a vertical axis through the centre of gravity
Given m = 15 kg l = 125 m x = y = 120 mm = 012 m Since the rod makes 20 oscillations in 40 s therefore frequency of
oscillation n = 2040 = 05 Hz
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
28Trifilar Suspension
Trifilar Suspension (Torsional Pendulum)It is also used to find the moment of inertia of a body experimentally The body (say a disc or flywheel) whose moment of inertia is to be determined is suspended by three long flexible wires A Band C as shown in Fig When the body is twisted about its axis through a small angle θ and then released it will oscillate with simple harmonic motion
Let m=Mass of the body in kg W=Weight of the body in newtons = mg kG=Radius of gyration about an axis through cg I=Mass moment of inertia of the disc about an axis through O and perpendicular to it = mk2 l=Length of each wire r=Distance of each wire from the axis of the disc θ=Small angular displacement of the disc φ=Corresponding angular displacement of the wires andα=Angular acceleration towards the equilibrium position
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
29
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
30Trifilar Suspension
For small angular deflection of disk
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
31Trifilar Suspension
For small angular deflection of disk bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Since the three wires are attached symmetrically with respect to the axis therefore the tension in each wire will be one-third of the weight of the bodythere4 Tension in each wire = mg3Component of the tension in each wire perpendicular to r
there4 Torque applied to each wire to restore the body to its initial equilibrium position ie restoring torque
Total restoring torque applied to three wires
disturbing torque
From above two equations
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
32Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
33Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
In order to find the radius of gyration of a car it is suspended with its axis vertical from three parallel wires 25 metres long The wires are attached to the rim at points spaced120deg apart and at equal distances 250 mm from the axis It is found that the wheel makes 50 torsional oscillations of small amplitude about its axis in170 seconds Find the radius of gyration of the wheelGiven l = 25 m r = 250 mm = 025 mn = 50170 = 517 Hz
kG = 0268 m = 268 mm
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
34Trifilar Suspension
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
We know that periodic time
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
kG = 0168 mI= 0198 kg-m2
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
35
A connecting rod of mass 55 kg is placed on a horizontal platform whose mass is 15 kg It is suspended by three equal wires each 125 m long from a rigid support The wires are equally spaced round the circumference of a circle of 125 mm radius When the cg of the connecting rod coincides with the axis of the circle the platform makes 10 angular oscillations in 30seconds Determine the mass moment of inertia about an axis through its cgGiven m1 = 55 kg m2 = 15 kg l = 125 m r = 125 mm = 0125 mn = 1030 = 13 Hz
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023
36Numericals
A small flywheel having mass 90 kg is suspended in a vertical plane as a compound pendulum The distance of centre of gravity from the knife edge support is 250 mm and the flywheel makes 50oscillations in 64 seconds Find the moment of inertia of the flywheel about an axis through the centre of gravity
[Ans 36 kg-m2] The connecting rod of a petrol engine has a mass 12 kg In order to find its moment of inertia it is suspended
from a horizontal edge which passes through small end and coincides with the small end centre It is made to swing in a vertical plane such that it makes 100 oscillations in 96 seconds If the point of suspension of the connecting rod is 170 mm from its cg find 1 radius of gyration about an axis through its cg 2 moment of inertia about an axis through its cg and 3 length of the equivalent simple pendulum
[Ans 101 mm 01224 kg-m2 023 m] A connecting rod of mass 40 kg is suspended vertically as a compound pendulum The distance between the
bearing centres is 800 mm The time for 60 oscillations is found to be 925 seconds when the axis of oscillation coincides with the small end centre and 884 seconds when it coincides with the big end centre Find the distance of the centre of gravity from the small end centre and the moment of inertia of the rod about an axis through the centre of gravity[Ans 0442 m 26 kg-m2]11The following data were obtained from an experiment to find the moment of inertia of a pulley by bifilar suspension Mass of the pulley = 12 kg Length of strings = 3 m Distance of strings on either side of centre of gravity = 150 mm Time for 20 oscillations about the vertical axis through cg = 468 seconds Calculate the moment of inertia of the pulley about the axis of rotation
[Ans 01226 kg-m2]
PROF K N WAKCHAURE
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
37Inertia force
The inertia force is an imaginary force which when acts upon a rigid body brings it in an equilibrium position It is numerically equal to the accelerating force in magnitude but opposite in direction
Mathematically Inertia force = ndash Accelerating force = ndash ma where m = Mass of the body and a=Linear acceleration of the centre of gravity of the body Similarly the inertia torque is an imaginary torque which when applied upon the
rigid body brings it in equilibrium position It is equal to the accelerating couple in magnitude but opposite in direction
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
38Analytical Method for Velocity and Acceleration of the Piston
Consider the motion of a crank and connecting rod of a reciprocating steam engine as shown in Fig
Let OC be the crank and PC the connecting rod
Let the crank rotates with angular velocity of ω rads and the crank turns through an angle θ from the inner dead centre (briefly written as IDC)
Let x be the displacement of a reciprocating body P from IDC after time t seconds during which the crank has turned through an angle θ
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Let l=Length of connecting rod between the centres r=Radius of crank or crank pin circle φ=Inclination of connecting rod to the line of stroke PO And n=Ratio of length of connecting rod to the radius of crank =
lr
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
39Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
40Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston From the geometry
From triangles CPQ and CQO CQ = l sin φ = r sin θ or lr = sin θsin φthere4 n = sin θsin φ or sin φ = sin θn
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
41Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Expanding the above expression by binomial theorem we get
Substituting the value of (1 ndash cos φ) in equation
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
42Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Velocity of the piston
Differentiating equation (iv) with respect to θ
there4 Velocity of P with respect to O or velocity of the piston P
Substituting the value of dxdθ from equation
Velocity of the piston
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
43Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Since the acceleration is the rate of change of velocity therefore acceleration of the piston P
Acceleration of the piston
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
44Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
45Analytical Method for Velocity and Acceleration of the Piston
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Acceleration of the piston
Velocity of the piston
Displacement of the piston
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
46Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular displacement of connecting rodConsider the motion of a connecting rod and a crank as shown in figure From the geometry of the figure we find that
CQ = l sin φ = r sin θ
Differentiating both sides with respect to time t
Since the angular velocity of the connecting rod PC is same as the angular velocity of point P with respect to C and is equal to dφdt therefore angular velocity of the connecting rod
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
47Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular velocity of connecting rod
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
48Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
αPC = Angular acceleration of P with respect to PC
differentiating equation of angular velocity of connecting rod
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
49Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular acceleration of connecting rod
Since is small as compared to therefore it may be neglected
unity is small as compared to n2 hence the term unity may be neglected
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
50Analytical Method for Velocity and Acceleration of the
connecting rod
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
51Formulae to be remember
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Acceleration of the piston
Velocity of the piston
Displacement of the piston
Angular velocity of connecting rod
Angular acceleration of connecting rod
Angular displacement of connecting rod
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
52Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
If the crank and the connecting rod are 300 mm and 1 m long respectively and the crank rotates at a constant speed of 200 rpm determine1 The crank angle at which the maximum velocity occurs and 2 Maximum velocity of the piston
Solution Given r = 300 mm = 03 m l = 1 m N = 200 rpm or ω = 2 π times 20060 = 2095 rads
n=lr = 103 = 333 For maximum velocity of the piston
Ansθ = 75ordmVpmax = = 654 ms
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
53Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The crank and connecting rod of a steam engine are 03 m and 15 m in length The crank rotates at 180 rpm clockwise Determine the velocity and acceleration of the piston when the crank is at 40 degrees from the inner dead centre position Also determine the position of the crank for zero acceleration of the piston
Given r = 03 l = 15 m N = 180 rpm or ω = π times 18060 = 1885 rads θ = 40deg n = lr = 1503 = 5
AnsVelocity of the piston =419msAcceleration of the piston =535ms2Position of the crank for zero acceleration of the piston ap=0θ1 = 7927deg or 28073deg
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
54Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are150 mm and 600 mm respectively The crank position is 60deg from inner dead centre The crank shaft speed is 450 rpm (clockwise) Using analytical method determine 1 Velocity and acceleration of the slider and 2 Angular velocity and angular acceleration of the connecting rod
Given r = 150 mm = 015 m l = 600 mm = 06 m θ = 60deg N = 400 rpm or ω = π times 45060 = 4713 rads
n = lr = 06015 = 4 Velocity of the slider =69ms acceleration of the slider= 12494 ms2 angular velocity of the connecting rod =59rads angular acceleration of the connecting rod =481 rads2
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
55Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a slider crank mechanism the length of the crank and connecting rod are 100 mm and 400 mm respectively The crank rotates uniformly at 600 rpm clockwise When the crank has turned through45deg from the inner dead centre find by analytical method 1 Velocity and acceleration of the slider2 Angular velocity and angular acceleration of the connecting rod
[Ans 52 ms 279 ms2 11 rads 698 rads2] A petrol engine has a stroke of 120 mm and connecting rod is 3 times the crank length The
crank rotates at 1500 rpm in clockwise direction Determine 1 Velocity and acceleration of the piston and 2 Angular velocity and angular acceleration of the connecting rod when the piston had travelled one-fourth of its stroke from IDC
[Ans 824 ms 1047 ms2 37 rads 5816 rads2] The stroke of a steam engine is 600 mm and the length of connecting rod is 15 m The crank
rotates at 180 rpm Determine 1 velocity and acceleration of the piston when crank has travelled through an angle of 40deg from inner dead centre and 2 the position of the crank for zero acceleration of the piston
[Ans 42 ms 854 ms2 793deg from IDC]
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
56Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The various forces acting on the reciprocating parts of a horizontal engine are shown in Fig The expressions for these forces neglecting the weight of the connecting rod may be derived as discussed below
Piston effort It is the net force acting on the piston or crosshead pin along the line of stroke It is denoted by FP in Fig
Let mR = Mass of the reciprocating parts eg piston crosshead pin or gudgeon pin etc in kg and WR = Weight of the reciprocating parts in newtons = mRg
We know that acceleration of the reciprocating parts
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
57Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Accelerating force or inertia force of the reciprocating parts
On the other hand the inertia force due to retardation of the reciprocating parts helps the force on the piston Therefore Piston effort
In a horizontal engine the reciprocating parts are accelerated from rest during the latter half of the stroke (ie when the piston moves from inner dead centre to outer dead centre)
It is then retarded during the latter half of the stroke (ie when the piston moves from outer dead centre to inner dead centre) The inertia force due to the acceleration of the reciprocating parts opposes the force on the piston due to of pressures in the cylinder on the two sides of the piston
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
58Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In a double acting reciprocating steam engine net load on the piston
where p1 A1=Pressure and cross-sectional area on the back end side of the pistonp2 A2=Pressure and cross-sectional area on the crank end side of the pistona=Cross-sectional area of the piston rodIf lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net load on the piston
In case of a vertical engine the weight of the reciprocating parts assists the piston effort during the downward stroke (ie when the piston moves from top dead centre to bottom dead centre) and opposes during the upward stroke of the piston (ie when the piston moves from bottom dead centre to top dead centre)there4 Piston effort
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
59Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Force acting along the connecting rod It is denoted by FQ in Fig From the geometry of the figure we find that
We know that
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
60Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Thrust on the sides of the cylinder walls or normal reaction on the guide bars It is denoted by FN in FigFrom the figure we find that
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
61Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The force acting on the connecting rod FQ may be resolved into two components bull one perpendicular to the crank bull other along the crank
The component of FQ perpendicular to the crank is known as crank-pin effort and it is denoted by FT in Fig
The component of FQ along the crank produces a thrust on the crank shaft bearings and it is denoted by FB in Fig
Resolving FQ perpendicular to the crank
Crank-pin effort and thrust on crank shaft bearings
and resolving FQ along the crank
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
62Dynamic force analysis of reciprocating engine mechanism
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Crank effort or turning moment or torque on the crank shaft The product of the crank-pin effort (FT) and the crank pin radius (r) is known as crank effort or turning moment or torque on the crank shaft
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
63Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
Consider a rigid body having its centre of gravity at G as Let m = Mass of the bodykG = Radius of gyration about its centre of gravity Gm1 and m2 = Two masses which form adynamical equivalent systeml1 = Distance of mass m1 from Gl2 = Distance of mass m2 from GL = Total distance between the masses m1 and m2
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
64Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
In order to determine the motion of a rigid body under the action of external forces it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart in such a way that
1 the sum of their masses is equal to the total mass of the body
2 the centre of gravity of the two masses coincides with that of the body
3 the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body
When these three conditions are satisfied then it is said to be an equivalent dynamical system
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
65Equivalent Dynamical System
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
From equations (i) and (ii)
Substituting the value of m1 and m2 in equation (iii) we have
This equation gives the essential condition of placing the two masses so that the system becomes dynamical equivalent The distance of one of the masses (ie either l1 or l2) is arbitrary chosen and the other distance is obtained from above equation
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
66Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
The connecting rod of a gasoline engine is 300 mm long between its centres It has a mass of 15 kg and mass moment of inertia of 7000 kg-mm2 Its centre of gravity is at 200 mm from its small end centre Determine the dynamical equivalent two-mass system of the connecting rod if one of the masses is located at the small end centre
l = 300 mm m = 15 kg I = 7000 kg-mm2l1 = 200 mm
m1 = Mass placed at the small end centre and m2 = Mass placed at a distance l2 from the centre of gravity G
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
67Numericals
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod is suspended from a point 25 mm above the centre ofsmall end and 650 mm above its centre of gravity its mass being 375 kg When permitted to oscil-late the time period is found to be 187 seconds Find the dynamical equivalent system constituted of two masses one of which is located at the small end centre
h = 650 mm = 065 m l1 = 650 ndash 25 = 625 mm= 0625 m m = 375 kg tp = 187 s
Frequency of oscillation
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
68Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
Consider two masses one at A and the other at D be placed arbitrarily as shown in Fig
Letl3=Distance of mass placed at D from GI1=New mass moment of inertia of the two massesk1=New radius of gyrationα=Angular acceleration of the bodyI=Mass moment of inertia of a dynamically equivalent systemk=Radius of gyration of a dynamically equivalent systemWe know that the torque required to accelerate the bodyT=Iα = m (k)2α Similarly the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
69Correction Couple to be Applied to Make Two Mass System
Dynamically Equivalent
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
when two masses are placed arbitrarily then the conditions (i) and (ii) of dynamic equivalent will only be satisfied But the condition (iii) is not possible to satisfy This means that the mass moment of inertia of these two masses placed arbitrarily will differ than that of mass moment of inertia of the rigid body
We know that the torque required to accelerate the bodyT=Iα = m (k)2 α(i) the torque required to accelerate the two-mass system placed arbitrarilyT1=I1α = m (k1)2 α(ii)
there4 Difference between the torques required to accelerate the two-mass system and the torque required to accelerate the rigid bodyT=T1ndashT = m (k1)2 α ndash m (k)2 α = m [(k1)2 ndash (k)2] α(iv)The difference of the torques T is known as correction couple This couple must be applied when the masses are placed arbitrarily to make the system dynamical equivalent This of course will satisfy the condition (iii)
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
70Correction Couple
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating enginesbull Two mass statically
and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
(analytical method only) bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
A connecting rod of an IC engine has a mass of 2 kg and the distance between the centre of gudgeon pin and centre of crank pin is 250 mm The CG falls at a point 100 mm from the gudgeon pin along the line of centres The radius of gyration about an axis through the CG perpendicular to the plane of rotation is 110 mm Find the equivalent dynamical system if only one of the masses is located at gudgeon pin If the connecting rod is replaced by two masses one at the gudgeon pin and the other at the crank pin and the angular acceleration of the rod is 23 000 rads2 clockwise determine the correction couple applied to the system to reduce it to a dynamically equivalent system
Equivalent dynamical systemIt is given that one of the masses is located at the gudgeon pin Let the other mass be located at a distance l2 from the centre of gravity We know that for an equivalent dynamical system
Since the connecting rod is replaced by two masses located at the two centres (ie one at the gudgeon pin and the other at the crank pin) therefore l = 01 m and l3 = l ndash l1 = 025 ndash 01 = 015 mLet k1 = New radius of gyration We know that (k1)2= l1l3 = 01 times 015 = 0015 m2
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
71Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
Inertia Forces in a Reciprocating Engine Considering the Weight of Connecting Rodmc=Mass of the connecting rodl=Length of the connecting rodl1=Length of the centre of gravity of the connecting rod from Pr= radius of crankS=2r =stroke of pistonƟ=crank angleⱷ= angle made by connecting rod ap= acceleration of pistonn=lr =obliquity ratioN= crank rotation in RPMω= angular velocity of crank radsαpc= angular acceleration of crank rads2=
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
72Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated and +ve sign is used when the piston is retarded
T1= Torque due to masses of reciprocating parts T2=Torque due correction couple acting on connecting rodT3= Torque due mass acting at crank pin
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
73Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
The ndashve sign is used when the piston is accelerated(0-180) and +ve sign is used when the piston is retarded(180-360)
T1= Torque due to masses of reciprocating parts
mR=reciprocating masses =mp+m1mp= mass of pistonm1= mass at piston pingudgeon pinap=acceleration of piston If lsquoprsquo is the net pressure of steam or gas on the piston and D is diameter of the piston then Net loadgas force on the piston
119898119903=119898119901+1198981
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
74Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T2=Torque due correction couple acting on connecting rod
119879 prime=119888119900119903119903119890119888119905119894119900119899119888119900119906119901119897119890
119931 120784=119931 primelowast119926119925119927119925
iquestiquest
α
119898119888=119898119886119904119904119900119891 119888119900119899119899119890119888119905119894119899119892 119903119900119889
11989612=1198971lowast1198973
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
75Crank Shaft Torque
bull Simple pendulumbull Theory and analysis of
Compound Pendulumbull Concept of equivalent
length of simple pendulum
bull Bifilar suspension bull Trifilar suspensionbull Dynamics of
reciprocating engines Two mass statically and dynamically equivalent system
bull correction couple bull static and dynamic
force analysis of reciprocating engine mechanism
bull (analytical method only)
bull Crank shaft torquebull Introduction to T-θ
diagram
THEORY OF MACHINES-I
UNIT-II
05012023PROF K N WAKCHAURE
When we consider mass of connecting rod to find torque acting on crank shaft then there are three torques acting on crank shaft
T3= Torque due mass acting at crank pin
1198983=119898119888 1198971
1198971+ 1198973
119931 120785=119950120785lowast119944lowast119926119925
iquest
119931 119957119952119957119938119949=119931 120783+119931 120784+119931 120785
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
76 TURNING MOMENT DIAGRAM
Turning Moment Diagram for a Single Cylinder Double Acting Steam Engine
The turning moment diagram (also known as crank effort diagram) is the graphical representation of the turning moment or crank-effort for various positions of the crank
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023
77 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a four stroke cycle internal combustion engine
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023
78 TURNING MOMENT DIAGRAM
PROF K N WAKCHAURE
Turning moment diagram for a multi-cylinder engine
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
79 Friction at every joint in a machine force of friction arises due to the
relative motion between two parts and hence some energy is wasted in overcoming the friction
The friction between the wheels and the road is essential for the car to move forward
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
80 Types of Friction 1 Static friction It is the friction experienced by a body when at rest 2 Dynamic friction It is the friction experienced by a body when in
motion The dynamic friction is also called kinetic friction and is less than the static friction
It is of the following three types (a) Sliding friction It is the friction experienced by a body when it
slides over another body (b) Rolling friction It is the friction experienced between the surfaces
which has balls or rollers interposed between them (c) Pivot friction It is the friction experienced by a body due to the
motion of rotation as in case of foot step bearings
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
81 Laws of Static Friction
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
82 Laws of Kinetic or Dynamic Friction
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
83 Coefficient of Friction
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
84 Limiting Angle of Friction
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023
85Angle of Repose
PROF K N WAKCHAURE
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
86 Friction in turning pairs
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
87 Friction axis
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
88 Friction axis
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
89Friction axis
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
-
05012023PROF K N WAKCHAURE
90
Thank youhellip
- Static and Dynamic Force Analysis
- Force
- Types of Forces
- Types of Forces (2)
- Laws of Motion
- Moment of Inertia
- Moment of Inertia (2)
- Moment of Inertia (3)
- Moment of Inertia (4)
- Slide 10
- Simple Pendulum
- Compound Pendulum
- Compound Pendulum (2)
- Compound Pendulum (3)
- Compound Pendulum (4)
- Compound Pendulum (5)
- Slide 17
- Compound Pendulum (6)
- Slide 19
- Equivalent length of simple pendulum
- Bifilar Suspension
- Slide 22
- Bifilar Suspension (2)
- Bifilar Suspension (3)
- Bifilar Suspension (4)
- Bifilar Suspension (5)
- Slide 27
- Trifilar Suspension
- Slide 29
- Trifilar Suspension (2)
- Trifilar Suspension (3)
- Trifilar Suspension (4)
- Trifilar Suspension (5)
- Trifilar Suspension (6)
- Slide 35
- Numericals
- Inertia force
- Analytical Method for Velocity and Acceleration of the Piston
- Analytical Method for Velocity and Acceleration of the Piston (2)
- Analytical Method for Velocity and Acceleration of the Piston (3)
- Analytical Method for Velocity and Acceleration of the Piston (4)
- Analytical Method for Velocity and Acceleration of the Piston (5)
- Analytical Method for Velocity and Acceleration of the Piston (6)
- Analytical Method for Velocity and Acceleration of the Piston (7)
- Analytical Method for Velocity and Acceleration of the Piston (8)
- Analytical Method for Velocity and Acceleration of the connecti
- Analytical Method for Velocity and Acceleration of the connecti (2)
- Analytical Method for Velocity and Acceleration of the connecti (3)
- Analytical Method for Velocity and Acceleration of the connecti (4)
- Analytical Method for Velocity and Acceleration of the connecti (5)
- Formulae to be remember
- Numericals (2)
- Numericals (3)
- Numericals (4)
- Numericals (5)
- Dynamic force analysis of reciprocating engine mechanism
- Dynamic force analysis of reciprocating engine mechanism (2)
- Dynamic force analysis of reciprocating engine mechanism (3)
- Dynamic force analysis of reciprocating engine mechanism (4)
- Dynamic force analysis of reciprocating engine mechanism (5)
- Dynamic force analysis of reciprocating engine mechanism (6)
- Dynamic force analysis of reciprocating engine mechanism (7)
- Equivalent Dynamical System
- Equivalent Dynamical System (2)
- Equivalent Dynamical System (3)
- Numericals (6)
- Numericals (7)
- Correction Couple to be Applied to Make Two Mass System Dynamic
- Correction Couple to be Applied to Make Two Mass System Dynamic (2)
- Correction Couple
- Crank Shaft Torque
- Crank Shaft Torque (2)
- Crank Shaft Torque (3)
- Crank Shaft Torque (4)
- Crank Shaft Torque (5)
- TURNING MOMENT DIAGRAM
- TURNING MOMENT DIAGRAM (2)
- TURNING MOMENT DIAGRAM (3)
- Friction
- Types of Friction
- Laws of Static Friction
- Laws of Kinetic or Dynamic Friction
- Coefficient of Friction
- Limiting Angle of Friction
- Angle of Repose
- Friction in turning pairs
- Friction axis
- Friction axis (2)
- Friction axis (3)
- Slide 90
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