me306-fall 2013- chapter (1)- introduction and basic concepts.pdf

Chapter (1)

Introduction and Basic Concept

Chapter (1): Introduction and Basic Concepts ME 306 - Fall 2013

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Chapter (1) - Introduction and Basic Concepts

1. Definitions

Kinematics: The branch of mechanics deals with the motion of a body without

discussing the cause of motion.

Kinetics: The branch of dynamics considering the forces causing the motion.

Dynamics: It is the combination of kinematics and kinetics.

Mechanism is a combination of rigid bodies joined together to provide a specific

absolute motion.

Machine is a device (consisting of fixed and moving parts) which receives energy in some

available form and utilizes it to do some particular type of work.

Engine is a machine which converts energy from one form to another.

Work: When a force is applied to a body so that it causes the body to move, the force is

said to be doing work. The work done is the product of the distance moved and applied

force in the direction of motion (Figure 1). The work done is measured in Joule (where 1

Joule = 1 Nm), i.e.,

Workdone= Force(F)Distance (d)

F F

x

Figure (1)

Example: Find the work done if the force is 100N and the distance is 4.0m for the shown

systems in Figure (2).


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F=100NF=100N

X=4.0m

(a)

F=100NF=100N

X=4.0m

(b)

30o 30o

Figure (2)

Energy is the capacity to do work which implies movement of a body by the application

of a force, as for example: raising of a weight, walking, moving of a car etc.

Types of Energies

Energy exists in many forms such as thermal energy, chemical energy, mechanical energy,

electrical energy. Examples:

1. An electric motor converts electrical energy into mechanical energy.

2. An internal combustion engine is one example of a machine that converts chemical

energy to mechanical energy.

3. Steam engine and Car engine convert heat energy to mechanical energy.

The mechanical energy can be kinetic energy (T) or potential energy (V).

where

221

2 2

mass velocityT mv

, and

V=mass(m)acceleration of gravity (g) height (h)=mgh

Law of Conservation of Energy

The law of conservation of energy states that Energy can be neither created nor

destroyed but is convertible from one form to another.

Power is the rate of doing work and can be expressed mathematically as:

Power (W) = work done (J)/time taken (sec) = force (N) velocity (m/sec)

The unit of the power is the Watt, where 1 Watt = 1 J/sec.

http://en.wikipedia.org/wiki/Electrical_energyhttp://en.wikipedia.org/wiki/Mechanical_energy


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Example: A body of weight 10kN is to be raised through a height of 8.0m in 5 sec.

Calculate the power required.

Solution:

Work done = force (weight) distance moved = 10kN 8.0m = 80 kNm = 80kJ

Power required = work done/time = 80 (kJ)/5 (sec) = 22 kW.

Efficiency is the measure of the usefulness of a machine, process or operation.

Efficiency () = power (or work) output/power (or work) input

2. Linear Velocity and Acceleration

Linear Velocity is the rate of change of linear displacement of a body with respect to

time. Linear velocity (v) can be expressed as:

Velocity(v) =dx

xdt

(m/sec)

where x is linear displacement,

Linear Acceleration is the rate of change of linear velocity of a body with respect to

the time. Mathematically, the linear acceleration is:

2

2Acceleration (a) =

dv d dx d xx

dt dt dt dt

(m/sec2)

3. Angular Velocity and Acceleration

When a rigid body rotates about a fixed axis all points on the

body are constrained to move in a circular path. Therefore in

any period of time all points on the body will complete the

same number of revolutions about the axis of rotation.

Angular Velocity is the rate of change of angular displacement with respect to time.

Angularvelocity( )d

dt

(rad/sec)


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in which = Angular displacement in radians, and 2

60N

(rad/sec)

where the speed N is given in rev/min.

Angular Acceleration is the rate of change of angular velocity with respect to time.

Angular acceleration ( )d d

dt dt

(rad/sec2)

Relation between Linear and Angular Motion

Consider a body moving along a circular path from A to B as shown in Figure (3).

From the geometry of Figure (3):

S = r

The relation between linear and angular velocities:

tdS d d

v r r rdt dt dt

Tangential velocity(v ) = Angular velocity ( ) radius(r)t

The relation between linear and angular accelerations:

The linear acceleration consists from two components;

Tangential component ttdv d d

a r r rdt dt dt

Radial component

2

2

n

d v va v v v

dt r r

r r

The total acceleration or resultant acceleration is given as:

2 2

t na a a

The angle of inclination with the tangential acceleration is given by:

1tan n

t

a

a

A

B

,

vt na

ta

S

at

ana

Total acceleration

O

Figure (3)


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Example: The speed of a flywheel increases from 150rpm to 200rpm in 5sec. Find its

angular acceleration in rad/sec.

Mass moment of Inertia

All rotating machines, such as pumps, turbines and engines have a moment of inertia

given as:

2I m k

Where m is the total mass and k is the radius of gyration is the radius.

(1) The mass moment of inertia of a thin disc of radius R ,

about an axis through its centre of gravity and

perpendicular to its length is:

2

2G

m RI .

(2) The mass moment of inertia of a thin rod of length , about an axis through its

centre of gravity and perpendicular to its length is:

2

12G

m LI .

The mass moment of inertia about any other parallel axis may be determined by using a

parallel axis theorem. Then the moment of inertia about a parallel axis is given by:

2O GI I m d

where GI is the moment of inertia of a body about an axis through its centre of gravity,

and d is the distance between two parallel axes.

Problem: Show that the mass moment of inertia about a parallel axis through the end of

the rod is given by: 2

3O

m LI .

Linear and Angular Momentum

R

G

Thin disc

G

L

Thin rod

O


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The linear momentum (LLm) of a system of particles is equal to the product of the total

mass (m) of the system and the velocity (v) of the center of mass.

LmL m v m r (kg.m/sec)

The moment of momentum (Lam) of whole body mass m about

fixed axis O is:

2am oL m v r m r I (kg.m

2/sec)

where Io =m r2 is the mass moment of inertia of whole body (m) about the point O.

Inertia Torque

The Newtons second law of motion states that the torque is directly proportional to the

rate of change of the angular momentum, i.e.,

oam d IdLT

dt dt

The torque required to rotate a body of angular acceleration is

o od

T I Idt

(N.m)

Chapter (2): Mechanisms and Machines

r

O

m

v

Fixed axis

me306-fall 2013- chapter (1)- introduction and basic concepts.pdf

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