kinematics of machinery notes.doc

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UNIT-1 (BASIC OF MECHANISMS)MechanismsA mechanism is a combination of rigid or restraining bodies so shaped and connected that

they move upon each other with a definite relative motion. A simple example of this is

the slider crank mechanism used in an internal combustion or reciprocating air

compressor.Machine

A machine is a mechanism or a collection of mechanisms which transmits force from the

source of power to the resistance to be overcome,and thus perform a mechanical work.Plane and Spatial Mechanisms

If all the points of a mechanism move in parallel planes, then it is defined as a plane

mechanism.If all the points do not move in parallel planes then it is called spatial mechanism.

Kinematic Pairs

A mechanism has been defined as a combination so connected that each moves

with respect to each other. A clue to the behaviors lies in the nature of connections,

known as kinetic pairs.The degree of freedom of a kinetic pair is given by the number independent coordinates

required to completely specify the relative movement.Lower Pairs

A pair is said to be a lower pair when the connection between two elements is

through the area of contact.Its 6 types are :Revolute Pair

Prismatic Pair

Screw Pair

Cylindrical Pair

Spherical Pair

Planar Pair

Higher PairsA higher pair is defined as one in which the connection between two elements has

only a point or line of contact. A cylinder and a hole of equal radius and with axis parallel

make contact along a surface. Two cylinders with unequal radius and with axis parallelmake contact along a line. A point contact takes place when spheres rest on plane orcurved surfaces (ball bearings) or between teeth of a skew-helical gears. in roller

bearings, between teeth of most of the gears and in cam-follower motion. The degree of

freedom of a kinetic pair is given by the number independent coordinates required tocompletely specify the relative movement.

Wrapping Pairs

Wrapping Pairs comprise belts, chains, and other such devices.
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To define a mechanism we define the basic elements as follows :

LinkA material body which is common to two or more kinematic pairs is called a link.

Kinematic ChainA kinematic chain is a series of links connected by kinematic pairs. The chain is said

to be closed chain if every u link is connected to atleast two other links, otherwise it is

called an open chain.A link which is connected to only one other link is known assingular link.If it is

connected to two other links, it is called binary link.

If it is connected to three other links, it is called ternary link, and so on. A chain

which consists of only binary links is called simple chain.A type of kinematic chain is one with constrained motion, which means that a

definite motion of any link produces unique motion of all other links. Thus motion of any

point on one link defines the relative position of any point on any other link.So it has one

degree of freedom.Kinematic inversions

The process of fixing different links of a kinematic chain one at a time to producedistinct mechanisms is called kinematic inversion. Here the relative motions of the links

of the mechanisms remain unchanged.

INVERSION OF 4-BAR KINEMATIC CHAIN

First, let us consider the simplest kinematic chain,i.e., achain consisting of four

binary links and four revolute pairs. The four different mechanisms can be obtained byfour different inversions of the chain.

First inversion: (double crank mechanism)In which shortest link is fixed, adjacent links having rotary motion.

Application: Shaping mechanism

Second inversion: (double rocker/lever mechanism)Link opposite to shortest link is fixed, both the links adjacent to fixed link

having oscillating motions only

Applications: Acckerman steering mechanism, pantograph.Third and fourth inversion :( Crank and rocker/lever mechanism)

Link adjacent to shortest link is fixed, having one rotary motion and one

oscillating motion.
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Applications: rotary beam engine

Slider Crank mechanism

It has four binary links, three revolute pairs, one prismatic pair. By fixing links 1,

2, 3 in turn we get various inversions.

A four-bar linkage with output cranks and ground member of infinite length. Aslider crank (see illustration) is most widely used to convert reciprocating to rotary

motion (as in an engine) or to convert rotary to reciprocating motion (as in pumps), but it

has numerous other applications. Positions at which slider motion reverses are calleddead centers. When crank and connecting rod are extended in a straight line and the slider

is at its maximum distance from the axis of the crankshaft, the position is top dead center

(TDC); when the slider is at its minimum distance from the axis of the crankshaft, theposition is bottom dead center (BDC).

Principal parts of slider-crank mechanism.

The conventional internal combustion engine employs a piston arrangement inwhich the piston becomes the slider of the slider-crank mechanism. Radial engines for

aircraft employ a single master connecting rod to reduce the length of the crankshaft. Themaster rod, which is connected to the wrist pin in a piston, is part of a conventional

slider-crank mechanism. The other pistons are joined by their connecting rods to pins on

the master connecting rod.To convert rotary motion into reciprocating motion, the slider crank is part of a

wide range of machines, typically pumps and compressors. Another use of the slider

crank is in toggle mechanisms, also called knuckle joints. The driving force is applied atthe crankpin so that, at TDC, a much larger force is developed at the slider. See also

Four-bar linkage.
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Double Slider Crank mechanism

It has four binary links, two revolute pairs, two sliding pairs.Its various types are :

Kinematics and Dynamics : Double Slider Crank mechanism

Scotch Yoke mechanism:

Here the constant rotation of the crank produces harmonic translation of the

yoke.Its four binary links are :Fixed Link

Crank

Sliding Block

Yoke

The four kinematic pairs are :

revolute pair (between 1 & 2)

revolute pair (between 2 & 3)

prismatic pair (between 3 & 4)

prismatic pair (between 4 & 1)Oldhams Coupling:

It is used for transmitting angular velocity between two parallel but eccentric

shafts

Elliptical Trammel:Here link 4 is fixed. Any point on the link 2 describes an ellipse as it moves.The

mid-point of the link 2 will obiviously describe a circle.

Very often a mechanism with higher pair can be replaced by an equivalent mechanism

with lower pair.This equivalence is valid for studying only the instantaneouscharacteristics.The equivalent lower-pairmechanism facilitates analysis as a certain

amount of sliding takes place between connecting links in a higher-pair mechanism.Another example of an equivalent lower-pair mechanism for a cam-follower system isshown.The sliding block is the additional link and thebhigher pair is replaced by two

lower pairs, one revoluteand other prismatic. C is the center of curvature of the cam

surface at the point of contact between the cam and the follower.

DEGREE OF FREEDOM:


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Let n be the no. of links in a mechanism out of which, one is fixed, and let j be the

no. of simple hinges(ie, those connect two links.) Now, as the (n-1) links move in a plane,

in the absence of any connections, each has 3 degree of freedom; 2 coordinates arerequired to specify the location of any reference point on the link and 1 to specify the

orientation of the link. Once we connect the links there cannot be any relative translation

between them and only one coordinate is necessary to specify their relative orientation.Thus, 2 degrees of freedom (translation) are lost, and only one degree of freedom

(rotational) is left. So, no. of degrees of freedom is:

F=3(n-1)-2jMost mechanisms are constrained, ie F=1. Therefore the above relation becomes,

2j-3n+4=0

this is called Grubler's Criterion.

Failure of Grubler's criterionA higher pair has 2 degrees of freedom .Following the same argument as before,

The degrees of freedom of a mechanism having higher pairs can be written as,

F=3(n-1)-2j-h

Often some mechanisms have a redundant degree of freedom. If a link can move withoutcausing any movement in the rest of the mechanism, then the link is said to have a

redundant degree of freedom.

Unit III (KINEMATICS OF CAM)

Cam and Follower Systems

A cam is a component on which a particular profile has been machined. The profile of

the cam imparts (causes) a follower to move in a particular way. This can be seen if we

examine the diagram below. As the shaft is rotated the cam rotates with it causing the

follower to move up and down.

Cams fall into two main categories:

1. edge, plate or face cams and

2. cylindrical cams.

Category No.1


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The plate cam is merely a flat disc that has had a certain shape (or profile) machined on

to it. The follower is placed in contact with this profile and as the cam is rotated the

profile of it translates into a particular movement of the follower usually up and down.This can be seen in the diagram above.

The face cam is a disc the has a groove machined into its face and a roller follower isused to follow the groove as the cam rotates.

Category No.2

The cylinder or drum cam is a cylinder which has had a profile machined onto it and asthe cam rotates the profile imparts a particular motion on its follower.

Types of follower:

There are three main types of follower:

1. the knife edge follower (seen in the cam and follower system table below)2. the roller follower (seen in the cam and follower system table below)

3. the flat follower (seen in the cam and follower system table below)

The table below shows three different system each containing a different type of

follower.

Knife edge

followerRoller follower

Flat

follower

Cam and Follower Systems


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As a cam rotates it imparts a particular motion on the follower in contact with it. From

the image shown below we can see that as the cam shaft rotates the follower will be

forced to move up and down. That is of course provided that there is an externaldownward force on the follower that makes it keep contact with the cam.

The next consideration is the type of the upward and downward motion of the cam andfollower system. The are three different types of motion that could have been imparted

on the follower. These are:

1. Uniform Velocity2. Simple Harmonic Motion

3. Acceleration/Retardation

Uniform Velocity:

Velocity is the rate of change of displacement with respect to time. This change may bea change of speed or a changein direction. From the definition uniform velocity therefore

means that the change in displacement is a steady change. Or to put it differently it

means that the change in displacement in the first second of the uniform velocity motionis the same as the change in displacement in any other second in that motion.


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Simple Harmonic Motion:

The motion of a follower is simple harmonic motion if its acceleration towards a

particular point if its acceleration towards a particular point is proportional to itsdisplacement from that point.

The best way to understand this non-uniform motion is to imagine a pendulum swinging.

If you examine the pendulum as it swings you can see that as it swings towards A it slowsdown until it finally stops at A. Then it starts to swing back in the other direction. As it

does so it starts to gain speed until it reaches it max speed at O. Once the pendulumpasses O it starts to slow down on its approach to B. At B the pendulum stops and begins

to swing back. Again it speed up as it approaches O, reaches its maximum speed at O,

then slows down on its approach to A, stops at A and then swings back in the otherdirection. This cycle then keeps repeating itself. If you watch the swinging pendulum

shown later in the site you should be able to notice the non-unifromity of its motion.

Displacement Diagram for Simple Harmonic Motion.


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Acceleration/Retardation:

Acceleration is the rate of change of velocity with respect to time. This motion is used

where the follower is required to rise or fall with uniform acceleration, that is it's velocityis changing at a constant rate.

Summary:

In cam and follower systems, the motion of the follower is usually perpendicular to the

axis around which the cam is rotating. Three types of motion that the follower cam

experience are;

1. Uniform velocity;

2. Simple harmonic motion;3. Uniform accelaration and retardation.


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Unit IV (GEARS)

GEAR CLASSIFICATION

Gears may be classified according to the relative position of the axes of revolution. The

axes may be

1. parallel,2. intersecting,

3. neither parallel nor intersecting.

Here is a brief list of the common forms. We will discuss each in more detail later.

Gears for connecting parallel shafts
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Gears for connecting intersecting shafts

Neither parallel nor intersecting shafts

Gears for connecting parallel shafts

1. Spur gears

The left pair of gears makes external contact, and the right pair of gears makes

internal contact

2. Parallel helical gears

3. Herringbone gears (or double-helical gears)

4. Rackandpinion (The rack is like a gear whose axis is at infinity.)
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Gears for connecting intersecting shafts

1. Straight bevel gears

2. Spiral bevel gears

Neither parallel nor intersecting shafts

1. Crossed-helical gears

2. Hypoid gears3. Worm and wormgear


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7.2 GEAR-TOOTH ACTION

7.2.1 Fundamental Law of Gear-Tooth Action

Figure 7-2 shows two mating gear teeth, in which

Tooth profile 1 drives tooth profile 2 by acting at the instantaneous contact pointK.

N1N2 is the common normal of the two profiles.

N1 is the foot of the perpendicular from O1 toN1N2

N2 is the foot of the perpendicular from O2 toN1N2.
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Figure 7-2 Two gearing tooth profiles

Although the two profiles have different velocities V1 and V2 at pointK, their velocities

alongN1N2 are equal in both magnitude and direction. Otherwise the two tooth profileswould separate from each other. Therefore, we have

(7-1)

or

(7-2)

We notice that the intersection of the tangencyN1N2 and the line of centerO1O2 is pointP, and

(7-3)

Thus, the relationship between the angular velocities of the driving gear to the driven

gear, orvelocity ratio, of a pair of mating teeth is

(7-4)

PointPis very important to the velocity ratio, and it is called the pitch point. Pitch pointdivides the line between the line of centers and its position decides the velocity ratio of

the two teeth. The above expression is the fundamental law of gear-tooth action.

7.2.2 Constant Velocity Ratio

For a constant velocity ratio, the position ofPshould remain unchanged. In this case, themotion transmission between two gears is equivalent to the motion transmission between

two imagined slipless cylinders with radiusR1 andR2 or diameterD1 andD2. We can gettwo circles whose centers are at O1 and O2, and through pitch pointP. These two circle

are termed pitch circles. The velocity ratio is equal to the inverse ratio of the diametersof pitch circles. This is the fundamental law of gear-tooth action.

The fundamental law of gear-tooth action may now also be stated as follow (for gears

with fixed center distance)(Ham 58):
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The common normal to the tooth profiles at the point of contact must always pass

through a fixed point (the pitch point) on the line of centers (to get a constant velocity

ration).

7.2.3 Conjugate Profiles

To obtain the expected velocity ratio of two tooth profiles, the normal line of theirprofiles must pass through the correspondingpitch point, which is decided by the velocity

ratio. The two profiles which satisfy this requirement are called conjugate profiles.

Sometimes, we simply termed the tooth profiles which satisfy thefundamental law ofgear-tooth action the conjugate profiles.

Although many tooth shapes are possible for which a mating tooth could be designed to

satisfy the fundamental law, only two are in general use: the cycloidaland involute

profiles. The involute has important advantages -- it is easy to manufacture and the centerdistance between a pair of involute gears can be varied without changing the velocity

ratio. Thus close tolerances between shaft locations are not required when using theinvolute profile. The most commonly used conjugate tooth curve is the involute curve(Erdman & Sandor 84).

7.3 INVOLUTE CURVE

The following examples are involute spur gears. We use the word involute because the

contour of gear teeth curves inward. Gears have many terminologies, parameters andprinciples. One of the important concepts is the velocity ratio, which is the ratio of the

rotary velocity of the driver gear to that of the driven gears.

The SimDesign file for these gears is simdesign/gear15.30.sim. The number of teeth

in these gears are 15 and 30, respectively. If the 15-tooth gear is the driving gear and the

30-teeth gear is the driven gear, their velocity ratio is 2.

Other examples of gears are in simdesign/gear10.30.sim andsimdesign/gear20.30.sim
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7.3.1 Generation of the Involute Curve

Figure 7-3 Involute curve

The curve most commonly used for gear-tooth profiles is the involute of a circle. This

involute curve is the path traced by a point on a line as the line rolls without slipping on

the circumference of a circle. It may also be defined as a path traced by the end of a stringwhich is originally wrapped on a circle when the string is unwrapped from the circle. The

circle from which the involute is derived is called the base circle.

In Figure 7-3, let lineMNroll in the counterclockwise direction on the circumference of acircle without slipping. When the line has reached the position M'N', its original point oftangentA has reached the positionK, having traced the involute curveAKduring the

motion. As the motion continues, the pointA will trace the involute curveAKC.

7.3.2 Properties of Involute Curves

1. The distanceBKis equal to the arcAB, because linkMNrolls without slipping onthe circle.

2. For any instant, the instantaneous centerof the motion of the line is its point of

tangent with the circle.

Note: We have not defined the term instantaneous centerpreviously. Theinstantaneous center orinstant center is defined in two ways (Bradford &

Guillet 43):1. When two bodies have planar relative motion, the instant center is a point

on one body about which the other rotates at the instant considered.

2. When two bodies have planar relative motion, the instant center is the

point at which the bodies are relatively at rest at the instant considered.3. The normal at any point of an involute is tangent to the base circle. Because of the

property (2) of the involute curve, the motion of the point that is tracing the

involute is perpendicular to the line at any instant, and hence the curve traced willalso be perpendicular to the line at any instant.

4. There is no involute curve within the base circle.

7.4 TERMINOLOGY FOR SPUR GEARS

Figure 7-4 shows some of the terms for gears.
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Figure 7-4 Spur Gear

In the following section, we define many of the terms used in the analysis of spur gears.

Some of the terminology has been defined previously but we include them here for

completeness. (See (Ham 58)for more details.)

Pitch surface : The surface of the imaginary rolling cylinder (cone, etc.) that the

toothed gear may be considered to replace.

Pitch circle: A right section of the pitch surface.

Addendum circle: A circle bounding the ends of the teeth, in a right section of

the gear.

Root (or dedendum) circle: The circle bounding the spaces between the teeth, in

a right section of the gear.

Addendum: The radial distance between the pitch circle and the addendum circle.

Dedendum: The radial distance between the pitch circle and the root circle.

Clearance: The difference between the dedendum of one gear and the addendum

of the mating gear.

Face of a tooth: That part of the tooth surface lying outside the pitch surface.

Flank of a tooth: The part of the tooth surface lying inside the pitch surface.

Circular thickness (also called the tooth thickness) : The thickness of the tooth

measured on the pitch circle. It is the length of an arc and not the length of a

straight line.

Tooth space: The distance between adjacent teeth measured on the pitch circle.

Backlash: The difference between the circle thickness of one gear and the toothspace of the mating gear.

Circular pitch p: The width of a tooth and a space, measured on the pitch circle.

Diametral pitch P: The number of teeth of a gear per inch of its pitch diameter. A

toothed gear must have an integral number of teeth. The circular pitch, therefore,

equals the pitch circumference divided by the number of teeth. The diametral

pitch is, by definition, the number of teeth divided by the pitch diameter. That is,
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(7-5)

and

(7-6)

Hence

(7-7)

where

p = circular pitch

P = diametral pitchN = number of teeth

D = pitch diameter

That is, the product of the diametral pitch and the circular pitch equals .

Module m: Pitch diameter divided by number of teeth. The pitch diameter is

usually specified in inches or millimeters; in the former case the module is theinverse of diametral pitch.

Fillet : The small radius that connects the profile of a tooth to the root circle.

Pinion: The smaller of any pair of mating gears. The larger of the pair is called

simply the gear.

Velocity ratio: The ratio of the number of revolutions of the driving (or input)

gear to the number of revolutions of the driven (or output) gear, in a unit of time.

Pitch point: The point of tangency of the pitch circles of a pair of mating gears.

Common tangent: The line tangent to the pitch circle at the pitch point.

Line of action: A line normal to a pair of mating tooth profiles at their point of

contact.

Path of contact: The path traced by the contact point of a pair of tooth profiles.

Pressure angle : The angle between the common normal at the point of tooth

contact and the common tangent to the pitch circles. It is also the angle between

the line of action and the common tangent.

Base circle :An imaginary circle used in involute gearing to generate the

involutes that form the tooth profiles.

Table 7-1lists the standard tooth system for spur gears. (Shigley & Uicker 80)
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Table 7-1 Standard tooth systems for spur gears

Table 7-2lists the commonly useddiametral pitches.

Coarse pitch 2 2.25 2.5 3 4 6 8 10 12 16

Fine pitch 20 24 32 40 48 64 96 120 150 200

Table 7-2 Commonly used diametral pitches

Instead of using the theoreticalpitch circle as an index of tooth size, thebase circle,

which is a more fundamental circle, can be used. The result is called the base pitchpb,

and it is related to the circular pitchp by the equation

(7-8)

7.5 CONDITION FOR CORRECT MESHING

Figure 7-5 shows two meshing gears contacting at pointK1 andK2.
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Figure 7-5 Two meshing gears

To get a correct meshing, the distance ofK1K2 on gear 1 should be the same as the

distance ofK1K2 on gear 2. AsK1K2 on both gears are equal to thebase pitch of theirgears, respectively. Hence

(7-9)

Since

(7-10)

and

(7-11)

Thus
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(7-12)

To satisfy the above equation, the pair of meshing gears must satisfy the following

condition:

(7-13)

7.6 ORDINARY GEAR TRAINS

Gear trains consist of two or more gears for the purpose of transmitting motion from one

axis to another. Ordinary gear trains have axes, relative to the frame, for all gears

comprising the train. Figure 7-6ashows a simple ordinary train in which there is onlyone gear for each axis. In Figure 7-6b a compound ordinary train is seen to be one in

which two or more gears may rotate about a single axis.

Figure 7-6 Ordinary gear trains

7.6.1 Velocity Ratio

We know that the velocity ratio of a pair of gears is the inverse proportion of the

diameters of theirpitch circle, and the diameter of the pitch circle equals to the number ofteeth divided by the diametral pitch. Also, we know that it is necessary for the to mating

gears to have the same diametral pitch so that to satisfy the condition of correct meshing.

Thus, we infer that the velocity ratio of a pair of gears is the inverse ratio of their numberof teeth.

For the ordinary gear trains in Figure 7-6a, we have
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(7-14)

These equations can be combined to give the velocity ratio of the first gear in the train to

the last gear:

(7-15)

Note:

The tooth number in the numerator are those of the driven gears, and the tooth

numbers in the denominator belong to the driver gears.

Gear 2 and 3 both drive and are, in turn, driven. Thus, they are called idler gears.Since their tooth numbers cancel, idler gears do not affect the magnitude of the

input-output ratio, but they do change the directions of rotation. Note thedirectional arrows in the figure. Idler gears can also constitute a saving of space

and money (If gear 1 and 4 meshes directly across a long center distance, their

pitch circle will be much larger.)

There are two ways to determine the direction of the rotary direction. The first

way is to label arrows for each gear as in Figure 7-6. The second way is to

multiple mth power of "-1" to the general velocity ratio. Where m is the number of

pairs ofexternal contact gears (internal contactgear pairs do not change the rotarydirection). However, the second method cannot be applied to the spatial gear

trains.

Thus, it is not difficult to get the velocity ratio of the gear train inFigure 7-6b:

(7-16)

7.7 PLANETARY GEAR TRAINS

Planetary gear trains, also referred to as epicyclic gear trains, are those in which oneor more gears orbit about the central axis of the train. Thus, they differ from an ordinary

train by having a moving axis or axes. Figure 7-8 shows a basic arrangement that is

functional by itself or when used as a part of a more complex system. Gear 1 is called a

sun gear , gear 2 is a planet, link H is an arm, orplanet carrier.
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Figure 7-8 Planetary gear trains

Figure 7-7 Planetary gears modeled using SimDesign

The SimDesign file is simdesign/gear.planet.sim. Since the sun gear (the largest

gear) is fixed, the DOF of the above mechanism is one. When you pull the arm or the

planet, the mechanism has a definite motion. If the sun gear isn't frozen, the relative

motion is difficult to control.

7.7.1 Velocity Ratio

To determine the velocity ratio of theplanetary gear trainsis slightly more complex an

analysis than that required forordinary gear trains. We will follow the procedure:

1. Invert the planetary gear train mechanism by imagining the application a rotary

motion with an angular velocity of H to the mechanism. Let's analyse the motionbefore and after the inversion withTable 7-3:
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Table 7-3 Inversion of planetary gear trains.

Note: H is the rotary velocity of gear i in the imagined mechanism.

Notice that in the imagined mechanism, the armHis stationary and functions as a

frame. No axis of gear moves any more. Hence, the imagined mechanism is anordinary gear train.

2. Apply the equation ofvelocity ratio of the ordinary gear trains to the imagined

mechanism. We get

(7-17)

or

(7-18)

7.7.2 Example

Take the planetary gearing train inFigure 7-8 as an example. Suppose N1 = 36, N2 = 18,1 = 0, 2 = 30. What is the value of N?

With the application of the velocity ratio equation for the planetary gearing trains, we

have the following equation:

(7-19)

From the equation and the given conditions, we can get the answer: N = 10.
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