KINEMATICS OF MACHINES-15ME42

KINEMATICS OF MACHINES-15ME42

Depart of Mechanical Engineering , ATMECE MYSORE Page 1

KINEMATICS OF MACHINES

10ME44/15ME42



UNIT 1 SIMPLE MECHANISMS

CONTENTS

1.1 Introduction

Objectives

1.2 Kinematics of Machines

1.3 Kinematic Link or an Element

1.4 Classification of Links

1.5 Degree of Freedom

1.6 Kinematic Pairs

1.7 Different Pairs

1.7.1 Types of Lower Pair

1.7.2 Higher Pair

1.7.3 Wrapping Pair

1.8 Kinematic Chains

1.9 Inversions of Kinematic Chain

1.10 Machine

1.11 Other Mechanisms

1.11.1 Pantograph

1.11.2 Straight Line Motion Mechanisms

1.11.3 Automobile Steering Gear

1.11.4 Hooks Joint or Universal Coupling

1.12 Key Words



Objectives:

1. Study about kinematic chain, machine & mechanisms

2. Get knowledge about various inversions of kinematic chains

1.1 INTRODUCTION

In our daily life, we come across a wide array of machines. It can be a sewing machine, a

cycle or a motor car. Power is produced by the engine which makes use of a mechanism

called slider crank mechanism. It converts reciprocating motion of a piston into rotary

motion of the crank. The power of the engine is transmitted to the wheels with the help

of different mechanisms. If you visit LPG gas filling plant or a bottling plant almost all

the functions are done by making use of mechanisms. These are only few examples.

Generally, manual handling in industries has been reduced to the minimum. In

engineering, mechanisms and machines are two very common and frequently used terms.

We shall start with simple definition of these terms.



Theory of Machines In this unit, you will also study about link, mechanism, machine, kinematic quantities,

different types of motion and planar mechanism. You will study about degree of

freedom, kinematic pairs and classification of links in this unit.

A moving body has to be assigned coordinates according to the axes assigned. The

motion of the bodies is constrained according to the requirement in a mechanism. The

links which are the basic elements of the mechanism are connected to form kinematic

pairs which are of different types. The links may further be connected to several links in

order to impart motion and they are classified accordingly.

In this unit, you will be explained how to get different mechanisms by using four bar

chain which is a basic kinematic chain. The four bar chain has four links which are

connected with each other with the help of four lower kinematic pairs. This chain

provides different mechanisms of common usage. For example, one mechanism,

provided by this, is used in petrol engine, diesel engine, steam engine, compressors, etc.

One mechanism makes possible to complete idle stroke in machine tool in lesser time

than cutting stroke which reduces machining time. This mechanism being termed as

quick return mechanism. Similarly, there are some mechanisms which can provide

rocking motion which can be used in materials handling. You will be explained

terminology and classification of cams and followers also.

Objectives After studying this unit, you should be able to

determine degrees of freedom for a link and kinematic pair,

describe kinematic pair and determine motion,

distinguish and categorise different type of links,

know inversions of different kinematic chains,

understand utility of various mechanisms of four bar kinematic chain,

make kinematic design of a mechanism,

know special purpose mechanisms,

know terminology of cams, and

know classification of followers and cams.

1.2 KINEMATICS OF MACHINES

The kinematics of machines deals with analysis and synthesis of mechanisms. Before proceeding to this, you are introduced to the kinematics.

Kinematics implies displacement, velocity and acceleration of a point of interest at a

particular time or with passage of time. A point or a particle may be displaced from its

initial position in any direction. The motion of a particle confined to move in a plane can

be defined by x, y or r, or some other pair of independent coordinates. The motion of a

particle constrained to move along a straight line can be defined by any one coordinate.

The concerned coordinate shall describe its location at any instant.

1.2.1 Displacement The distance of the position of the point from a fixed reference point is called displacement. In rectilinear motion the displacement, is along one axis say x-axis, therefore,

Displacement ‘s’ = x In a general plane motion,

Displacement ‘s’ = x + i y



1.2.2 Velocity The velocity of a particle is defined as the rate of change of displacement, therefore, the velocity

V S 2 S1 s

t t

t 2

1

where, s S 2 S1

and t t 2 t1

s is the distance traveled in time t. The direction of velocity shall be tangent to the

path of motion. Y

S2

S1

X

Figure 1.1 : Plane Motion

1.2.3 Acceleration The acceleration of a particle is defined as the rate of change of velocity, therefore,

Acceleration ‘a’ V2 V1 V

t t

t 2

1

where V V2 V1

and t t 2 t1

V is the change in velocity in time t.

1.3 KINEMATIC LINK OR AN ELEMENT

Machines consist of several material bodies, each one of them being called link or

kinematic link or an element. It is a resistant body or an assembly of resistant bodies.

The deformation, if any, due to application of forces is negligible. If a link is made of

several resistant bodies, they should form one unit with no relative motion of parts with

respect to each other.

For example, piston, piston rod and cross head in steam engine consist of different parts

but after joining together they do not have relative motion with respect to each other and

they form one link. Similarly, ropes, belts, fluid in hydraulic press, etc. undergo small

amount of deformation which, if neglected, will work as resistant bodies and, thereby,

can be called links.

SAQ 1

(a) What is a resistant body?

(b) Define link.

Simple Mechanisms



Theory of Machines

1.4 CLASSIFICATION OF LINKS

A resistant body or group of resistant bodies with rigid connections preventing their relative movement is known as a link. The links are classified depending on number of joints.

Singular Link

A link which is connected to only one other link is called a singular link (Figure 1.2).

Figure 1.2 : Singular Link

Binary Link

A link which is connected to two other links is called a binary link (Figure 1.3).

2 3

1

Figure 1.3 : Binary Link

Ternary Link

A link which is connected to three other links is called a ternary link (Figure 1.4).

Figure 1.4 : Ternary Link Quarternary Link

A link which is connected to four other links is called quarternary link (Figure 1.5).

Figure 1.5 : Quarternary Link



1.5 DEGREE OF FREEDOM

The degree of freedom of a body is equal to the number of independent coordinates

required to specify the movement. For a cricket ball when it is in air, six independent

coordinates are required to define its motion. Three independent displacement

coordinates along the three axes (x, y, z) and three independent coordinates for rotations

about these axes are required to describe its motion in space. Therefore, degrees of

freedom for this ball is equal to six. If this cricket ball moves on the ground, this

movement can be described by two axes in the plane.

When the body has a plane surface to slide on a plane, the rotation about x and y-axes

shall be eliminated but it can rotate about an axis perpendicular to the plane, i.e. z-axis.

At the same time, while executing plane motion, this body undergoes displacement

which can be resolved along x and y axis. The rotation about z-axis and components of

displacement along x and y axes are independent of each other. Therefore, a sliding body

on a plane surface has three degrees of freedom.

These were the examples of unconstrained or partially constrained bodies. If a cylinder

rolls without sliding along a straight guided path, the degree of freedom is equal to one

only because rotation in case of pure rolling is dependent on linear motion. This is a case

of completely constrained motion.

The angle of rotation rx

where, r is radius of cylinder and x is linear displacement. z z

Z y

x x

Y

x o x

O

Y Figure 1.6 : Degree of Freedom

o

x

Figure 1.7 : Completely Constrained Motion

1.6 KINEMATIC PAIRS

In a mechanism, bodies or links are connected such that each of them moves with respect

to another. The behaviour of the mechanism depends on the nature of the connections of

the links and the type of relative motion they permit. These connections are known as

Simple Mechanisms



kinematic pairs. Hence kinematic pair is defined as a joint of two links having relative

motion between them.

Broadly, kinematic pairs can be classified as :

(a) Lower pair,

(b) Higher pair, and

(c) Wrapping pair.

1.7 DIFFERENT PAIRS

When connection between two elements is through the area of contact, i.e. there is

surface contact between the two links, the pair is called lower pair. Examples are motion

of slider in the cylinder, motion between crank pin and connecting rod at big end.

1.7.1 Types of Lower Pairs

There are six types of lower pairs as given below :

(a) Revolute or Turning Pair (Hinged Joint)

(b) Prismatic of Sliding Pair

(c) Screw Pair

(d) Cylindrical Pair

(e) Spherical Pair

(f) Planar Pair

Revolute or Turning Pair (Hinged Joint)

A revolute pair is shown in Figure 1.8. It is seen that this pair allows only one

relative rotation between elements 1 and 2, which can be expressed by a single

coordinate ‘’. Thus, a revolute pair has a single degree of freedom.

1

2

Figure 1.8 : Revolute or Turning Pair

Prismatic or Sliding Pair

As shown in Figure 1.9, a prismatic pair allows only a relative translation between elements 1 and 2, which can be expressed by a single coordinate ‘s’, and it has one degree of freedom.



S

1

2

Figure 1.9 : Prismatic or Sliding Pair

Screw Pair

As shown in Figure 1.10, a screw pair allows rotation as well as translation but these two movements are

related to each other. Therefore, screw pair has one degree of freedom because the relative movement

between 1 and 2 can be expressed by a single coordinate ‘’ or ‘s’. These two coordinates are related by the

following relation :

2L

s

where, L is lead of the screw.

2

S

1

Figure 1.10 : Screw Pair Cylindrical Pair

As shown in Figure 1.11, a cylindrical pair allows both rotation and translation parallel to the axis of

rotation between elements 1 and 2. These relative movements can be expressed by two independent

coordinates ‘’ or ‘s’ because they are not related with each other. Degrees of freedom in this case are equal

to two.

S

1

2

Figure 1.11 : Cylindrical Pair Spherical Pair

A ball and socket joint, as shown in Figure 1.12, forms a spherical pair. Any rotation of element 2 relative

to 1 can be resolved in the three components. Therefore, the complete description of motion requires three

independent coordinates. Two of these coordinates ‘’ and ‘’ are required to specify the position of axis

OA and the third coordinate ‘’ describes the rotation about the axis of OA. This pair has three degrees of

freedom.



Theory of Machines A

A

2

o 1

o

Figure 1.12 : Spherical Pair

Planar Pair

A planar pair is shown in Figure 1.13. The relative motion between 1 and 2 can be described by x and y coordinates in x-y plane. The x and y coordinates describe

relative translation and describes relative rotation about z-axis. This pair has three degrees of freedom.

Z

Y

1

X 2

Figure 1.13 : Planar Pair

1.7.2 Higher Pair

A higher pair is a kinematic pair in which connection between two elements is only a

point or line contact. The cam and follower arrangement shown in Figure 1.14 is an

example of this pair. The contact between cam and flat faced follower is through a line.

Other examples are ball bearings, roller bearings, gears, etc. A cylinder rolling on a flat

surface has a line contact while a spherical ball moving on a flat surface has a point

contact.

Follower

B A Cam

Rotating

with Shaft

Figure 1.14 : Higher Pair

1.7.3 Wrapping Pair



Wrapping pairs comprise belts, chains and such other devices. Belt comes from one side of the pulley and moves over to other side through another pulley as shown in Figure 1.15.

Figure 1.15 : Wrapping Pair

1.8 KINEMATIC CHAINS

In a kinematic chain, four links are required which are connected with each other with the help of lower pairs.

These pairs can be revolute pairs or prismatic pairs. A prismatic pair can be thought of as the limiting case of a

revolute pair. Before going into the general theory of mechanisms it may be observed that to form a simple closed chain we need

at least three links with three kinematic pairs. If any one of these three links is fixed, there cannot be relative

movement and, therefore, it does not form a mechanism but it becomes a structure which is completely rigid.

Thus, a simplest mechanism consists of four links, each connected by a kinematic lower pair (revolute etc.), and it

is known as four bar mechanism.

3

3 2

4

2

1

A

1

2

3

B

Q

1

Figure 1.16 : Planar Mechanism

For example, reciprocating engine mechanism is a planner mechanism in which link 1 is fixed, link 2 rotates and

link 4 reciprocates. In internal combustion engines, it converts reciprocating motion of piston into rotating motion

of crank. This mechanism is also used in reciprocating compressors in which it converts rotating motion of crank

into reciprocating motion of piston. This was a very common practical example and there are many other examples

like this. More about planar mechanisms shall follow in following sections. Let us consider the two mechanisms shown in Figure 1.17. The curved slider in figure acts similar to the revolute

pair. If radius of curvature ‘’ of the curved slider becomes infinite, the angular motion of the slider changes into

linear displacement and the revolute pair R4 transforms to a prismatic pair. Depending on different type of

kinematic pairs, four bar kinematic chain can be classified into three categories :



4R-kinematic chain which has all the four kinematic pairs as revolute pairs. (a) 3R-1P kinematic chain which has three revolute pairs and one prismatic pair. This is also called as single

slider crank chain. (b) 2R-2P kinematic chain which has two revolute pairs and two prismatic pairs. This is also called as

double slider crank chain.

3 B

R3

B

3 R3 4

R2

A

R2

2 4

2

R1 R4 O4

O2 R1 R4 O4

O2

1 1

1 1

Figure 1.17 : Kinematic Chain

SAQ 2

Form a kinematic chain using three revolute pairs and one prismatic pair.

1.9 INVERSIONS OF KINEMATIC CHAIN

If in a four bar kinematic chain all links are free, motion will be unconstrained. When

one link of a kinematic chain is fixed, it works as a mechanism. From a four link

kinematic chain, four different mechanisms can be obtained by fixing each of the four

links turn by turn. All these mechanisms are called inversions of the parent kinematic

chain. By this principle of inversions of a four link chain, several useful mechanisms can

be obtained.

3 3 3

2 4 2 4 2 4

1 1 1

3 3

2 4

2 4

1

1

Figure 1.18 : Inversion of Kinematic Chain

1.9.1 Inversions of 4R-Kinematic Chain

Kinematically speaking, all four inversions of 4R-kinematic chain are identical.

However, by suitably altering the proportions of lengths of links 1, 2, 3 and 4

respectively several mechanisms are obtained. Three different forms are illustrated here.

In Figure 1.19, links have been shown by blocks and lines connecting them represent

pairs.

1 T1

2 T2

3 T3

4

T4

Figure 1.19 : First Inversion



Crank-lever Mechanism or Crank-rocker Mechanism Simple Mechanisms

This mechanism is shown in Figure 1.20. In this case for every complete rotation of link 2 (called a crank), the link 4 (called a lever or rocker), makes oscillation

between extreme positions O4B1 and O4B2. B

B1 B2

A 3

4

2

A2

A1 O2 O4

Figure 1.20 : Crank-rocker Mechanism

The position of O4B1 is obtained when point A is A1 whereas position O4B2 is

obtained when A is at A2. It may be observed that crank angles for the two strokes

(forward and backward) of oscillating link O4B are not same. It may also be noted

that the length of the crank is very short. If l1, l2, l3 and l4 are lengths of links 1, 2, 3 and 4 respectively, the proportions of the link may be as follows :

(l1 l2 ) (l3 l4 )

(l2 l3 ) (l1 l4 )

Double-leaver Mechanism or Rocker-Rocker Mechanism

In this mechanism, both the links 2 and 4 can only oscillate. This is shown in Figure 1.21.

A1 A I3 B

I4

B2

B1

2

I2 A2 O4

O2

Figure 1.21 : Double-lever Mechanism

Link O2A oscillates between positions O2A1 and O2A2 whereas O4B oscillates

between positions O4B1 and O4B2. Position O4B2 is obtained when O2A and AB

are along straight line and position O2A1 is obtained when AB and O4B are along straight line. This mechanism must satisfy the following relations.

(l3 l4 ) (l1 l2 )

(l2 l3 ) (l1 l4 )

It may be observed that link AB has shorter length as compared to other links.

If links 2 and 4 are of equal lengths and l1 > l3, this mechanism forms automobile

steering gear.

Double Crank Mechanism

The links 2 and 4 of the double crank mechanism make complete revolutions.

There are two forms of this mechanism.

Parallel Crank Mechanism

In this mechanism, lengths of links 2 and 4 are equal. Lengths of links 1 and

3 are also equal. It is shown in Figure 1.22.



Theory of Machines

A 3 B

2

4

O2 O4

1 1

Figure 1.22 : Double Crank Mechanism

The familiar example is coupling of the locomotive wheels where wheels

act as cranks of equal length and length of the coupling rod is equal to

centre distance between the two coupled wheels.

Drag Link Mechanism

In this mechanism also links 2 and 4 make full rotation. As the link 2 and 4

rotate sometimes link 4 rotate faster and sometimes it becomes slow in

rotation.

A1 B1

A B

B2 A4

O O B4

A2

A3 B3

Figure 1.23 : Drag Link Mechanism

The proportions of this mechanism are

l3 l1 ; l4 l2 l3

(l1 l4 l2 )

and l3 (l2 l4 l1)

It may be observed from Figure 1.23 that length of link 1 is smaller as compared to other links.

SAQ 3

Why 4R-kinematic chain does not provide four different mechanisms?

SAQ 4

In this mechanism, if length of link 2 is equal to that of link 4 and link 4 has lengths equal to that of link 2 which mechanism will result and analyse motion.



1.9.2 Inversions of 3R-1P Kinematic Chain or Inversions of

Slider Crank Chain In this four bar kinematic chain, four links shown by blocks are connected through three

revolute pairs T1, T2 and T3 and one prismatic pair.

T1

T2

T3

1 2 3 4

S

Figure 1.24 ; Inversion of Slider Crank Chain

First Inversion

In this mechanism, link 1 is fixed, link 2 works as crank, link 4 works as a slider

and link 3 connects link 2 with 4. It is called connecting rod. Between links 1 and

4 sliding pair has been provided.

1 T1

2 T2

3 T3

4

S

2 3

4

1

Figure 1.25 : First Inversion

This mechanism is also known as slider crank chain or reciprocating engine

mechanism because it is used in internal combustion engines. It is also used in

reciprocating pumps as it converts rotatory motion into reciprocating motion and

vice-versa.

Second Inversion

In this case link 2 is fixed and link 3 works as crank. Link 1 is a slotted link which facilitates movement of link 4 which is a slider. This arrangement gives quick return motion mechanism. The motion of link 1 can be taped through a link and provided to ram of shaper machine. Figure 1.26 shows this mechanism and it is called Whitworth Quick Return Motion Mechanism. The forward stroke starts

when link 3 occupies position O4Q. At that time, point A is at A1. The forward

stroke ends when link 3 occupies position O4P and point A occupies position A2.

The return stroke takes place when link 3 moves from position O4P to O4Q. The stroke length is distance between A1 and A2 along line of stroke. If

acute angle < PO4 Q = and crank rotates at constant speed .

Quick Return Ratio = Time taken in forward stroke

Time taken in return stroke

(2 )

2

Simple Mechanisms



Theory of Machines

1 T1

2 T2

3 T3

4

3 4

S

2

1

4

B

3

O4

1

2 Q Ram

P

6

A1 O2 A2

5

A

Figure 1.26 : Second Inversion

Third Inversion

This inversion is obtained by fixing link 3. Some applications of this inversion are

oscillating cylinder engine and crank and slotted lever quick return motion

mechanism of a shaper machine. Link 1 works as a slider which slides in slotted or

cylindrical link 4. Link 2 works as a crank. The oscillating cylinder engine is

shown in Figure 1.27(a).

1

T1 2

T2 3

T3 4

2

4 1

3

S

3

(a) Oscillating Cylinder Engine

6 Ram

5

1

A

2

O2

A1 4

A2

3

O4

(b) Crank and Slotted Lever Mechanism

Figure 1.27 : Third Inversion

The motion of link 4 in crank and slotted lever quick return motion mechanism

can be taped through link 5 and can be transferred to ram. O2A1 and O2A2 are two positions of crank when link 4 will be tangential to the crank circle and corresponding to which ram will have extreme positions. When crank travels from

position O2A1 and O2A2 forward stroke takes place. When crank moves further

from position O2A2 to O2A1 return stroke takes place. Therefore, for constant

angular velocity for crank ‘’.



(2)

2

Time for forward stroke

Quick Return Ratio =

Time for return stroke

Fourth Inversion – Pendulum Pump

Simple Mechanisms

It is obtained by fixing link 4 which is slider. Application of this inversion is

limited. The pendulum pump and hand pump are examples of this inversion. In

pendulum pump, link 3 oscillates like a pendulum and link 1 has translatory

motion which can be used for a pump.

1 4

3 1

2

Figure 1.28(a) : Pendulum Pump

2

1

3

1 T1

2 T2

3 T3

4

S

4

Figure 1.28(b) : Hand Pump

1.9.3 Inversions of 2R-2P Kinematic Chain or Double

Slider Crank Chain

This four bar kinematic chain has two revolute or turning pairs – T1 and T2 and two

prismatic or sliding pairs – S1 and S2. This chain provides three different mechanisms.

T1 T2 S1 1 2 3 4

S2

Figure 1.29 : Inversion of 2R-2P Kinematic Chain

First Inversion

The first inversion is obtained by fixing link 1. By doing so a mechanism called

Scotch Yoke is obtained. The link 1 is a slider similar to link 3. Link 2 works as a

crank. Link 4 is a slotted link. When link 2 rotates, link 4 has simple harmonic

motion for angle ‘’ of link 2, the displacement of link 4 is given by

x OA cos



Theory of Machines

1 T1

2 T2

3 S1

4

S2

A T2

3 2

s2

S T1

4

O

1 1

Figure 1.30 : Scotch Yoke Mechanism Second Inversion

In this case, link 2 is fixed and a mechanism called Oldham’s coupling is obtained.

This coupling is used to connect two shafts which have eccentricity ‘’. The axes

of the two shafts are parallel but displaced by distance . The link 4 slides in the

two slots provided in links 3 and 1. The centre of this link will move on a circle

with diameter equal to eccentricity.

Fixed Link

1 T1

2 T2

3 S1

4

S2

S1 1 2

3 T1

2 4 S2

T2

Figure 1.31 : Oldham’s Coupling Third Inversion

This inversion is obtained by fixing link 4. The mechanism so obtained is called

elliptical trammel which is shown in Figure 1.32. This mechanism is used to draw

ellipse. The link 1, which is slider, moves in a horizontal slot of fixed link 4. The

link 3 is also a slider moves in vertical slot. The point D on the extended portion

of link 2 traces ellipse with the system of axes shown in the figure, the position

coordinates of point D are as follows :

x AD sin or sin xD

D AD

y D CD cos or cos

yD

CD

Since, sin 2 cos

2 1

Substituting for sin and cos in this equation, the following equation of ellipse is

obtained.

x D2

y1

D 1

AD 2 CD

2



The semi-major axis of the ellipse is AD and semi-minor axis is CD. Simple Mechanisms

T1

T2 Y

1 2 3 S1

4

S2 4 T1

1 D

O X

4

S2

3

2

B C

A

T2 S1

Figure 1.32 : Elliptical Trammel

SAQ 5

(a) Explain why only three different mechanisms are available from 2R-2P kinematic chain.

(b) If length of crank in the reciprocating mechanism is 15 cm, find stroke

length of the slider.

(c) If length of fixed link and crank in crank and slotted lever quick return mechanism are 30 cm and 15 cm respectively, determine quick return ratio.

(d) If an ellipse of semi-major axis 30 cm and semi-minor axis 20 cm is to be

drawn, what should be the length of link ACD in elliptical trammel.

1.10 MACHINE

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

from power source to the resistance to be overcome and, thereby, it performs useful

mechanical work. A common type of example is the commonly used internal-combustion

engine. The burning of petrol or diesel in cylinder results in a force on the piston which is

transmitted to the crank to result in driving torque. This driving torque overcomes the

resistance due to any external agency or friction, etc. at the crankshaft and thereby doing

useful work.

1.10.1 Difference between Machine and Mechanism A system can be defined as a mechanism or a machine on the basis of primary objective.

Sl. No. Machine Mechanism

1 If the system is used with the If the objective is to transfer or objective of transforming transform motion without

mechanical energy, then it is considering forces involved, the

called a machine system is said to be a mechanism

2 Every machine has to transmit It is concerned with transfer of motion because mechanical motion only

work is associated with the

motion, and thus makes use of

mechanisms

3 A machine can use one or It is not the case with mechanisms. more than one mechanism to A mechanism is a single system to

perform the desired function, transfer or transform motion

e.g. sewing machine has

several mechanisms



Theory of Machines

1.11 OTHER MECHANISMS

Geometry of motion of well known lower pair mechanisms will be examined and their

actual working and application will be dealt with. On the strength of analytical study of

these mechanisms, new mechanisms can be developed for specific requirements, for

modern plants and machinery.

1.11.1 Pantograph Pantograph is a geometrical instrument used in drawing offices for reproducing given

geometrical figures or plane areas of any shape, on an enlarged or reduced scale. It is

also used for guiding cutting tools. Its mechanism is utilised as an indicator rig for

reproducing the displacement of cross-head of a reciprocating engine which, in effect,

gives the position of displacement. There could be a number of forms of a pantograph. One such form is shown in Figure 1.33. It comprises of four links : AB, BC, CD, DA, pin-jointed at A, B, C and D.

Link BA is extended to a fixed pin O. Suppose Q is a point on the link AD of which the

motion is to be enlarged, then the link BC is extended to P such that O, Q, P are in a

straight line. It may be pointed out that link BC is parallel to link AD and that AB is

parallel to CD as shown. Thus, ABCD is a parallelogram.

P1 P

C D

C1 D1

Q

Q1

B A

O

A1

B1 Figure 1.33 : Pantograph Mechanism

Suppose a point Q on the link AD moves to position Q1 by rotating the link OAB

downward. Now all the links and the joints will move to the new positions : A to A1, B to B1, C to C1, D to D1 and P to P1 and the new configuration of the mechanism will

be as shown by dotted lines. The movement of Q (QQ1) will stand enlarged to PP1 in a definite ratio and in the same form as proved below : Triangles OAQ and OBP are similar. Therefore,

OBOA OQ

OP

In the dotted position of the mechanism when Q has moved to position Q1 and

correspondingly P to P1, triangles OA1Q1 and OB1P1 are also similar since length of

the links remain unchanged.

OA1

OQ1

OB

OP

1 1

But OB1 OB

OA1 OA

OA

OQ1

OB OP

1

OQ

QQ1

OP OP

1



As such triangle OQQ1 and OPP1 are similar, and PP1 and QQ1 are parallel and further, Simple Mechanisms

PP QQ

1

1

OP OQ

PP OQ QQ1

1 1 OQ

QQ1 OB

OA

Therefore, PP1 is a copied curve at enlarged scale.

1.11.2 Straight Line Motion Mechanisms A mechanism built in such a manner that a particular point in it is constrained to trace a

straight line path within the possible limits of motion, is known as a straight line motion

mechanism.

The Scott Russel Mechanism

This mechanism is shown in Figure 1.34. It consists of a crank OC, connecting rod

CP, and a slider block P which is constrained to move in a horizontal straight line

passing through O. The connecting rod PC is extended to Q such that

PC CQ CO

It will be proved that for all horizontal movements of the slider P, the locus of point Q will be a straight line perpendicular to the line OP.

Q

Locus o

f Q

O

Extension

Rod CD

Crank C

P

Figure 1.34 : Scott Russel Mechanism

Draw a circle of diameter PQ as shown. It is will known that diameter of a circle

always subtends a right angle or any point on the circle. Thus, at point O, the

angle QOP is a right angle. For any position of P, the line connecting O with P

will always be horizontal. Therefore, line joining the corresponding position of Q

with O will always a straight line perpendicular to OP. Thus, the locus of point Q

will be straight line perpendicular to OP. Thus, a horizontal straight line motion of

slider block P will enable point Q to generate a vertical straight line, both passing

through O.

Generated Straight Line Motion Mechanisms

Principle

The principle of working of an accurate straight line mechanism is based

upon the simple geometric property that the inverse of a circle with

reference to a pole on the circle is a straight line. Thus, referring to

Figure 1.35, if straight line OAB always passes through a fixed pole O and

the points A and B move in such a manner that : OA OB = constant, then

the end B is said to trace an inverse line to the locus of A moving on the



Theory of Machines

circle of diameter OC. Stated otherwise if O be a point on the circumference

of a circle diameter OP, OA by any chord, and B is a point on OA produced,

such that OA OB = a constant, then the locus of a point B will be a straight

line perpendicular to the diameter OP. All this is proves as follows :

B

A

O Pole O1 C D

Figure 1.35 : Straight Line Motion Mechanism

Draw a horizontal line from O. From A draw a line perpendicular to OA

cutting the horizontal at C. OC is the diameter of the circle on which the

point A will move about O such that OA OB remains constant.

Now, s OAC and ODB are similar.

Therefore, OA

OD

OC

OB

OD OA OB

OC

But OC is constant and so that if the product OA OB is constant, OD will

be constant, or the position of the perpendicular from B to OC produced is

fixed. This is possible only if the point B moves along a straight path BD

which is perpendicular to OC produced.

A number of mechanisms have been innovated to connect O, B and A in

such a way as to satisfy the above condition. Two of these are given as

follows :

The Peaucellier Mechanism

The mechanism consists of isosceles four bar chain OKBM (Figure 1.36). Additional links AK and AM from, a rhombus AKBM. A is constrained to move on a circular path by the radius bar CA which is equal to the length of the fixed link OC.

CA K

B

L

A

M

O C D

Figure 1.36 : The Peaucellier Mechanism



From the geometry of the figure, it follows that Simple Mechanisms

OA OB (OL AL) (OL LB ) OL2 AL

2 [ AL LB]

(OK 2 KL

2 ) ( AK

2 KL

2 ) OK

2 AK

2 constant

Hence, OA OB is constant for a given configuration and B

traces a straight path perpendicular to OC produced.

The Hart’s Mechanism

This is also known as crossed parallelogram mechanism. It is

an application of four-bar chain. PSQR is a four-bar chain in

which

SP = QR

and SQ = PR (Figure 1.37)

On three links SP, SQ and PQ, then it can be proved that for

any configuration of the mechanism :

OA OB = Constant

Q

a

x b

B R

P M

y

A

a

n

d

T

h

e

n

B

u

t

Hence,

S

Q

P

R

b

P

Q

x

SR y



N

OA OB OS

a x OP

a y xy constant

x SM NM [QK SR]

y SM NM [QN || PS]

xy ( SM 2 NM

2 )

(b 2 QM

2 ) ( a

2 QM

2 ) (b

2 a

2 )

OA OB xy constant = constant



It is therefore, concluded if the mechanism is pivoted at Q as a

fixed point and the point A is constrained to move on a circle

through O the point B will trace a straight line perpendicular to

the radius OC produced.

Approximate Straight-line Mechanisms

With the four-bar chain a large number of mechanisms can be devised which give a path which is approximately straight line. These are given as follows :

The Watt Mechanism

OABO is the mechanism used for Watt for obtaining approximate straight

line motion (Figure 1.38). It consists of three links : OA pivoted at O. OB

pivoted at O and both connected by link AB. A point P can be found on the

link AB which will have an approximate straight line motion over a limited

range of the mechanism. Suppose in the mean position link OA and OB are

in the horizontal position an OA and OB are the lower limits of movement

of these two links such that the configuration is OABO. Let I be the

instantaneous centre of the coupler link AB, which is obtained by

producing OA and BO to meet at I. From I draw a horizontal line to meet

AB at P. This point P, at the instant, will move vertically.

O A

a

A1

P

P1

B O

1

b

B1

Figure 1.38 : The Watts Mechanism

Considering angles and being exceedingly small, as an approximation,

AP

AA

BB

b

BP a b a

Where a and b are the lengths of OA and OB respectively. Since, both OA

and OB are horizontal in the mean or mid-position ever point in the

mechanism then moves vertically.

Hence, if P divides AB in the ratio

AP : BP = b : a

then P will trace a straight line path for a small range of movement on either side of the mean position of AB.

The Grass-hopper Mechanism

It is shown in Figure 1.39. It is a modification of Scott-Russel mechanism. It

consists of crank OC pivoted at O, link OP pivoted at O and a link PCR as

shown. It is, in fact, a laid out four-bar mechanism. Line joining O and P is

horizontal in middle position of the mechanism. The lengths of the link are

so fixed such that :



OC (CP)

2 Simple Mechanisms

CR

If this condition is satisfied, it is found that for a small angular displacement

of the link OP, the point R on the link PCR will trace approximately a straight line path, perpendicular to line PQ.

R

D

R1 C

C1 P P1

O P2

C2

R2

O1

Figure 1.39 : Grass-hopper Mechanism

In Figure 1.39, the positions both of P and R have been shown for three

different configuration of links. It may be noted that the pin at C is slidable

along with link RP such that at each position the above equation is satisfied. Robert’s Mechanism

This is also a four-bar chain ABCD in which links AD = BC (Figure 1.40).

The tracing point P is obtained by intersection of the right bisector of the

couple CD and a perpendicular on the horizontal from the instantaneous

centre I. Thus, an additional link E is connected to the coupler link BC and

the path of point P is approximately horizontal in this Robert’s mechanism.

I

E

D

C

A B

P

Figure 1.40 : Robert’s Mechanism

Tchebicheff’s Mechanism

This consists of four-bar chain in which two links AB and CD of equal

length cross each other; the tracing point P lies in the middle of the



Theory of Machines

connecting link BC (Figure 1.41). The proportions of the links are usually

such that P is directly above A or D in the extreme position of the

mechanism, i.e. when CB lies along AB or when CB lies along CD. It can be

shown that in these circumstances the tracing point P will lies on a straight

line parallel to AD in the two extreme positions and in the mid position if

BC : AD : AB : : 1 : 2 : 2.5

B2 C1

C

P2 P

P1

B

C2 B1

A D

Figure 1.41 : Tchebicheff’s Mechanism

SAQ 6

Which mechanisms are used for

(a) exact straight line, and

(b) approximate straight line.

1.11.3 Automobile Steering Gear

In an automobile vehicle the relative motion between its wheels and the road surface

should be one of pure rolling. To satisfy this condition, the steering gear should be so

designed that when the vehicle is moving along a curved path, the paths of points of

contact of each wheel with the road surface should be concentric circular arcs. The

steering or turning of a vehicle to one side or the other is accomplished by turning the

axis of rotation of the front two wheels. Each front wheel has a separate short axle of

rotation, known as stub axle such as AB and CD. These sub axles are pivoted to the

chassis of the vehicle. To satisfy the condition of pure rolling during turning, the design

of the steering gear should be such that at any instant while turning the axes of rotation

of the front and the rear wheels must intersect at one point which is known as

instantaneous centre denoted by I. The whole vehicle is assumed to be revolving about

this point at the instant considered. In Figure 1.42, AB and CD are the short axles of the

front wheel and EF for the rear wheels.



Outer Front

Right turn Simple Mechanisms

Wheel

B

C

Inner Front

A D Wheel

a

E

F

I

Rear Wheels

Figure 1.42 : Automobile Steering Gear

While turning to the right side, axes of the front and the rear wheels meet at I.

Suppose = The angle by which the inner wheel is turned;

= The angle by which the outer wheel is turned;

A = Distance between the points of the front axles; and

l = Wheel base AE.

As may be seen from the geometry of the Figure 1.42, the angle of turn of the inner

front wheel is always more than the angle of turn of the outer front wheel. From Figure 1.42,

a AC EF EI FI l (cot cot )

a l (cot cot )

This is the fundamental equation of steering. If this equation is satisfied in a vehicle,

it is assured that the vehicle while taking a turn of any angle would not slip but would

have pure rolling motion between its wheels and the road surface.

Types of Steering Gears

There are mainly two types of steering gear mechanisms :

(a) Davis steering gear,

(b) Ackerman’s steering gear,

Both these mechanisms are described separately as follows :

Davis Steering Gear

This steering gear mechanism is shown in Figure 1.43(a). It consists of the

main axle AC having a parallel bar MN at a distance h. The steering is

accomplished by sliding bar MN within the guides (shown) either to left or

to the right hand side. KAB and LCD are two bell-crank levers pivoted with

the main axle at A and C respectively such that BAK and DCL remain

always constant. Arms AK and CL have been provided with slots and these

house die-blocks M and N. With the movement of bar MN at the fixed

height, it is the slotted arms AK and CL which side relative to the die-blocks

M and N.

In Figure 1.43(a), the vehicle has been shown as moving in a straight path

and both the slotted arms are inclined at an angle as shown.



Theory of Machines Now suppose, for giving a turn to the right hand side, the base MN is moved

to the right side by distance x. The bell-crank levers will change to the

positions shown by dotted lines in Figure 1.43(b). The angle turned by the

inner wheel and the outer wheels are and respectively. The arms BA and

CD when produced will meet say at I, which will be the instantaneous

centre.

K L

M h N

B D

A C

(a)

b b

x x

B K

L

h

A

C

D

Suppose

Now,

or

(b) I

Figure 1.43 : Davis Steering Gear

2b = Difference between AC and MN, and

= Angle AK and CL make with verticals in normal position.

tan b

. . . (1.1(a))

h

tan () (b x)

[considering point A] . . . (1.2(b))

h

tan () (b x)

[considering point C] . . . (1.2(c))

h

tan () tan tan

1 tan tan

b

b x

tan

h

h

1

b

tan

h

b h tan (b x) (h

b

tan

)

h

hb h 2 tan (b x ) ( h b tan )

bh hx b 2 tan xb tan



h 2 tan b

2 tan xb tan bh hx bh hx

tan ( h 2 b

2 xb) hx

tan hx

. . . (1.2(d))

(h 2

b 2 xb)

Similarly, tan () tan tan

b x

1 tan tan h

Studying for tan and simplifying :

Simple Mechanisms

tan hx

(h 2

b 2 xb)

After obtaining the expressions for tan and

fundamental equation of steering :

cot cot h

2

b

2

xb

hx

cot cot 2h

b 2 tan

But for correct steering,

. . . (1.2(e))

tan , let us not take up the

( h 2 b

2 xb)

xh

. . . (1.2(f))

cot cot a

l

2 tan a

l

tan a

. . . (1.2(g))

2l

The ratio al varies from 0.4 to 0.5 and correspondingly to 14.1

o.

The demerits of the Davis gear are that due to number of sliding pairs,

friction is high and this causes wear and tear at contact surfaces rapidly,

resulting in in-accuracy of its working.

Ackermann Steering Gear

The mechanism is shown in Figure 1.44(a). This is simpler than that of the

Davis steering gear system. It is based upon four-bar chain. The two

opposite links AC and MN are unequal; AC being longer than MN. The other

two opposite links AM and CN are equal in length. When the vehicle is

moving on a straight path link AC and MN are parallel to each other. The

shorter links AM and CN are inclined at angle to the longitudinal axis of

the vehicle as shown. AB and CD are stub axles but integral part of AM and

CN such that BAM and DCN are bell-crank levers pivoted at A and C. Link

AM and CN are known as track arms and the link MN as track rod. The track

rod is moved towards left or right hand sides for steering. For steering a

vehicle on right hand side, link NM is moved towards left hand side with the

result that the link CN turns clockwise. Thus, the angle is increased and

that on the other side, it is decreased. From the arrangement of the links it is

clear that the link CN or the inner wheel will turn by an angle which is

more than the angle of turn of the outer wheel or the link AM.



Theory of Machines

a

A C

B B

M N

(a)

A C O

B

D

N

M

a

0.3t

I

(b)

Figure 1.44 : Ackermann Steering Gear

To satisfy the basic equation of steering :

cot cot a

l ,

the links AM and MN are suitably proportioned and the angle is suitable

selected. In a given automobile, with known dimensions of the four-bar

links, angle is known. For different angle of turn , the corresponding

value of are noted. This is done by actually drawing the mechanism to a

scale. Thus, for different values of , the corresponding value of and

(cot cot ) are tabulated. As given above, for correct steering,

cot cot a

l

For approximately correct steering, value of al should be between 0.4

and 0.5. Generally, it is 0.455. In fact, there are three values of which give correct

steering; one when = 0, second and third for corresponding turning to the

right and the left hand.

Now there are two values of corresponding to given values of . The

value actually determined graphically by drawing the mechanism and

tabulating corresponding to different values of is known as actual



or a. But the value of obtained from the fundamental equation

cot cot a

l corresponding to different values of is known as

correct or c. On making comparison between the two values it is found

that a is higher than c for small values of , a are lower than c. The

difference is negligible for small value of but for large values of , it is substantial. This would reduce the life of the tyres due to greater wear on

account of slipping but then for large value of , the vehicle takes a sharper turn as such its speed is reduced and accordingly the wear is also reduced.

Thus, the greater difference between a and c for large value of will not

matter much.

As a matter of fact the position for correct steering is only an arbitrary

condition. In Ackermann steering, for keeping the value of angle on the

lower side, the instantaneous centre of the front wheels does not lie on the

line of axis of the rear wheel as shown in Figure 1.44(b).

1.11.4 Hook’s Joint or Universal Coupling It is shown in Figure 1.45, it is also known as universal joint. It is used for connecting

two shafts whose axes are non-parallel but intersecting as shown in Figure 1.45. Both the

shafts, driving and the driven, are forked at their ends. Each fork provides for two

bearings for the respective arms of the cross. The cross has two mutually perpendicular

arms. In fact, the cross acts as an intermediate link between the two shafts. In the figure,

the driven shaft has been shown as inclined at an angle with the driving shaft.

Driven Shaft Forks

Driving Shaft

Cross

Figure 1.45 : Hook’s Joint

The Hook’s joint is generally found being used for transmission of motion from the gear

box to the back axle of automobile and in the transmission of drive to the spindles in a

multi-spindle drilling machines. There are host of other applications of the Hook’s joint

where motion is required to be transmitted in non-parallel shafts with their axes

intersecting.

Figure 1.46(a) gives the end of the driving shafts. AB and CD are the mutually

perpendicular arms of the cross in the initial position. Arm AB is of the driving shaft and

CD for the driven shaft. The plan of rotation of the driving shaft and its arm AB will be

represented in the plane of the paper in elevation. In Figure 1.46(b), i.e. in the plan the direction of driving and driven shaft and that of the

cross arms are given. The driven shaft is inclined at an angle with the axis of the

driving shaft. PP gives the direction of the arm connected to the driving shaft and QQ gives the

direction of the arm connected to the driven shaft. In fact, the traces PP and QQ give the

plane of rotation of the arms of the cross, as seen in the plan view. Now, suppose, the driving shaft turns by angle . The arm AB will also turn by and

will take the position A1B1 as shown in the elevation. Suppose, correspondingly the

driven shaft and its arm CD are rotated by . The new position of CD is C2O. With the

Simple Mechanisms



Theory of Machines

rotation of AB by , it is the projection C1D1 of CD which will rotate through angle .

OC1 is the projection of OC and its sure length is given by OC2 and accordingly the

angle of rotation of the arm CD of the driven shaft, is known.

C

C1 C2

A1

A M N

O B

B1

D1

D

(a) End Elevation

Driving Shaft

O

M N

P

P

O

N1

Drivin

g

Shaft

O

(b) Plan

Figure 1.46

Ratio between and

As given above,

= The angle through which the driving shaft is rotated, and

= The corresponding angle through which the driven shaft is rotated.

Refer Figures 1.46(a) and (b).

tan OM

OM

. . . (1.3)

OC NC

1 2

tan

ON

NC

2

tan OM NC2 OM . . . (1.4)

NC

ON ON

tan

2

But from Figure 1.46(b)

OM cos

ON

tan 1 . . . (1.5)

cos

tan



Ratio between Speed of Driven and Driving Shafts Simple Mechanisms

= Angular speed of the driving shaft, and

1 = Angular speed of the driven shaft.

d

dt

d

1 dt

By Eq. (1.4)

tan

1

tan

cos

tan cos tan

Differentiating both sides,

sec2 d cos sec

2 d

dt

dt

sec 2 cos sec

2

1

cos sec2

cos cos2 sec

2 . . . (1.4(a))

1

sec

2

But sec2 1 tan

2

From Eq. (1.4),

tan

tan

cos

tan

2

tan2

cos

sec2

1 tan

2

1 sin

2

cos2

cos2 cos

2

cos

2 cos

2 sin

2

cos2 cos

2

cos

2 (1 sin

2 ) 1 cos

2

cos2 cos

2

cos

2 cos

2 sin

2 1 cos

2

cos2 cos

2

Hence, sec2

1 cos

2 sin

2

cos

2 cos

2

But as per Eq. (1.4(a)),

cos cos 2 sec

2

1



Theory of Machines Substituting for sec

2

(1 cos2

sin2 ) cos

2 sec

2

1

sec2 cos

2

1 cos2 sin

2

cos

Speed of driven Speed of driver

cos

1 1 cos 2 cos

2

Condition for Maximum and Minimum Speed Ratio

For a given value of :

will be a maximum

1

2 7

c o s

6

3

/ cos

. . . (1.5)

Speed of

Driver Speed of

1 Driven

5

4

Figure 1.47 : Polar Velocity Diagram

when in the equation :

cos ;

1 1 cos2 sin

2

The denominator (1 cos 2 sin

2 ) is minimum, i.e.

cos 2 1 or when cos 1

or when = 0 or 180o, corresponding to points 5 and 6 in Figure 1.47 and the

expression for the maximum speed ratio would be

cos 1 . . . (1.6)

1 sin2

1 cos

or [Represented by points 5 and 6 in Figure 1.47] . . . (1.7)

1 cos

Similarly for a given angle ,

will be minimum when in the equation.

1

cos

1 1 cos2 sin

2

the denominator is maximum.



It will be so when = 90o or 270

o.

In that case cos . . . (1.8)

1

For maximum speed ratio, 1 = cos . . . (1.9)

Represented by points 7 and 8 in Figure 1.47.

Condition for the Same Speed

Simple Mechanisms

cos

1 1 sin

2 cos

2

For

to be unity

1

1

cos

1 sin2 cos

2

cos (1 sin 2 cos

2 )

cos2

1

cos

(1

cos

)

sin2 (1 cos

2 )

For the same speed,

cos

1

1 cos

1

1 cos

. . . (1.10)

Condition for Maximum Variation of Driven Speed

Maximum variation of speed of driven shaft

(

1 max

1 min

)

of driven shaft

mean

where mean =

cos

(1 cos 2 )

Maximum variation cos

cos

sin2

tan sin

. . . (1.11)

cos

Maximum variation = tan sin

If is small, tan = as well as sin =

Maximum variation 2 if is very small . . . (1.12)

Generally, the speed of the driving shaft is constant. As such it can be represented

by a circle of radius . In that case the maximum and minimum speed of the

driven shaft will be

and cos respectively, represented by an ellipse of

cos

major axis

and minor axis cos . This is shown in Figure 1.47 which is

cos

known as polar velocity diagram.

37



Angular Acceleration of the Driven Shaft

That angular acceleration of the driven shaft is given by

d 1 d 1 d d1

1 dt d dt d

But by Eq. (1.4),

1 cos2 sin

2

cos

1

1

cos

1 cos2 sin

2

d cos sin2

sin 2

1

d

(1 cos2 sin

2 )

2

By Eq. (1.13), 1 d1

d

Angular acceleration of driven shaft :

2 cos sin

2 sin 2

1 (1 cos

2 sin

2 )

2

For determining conditions for maximum acceleration, differentiate 1, w.r.t. and equate it to zero. The

resulting expression is, however, very complicated, and it will be found that the following expression

which is derived from the exact expression by a simple approximation, gives results which are sufficiently

close for most practical purpose.

For maximum 1, cos 2 2sin2 . . . (1.15)

sin2

2

This equation gives the value of almost accurate upto a maximum value of as 30o. It should be noted

that the angular acceleration of the driven shaft is a maximum when is approximate 45o, 135

o, etc., i.e.

when the arms of the cross are inclined at 45o to the plane contacting the axes of the two shafts.

MECHAICAL ADVANTAGE

The mechanical advantage of a mechanical is the ratio of the output force or torque at any instant to the input force or torque.

If friction and inertia forces are ignored.

Power input = Power output

If T2 be input torque,

2 be input angular velocity,

T4 be output torque, and

4 be output angular velocity.



T2 2 T4 4

or T4

2

T

2 4

Mechanical Advantage T4

2

T2

4

Thus, mechanical advantage is the reciprocal of the velocity ratio.

1.12 KEY WORDS 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 performs useful

mechanical work.

Mechanism : A mechanism is a combination of rigid or

restraining bodies so shaped and connected that

they move upon each other with definite

relative motion. slider-crank mechanism used

in internal combustion engine or reciprocating

air compressor is the simplest example.

Kinematic Pair : A pair is a joint of two elements that permits

relative motion. The relative motion between

the elements of links that form a pair is

required to be completely constrained or

successfully constrained.

Kinematic Link or Element : Kinematic link is a resistant body or an

assembly of resistant bodies which go to make

a part or parts of a machine connecting other

parts which have motion relative to it.

Resistant Body : A body is said to be a resistant body if it is

capable of transmitting the required forces with

negligible deformation.

Completely Constrained Motion : When the motion between a pair is limited to a

definite direction, irrespective of direction of

force applied and only one independent

variable is required to define motion. Such a

motion is said to be a completely constrained

motion.

Incompletely Constrained Motion : When a connection between the elements,

forming a pair, is such that the constrained

motion is not completed by itself, but by some

other means. Such a motion is said to be a



successfully constrained motion.

Kinematic Chain : A kinematic chain is a combination of

kinematic pairs, joined in such a way, that each

link forms a part of two pairs and the relative

motion between the links or elements is

completely constrained.

Kinematic Inversions : In a mechanism, one of the links in a kinematic

chain is fixed. One may obtain different

mechanisms by fixing in different links in a

kinematic chain. This method of obtaining

different mechanisms is known as kinematic

inversions.

Crank : The link which makes complete rotation.

Rocker : The link oscillates between two extreme

positions.

Coupler : It transfers motion from input link to output

link or vice-versa.



OUT COMES

1) Students understand link, Mechanism and various mechanisms

2) Students able to solve mobility of mechanisms by applying Grubler’s criterion

3) Demonstrate various inversions of kinematic chain and their applications

Exercise

1. Define link, Inversion, Kinematic pair, Kinematic chain, Machine 2. List inversions of a four bar chain and explain whit worth quick return motion

mechanism 3. Explain Pantograph 4. Explain Ackerman steering gear mechanism with a neat sketch 5. Explain ratchet and pawl mechanism

FURTHER READING

1. "Theory of Machines”, Rattan S.S, Tata McGraw-Hill Publishing Company Ltd., New

Delhi, and

3rd edition -2009.

2. "Theory of Machines”, Sadhu Singh, Pearson Education (Singapore) Pvt. Ltd, Indian

Branch New

Delhi, 2nd Edi. 2006

3. “Theory of Machines & Mechanisms", J.J. Uicker, , G.R. Pennock, J.E. Shigley.

OXFORD 3rd

Ed. 2009.

4. Mechanism and Machine theory, Ambekar, PHI, 2007



UNIT 4 Velocity Analysis by Instantaneous Center

Method

CONTENTS

4.1 Definition, Kennedy's Theorem,

4.2 Determination of linear and angular velocity using instantaneous center method

4.3 Klein's Construction: Analysis of velocity and acceleration of single slidercrank

mechanism.

OBJECTIVES:

Learn Instantaneous centre method to do velocity analysis of a kinematic

chain

Study Klein’s construction method to do velocity and acceleration analysis

of slider crank mechanisms

4.1 Definition, Kennedy's Theorem

II Method Instantaneous

Method

To explain instantaneous centre let us consider a plane body P having a nonlinear

motion relative to another body q consider two points A and B on body P having velocities

as Va and Vb respectively in the direction shown.

A V a

B V b

I P

q



If a line is drawn r to Va, at A the body can be imagined to rotate about some point on

the line. Thirdly, centre of rotation of the body also lies on a line r to the direction of Vb at B.

If the intersection of the two lines is at I, the body P will be rotating about I at that

instant. The point I is known as the instantaneous centre of rotation for the body P. The

position of instantaneous centre changes with the motion of the body.

A V a P

B V b

I q

Fig. 2

In case of the r

lines drawn from A and B meet outside the body P as shown in Fig 2.

A V a

B V b

I at

Fig. 3

If the direction of Va and Vb are parallel to the r at A and B met at . This is the case when the

body has linear motion.

• Number of Instantaneous Centers

The number of instantaneous centers in a mechanism depends upon number of

links. If N is the number of instantaneous centers and n is the number of links. • Types of Instantaneous Centers

There are three types of instantaneous centers namely fixed, permanent and neither

fixed nor permanent.

Example: Four bar mechanism. n = 4. n n 1

4 4 1

N = = 6



Fixed instantaneous center I12, I14

Permanent instantaneous center I23, I34

Neither fixed nor permanent instantaneous center I13, I24 • Arnold Kennedy theorem of three centers: Statement: If three bodies have motion relative to each other, their instantaneous

centers should lie in a straight line. Proof:

1 V A3 V A2

I 12

I 13

2 3

I 23 A

Consider a three link mechanism with link 1 being fixed link 2 rotating about

I12 and link 3 rotating about I13. Hence, I12 and I13 are the instantaneous centers for link 2

and link 3. Let us assume that instantaneous center of link 2 and 3 be at point A i.e. I23.

Point A

is a coincident point on link 2 and link 3.

Considering A on link 2, velocity of A with respect to I12 will be a vector VA2

link A I12. Similarly for point A on link 3, velocity of A with respect to I13 will be I13. It is seen that velocity vector of VA2 and VA3 are in different directions which is impossible. Hence, the instantaneous center of the two links cannot be at the

assumed position.

It can be seen that when I23 lies on the line joining I12 and I13 the VA2 and VA3

will be same in magnitude and direction. Hence, for the three links to be in relative

motion all the three centers should lie in a same straight line. Hence, the proof.

4.2 Determination of linear and angular velocity using instantaneous

center method

r to

r to A



Steps to locate instantaneous centers:

Step 1: Draw the configuration

diagram. Step 2: Identify the number of instantaneous centers by using the relation n 1 n N = .

2 Step 3: Identify the instantaneous centers by circle diagram.

Step 4: Locate all the instantaneous centers by making use of Kennedy’s theorem.

To illustrate the procedure let us consider an example.

A slider crank mechanism has lengths of crank and connecting rod equal to 200 mm

and 200 mm respectively locate all the instantaneous centers of the mechanism for the

position of the crank when it has turned through 30o from IOC. Also find velocity of

slider and angular velocity of connecting rod if crank rotates at 40 rad/sec. Step 1: Draw configuration diagram to a suitable scale.

Step 2: Determine the number of links in the mechanism and find number of instantaneous

centers. n 1 n N =

2

4 4 1

n = 4 linksN = = 6

2

I 13

I 24

A

3

2

200 I 23 800 B

I 12

30o

4

O 1 1 I 12

I 14 to I 14 to

Step 3: Identify instantaneous centers. o Suit it is a 4-bar link the



resulting figure will be a square.

1 I12 2



I 24

1 2 3 4

I 12 I 23

I 34

I 13 I 24

I 13 I 14

I41

I23 OR

4 I34 3

o Locate fixed and permanent instantaneous centers. To locate neither fixed nor

permanent instantaneous centers use Kennedy’s three centers theorem.

Step 4: Velocity of different points. Va = 2 AI12 = 40 x 0.2 = 8 m/s

also Va = 2 x A13

3 = Va

AI13

Vb = 3 x BI13 = Velocity of slider.

OUT COME

Analyze the given the simple mechanisms for calculating velocity and acceleration

at various points

Exercise

1. Explain Kennedy’s theorem with proof.

2. A reciprocating engine mechanism has connecting rod 200mm long and

crank 50mm long. By KLEIN’s construction, determine the velocity and acceleration of piston,

and angular acceleration of connecting rod, when the crank has turned through 450 from IDC

clockwise and is rotating at 240rpm.



UNIT 6 SPUR GEARS

CONTENTS

6.1 Introduction

6.2 Gear terminology,

6.3 Law of gearing,

6.4 Path of contact, arc of contact, contact ratio of spur gear.

6.5 Interference in involute gears, methods of avoiding interference,

6.6 Back lash, condition for minimum number of teeth to avoid

interference,

6.7 Expressions for arc of contact and path of contact

OBJECTIVES

1. Overview the various types gears and terminologies used

2. Comprehend the theorems and expressions used for design of gears

6.1 Introduction

Spur gears: Spur gears are the most common type of gears. They have straight teeth, and

are mounted on parallel shafts. Sometimes, many spur gears are used at once to create

very large gear reductions. Each time a gear tooth engages a tooth on the other gear,

the teeth collide, and this impact makes a noise. It also increases the stress on the gear

teeth. To reduce the noise and stress in the gears, most of the gears in your car are

helical.

Spur gears (Emerson Power Transmission Corp)

External contact Internal contact



Spur gears are the most commonly used gear type. They are characterized by teeth,

which are perpendicular to the face of the gear. Spur gears are most commonly available,

and are generally the least expensive. • Limitations: Spur gears generally cannot be used when a direction change between

the two shafts is required. • Advantages: Spur gears are easy to find, inexpensive, and efficient.

2. Parallel helical gears: The teeth on helical gears are cut at an angle to the face of the

gear. When two teeth on a helical gear system engage, the contact starts at one end of

the tooth and gradually spreads as the gears rotate, until the two teeth are in full

engagement.

Helical gears



(Emerson Power Transmission Corp) Herringbone gears

This gradual engagement makes helical gears operate much more smoothly and quietly than

spur gears. For this reason, helical gears are used in almost all car transmission. Because of the angle of the teeth on helical gears, they create a thrust load on the gear when

they mesh. Devices that use helical gears have bearings that can support this thrust load. One interesting thing about helical gears is that if the angles of the gear teeth are correct,

they can be mounted on perpendicular shafts, adjusting the rotation angle by 90 degrees.

Helical gears to have the following differences from spur gears of the same size:

o Tooth strength is greater because the teeth are longer,

o Greater surface contact on the teeth allows a helical gear to carry more load than a spur

gear o The longer surface of contact reduces the efficiency of a helical gear relative to a spur

gear Rack and pinion (The rack is like a gear whose axis is at

infinity.): Racks are straight gears that are used to convert

rotational motion to translational motion by means of a

gear mesh. (They are in theory a gear with an infinite pitch

diameter). In theory, the torque and angular velocity of the pinion gear are related to the Force and the velocity of the rack by the radius of the

pinion gear, as is shown. Perhaps the most well-known application of a rack is the rack and pinion steering system

used on many cars in the past Gears for connecting intersecting shafts: Bevel gears are useful when the direction of

a shaft's rotation needs to be changed. They are usually mounted on shafts that are 90

degrees apart, but can be designed to work at other angles as well. The teeth on bevel gears can be straight, spiral or hypoid. Straight bevel gear teeth actually

have the same problem as straight spur gear teeth, as each tooth engages; it impacts the

corresponding tooth all at once. Just like with spur gears, the solution to this problem is to curve the gear teeth. These spiral

teeth engage just like helical teeth: the contact starts at one end of the gear and

progressively spreads across the whole tooth.



Straight bevel gears Spiral bevel gears

On straight and spiral bevel gears, the shafts must be perpendicular to each other, but they

must also be in the same plane. The hypoid gear, can engage with the axes in different

planes. This feature is used in many car differentials. The ring gear of the differential and the

input pinion gear are both hypoid. This allows the input pinion to be mounted lower than

the axis of the ring gear. Figure shows the input pinion engaging the ring gear of the

differential. Since the driveshaft of the car is connected to the input pinion, this also

lowers the driveshaft. This means that the driveshaft

Hypoid gears (Emerson Power Transmission Corp)

doesn't pass into the passenger compartment of the car as much, making more room for people

and cargo. Neither parallel nor intersecting shafts: Helical gears may be used to mesh two shafts that

are not parallel, although they are still primarily use in parallel shaft applications. A special

application in which helical gears are used is a crossed gear mesh, in which the two shafts are

perpendicular to each other.



Crossed-helical gears

Worm and worm gear: Worm gears are used when large gear reductions are

needed. It is common for worm gears to have reductions of 20:1, and even up

to 300:1 or greater. Many worm gears have an interesting property that no other gear set has: the

worm can easily turn the gear, but the gear cannot turn the worm. This is

because the angle on the worm is so shallow that when the gear tries to spin

it, the friction between the gear and the worm holds the worm in place. This feature is useful for machines such as conveyor systems, in which the

locking feature can act as a brake for the conveyor when the motor is not

turning. One other very interesting usage of worm gears is in the Torsen

differential, which is used on some high-performance cars and trucks.

6.2 Gear terminology,



Addendum: The radial distance between the Pitch Circle and the top of the teeth.

Arc of Action: Is the arc of the Pitch Circle between the beginning and the end of

the engagement of a given pair of teeth.

Arc of Approach: Is the arc of the Pitch Circle between the first point of contact of the gear

teeth and the Pitch Point.

Arc of Recession: That arc of the Pitch Circle between the Pitch Point and the last point of

contact of the gear teeth.

Backlash: Play between mating teeth.

Base Circle: The circle from which is generated the involute curve upon which the tooth profile

is based.

Center Distance: The distance between centers of two gears.

Chordal Addendum: The distance between a chord, passing through the points where the Pitch

Circle crosses the tooth profile, and the tooth top.

Chordal Thickness: The thickness of the tooth measured along a chord passing through the

points where the Pitch Circle crosses the tooth profile.

Circular Pitch: Millimeter of Pitch Circle circumference per tooth.

Circular Thickness: The thickness of the tooth measured along an arc following the Pitch Circle

Clearance: The distance between the top of a tooth and the bottom of the space into which it fits

on the meshing gear.

Contact Ratio: The ratio of the length of the Arc of Action to the Circular Pitch.

Dedendum: The radial distance between the bottom of the tooth to pitch circle.

Diametral Pitch: Teeth per mm of diameter.

Face: The working surface of a gear tooth, located between the pitch diameter and the top of the

tooth.

Face Width: The width of the tooth measured parallel to the gear axis.

Flank: The working surface of a gear tooth, located between the pitch diameter and the bottom

of the teeth

Gear: The larger of two meshed gears. If both gears are the same size, they are both called

"gears".

Land: The top surface of the tooth.



Line of Action: That line along which the point of contact between gear teeth travels, between

the first point of contact and the last.

Module: Millimeter of Pitch Diameter to Teeth.

Pinion: The smaller of two meshed gears.

Pitch Circle: The circle, the radius of which is equal to the distance from the center of the gear

to the pitch point.

Diametral pitch: Teeth per millimeter of pitch diameter.

Pitch Point: The point of tangency of the pitch circles of two meshing gears, where the Line of

Centers crosses the pitch circles.

Pressure Angle: Angle between the Line of Action and a line perpendicular to the Line of

Centers.

Profile Shift: An increase in the Outer Diameter and Root Diameter of a gear, introduced to

lower the practical tooth number or acheive a non-standard Center Distance.

Ratio: Ratio of the numbers of teeth on mating gears.

Root Circle: The circle that passes through the bottom of the tooth spaces.

Root Diameter: The diameter of the Root Circle.

Working Depth: The depth to which a

tooth extends into the space between teeth

on the mating gear.

6.3 Law of gearing

Gear-Tooth Action

Figure 5.2 shows two mating gear teeth, in

which • Tooth profile 1 drives tooth profile 2 by acting

at the instantaneous contact point K. • N1N2 is the common normal of the two profiles.

• N1 is the foot of the perpendicular from O1 to

N1N2

• N2 is the foot of the perpendicular from O2 to

N1N2.



Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are

equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other.

Constant Velocity Ratio For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion

transmission between two gears is equivalent to the motion transmission between two imagined slip-less

cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centers are at O1 and O2, and through pitch point P. These two circles are termed pitch circles. The velocity ratio is equal to the

inverse ratio of the diameters of 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) A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions

of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves,

and the relative rotation speed of the gears will be constant(constant velocity ratio). Conjugate Profiles

To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must pass

through the corresponding pitch 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 the fundamental law of gear-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 cycloidal andinvolute profiles. The involute has

important advantages; it is easy to manufacture and the center distance 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 the involute profile. The most commonly used conjugate tooth curve is the involute

curve. (Erdman & Sandor). conjugate action : It is essential for correctly meshing gears, the size of the teeth ( the module ) must be

the same for both the gears. Another requirement - the shape of teeth necessary for the speed ratio to remain constant during an

increment of rotation; this behavior of the contacting surfaces (ie. the teeth flanks) is known as conjugate action.

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 and principles. 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.



Generation of the 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 string, which 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.

Figure 4.3 Involute curve

Properties of Involute Curves

1. The line rolls without slipping on the circle.

2. For any instant, the instantaneous center of the motion of the line is its point of tangent with the circle.

Note: We have not defined the term instantaneous center previously. The instantaneous center or

instant center is defined in two ways.

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 will also be perpendicular to the line at any instant.

There is no involute curve within the base circle.

Cycloidal profile:



Epicycliodal Profile:

Hypocycliodal Profile:

The involute profile of gears has important advantages;

• It is easy to manufacture and the center distance between a pair of involute gears can be varied

without changing the velocity ratio. Thus close tolerances between shaft locations are not

required. The most commonly used conjugate tooth curve is the involute curve. (Erdman &

Sandor). 2. In involute gears, the pressure angle, remains constant between the point of tooth engagement

and disengagement. It is necessary for smooth running and less wear of gears.

But in cycloidal gears, the pressure angle is maximum at the beginning of engagement, reduces to zero

at pitch point, starts increasing and again becomes maximum at the end of engagement. This results in

less smooth running of gears. 3. The face and flank of involute teeth are generated by a single curve where as in cycloidalgears, double

curves (i.e. epi-cycloid and hypo-cycloid) are required for the face and flank respectively. Thus the

involute teeth are easy to manufacture than cycloidal teeth.

In involute system, the basic rack has straight teeth and the same can be cut with simple tools.

Advantages of Cycloidal gear teeth:

1. Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than the involute

gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred specially for cast teeth.



2. In cycloidal gears, the contact takes place between a convex flank and a concave surface, where as in

involute gears the convex surfaces are in contact. This condition results in less wear in cycloidal gears

as compared to involute gears. However the difference in wear is negligible 3. In cycloidal gears, the interference does not occur at all. Though there are advantages of cycloidal gears

but they are outweighed by the greater simplicity and flexibility of the involute gears. Properties of involute teeth:

1. A normal drawn to an involute at pitch point is a tangent to the base circle. 2. Pressure angle remains constant during the mesh of an involute gears. 3. The involute tooth form of gears is insensitive to the centre distance and depends only on the

dimensions of the base circle. 4. The radius of curvature of an involute is equal to the length of tangent to the base circle. 5. Basic rack for involute tooth profile has straight line form. 6. The common tangent drawn from the pitch point to the base circle of the two involutes is the line of

action and also the path of contact of the involutes. 7. When two involutes gears are in mesh and rotating, they exhibit constant angular velocity ratio and is

inversely proportional to the size of base circles. (Law of Gearing or conjugate action) 8. Manufacturing of gears is easy due to single curvature of profile.

System of Gear Teeth

The following four systems of gear teeth are commonly used in practice:

1. 14 ½O

Composite system 2. 14 ½

O Full depth involute system

3. 20

O Full depth involute system

4. 20

O Stub involute system

The 14½O

composite system is used for general purpose gears. It is stronger but has no interchangeability. The tooth profile of this system has cycloidal curves at the top

and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs.

The tooth profile of the 14½O

full depth involute system was developed using gear hobs for spur and

helical gears.

The tooth profile of the 20o full depth involute system may be cut by hobs.

The increase of the pressure angle from 14½o to 20

o results in a stronger tooth, because the tooth acting

as a beam is wider at the base.

The 20o stub involute system has a strong tooth to take heavy loads.



Constant Velocity Ratio

For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion

transmission between two gears is equivalent to the motion transmission between two imagined

slip-less cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose

centers are at O1 and O2, and through pitch point P. These two circles are termed pitch circles.

The velocity ratio is equal to the inverse ratio of the diameters of 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) A common normal (the line of action) to the tooth profiles at their point of contact must, in all

positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch

point Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate

curves, and the relative rotation speed of the gears will be constant(constant velocity ratio). Conjugate Profiles

To obtain the expected velocity ratio of two tooth profiles, the normal line of their profiles must

pass through the corresponding pitch 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 the fundamental law of gear-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 cycloidal andinvolute profiles. The involute

has important advantages; it is easy to manufacture and the center distance 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 the involute profile. The most commonly used

conjugate tooth curve is the involute curve. (Erdman & Sandor). conjugate action : It is essential for correctly meshing gears, the size of the teeth ( the module )

must be the same for both the gears. Another requirement - the shape of teeth necessary for the speed ratio to remain constant during

an increment of rotation; this behavior of the contacting surfaces (ie. the teeth flanks) is known

as conjugate action.



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 and principles. 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.

Generation of the 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 string,

which 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. Properties of Involute Curves 3. The line rolls without slipping on the circle. 4. For any instant, the instantaneous center of the motion of the line is its point of tangent with

the circle.

Note: We have not defined the term instantaneous center previously. The instantaneous center

or instant center is defined in two ways.

4. 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. 5. 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.



6. 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 will also be perpendicular to the line at any

instant.

There is no involute curve within the base circle.

Cycloidal profile:

Epicycliodal Profile:

Hypocycliodal Profile:

The involute profile of gears has important advantages;

• It is easy to manufacture and the center distance between a pair of involute gears can be

varied without changing the velocity ratio. Thus close tolerances between shaft locations

are not required. The most commonly used conjugate tooth curve is the involute curve.

(Erdman & Sandor).



4. In involute gears, the pressure angle, remains constant between the point of tooth

engagement and disengagement. It is necessary for smooth running and less wear of gears.

But in cycloidal gears, the pressure angle is maximum at the beginning of engagement,

reduces to zero at pitch point, starts increasing and again becomes maximum at the end of engagement. This results in less smooth running of gears.

5. The face and flank of involute teeth are generated by a single curve where as in

cycloidalgears, double curves (i.e. epi-cycloid and hypo-cycloid) are required for the face

and flank respectively. Thus the involute teeth are easy to manufacture than cycloidal teeth.

In involute system, the basic rack has straight teeth and the same can be cut with simple tools.

Advantages of Cycloidal gear teeth:

9. Since the cycloidal teeth have wider flanks, therefore the cycloidal gears are stronger than

theinvolute gears, for the same pitch. Due to this reason, the cycloidal teeth are preferred

specially for cast teeth. 10. In cycloidal gears, the contact takes place between a convex flank and a concave surface,

where as in involute gears the convex surfaces are in contact. This condition results in less

wear in cycloidal gears as compared to involute gears. However the difference in wear is

negligible 11. In cycloidal gears, the interference does not occur at all. Though there are advantages of

cycloidal gears but they are outweighed by the greater simplicity and flexibility of the involute

gears. Properties of involute teeth:

1. A normal drawn to an involute at pitch point is a tangent to the base circle. 2. Pressure angle remains constant during the mesh of an involute gears. 3. The involute tooth form of gears is insensitive to the centre distance and depends only on

thedimensions of the base circle. 12. The radius of curvature of an involute is equal to the length of tangent to the base circle. 13. Basic rack for involute tooth profile has straight line form. 14. The common tangent drawn from the pitch point to the base circle of the two involutes is

theline of action and also the path of contact of the involutes. 15. When two involutes gears are in mesh and rotating, they exhibit constant angular velocity

ratioand is inversely proportional to the size of base circles. (Law of Gearing or conjugate

action)



16. Manufacturing of gears is easy due to single curvature of profile.

System of Gear Teeth

The following four systems of gear teeth are commonly used in practice:

5. 14 ½O

Composite system

6. 14 ½O

Full depth involute system

7. 20O

Full depth involute system

8. 20O

Stub involute system

The 14½O

composite system is used for general purpose gears.

It is stronger but has no interchangeability. The tooth profile of this system has cycloidal

curves at the top and bottom and involute curve at the middle portion. The teeth are produced by formed milling cutters or hobs.

The tooth profile of the 14½O

full depth involute system was developed using gear hobs for

spur and helical gears.

The tooth profile of the 20o full depth involute system may be cut by hobs.

The increase of the pressure angle from 14½o to 20

o results in a stronger tooth, because the

tooth acting as a beam is wider at the base.

The 20o stub involute system has a strong tooth to take heavy loads.


The study of the geometry of the involute profile for gear teeth is called involumetry. Consider

an involute of base circle radius ra and two points B and C on the involute as shown in figure.

Draw normal to the involute from the points B and C. The normal BE and CF are tangents to

the Base circle.



Let ra= base circle radius of gear rb= radius of point B on the involute rc= radius of point C on the

involute and From the properties of the Involute:

Arc AE = Length BE and

Arc AF = Length CF

2rc

Using this equation and knowing tooth thickness at any point on the tooth, it is possible

to calculate the thickness of the tooth at any point Number of Pairs of Teeth in Contact

Continuous motion transfer requires two pairs of teeth in contact at the ends of the path

of contact, though there is only one pair in contact in the middle of the path, as in Figure.

R

r

The average number of teeth in contact is an important parameter - if it is too low due to the use

of inappropriate profile shifts or to an excessive centre distance.The manufacturing

inaccuracies may lead to loss of kinematic continuity - that is to impact, vibration and noise. The average number of teeth in contact is also a guide to load sharing between teeth; it is

termed the contact ratio

6.4 Path of contact, arc of contact, contact ratio of spur gear.

Length of path of contact for Rack and Pinion:



Let

r = Pitch circle radius of the pinion = O1P

= Pressure angle ra. = Addendu m

radius of the pinion a = Addendum of

rack

EF = Length of path of contact

EF = Path of approach EP + Path of recess PF

From AP a

sin

triang EP EP

(1)

leO1N

a

P :Path of approach EP

NP O1Psin

rsin

sin (2)

O1N O1Pcos Path of recess PF NF NP (3)

rcos

From triangle O1NF:

NF O1F2 O1N2 12 ra2 r2 cos2 12

Substituting NP and NF values in theequation (3)



Path of racess PF ra2

r2 cos

2 1

2 r sin

Path of

length

of

contact EF EP PF a

ra2

r2 cos

2

12 r sin sin

Exercise problems refer presentation slides


Pitch

Pitch

Circle

Figure shows a pinion and a gear in mesh with their center as O1andO2 respectively. MN is the

common tangent to the basic circles and KL is the path of contact between the two mating teeth.

Consider, the radius of the addendum circle of pinion is increased to O1N, the point of contact L

will moves from L to N. If this radius is further increased, the point of contact L will be inside

of base circle of wheel and not on the involute profile of the pinion. The tooth tip of the pinion will then undercut the

tooth on the wheel at the root and damages Wheel

part

of the involute profile. This effect is known as

interference, and occurs when the teeth are

being cut and weakens the tooth at its root.

In general, the phenomenon, when the tip of

Undercut Pinion

tooth undercuts the root on its mating gear is

known as interference.



Similarly, if the radius of the addendum circles of the wheel increases beyond O2M, then the tip

of tooth on wheel will cause interference with the tooth on pinion. The points M and N are

called interference points. Interference may be avoided if the path of the contact does not extend beyond interference

points. The limiting value of the radius of the addendum circle of the pinion is O1N and of the

wheel is O2M. The interference may only be prevented, if the point of contact between the two teeth is

always on the involute profiles and if the addendum circles of the two mating gears cut the

common tangent to the base circles at the points of tangency. When interference is just prevented, the maximum length of path of contact is MN.

Methods to avoid Interference

1. Height of the teeth may be reduced. 2. Under cut of the radial flank of the pinion. 3. Centre distance may be increased. It leads to increase in pressure angle. 4. By tooth correction, the pressure angle, centre distance and base circles remain unchanged,

buttooth thickness of gear will be greater than the pinion tooth thickness. Minimum

number of teeth on the pinion avoid Interference

The pinion turns clockwise and drives the gear as shown in Figure.

Points M and N are called interference points. i.e., if the contact takes place beyond M and

N, interference will occur. The limiting value of addendum circle radius of pinion is O1N and the limiting value of

addendum circle radius of gear is O2M. Considering the critical addendum circle radius of

gear, the limiting number of teeth on gear can be calculated. Let

Ф = pressure angle

R = pitch circle radius of gear = ½mT r =

pitch circle radius of pinion = ½mt T & t =

number of teeth on gear & pinion

m = module

aw = Addendum constant of gear (or) wheel ap = Addendum

constant of pinion aw. m = Addendum of gear ap. m = Addendum

of pinion



6.6 Back lash, condition for minimum number of teeth to avoid interference,

Minimum number of teeth on the wheel avoid Interference

From triangle O2MP, applying cosine rule and simplifying, The limiting radius of wheel

addendum circle: Minimum number of teeth on the pinion for involute rack to avoid Interference

The rack is part of toothed wheel of infinite

diameter. The base circle diameter and profile of the involute teeth are straight lines.

PITCH LINE

RACK

Let t= Minimum number of teeth on the pinion r=

Pitch circle radius of the pinion = ½ mt

= Pressure angle

AR.m= Addendum of rack

The straight profiles of the rack are tangential to the pinion profiles at the point of contact

and perpendicular to the tangent PM. Point L is the limit of interference.

Backlash:

The gap between the non-drive face of the pinion tooth and the adjacent wheel tooth is known

as backlash.



If the rotational sense of the pinion were to reverse, then a period of unrestrained pinion

motion would take place until the backlash gap closed and contact with the wheel tooth re-

established impulsively. Backlash is the error in motion that occurs when gears change direction. The term "backlash"

can also be used to refer to the size of the gap, not just the phenomenon it causes; thus, one could

speak of a pair of gears as having, for example, "0.1 mm of backlash." A pair of gears could be designed to have zero backlash, but this would presuppose perfection

in manufacturing, uniform thermal expansion characteristics throughout the system, and no

lubricant. Therefore, gear pairs are designed to have some backlash. It is usually provided by reducing

the tooth thickness of each gear by half the desired gap distance. In the case of a large gear and a small pinion, however, the backlash is usually taken entirely

off the gear and the pinion is given full sized teeth. Backlash can also be provided by moving the gears farther apart. For situations, such as

instrumentation and control, where precision is important, backlash can be minimised

through one of several techniques.

Let r = standard pitch circle radius of pinion

R = standard pitch circle radius of wheel c =

standard centre distance = r +R r’ =

operating pitch circle radius of pinion



R’ = operating pitch circle radius of wheel

c’ = operating centre distance = r’ + R’

Ф = Standard pressure angle

Ф’ = operating pressure angle h = tooth thickness of pinion on

standard pitch circle= p/2 h’ = tooth thickness of pinion on

operating pitch circle Let

H = tooth thickness of gear on standard pitch circle

H1 = tooth thickness of gear on operating pitch circle p =

standard circular pitch = 2п r/ t = 2пR/T

p’ = operating circular pitch = 2п r1/t = 2пR1/T

∆C = change in centre distance B

= Backlash t = number of teeth

on pinion

T = number of teeth on gear.

Involute gears have the invaluable ability of providing conjugate action when the gears' centre

distance is varied either deliberately or involuntarily due to manufacturing and/or mounting

errors. There is an infinite number of possible centre distances for a given pair of profile shifted gears,

however we consider only the particular case known as the extended centre distance. Non Standard Gears:

The important reason for using non standard gears are to eliminate undercutting, to prevent

interference and to maintain a reasonable contact ratio.

The two main non- standard gear systems:

(1) Long and short Addendum system and

(2) Extended centre distance system.

Long and Short Addendum System: The addendum of the wheel and the addendum of the pinion are generally made of equal lengths.

Here the profile/rack cutter is advanced to a certain increment towards the gear blank and the same quantity of increment will be withdrawn from the pinion blank.



Therefore an increased addendum for the pinion and a decreased addendum for the gear

is obtained. The amount of increase in the addendum of the pinion should be exactly

equal to the addendum of the wheel is reduced. The effect is to move the contact region from the pinion centre towards the gear centre,

thus reducing approach length and increasing the recess length. In this method there is

no change in pressure angle and the centre distance remains standard. Extended centre distance system:

Reduction in interference with constant contact ratio can be obtained by increasing the

centre distance. The effect of changing the centre distance is simply in increasing the

pressure angle. In this method when the pinion is being cut, the profile cutter is withdrawn a certain

amount from the centre of the pinion so the addendum line of the cutter passes through

the interference point of pinion. The result is increase in tooth thickness and decrease in

tooth space. Now If the pinion is meshed with the gear, it will be found that the centre distance has

been increased because of the decreased tooth space. Increased centre distance will have

two undesirable effects.

OUT COME

Students able to Analyse & Calculate the design specifications of

simple involute gears with given inputs.

Exercise

1. With a neat sketch derive the Length of arc of contact Derivation

2. Two gears in mesh have a module of 8mm and a pressure angle of 200. The larger

gear has 57 teeth while the pinion has 23 teeth. If the addendum on pinion and gear

wheel are equal to one module, find

a) The number of pairs of teeth in contact

b) The angle of action of the pinion and the gear wheel



UNIT 7 Gear Trains

CONTENTS

7.1 Simple gear trains,

7.2 Compound gear trains for large speed. reduction,

7.3 Epicyclic gear trains,

7.4 Algebraic and tabular methods of finding velocity ratio of epicyclic gear trains.

7.5 Torque calculations in epicyclic gear trains

OBJECTIVES:

Outline the procedure for analyzing gear train to calculate speeds and

velocities of gears

7.1 Simple gear trains

A gear train is two or more gear working together by meshing their teeth and

turning each other in a system to generate power and speed. It reduces speed and

increases torque. To create large gear ratio, gears are connected together to form

gear trains. They often consist of multiple gears in the train. The most common of the gear train is the gear pair connecting parallel shafts. The

teeth of this type can be spur, helical or herringbone. The angular velocity is

simply the reverse of the tooth ratio. Any combination of gear wheels employed to transmit motion

from one shaft to the other is called a gear train. The meshing of

two gears may be idealized as two smooth discs with their edges

touching and no slip between them. This ideal diameter is called

the Pitch Circle Diameter (PCD) of the gear.



Simple Gear Trains

The typical spur gears as shown in diagram. The direction of rotation is reversed

from one gear to another. It has no affect on the gear ratio. The teeth on the gears

must all be the same size so if gear A advances one tooth, so does B and C. t = number of teeth on the gear,

D = Pitch circle diameter, N = speed in rpm

D

m = module =

t

and module must be the

same for all gears

otherwise they would not

mesh.

DA

DB

D m = = = C

tA tB tC

DA = m tA; DB = m tB and DC = m tC

= angular velocity.

GEAR 'A' GEAR 'B' GEAR 'C'

D

v = linear velocity on the circle. v = = r (Idler gear)

2

Application:

a) to connect gears where a large center distance is required b) to obtain desired direction of motion of the driven gear ( CW or CCW) c) to obtain high speed ratio

Torque & Efficiency

The power transmitted by a torque T N-m applied to a shaft rotating at N rev/min is

given by:

P 2 NT

60



In an ideal gear box, the input and output powers are the same so;

P2 N1T1 2 N2 T2

60 60

T2 N1 GR

N1T1 N2 T2

T1 N2

It follows that if the speed is reduced, the torque is increased and vice versa. In a real

gear box, power is lost through friction and the power output is smaller than the power

input. The efficiency is defined as:

Power out 2 N2T2 60 N 2T2

Power In 2 N1T1 60 N1T1

Because the torque in and out is different, a gear box has to be

clamped in order to stop the case or body rotating. A holding

torque T3 must be applied to the body through the clamps. The total torque must add up to zero.

T1 + T2 + T3 = 0

If we use a convention that anti-clockwise is positive and clockwise is negative we

can determine the holding torque. The direction of rotation of the output shaft

depends on the design of the gear box.

7.2 Compound gear trains for large speed. reduction, Compound Gear train

Compound gears are simply a chain of simple gear trains with the input of the second

being the output of the first. A chain of two pairs is shown below. Gear B is the output

of the first pair and gear C is the input of the second pair. Gears B and C are locked to

the same shaft and revolve at the same speed. For large velocities ratios, compound gear

train arrangement is preferred.

The velocity of each tooth on A and B are the same

GEAR 'D' so: A tA = B tB -as they are simple gears.

Likewise for C and D, C tC = D tD.



Reverted Gear train

The driver and driven axes lies on the same line. These

are used in speed reducers, clocks and machine tools.

GR NA tB tD

ND tA tC

If R and T=Pitch circle radius & number of teeth of the gear

RA + RB = RC + RD and tA + tB = tC + tD



7.3 Epicyclic gear trains

Observe point p and you will see that gear B also revolves once on its own axis. Any object

orbiting around a center must rotate once. Now consider that B is free to rotate on its shaft

and meshes with C.



Suppose the arm is held stationary and gear C is rotated once. B spins about its own center

and the Now consider that C is unable to rotate and the arm A is revolved once. Gear B will revolve

because of the orbit. It is this extra rotation that causes confusion. One way to get round this

is to imagine that the whole system is revolved once. Then identify the gear that is fixed and

revolve it back one revolution. Work out the revolutions of the other gears and add them up.

The following tabular method makes it easy. Suppose gear C is fixed and the arm A makes one revolution. Determine how

many revolutions the planet gear B makes. Step 1 is to revolve everything once about the center.

Step 2 identify that C should be fixed and rotate it backwards one revolution keeping the

arm fixed as it should only do one revolution in total. Work out the revolutions of B. Step 3 is simply add them up and we find the total revs of C is zero and for the arm is 1.

Opposite

Step Action A B C

1 Revolve all once 1 1 1

2

Revolve C by –1 revolution, 0

-1

keeping the arm fixed tC tB

3

Add

1 1

0

tC tB

tC

The number of revolutions made by B is 1 tB Note that if C revolves -1, then

the direction of B is



7.4 Algebraic and tabular methods of finding velocity ratio of epicyclic

gear trains.

Example: A simple epicyclic gear has a fixed sun gear with 100 teeth and a planet gear with

50 teeth. If the arm is revolved once, how many times does the planet gear revolve? Solution:

Step Action A B C

1 Revolve all once 1 1 1

2 Revolve C by –1 revolution,

0

-1

keeping the arm fixed

3 Add 1 3 0

Gear B makes 3 revolutions for every one of the arm.

The design so far considered has no identifiable input and output. We need a design that

puts an input and output shaft on the same axis. This can be done several ways. Problem 1: In an ecicyclic gear train shown in figure, the arm A is fixed to the shaft S. The

wheel B having 100 teeth rotates freely on the shaft S. The wheel F having 150 teeth driven

separately. If the arm rotates at 200 rpm and wheel F at 100 rpm in the same direction; find

(a) number of teeth on the gear C and (b) speed of wheel B.



Solution:

TB=100; TF=150; NA=200rpm; NF=100rpm:

Since the module is same for all gears : the number

of teeth on the gearsis proportional to the pitch cirlce: The gear B and gear F rotates in the opposite directions:

The Gear B rotates at 350 rpm in the same direction of gears F and Arm A.

Problem 2: In a compound epicyclic gear train as shown in the figure, has gears A and an

annular gears D & E free to rotate on the axis P. B and C is a compound gear rotate about

axis Q. Gear A rotates at 90 rpm CCW and gear D rotates at 450 rpm CW. Find the speed

and direction of rotation of arm F and gear E. Gears A,B and C are having 18, 45 and 21

teeth respectively. All gears having same module and pitch.

Solution:

TA=18 ; TB=45; TC=21; NA= -90rpm; ND=10rpm:

Since the module and pitch are same for all gears:

the number of teeth on the gearsis proportional to the pitch cirlce:

rD rA rB rC

TD TA TB TC

TD 18 45 21 84 teeth on gear D

Gears A and D rotates in the opposite directions:

TB TD NA NF



Problem 3: In an epicyclic gear of sun and planet type shown in figure 3, the pitch circle

diameter of the annular wheel A is to be nearly 216mm and module 4mm. When the annular

ring is stationary, the spider that carries three planet wheels P of equal size to make one

revolution for every five revolution of the driving spindle carrying the sun wheel. Determine the number of teeth for all the wheels and the exact pitch circle diameter of the

annular wheel. If an input torque of 20 N-m is applied to the spindle carrying the sun wheel, determine the fixed torque on the annular wheel.

Solution: Module being the same for all the meshing gears:

Operation Spider Sun Wheel S Planet wheel P Annular wheel A

arm L TS

TP

TA = 54

Arm L is fixed & TS TP

Sun wheel S is 0 +1

TS TP TS

given +1

TP TA TA

revolution

Multiply by m

0 m

TS m TS m

(S rotates through TP TA

m revolution)

Add n revolutions

n m+n

n TS m n TS m

to all elements

TP

TA



Problem 4: The gear train shown in figure 4 is used in

an indexing mechanism of a milling machine. The

drive is from gear wheels A and B to the bevel gear

wheel D through the gear train. The following table

gives the number of teeth on each gear. How many revolutions does D makes for one Figure 4

revolution of A under the following situations:

a. If A and B are having the same speed

and same direction

b. If A and B are having the same speed and opposite direction

c. If A is making 72 rpm and B is at rest

d. If A is making 72 rpm and B 36 rpm in the same direction

Solution:

Gear D is external to the epicyclic train and thus C and D constitute an ordinary train.

Operation

Arm E (28) F (24)

A (72) B (72) G (28) H (24)

C (60)

Arm or C is fixed 28 7 28 7

24

6

& wheel A is

0 -1

+1 -1 +1

24 6

given

+1 revolution

Multiply by m

(A rotates through 0 -m m

+m -m +m m

m revolution)

Add n revolutions n n - m

n + m n - m n + m

to all elements n m

nm

(i) For one revolution of A: n + m = 1 (1)



For A and B for same speed and direction: n + m = n – m (2)

From (1) and (2): n = 1 and m = 0

If C or arm makes one revolution, then revolution made by D is given by:

ND TC 60

2

NC TD 30

ND 2NC

(ii)A and B same speed, opposite direction: (n + m) = - (n – m)(3) n = 0; m

= 1

When C is fixed and A makes one revolution, D does not make any revolution.

(iii) A is making 72 rpm: (n + m) = 72

B at rest (n – m) = 0 n = m = 36 rpm

C makes 36 rpm and D makes 36 72 rpm

(iv) A is making 72 rpm and B making 36 rpm

(n + m) = 72 rpm and (n – m) = 36

rpm

(n + (n – m)) = 72; n = 54

D makes 54 108 rpm

Problem 5: Figure 5 shows a compound

epicyclic gear train, gears S1 and S2 being

rigidly attached to the shaft Q. If the shaft P

rotates at 1000 rpm clockwise, while the

Q

annular A2 is driven in counter clockwise

direction at 500 rpm, determine the speed and

direction of rotation of shaft Q. The number of

teeth in the wheels are S1 = 24; S2 = 40; A1 = 100; A2 = 120.

Solution: Consider the gear train P A1 S1:



Operation Arm A1

S1 (24)

P (100)

100 P1

Arm P is fixed &

P1 24

wheel A1 is given 0 +1

+1 revolution

Multiply by m

(A1 rotates through 0 +m m

m revolution)

Add n revolutions n n+ m

to all elements n

m

If A1 is fixed: n+ m; gives n = - m

Now consider whole gear train:

OR Operation

Arm A1 S1 (24)

P (100)

Arm P is fixed

A1

P1

& wheel A1 is

0 -1

P1 S1

given -1

A1

revolution

S1

100 25

0 -1

24 6

Add +1

revolutions to +1 0 1

all elements

Operation A1 A2 S1 (24), S2 (40)

(100)

(120)

and Q Arm P

A1 is fixed & 120 P2

3

wheel A2 is given 0

+1 P2 40

+1 revolution 3

Multiply by m

(A1 rotates through 0

+m

3m m

m revolution)

Add n revolutions n

n+ m

n 3m

to all elements

n m

When P makes 1000 rpm: n m = 1000

(1) and A2 makes – 500 rpm: n+ m = -500 (2)

from (1) and (2):

500 m m 1000



31 1000 500 31 49m

m 949rpm

and n 949 500 449rpm

NQ = n – 3 m = 449 – (3 -949) = 3296 rpm

OUT COME

Students able to Calculate the velocities and speeds of various gears in gear trains .

Exercise

1. An epicyclic gear train is shown in fig 3. The wheel A is fixed and the input

at the arm R is 3kw at 600 rpm. Find the speed of wheel D and the torque on

it and the torque required to hold the wheel ‘A’ neglect frictional losses. 2. Explain torque in an epicyclic gear train?



UNIT 8 CAMS

CONTENTS

8.1 Types of cams, types of followers

8.2 displacement, velocity and acceleration curves for uniform velocity, Simple Harmonic

Motion, Uniform Acceleration Retradation,

8.3 . Cam profiles: disc cam with reciprocating / oscillating follower having knife-edge,

roller and flat-face follower inline and offset.

OBJECTIVES:

1. Outline the classification various types of cams and followers and their

Arrangements.

2. Comprehend the procedure of developing Cam profiles.



8.1 Types of cams, types of followers

Definition A cam may be defined as a rotating, reciprocating or oscillating machine part, designed to impart

reciprocating and oscillating motion to another mechanical part, called a follower. A cam and follower have, usually, a line contact between them and as such they constitute a higher

pair. The contact between them is maintained by an external force which is generally, provided by a

spring or sometimes by the sufficient weight of the follower itself.

Classification of Cams

Cams are classified according to :

(a) Shape

(b) Follower movement

(c) Type of constraint of the follower

According to Shape

Wedge and Flat Cams

It is shown in Figures 1.48(a), (b), (c) and (d).

In Figure 1.48(a), on imparting horizontal translatory motion to wedge, the follower also

translates but vertically in Figure 1.48(b), the wedge has curved surface at its top. The

follower gets a oscillatory motion when a horizontal translatory motion is given to the

wedge.

In Figure 1.48(c), the wedge is stationary, the guide is imparted translatory motion within the

constraint provided. This results in translatory motion of the follower in Figure 1.48(d),

instead of a wedge, a rectangular block or a flat plate with a groove is provided. When

horizontal translatory motion is imparted to the block, the follower is constrained to have a

vertical translatory motion.



Follower Follower Oscillates

Guide

(a) Wedge Cam (b) Wedge Cam

Follower Follower

Constraint

Guide

Guide

Block / Flat

Plat

Fixed Groove

Wedge Wedge

(c)

Fixed

Wedge

Cam (d) Flate Wedge Cam

F

i

g

u

r

e

1

.

4

8

Guide

Oscillating Foll

ower

(a) Follower Reciprocating (b) Follower Oscillating

Guide Offset

Cam

Offset

(c) Offset Follower Reciprocating (d) Offset Follower Oscillating

(e)

Figure 1.49 : Radial or Disc Cams

Wedge Wedge



It is pointed out that the radial cams are very popular due to their simplicity and compactness,

Cylindrical Cams

Cylindrical cams have been shown in Figures 1.50(a) and (b). In

Figure 1.50(a) the follower reciprocates whereas in Figure 1.50(b) the

follower oscillates. Cylindrical cams are also known as barrel or drum

cams.

Follower

Follower

(a) Follower Reciprocates (b) Follower Oscillates

Figure 1.50 : Cylindrical Cams

Spiral Cams

It is shown in Figure 1.51. The cam comprises of a plate on the face of

which a groove of the form of a spiral is cut. The spiral groove is provided

with teeth which mesh with pin gear follower.

This cam has a limited use because it has to reverse its direction to reset the position of the follower. This cam has found its use in computers.

Follower

Guide

Spiral groove

Cam (plate)

Figure 1.51 : Spiral Cam

Conjugate Cams

As the name implies, the cam comprises of two discs, keyed together and

remain in constant touch with two rollers of a follower as shown in

Figure 1.52.

Cam disc Roller

Follower

Roller

Figure 1.52 : Conjugate Cam

Simple Mechanisms

41



Theory of Machines This cam is used where the requirement is of high dynamic load, low wear,

low noise, high speed and better control of follower. Globoidal Cams

This cam has two types of surfaces : convex and concave. A helical contour

is cut on the circumference of the surface of rotation of the cam as shown in

Figures 1.53(a) and (b). The end of the follower is constrained to move

along the contour and then oscillatory motion is obtained. In this cam, a

large angle of oscillation of the follower is obtained.

Concave Follower

Follower Surface

Convex surface Support

Support

(a) (b)

Figure 1.53 : Globoidal Cams Spherical Cams

In this cam, as shown in Figure 1.54, the cam is of the shape of a sphere on

the peripheral of which a helical groove is cut. The roller provided at the

end of the follower rolls in the groove causing oscillatory motion to the

follower in an axis perpendicular to the axis of rotation of the cam.

Cam Bearing

Follower

Figure 1.54 : Spherical Cam



8.2 displacement, velocity and acceleration curves for uniform velocity, Simple Harmonic Motion,

Uniform Acceleration Retradation,

Rise-return-rise (RRR)

In this type of cam, its profile or contour is such that the cam rises, returns

without rest or dwell, and without any dwell or rest, it again rises. Follower

displacement and cam angle diagram for this type of cam is shown in

Figure 1.55(a).

Dwell, Rise-return Dwell (DRRD)

In this type of cam after dwell, there is rise of the follower, then it returns to

its original position and dwells for sometimes before again rising.

Generally, this type of cam is commonly used. Its displacement cam angle

diagram is shown in Figure 1.55(b).

Dwell-rise-dwell-return

It is the most widely used type of cam. In this, dwell is followed by rise.

Then the follower remains stationary in the dwell provided and then returns

to its original position [Figure 1.55(c)].



Dwell-rise-dwell

As may be seen in the follower-displacement verses cam angle diagram,

shown in Figure 1.55(d) in this cam, the fall is sudden which necessities

enormous amount of force for this to take place.

Y Y

Fo

llow

erdi

spla

cem

ent

Follo

wer

disp

lace

men

t Return

Rise

Rise

X

O X Dwell

Cam angle, 360o

(a) [R – R – R]

(b) [D – R – R – D]

Y Dwell

Rise

Fa

ll

Fol

low

erdi

spl

acem

ent

F o l l o w e r d i s p l a c e m e n t

Rise Return

X

Dwell X Dwell

Cam angle, 360o

(c) [D – R – D – R – D]

(d) [D – R – D]

Figure 1.55 : Dwell-rise-dwell

According to Type of Constraint of the Follower

Pre-loaded Spring Cam

For its proper working there should be contact between the cam and the

follower throughout its working, and it is achieved by means of a pre-loaded

spring as shown in Figures 1.48(a) and (b), etc.

Positive Drive Cam

In this case, the contact between the cam and the follower is maintained by

providing a roller at the operating end of the follower. This roller operates

in the groove provided in the cam. The follower cannot come out of the

groove, as shown in Figures 1.52 to 1.54.

Gravity Drive Cam

In this type of cam, the lift or rise of the follower is achieved by the rising

surface of the cam (Figure 1.48(c)) and the follower returns or falls due to

force of gravity of the follower. Such type of cams cannot be relied upon

due to their uncertain characteristics.

Classification of Followers

Followers may be classified in three different ways :

(a) Depending upon the type of motion, i.e. reciprocating or oscillating.

(b) Depending upon the axis of the motion, i.e. radial or offset.

(c) Depending upon the shape of their contacting end with the cam.

Those of followers falling under classification (a) and (b) have already been dealt with as indicated above. Followers of type (c) will be taken up now.

Simple Mechanisms

43



Theory of Machines Depending upon the Shape of their Contacting End with the Cam

Under this classification followers may be divided into three types :

(a) Knife-edge Follower (Figure 1.55(a))

(b) Roller Follower (Figure 1.55(b))

(c) Flat or Mushroom Follower (Figure 1.56(c))

Knife-edge Follower

Knife-edge followers are generally, not used because of obvious high rate of

wear at the knife edge. However, cam of any shape can be worked with it.

During working, considerable side thrust exist between the follower and the

guide.

Roller Follower

In place of a knife edge, a roller is provided at the contacting end of the

follower, hence, the name roller follower. Instead of sliding motion between

the contacting surface of the follower and the cam, rolling motion takes

place, with the result that rate of wear is greatly reduced. In roller followers

also, as in knife edge follower, side thrust is exerted on the follower guide.

Roller followers are extensively used in stationary gas and oil engines. They

are also used in aircraft engines due to their limited wear at high cam

velocity.

While working on concave surface of a cam the radius of the surface must

be at least equal to radius of the roller.

Roller

(a) Knife-edge Follower (b) Roller Follower

Offset

Flat

(c) Flat or Mushroom Follower (i)

Spherical (d) : Spherical Follower

44 Figure 1.56 : Classification of Followers



Flat or Mushroom Follower

At the name implies the contacting end of the follower is flat as shown. In

mushroom followers there is no side thrust on the guide except that due to

friction at the contact of the cam and the follower. No doubt that there will

be sliding motion between the contacting surface of the follower and the

cam but the wear can be considerably reduced by off-setting the axis of the

followers as shown in Figure 1.56(c)(i). The off-setting provided causes the

follower to rotate about its own axis when the cam rotates.

Flat face follower is used where the space is limited. That is why it is used

to operate valves of automobile engines. Where sufficient space is available

as in stationary gas and oil engines, roller follower is used as mentioned

above. The flat faced follower is generally preferred to the roller follower

because of the compulsion of having to use small diameter of the pin in the

roller of the roller follower.

In flat followers, high surface stresses are produced in the flat contacting

surface. To minimise these stresses, spherical shape is given to the flat end,

as shown in Figure 1.56(d). The curved faced or spherical faced followers

are used in automobile engines.

With flat followers, it is obviously, essential that the working surface of the cam should be convex everywhere.

8.3 Terminology of Cam and Follower

The Cam Profile

The working contour of a cam which comes into contact with the follower to

operate it, is known as the cam profile. In Figure 1.57, A-B-C-D-A is the cam

profile or the working contour.

Simple Mechanisms

Roller

Prime circle Base circle

-A)

4

Dw

ell(

D

D

Pitch Curve

Tracing point for roller

Tracing point for knife edge

Cam profile Normal

Out stroke (A-B)

A

1

B’

2 B

Dw

ell

C’

C)-(B

3 C

(C-D) Instroke

Pressure Angle

Tangent

Figure 1.57 : Cam Profile

The Base Circle

The smallest circle, drawn from the centre of rotation of a cam, which forms part

of the cam profile, is known as the base circle and its radius is called the least

radius of the cam. A circle with centre O and of radius OA forms the base circle.

Size of a cam depends upon the size of the base circle.

The Tracing Point

The point of the follower from which the profile of a cam is determined is called

the tracing point. In case of a knife-edge follower, the knife edge itself is the

tracing point. In roller follower, the centre of roller is the tracing point.

45

Theory of Machines The Pitch Curve



The locus or path of the tracing point is known as the pitch curve. In knife-edge

follower, the pitch curve itself will be the cam profile. In roller follower, the

cam profile will be determined by subtracting the radius of the roller radially

throughout the pitch curve.

The Prime Circle

The smallest circle drawn to the pitch curve from the centre of rotation of the

cam is called as the prime circle. In knife-edge follower, the base circle and the

prime circle are the same. In roller follower, the radius of the prime circle is the

base circle radius plus the radius of the roller.

The Lift or Stroke

It is the maximum displacement of the follower from the base circle of the cam.

It is also called as the throw of the cam. In Figure 1.57, distance BB and CC is

the lift, for the roller follower.

The Angles of Ascent, Dwell, Descent and Action

Refer Figure 1.57, the angle covered by a cam for the follower to rise from its lowest position to the highest position is called the angle of ascent denoted as

1.

The angle covered by the cam during which the follower remains at rest at

its highest position is called the angle of dwell, denoted by 2.

The angle covered by the cam, for the follower to fall from its highest position

to the lowest position is called the angle of descent denoted as 3.

The total angle moved by the cam for the follower to return to its lowest position after the period of ascent, dwell and descent is called the angle of

action. It is the sum of 1, 2 and 3.

The Pressure Angle

The angle included between the normal to the pitch curve at any point and the

line of motion of the follower at the point, is known as the pressure angle. This

angle represents the steepness of the cam profile and as such it is very important

in cam design.

The Pitch Point

The point on the pitch curve having the maximum pressure angle is known as the pitch point.

The Cam AngleIt is the angle of rotation of the cam for a certain displacement of the follower.



OUT COME

Students able to Develop cam profiles for given input parameters

Exercise

1. With a neat sketch list the classification of followers.

2. A cam rotating clockwise at uniform speed of 300 rpm operates a reciprocating follower through a

roller 1.5cm diameter. The follower motion is defined as below

a) Outward during 1500 with UARM

b) Dwell for next 300

c) Return during next 1200 with SHM

d) Dwell for the remaining period

Stroke of the follower is 3cm.minimum radius of the cam is 3cm.Draw the cam profile the

follower axis passes through the cam.

FURTHER READING

1. "Theory of Machines”, Rattan S.S, Tata McGraw-Hill Publishing Company Ltd., New Delhi, and

3rd edition -2009.

2. "Theory of Machines”, Sadhu Singh, Pearson Education (Singapore) Pvt. Ltd, Indian Branch New

Delhi, 2nd Edi. 2006

3. “Theory of Machines & Mechanisms", J.J. Uicker, , G.R. Pennock, J.E. Shigley. OXFORD 3rd

Ed. 2009.

4. Mechanism and Machine theory, Ambekar, PHI, 2007

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