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Theory of Machines and Mechanisms (9050)

1

Chapter 1Fundamental and Types of

Mechanisms

Content-

1.1 Kinematics of Machines: - Definition of Kinematics, Dynamics, Statics,

Kinetics, Kinematic link, Kinematic Pair and its types, constrained motion

and its types, Kinematic chain and its types, Mechanism, inversion,machine and structure.

1.2 Inversions of Kinematic Chain.

1.2.1 Inversion of four bar chain, coupled wheels of Locomotive &

Pantograph.

1.2.2 Inversion of Single Slider Crank chain- Rotary I.C. Engines mechanism,

Whitworth quick return mechanism, Crank and Slotted lever quick return

mechanism.

1.2.3 Inversion of Double Slider Crank Chain- Scotch Yoke Mechanism &

Oldhams coupling.

1.3 Common Mechanisms

1.3.1 Bicycle free wheel Sprocket mechanism.

1.3.2 Geneva Mechanism.

1.3.3Ackermans Steering gear mechanism.

1.3.4 Foot operated air pump mechanism.


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Introduction

The subject Theory of Machines is an applied science. Theory of

Machines takes care of motion and strength aspect of a machine and uses

principles from physics, kinematics, static and kinetics.Machines are mechanical devices used to accomplish work. A mechanism is

a heart of a machine. It is the mechanical portion of the machine that has

the function of transferring motion and forces from a power source to an

output.

Mechanism is a system of rigid elements (linkages) arranged and

connected to transmit motion in a predetermined fashion.

Mechanism consists of linkages and joints.

Kinematics of Machines:-

Kinematics is the branch of Theory of Machines which deals with

relative motion between the various parts of the machine. It is related with

study of motion characteristics i.e. velocity and acceleration.

Dynamics-

Dynamics is the branch of Theory of Machines which deals with

forces acting on the machine parts while in motion.

Force in a machine part can be either from outside or from within

the body.

Statics-

Statics is the branch of Theory of Machines which deals with forces

and their effect while the machine parts are at rest. When dynamics of

machine neglects mass effect and studies only external forces that branch

of Dynamics is known as Statics i.e. mass of the part is assumed to be

negligible

Kinetics

When Dynamics of machine neglects external forces and studies

forces only on account of mass of the machine components, then the force

study is known as Kinetics.


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Kinematic link-

Each part of the machine which moves relative to some other part is

known as link or Kinematic link or Element. The link element need not to

be a rigid body but must be a resistant body.

Types of links-

i) Rigid link- Rigid link is one which does not undergo any

deformation while transmitting motion. Rigid link do not exists. But

deformation of connecting rod or other element is negligible, so it is

considered as rigid link

ii) Flexible link- A flexible link is one which is partly deformed in a

manner not to affect the transmission of motionExample-Belts, ropes, Wires & Chains

iii) Fluid link- A fluid link is one which is formed by having a fluid in a

receptable and the motion is transmitted through fluid by pressure

or compression only.

Example- Hydraulic press, jack etc

Kinematic Pair

There is always a relative motion between existing between twolinks. If this relative motion between the pair of links is constrained type

then the pair is called as Kinematic pair.

Types of Kinematic Pair

1. According to type of relative motion between parts

i) Sliding Pair-When two elements of pair are connected in such a way

thatone can only slide relative to other the pair is known as sliding

pair...

Example- Piston & Cylinder, Tailstock.


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ii) Turning Pair- When the two elements of pair are connected in such

a way thatone can only turn or revolve about a fixed axis of another

link the pair is known as turning pair.

Example- Cycle wheel

iii) Rolling pair- When the two elements of pair are connected in such a

way that one roll over another fixed link, the pair is known as

rolling pair.

Example- Ball Bearing

iv) Screw Pair- When the two elements of pair are connected in such a

way that one element can turn about the other by screw threads,

pair is known as screw pair.Example- Nut and Bolt


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v) Spherical Pair- When the two elements of pair are connected in

such a way that only one element (with spherical shape) turns or

swivels about other fixed element the pair formed is called as

Spherical pair.

Example-Attachment of car mirror

2. According to type of contact between Elements-

i) Lower pair- When the two elements of pair have surface contact

when relative motion takes place and surface of one element slides

over surface of another the pair formed is Lower pair.

Example- Sliding pair, turning Pair

ii) Higher pair- When the two elements of pair have a line or point

contactwhen relative motion takes place and the motion between

two elements is partly turning and partly sliding then the pair

formed is known as higher pair.

Example- Belt or rope drive, Cam and Follower

3. According to the type of closure


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i) Self Closed pair- When the two elements of pair are connected

together mechanically in such a way that only required kind of

relative motion occur, it is known as self closed pair.

ii) Force closed pair- When the two elements of pair are connected

mechanically but are kept in contact by the action of external forces

the pair is said to be force closed pair.

Constrained motion and its types-

Two links are connected with each other by various means and this

method of connection decides the type of relative motion between the

links. If this relative motion is one and only type then it is said to be

constrained motion.

i) Completely constrained motion- When the motion between the pair

is limited to a defined direction irrespective of the direction of

force applied, then the motion is said to be completely constrained

motion.

ii) Incompletely constrained motion- When motion between pair can

take place in more than one direction then the motion is called

incompletely constrained motion.

iii) Successfully constrained motion- When the motion between the

elements forming a pair is such that the constrained motion is not

completed by itself but by some other means then the motion is

said to be successfully constrained motion.


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Kinematic chain and its types

Link is the smallest possible member in a machine. When such two

links come together and show constrained motion then they form a

Kinematic pair. When Kinematic pair are coupled in such a way that the

last link is joined to the first link to transmit definite motion, it is called as

Kinematic chain.

If each link is assumed to form two pairs with two adjacent link

then relation between number of pairs (P) forming a chain and number of

links (l) is

L= 2p-4. (i)

In a Kinematic chain each link forms a part of two pairs; therefore there

will be as many links as number of pairs

J=3/2l-2 (ii)

Equation I and ii are applicable to Kinematic chains in which lower pairs

are used.

Types of Kinematic chain

I) Locked Chain-

Number of links l=3; Number of pairs p=3; Number of joints j=3

Equation I gives L= 2p-4

3=2*3-4 = 2


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LHS > RHS

Equation II gives J=3/2l-2

3 = 3/2*3 2 = 2.5

LHS > RHS

As this arrangement does not satisfy Equation I and II, therefore it

is not a Kinematic chain and hence relative motion is not possible.

Such type of chain is called as locked chain.

II) Constrained Kinematic chain-

(Kinematic chain of one degree of freedom)

L = 4; p = 4; j = 4

Equation I gives L= 2p-4

4 = 2*4 -4 = 4

LHS =RHS

Equation Ii gives J=3/2l-2

4 = 3/2 * 4 4 = 4

LHS =RHS

Since this arrangement satisfy Equation I and II this is called

Kinematic chain of one degree of freedom.Now if link AB is fixed and a definite displacement O is given

to AD then resulting displacement of the two links BC and CD

are also perfectly definite. Thus in four bar chain relative

motion is completely constrained. Hence it may be called as

constrained Kinematic chain.

III) Unconstrained kinematic chain-

L = 5; p = 5; j = 5Equation I gives L= 2p-4


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5 = 2*5 4 = 6

LHS < RHS

Equation Ii gives J=3/2l-2

5 = 3/2 * 5 2= 5.5

LHS


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As this arrangement satisfies equation I and II therefore it is

kinematic chain.

A chain having more than four links is called as compound kinematic chain

Mechanism

When one of the links of a kinematic chain is fixed then the chain is

known as mechanism.

It may be used for transmitting or transforming the motion.

Mechanism with four links is called simple mechanism and mechanism

with more than four links is called compound mechanism.

Inversion-

In a kinematic chain if one of the link is fixed it is called as

mechanism. So we can obtain as many mechanisms as number of links. This

method of obtaining different mechanisms by fixing different links in a

kinematic chain is called Inversion of mechanism

Machine and structure

Machine-

When a mechanism is required to transmit a power or to do some

particular type of work it then becomes a machine

Structure-It is an assemblage of a number of resistant bodies (known as

members) having no relative motion between them and meant for carrying

loads having straining action.

Example- a Railway Bridge, a roof or truss, machine frame, etc

Machine Structure

1 Parts of Machine move

relative to each other

Members of structure do not move

relative to each other2 A machine transforms No energy is transformed into work in


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available energy into some

useful work

case of structure, energy is stored

within it.

3 Links of machine transmit

both power and motion

Members of structure transmit forces

only

4 Example-Screw jack, lathe

machine etc

Example- A railway bridge, a roof or

truss, machine frame, etc

Inversions of Kinematic Chain

In a kinematic chain if one of the link is fixed it is called as

mechanism. So we can obtain as many mechanisms as number of links. This

method of obtaining different mechanisms by fixing different links in akinematic chain is called Inversion of mechanism

Grashofs law-

For a four bar mechanism the sum of the shortest and the longest

links length should not be greater than the sum of the remaining to links

lengths if there is to be continuous relative motion between two links.

Crank (link AD) - A link making complete revolution (DRIVER)

Lever (link BC) - A link oscillating or partially revolving

(ROCKER/FOLLOWER)

Connecting rod (link DC) - A link connecting crank and lever (COUPLER)

Frame (link AB) - a Fixed link

Inversion of four bar chain,1. Beam Engine (Crank and lever mechanism)


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When crank rotates about fixed center A , lever oscillates about

center D and end E of lever CDE is connected to piston rod which

reciprocates due to rotation of crank. This mechanism converts

rotary motion into reciprocating motion.

2. coupled wheels of Locomotive (Double crank Mechanism)

This mechanism is having two cranks AD and BC respectively. Link

CD acts as coupling rod and AB is fixed in order to maintain constant

center to center distance between wheels. Thus this mechanism is

meant to transmit rotary motion from one wheel to another.


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3. Pantograph

A Pantograph is a device based on kinematic chain with turning

pairs used to reproduce a drawing exactly either on an enlarged or

on a reduced scale. It consists of a jointed parallelogram ABCD, Bars

BA and BC are extended to O and E respectively

OA/OB=AD/BE

Thus for all relative motions triangle OAD and OBE are similar and

points O, D, E are in straight line.

For similar triangles OAD and OBE

OD/OE = AD/BE

Point O is fixed and point D and E moves to new positions D and E

Straight line DD is parallel to line EE. Hence if O is fixed to frame of

machine by means of turning pair and D is attached to point in the machine

which has rectilinear motion relative to frame then E will also trace the

straight line path.

Similarly if E is constrained to move in straight line then D will trace

out straight line path parallel to EE.


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Inversion of Single Slider Crank chain-

A single slider crank chain is a modification of a basic four bar chain

mechanism. It consists of one sliding pair and three turning pairs.

Link 1- Frame of engine; link 2-crank; link 3-connecting rod; link 4-cross

headAs the crank rotates the cross head reciprocates in the guides and thus

the piston reciprocates in the cylinder

Rotary I.C. Engines mechanism-

Link 1-Cyllinderlink 2-crank; link 3- connecting rod; link 4-piston

It consist of seven cylinders in one plane and all revolve about fixed center

D. Crank (link 2) is fixed. When connecting rod (link 4) rotates piston

reciprocates inside cylinder. This engine is used in aviation.

Whitworth quick return mechanism-

This mechanism is used in shaping and slotting machines.


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Link 1-Slotted bar oscillating at pivoted point d;

Link 2- Fixed link CD

Link 3- Crank CA, rotating at uniform angular speed

Link 4- slider, slides along slotted bar

Connecting rod carries ram R at which tool is fixed and its motion is

constrained along a line passing through D and perpendicular to CD.

Forward stroke/Cutting stroke-

When the driving crank CA moves from position CA1 to CA2 through

an angle in the clockwise direction, tool moves from left hand end of

stroke to right hand end through distance 2PD.

Return stroke/Idle stroke-

When the driving crank moves from the position CA2 to CA1 through

an angle in the clockwise direction tool moves back from right hand end

of its stroke to left hand end

Time taken during forward stroke is more than time taken during

return stroke.

Time of cutting stroke = =

Time of return stroke 360-

is always greater than 180

Crank and Slotted lever quick return mechanism-

This mechanism is used in shaping and slotting machines.


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Link 1- Slider; link 2- Crank; link 3-Fixed link; link 4-slotted bar

In this mechanism the link AC (link 3) forming turning pair is fixed. The

driving crank CB revolves with uniform angular speed about fixed center C.

A sliding block attached to the crank pin at B slides along the slotted bar AP

and thus causes it to oscillate about the pivoted point A. A short link PR

transmits the motion from AP to the ram which carries the tool and

reciprocates along the line of stroke R1R2.

Forward stroke/Cutting stroke-

It occurs when crank rotates from CB1 to CB2 at an angle in the

clockwise direction.

Return stroke/Idle stroke-

It occurs when crank rotates from CB2 to CB1 through an angle inclockwise direction.

Time of cutting stroke = = 360-

Time of return stroke

is always greater than and since crank rotates with uniform angular

velocity therefore return stroke is completed within short time.

Inversion of Double Slider Crank Chain-

A kinematic chain which consists of two turning pairs and two slidingpairs is known as double slider crank chain.


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1. Scotch Yoke Mechanism

This mechanism is used for converting rotary motion into a reciprocating

motion. The inversion is obtained by fixing either the lank 1 or link 3.as

shown in figure link 1 is fixed.

In this mechanism when link 2 (which corresponds to crank) rotates

about B as center, the link 4(which corresponds to a frame)

reciprocates. The fixed link 1 guides the frame.

2. Oldhams Coupling-

An Oldhams coupling is used for connecting two parallel

shafts whose axes are at small distance apart. The shafts to be

connected have two flanges rigidly fastened at their ends by forging.

Link 1-Flange; link 2-supporting frame; link 3- Flange 2; link 4-

Intermediate piece


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The flanges have diametrical slot cut in their inner faces. The

intermediate piece is a circular disc having two tongues (diametrical

projections at right angle to each other)

The link 4 slides or reciprocates in the slot in flanges. When thedriving shaft A is rotated the flange C (link 1) causes the intermediate piece

(link 4) to rotate. Hence link 1, link 3, link4 have same angular speed at

every instant.

Common Mechanisms

1. Bicycle free wheel Sprocket mechanism.

Mechanical or automotive engineering, a freewheel oroverrunning clutch is a device in a transmission that disengages the

driveshaftfrom the driven shaft when the driven shaft rotates faster than

the driveshaft. An overdrive is sometimes mistakenly called a freewheel,

but is otherwise unrelated.

The condition of a driven shaft spinning faster than its driveshaft

exists in mostbicycles when the rider holds his or her feet still, no longer

pushing the pedals. Without a freewheel the rear wheel would drive the

pedals around.

In the past, such freewheel mechanisms have included an inner

freewheel body which engages threads on a rear wheel hub, and an outer

freewheel body, including an integral sprocket for engagement with the

roller chain. A pair of pawls, and at least one pawl spring have been

disposed between said inner and outer freewheel bodies, whereby forward

rotation of the outer freewheel body would cause the pawls to engage and
http://en.wikipedia.org/wiki/Mechanical_engineeringhttp://en.wikipedia.org/wiki/Automotive_engineeringhttp://en.wikipedia.org/wiki/Transmission_%28mechanics%29http://en.wikipedia.org/wiki/Driveshafthttp://en.wikipedia.org/wiki/Overdrive_%28mechanics%29http://en.wikipedia.org/wiki/Bicyclehttp://en.wikipedia.org/wiki/Pedalhttp://en.wikipedia.org/wiki/Pedalhttp://en.wikipedia.org/wiki/Bicyclehttp://en.wikipedia.org/wiki/Overdrive_%28mechanics%29http://en.wikipedia.org/wiki/Driveshafthttp://en.wikipedia.org/wiki/Transmission_%28mechanics%29http://en.wikipedia.org/wiki/Automotive_engineeringhttp://en.wikipedia.org/wiki/Mechanical_engineering


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drive the inner freewheel body and rear wheel. Also, the pawls would allow

the rear wheel to rotate in a forward direction when the outer freewheel

body was rotating more slowly or was stopped.

2. Geneva Mechanism.


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The Geneva mechanism is a timing device.

Geneva mechanism consists of a rotating disk with a pin and another

rotating disk with slots (usually four) into which the pin slides

In the most common arrangement, the driven wheel has four slots

and thus advances for each rotation of the drive wheel by one step of 90. If

the driven wheel has n slots, it advances by 360/n per full rotation of the

drive wheel.

One application of the Geneva drive is in movie projectors. Geneva

wheels having the form of the driven wheel were also used in mechanical

watches. Other applications of the Geneva drive include the pen change

mechanism in plotters, automated sampling devices, indexing tables in

assembly lines, tool changers for CNC machines, and so on.
http://en.wikipedia.org/wiki/Degree_%28angle%29http://en.wikipedia.org/wiki/Degree_%28angle%29http://en.wikipedia.org/wiki/Movie_projectorhttp://en.wikipedia.org/wiki/Plotterhttp://en.wikipedia.org/wiki/CNChttp://en.wikipedia.org/wiki/CNChttp://en.wikipedia.org/wiki/Plotterhttp://en.wikipedia.org/wiki/Movie_projectorhttp://en.wikipedia.org/wiki/Degree_%28angle%29


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The Iron Ring Clock uses a Geneva mechanism to provide intermittent

motion to one of its rings.

3.Ackermans Steering gear mechanism.

The steering gear mechanism is used for changing the direction of

two or more of the wheel axles with reference to the chassis, so as to move

the automobile in any desired path.

When the vehicle takes a turn the front wheels along with the respective

axles turn about the respective pivoted points. The back wheels remain straight

and do not turn. Therefore steering is done by front wheels only.

In order to avoid skidding the two front wheels must turn about the same

Instantaneous center I which lies on the axis of the back axle. If the ICR of the twofront wheels do not coincide with the ICR of the back wheels skidding will take

place, which causes wear and tear of tires.

Thus the condition for correct steering is that all the four wheel

must turn about the same ICR. The axis of the inner wheel makes a larger

turning angle than the angle subtended by the axis of outer wheel.

a= wheel track

b=wheel base

c=distance between the pivots A and b of the front axle

Now from triangle IBP

Cot =BP

IP

And from triangle IAP

Cot =AP = AB + BP = AB + BP = c +cot

IP IP IP IP b

Cot - cot =c/b
http://en.wikipedia.org/wiki/Iron_Ring_Clockhttp://en.wikipedia.org/wiki/Iron_Ring_Clock


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This is the fundamental equation for correct steering. If this condition is

satisfied there will be no skidding of the wheels when vehicle takes a turn.

In Ackerman steering gear the mechanism ABCD is a four bar crank chain.

The shorter link BC and AD are of equal length and are connected by hinge joints

with front wheel axle. The longer link AB and CD are of unequal length. The

following are three positions for correct steering

i) When vehicle moves along a straight path, the longer link AB and

CD are parallel and shorter link BC and AD are equally inclined to

the longitudinal axis of the vehicle.

ii) When the vehicle is steering to the left, the position of the gear as

shown by dotted lines. In this position the lines of the front wheel

axle intersect on the back wheel axle at I for correct steering.

iii) When the vehicle is steering to the right the similar position may beobtained.

4. Foot operated air pump mechanism.

It consists of a cylinder which can oscillate. A piston is mounted in

the cylinder. The cylinder is connected to the foot rest. The arms connected

to the foot rest can oscillate.

A retrieving spring can bring back the foot rest back to initial

positions the foot rest is pressed the cylinder oscillates.


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It creates reciprocating motion of the piston in the cylinder.

Therefore suction and delivery stroke can be obtained.

This is also called as oscillating cylinder mechanism.


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Chapter 2Velocity and Acceleration in Mechanism

2.1 Concept of relative velocity and relative acceleration of a

point on link, angular velocity and angular acceleration, inter-

relation between linear and angular velocity and acceleration.

2.2 Drawing of velocity and acceleration diagram of a given

configuration, diagrams of simple mechanisms. Determinationof velocity and acceleration of a point on link by relative velocity

Method [Excluding coriollis components of acceleration].

2.3 Analytical method [no derivation] and Kleins construction

to determine velocity and acceleration of different links in single

slider crank mechanism.


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Introduction-

Kinematics deals with study of relative motion between the various

parts of the machines. Kinematics does not involve study of forces. Thus

motion leads study of displacement, velocity and acceleration of a part of

the machine.

Study of Motions of various parts of a machine is important for

determining their velocities and accelerations at different moments. As

dynamic forces are a function of acceleration and acceleration is a function

of velocities, study of velocity and acceleration will be useful in the design

of mechanism of a machine. The mechanism will be represented by a line

diagram which is known as configuration diagram. The analysis can be

carried out both by graphical method as well as analytical method.

Concept of relative velocity of a point on link

Some important Definitions

Displacement: All particles of a body move in parallel planes and travel by

same distance is known, linear displacement and is denoted by x.

A body rotating about a fired point in such a way that all particular move in

circular path angular displacement and is denoted by.

Velocity:Rate of change of displacement is velocity. Velocity can be linear

velocity of angular velocity.

Linear velocity is Rate of change of linear displacement= V =

Angular velocity is Rate of change of angular displacement ==

Relation between linear velocity and angular velocity

x = r

=r

V = r

Acceleration: Rate of change of velocity

Linear Acceleration (Rate of change of linear velocity)

a=


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Absolute velocity: Velocity of a point with respect to a fixed point (zerovelocity point).

Va = 2 x r

Va = 2 x O2A

Ex: Vao2 is absolute velocity.

Angular Acceleration (Rate of change of angular velocity)

=

Relative velocity: Velocity of a point with respect to another point x

Note: Capital letters are used for configuration diagram. Small letters are

used for

velocity vector diagram.

This is absolute velocity

Velocity of point A with respect to O2 fixed point, zero velocity point.

Vba = or Vab

Vba = or Vab Equal in magnitude but opposite in direction.

Vb Absolute velocity is velocity of B with respect to O4 (fixed point, zerovelocity point


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Vector O a 2 = Va= Absolute velocity

Vector ab = Vab

ba = Va

Vab is equal magnitude with Vba but is apposite in direction

Vector O b 4 = Vb absolute velocity.

To illustrate the difference between absolute velocity and relative velocity.

Let, us consider a simple situation.

A link AB moving in a vertical plane such that the link is inclined at 30o to

the horizontal with point A is moving horizontally at 4 m/s and point B

moving vertically upwards. Find velocity of B.

Va = 4 m/s ab Absolute velocity Horizontal direction (known in

magnitude and directors)

Vb = ? ab Absolute velocity Vertical direction (known in

directors only)

Velocity of B with respect to A is equal in magnitude to velocity of A with

respect to B but opposite in direction.


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Velocity analysis of any mechanism can be carried out by various methods.

1. Graphical method

2. Relative velocity method

3. Instantaneous method

By Graphical Method

The following points are to be considered while solving problems by this

method.

1. Draw the configuration design to a suitable scale.

2. Locate all fixed point in a mechanism as a common point in velocity

diagram.

3. Choose a suitable scale for the vector diagram velocity.

4. The velocity vector of each rotating link is ^r to the link.

5. Velocity of each link in mechanism has both magnitude and direction.

Start from a point whose magnitude and direction is known.

6. The points of the velocity diagram are indicated by small letters.

To explain the method let us take a few specific examples.

1. Four Bar Mechanism: In a four bar chain ABCD link AD is fixed and in 15

cm long. The crank AB is 4 cm long rotates at 180 rpm (cw) while link CD

rotates about D is 8 cm long BC = AD and | BAD = 60o. Find angular velocity

of link CD.

Velocity vector diagram

Vb = r = ba x AB =

50.24 cm/sec

Choose a suitable scale1 cm = 20 m/s = ab


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Vcb = bcVc = dc = 38 cm/s = VcdWe know that V = R

Vcd = CD x CDCD=

=

=4.75 rad/sec

2.Slider Crank Mechanism:

In a crank and slotted lover mechanism crank rotates of 300 rpm in a

counter clockwise direction. Find

(i) Angular velocity of connecting rod and

(ii) Velocity of slider.

Step 1: Determine the magnitude and velocity of point A with respect to 0,

VA = O1A x O2A=

Step 2: Choose a suitable scale to draw velocity vector diagram


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Vab = ab =1300mm/sec

ab=

Vb = ob velocity of slider

Note: Velocity of slider is along the line of sliding

Problem 2:

In a slider crank mechanism the crank is 200 mm long and rotates at 40

rad/sec in a CCW direction. The length of the connecting rod is 800 mm.

When the crank turns through 60o from Inner-dead centre.

Determine,i) The velocity of the slider

ii) Velocity of point E located at a distance of 200 mm on the connecting rod

extended.

iii) The position and velocity of point F on the connecting rod having the

least absolute velocity.

iv) The angular velocity of connecting rod.

Va = Woa x OA

Va = 40 x 0.2


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Va = 8 m/s

Step 2: Choose a suitable scale for velocity vector diagram and draw the

velocity vector diagram.

Mark zero velocity point o, g.

Draw oa ^r to link OA equal to 8 m/s

From a draw a line ^r to AB and from o, g draw a horizontal line(representing the line of motion of slider B) to Xseet the previously drawn

line at b.

ab give Vba=4.8 m/sec

Step 3: To mark point e since E is on the extension of link AB drawn

be=

Mark the point e on extension of vector ba. Join e to o, g. ge will give

velocity of point E.

Ve = ge =8.4 m/sec

Step 4: To mark point F on link AB such that this has least velocity

(absolute).

Draw a line ^r to ab passing through o, g to cut the vector ab at f. From f to

o, g.

gf will have the least absolute velocity.

To mark the position of F on link AB.

Find BF by using the relation.

BF=

=200mmStep 5: To determine the angular velocity of connecting rod.

We know that Vab = wab x AB

ab=


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]

Acceleration of a point on a Link-Acceleration of a point has two components

1. The centripetal or Radial component- It is perpendicular to the

velocity of the particle at the given instant

2. The tangential component- It is parallel to velocity of particle i.e.

perpendicular to link

= Angular velocity of link

= Angular acceleration of link

The centripetal or Radial component = * length of link

The tangential component = * length of link

Angular acceleration=

=

Inter- relation between linear and angular velocity andacceleration.


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Angular velocity-

It is defined as ratio of change of angular displacement with respect to

time

=

Angular acceleration-

It is defined as rate of change of angular velocity with respect to time.

= =

=

=

VELOCITY DIAGRAMS

This section involves the construction of diagrams which needs to be

done accurately and to a suitable scale. Students should use a drawingboard, ruler, compass, protractor and triangles and possess the necessary

drawing skills.

ABSOLUTE AND RELATIVE VELOCITY

An absolute velocity is the velocity of a point measured from a fixed

point (normally the ground or anything rigidly attached to the ground and

not moving). Relative velocity is the velocity of a point measured relative to

another that may itself be movingTANGENTIAL VELOCITY

Consider a link A B pinned at A and revolving about A at angular

velocity. Point B moves in a circle relative to point A but its velocity is

always tangential and hence at 90 to the link. A convenient method of

denoting this tangential velocity is (vba) meaning the velocity of B relative

to A. This method is not always suitable

CRANK, CONNECTING ROD AND PISTONConsider this mechanism again. Lets freeze the motion (snap shot) at

the position shown. The diagram is called a space diagram.


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Every point on every link has a velocity through space. First we label

the centre of rotation, often this is the letter O. Point A can only move in a

tangential direction so the velocity of A relative to O is also its absolute

velocity and the vector is normal to the crank and it is designated (vA)O.

(Note the rotation is anticlockwise).

Now suppose that you are sat at point A and everything else moves

relative to you. Looking towards B, it would appear the B is rotating relative

to you (in reality it is you that is rotating) so it has a tangential velocity

denoted (Vba) The direction is not always obvious except that it is normal

to the link.

Consider the fixed link OC. Since both points are fixed there is no

velocity between them so (vC)O = 0

Next consider that you at point C looking at point B. Point B is a

sliding link and will move in a straight line in the direction fixed by the

slider guides and this is velocity (vB) C. It follows that the velocity of B seen

from O is the same as that seen from C so (vB)C = (vB)O

The absolute velocity of B is (vB) C = (vB) O and this must be the

vector sum of (VA) O and (vB)A and the three vectors must form a closed

triangle as shown. The velocity of the piston must be in the direction in

which it slides (conveniently horizontal here). This is a velocity diagram


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METHODOLOGY

First calculate the tangential velocity (vA)O from v = x radius = x

OA

Draw the vector o - a in the correct direction (note lower case letters) We

know that the velocity of B relative to A is to be added so the next vector ab

starts at point a. At point a draw a line in the direction normal to the

connecting rod but of unknown length we know that the velocity of B

relative and absolute to O is horizontal so the vector ob must start at a.

Draw a horizontal line (in this case) through o to intersect with the other

line. This is point b. The vectors ab and ob may be measured or calculated.

Usually it is the velocity of the slider that is required.

Remember that the slider direction is not always horizontal and

the direction of o - b must be the direction of sliding

Numericals-

1.The mechanism shown has a crank 50 mm radius which rotates at

2000 rev/min. Determine the velocity of the piston for the position

shown. Also determine the angular velocity of link AB about A.

Note the diagrams are not drawn to scale. The student should do this using a

suitable scale for example 1 cm = 1 m/s. This is important so that the

direction at 90 to the link AB can be transferred to the velocity diagram

Angular speed of the crank = 2N/60 = 2 x 2000/60 = 209.4 rad/s(vA)O = x radius = 209.4 x 0.05 = 10.47 m/s.

First draw vector oa. (Diagram a)

Next add a line in the direction ab (diagram b)

Finally add the line in the direction of ob to find point b and measure ob to

get the velocity (diagram C).


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The velocity of B relative to O is 7 m/s

The tangential velocity of B relative to A is the vector ab and this gives 9.2

m/s. The angular velocity of B about A is found by dividing by the radius

(length of AB). For AB is then 9.2/0.09 = 102.2 rad/s. (note this is relative to

A and not an absolute angular velocity)

2. In a four bar chain ABCD, AD is fixed and is 150mm long. The crank

AB is 40mm long and rotates at 120 rpm clockwise. While the link CD

= 80mm oscillates about D. BC and AD are of equal length. Find the

angular velocity of link CD when angle BAD =60.

Solution- NAB= 120 rpm or AB =

= 12.568 rad/sec

Since length of link AB is 40mm

VAB = AB AB = 12.568 0.04 = 0.503 m/sec


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Assignment-

1. In the mechanism shown in figure the slider D is constrained to

move on a horizontal path. The crank OA is rotating in counter

clockwise direction at a speed of 180 rpm.

For the given configuration find

i) Velocity of slider

ii) Angular velocity of links AB, CB, BD.


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ACCELERATION DIAGRAMSIt is important to determine the acceleration of links because

acceleration produces inertia forces in the link which stress the component

parts of the mechanism. Accelerations may be relative or absolute in the

same way as described for velocity.

CENTRIPETAL ACCELERATION

A point rotating about a centre at radius R has a tangential velocity

v and angular velocity and it is continually accelerating towards the centre

even though it never moves any closer. This is centripetal acceleration and

it is caused by the constant change in direction. It follows that the end of any

rotating link will have a centripetal acceleration towards the opposite end.

ar= v2/AB.

Note the direction is towards the centre of rotation but the vector starts at a

and ends at b

It is very important to get this the right way round otherwise the complete

diagram will be wrong


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TANGENTIAL ACCELERATION

Tangential acceleration only occurs if the link has an angular

acceleration a rad/s2. Consider a link AB with an angular acceleration about

A. Point B will have both radial and tangential acceleration relative to point

A. The true acceleration of point B relative to A is the vector sum of them.

This will require an extra point. We will use b1 and b on the vector diagram

as shown. Point B is accelerating around a circular path and its direction is

tangential (at right angles to the link). It is designated aT and calculated

using aT= x AB. The vector starts at b1 and ends at b. The choice of lettersand notation are arbitrary but must be logical to aid and relate to the

construction of the diagram

EXAMPLE

1. A piston, connecting rod and crank mechanism is shown in the

diagram. The crank rotates at a constant velocity of 300 rad/s. Find

the acceleration of the piston and the angular acceleration of the link

BC. The diagram is not drawn to scale.


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First calculate the tangential velocity of B relative to A.

(vB)A = x radius = 300 x 0.05 = 15 m/s.

Next draw the velocity diagram and determine the velocity of C relative to B

From the velocity diagram (vC)B = 7.8 m/s

Next calculate all accelerations possible and construct the acceleration

diagram to find the acceleration of the piston.

The tangential acceleration of B relative to A is zero in this case since the

link has no angular acceleration (a = 0).

The centripetal acceleration of B relative to A

aR== 2x AB = 3002 x 0.05 = 4500 m/s2.

The tangential acceleration of C relative to B is unknown

The centripetal acceleration of C to B

aR= v2/BC = 7.82 /0.17 = 357.9 m/s2.

The stage by stage construction of the acceleration diagram is as

follows

First draw the centripetal acceleration of link AB (Fig.a). There is no

tangential acceleration so designate it ab. Note the direction is the same as

the direction of the link towards the centre of rotation but is starts at a and

ends at b.


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Next add the centripetal acceleration of link BC (Figure b). Since

there are two accelerations for point C designate the point c1. Note the

direction is the same as the direction of the link towards the centre of

rotation.

Next add the tangential acceleration of point C relative to B (Figure c).

Designate it c1 c. Note the direction is at right angles to the previous vector

and the length is unknown. Call the line a c line.

Next draw the acceleration of the piston (figure d) which is

constrained to be in the horizontal direction. This vector starts at a and

must intersect the c line. Designate this point c.

The acceleration of the piston is vector ac so (a c)B= 1505 m/s2.

The tangential acceleration of C relative to B is c1 c = 4000 m/s2.

At the position shown the connecting rod has an angular velocity and

acceleration about its end even though the crank moves at constant speed

The angular acceleration of BC is the tangential acceleration divided by the

length BC

a (BC) = 4000 / 0.17 = 23529 rad/s2

Analytical method to determine velocity and acceleration of

different links in single slider crankmechanism


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Take r= length of crank

L= length of connecting rod

= Inclination of crank from Inner dead centern=

= Obliquity ratio

Vb = Velocity of Piston/slider

ab= Acceleration of slider

ab= angular velocity of connecting rodab = angular acceleration of connecting rod

i) Velocity of Piston (Vb)

Vb= .r (

)

ii) Acceleration of Piston/Slider (ab)

ab =2 .r (cos + )iii) Angular velocity of connecting rod (ab)

ab =

iv) angular acceleration of connecting rod

ab =

Example-

In a slider crank mechanism, the length of crank and connecting rod are

150mm and 600mm respectively. The crank position is 60 from inner


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dead centre. The crank shaft speed is 450 rpm (clockwise). Using analytical

method determine

1. Velocity and acceleration of connecting rod and 2. Angular

velocity and angular acceleration of connecting rod.

Solution- Given: r= 150mm= 0.15m; l= 600mm= 0.6m; = 60; N= 450 rpm=

=

= 47.13 rad/sec

1. Velocity and acceleration of slider

Obliquity ratio n=

=

= 4

Vb= .r (

)

= 47.13*0.15 (sin 60 +

)

= 6.9 m/s

Acceleration of slider

ab =2 .r (cos + )

= 47.13* 0.15 (cos 60 +

)

= 124.94 m/s

2. Angular velocity and acceleration of connecting rod

ab =

=

= 5.9rad/sec

Angular acceleration

ab =

=

= 481 rad/sec


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Chapter 3

Cams and Followers

3.1 Concept, definition and application of Cams and Followers.

3.2 Classification of Cams and Followers.

3.3 Different follower motions and their displacement diagrams

like uniform velocity, SHM, uniform acceleration andRetardation.

3.4 Drawing of profile of radial cam with knife-edge and roller

follower with and without offset with reciprocating motion

(graphical method).

Introduction

Definition

A cam is a mechanical component of a machine that is used to transmit

motion to another component, called the follower, through a prescribed

motion program by direct contact

Concept of cam and follower mechanism


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A cam mechanism consists of three elements: the cam, the follower (orfollower system), and the frame. The follower is in direct contact with the

cam. The cam may be of various shapes. The follower system includes all of

the elements to which motion is imparted by the cam. This may be

connected directly to the follower, or connected through linkages and

gearing. The frame of the machine supports the bearing surfaces for the

cam and for the follower.

Uses for cams:

The cam mechanism is a versatile one. It can be designed to produce almost

unlimited types of motioning the follower. It is used to transform a rotary

motion into a translating or oscillating motion. On certain occasions, it is

also used to transform one translating or oscillating motion into a different

translating or oscillating motion.

The cam operated valve system:


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Cam follower systems are used in a wide variety of daily applications,

including motor vehicles, moving lawn ornaments, and pumping devices

Classification of Cams and Followers

1. Radial or Disc camIn radial cams, the follower reciprocates or oscillates in a direction

perpendicular to cam axis.
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2. Cylindrical cam

Cylindrical cams are used when motion has to be transmitted

parallel to the axis of rotation of the cam. The cylindrical or barrel

cam consists of a rotating cylinder with a helical (screw shaped)

groove in its curved surface. A follower with a tapered roller end is

located in the groove. As the cylinder turns, the follower moves in a

straight line parallel to the axis of the rotation barrel cam. This type

of cam is often used to guide thread on sewing machines, looms and

fabric making machines.

Classification of Followers

1. According to surface of contact-

a) Knife edge follower-

When the contacting end of the follower has sharp knife edge it is

called knife edge follower.

b) Roller follower-

When the contacting end of the follower is a roller, it is called a

roller follower.c) Flat faced or mushroom follower-

When the contacting end of the follower is perfectly flat face, it is

called a flat faced follower.

d) Spherical shaped follower-

When the contacting end of the follower is of spherical shape, it is

called a spherical faced follower.


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2. According to the motion of the follower-

a) Reciprocating or translating follower-

When the follower reciprocates in guides as cam rotates

uniformly, it is known as reciprocating or translating follower.

b) Oscillating or rotating follower-

When the uniform rotary motion of the cam is converted into

predetermined oscillatory motion of the follower, it is called

oscillating or rotating follower.

3. According to the path of motion of the follower-a) Radial follower- When the motion of the follower is along an axis

passing through the center of the cam, it is known as radial cam

b) Offset follower- when the motion of the follower is along an axis

away from the axis of the cam center, it is called offset follower.

Terms used in radial cam-

Following terms are important in drawing the cam profile

a) Base circle- The smallest circle that can be drawn to the camprofile

b) Trace point- It is the reference point on the follower and it is

used to generate pitch curve. In case of knife edge follower,

knife edge represents it and in case of roller follower it

corresponds to center of roller.


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c) Pressure angle- It is the angle between the direction of

follower motion and a normal to the pitch curve.

d) Pitch point- it is the point on the pitch curve having themaximum pressure angle.

e) Pitch circle- it is the circle drawn from the center of the cam

through the pitch point.

f) Pitch curve- It is the curve generated by the trace point as the

follower moves relative to the cam.

g) Prime circle- It is the smallest circle that can be drawn from

the center of the cam and tangent to the pitch curve.h) Lift or stroke- It is the maximum travel of the follower from its

lower position to the topmost position.

Motion of the follower-

Displacement diagrams:

Design requirements in the part of the machine under

consideration will dictate the type of movement required in the cam


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follower. This is then translated into the profile of a cam, which will

give the follower the required motion. When designing this profile

the movement of the follower is usually considered in four separate

sections: the period when the follower is at the bottom of its

movement, called the bottom dwell; the movement required during

the rise or lift of the follower; the period when the follower remains

at the top of its movement, called the top dwell; and the movement

required when the follower returns to the bottom position. There

are three different types of follower motion in standard use, which

are shown below.

The follower during its travel may have any one of the type of motion

as below

i) Uniform velocity

j) Simple harmonic motion

k) Uniform acceleration and retardation

Displacement diagram and cam profile when follower moves

with uniform velocity-Uniform (constant) velocity:

Since the velocity is constant, the displacement diagram will be a straight

line with constant slope and the velocity diagram rectangular with zero

acceleration.

However, to achieve this velocity immediately at the commencement

of the motion, and maintain it until the very end of the stroke, would

require infinitely high accelerations and declarations for infinitely shortperiods of time at the beginning and end of the stroke. This of course is

impossible. To reduce these peak accelerations and declarations and to

make the motion possible the conditions are modified to include a short

period of uniform acceleration and deceleration at the beginning and end

of the motion. This means that the follower moves with uniform velocity

for most of the stroke, parabolic or circular arcs being introduced at the

beginning and end of the displacement diagram. Despite thesemodifications it can be seen that, considering the conditions previously laid


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down, the high accelerations, particularly those at the end of the outgoing

stroke and the beginning of the fall stroke, require heavy springing to

ensure continuous contact between edge cam and follower.

Simple harmonic motion

The displacement diagram is a sine curve and if a cam is produced from

this curve only (i.e. devoid of top and bottom dwell) it will have lobes of

circular form. Consideration shows that this type of cam will give the

smoothest change of motion in the follower. An eccentric cam transmits

simple harmonic motion to the follower. Examples of simple harmonic

motion from everyday life are the up and down motion of a cork bobbing

on the waves on a pond, and the oscillating motion of a pendulum weight as

it swings from side to side, as shown below.


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Uniform acceleration and retardation

This displacement curve is parabolic. It gives a uniform rate of acceleration

from the start to the midpoint and a similar uniform rate of retardation

from the midpoint to the end of the movement.


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Drawing cam profiles

Examples-

1. uniform velocity with a knife-edge follower

Cam dataIn-line knife edge follower,

50 mm minimum diameter,

40 mm lift (rise) with uniform velocity,

0 degrees to 90 degrees bottom dwell, 90 degrees to 180 degrees

rise,

180 degrees to 270 degrees top dwell, 270 degrees to 360 fall,

clockwise rotation.

2. simple harmonic motion with a roller follower

Cam data

In-line roller follower, diameter 12 mm,

minimum cam diameter 50 mm,

total rise 42 mm, both rise and fall have simple harmonic motion,

0 to 90 degrees bottom dwell,

90 to 180 degrees rise with simple harmonic motion,

180 to 270 degrees top dwell,


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270 to 360 degrees fall with simple harmonic motion,

because this is a symmetrical cam it can rotate in either direction.

3. uniform acceleration and retardation + uniform velocity,

with a knife-edge follower

Cam data,

In-line knife edge follower,

minimum cam diameter 50 mm, rise 42 mm through 180 degrees

with uniform acceleration and retardation, fall 42 mm through 180

degrees with uniform velocity,

clockwise rotation.


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4. Uniform velocity with offset roller follower

Follower type = Knife edged, in-line; lift = 50mm; base circle radius =

50mm; out stroke with SHM, for 60 cam rotation; dwell for 45camrotation; return stroke with SHM, for 90 cam rotation; dwell for theremaining period.

Displacement diagram:


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Cam profile: Construct base circle. Mark points 1, 2, 3..in direction

opposite to the direction of cam rotation. Transfer points a,b,c..l from

displacement diagram to the cam profile and join them by a smooth free

hand curve. This forms the required cam profile.

(2) Draw the cam profile for the same operating conditions of problem (1),

with the follower offset by 10 mm to the left of cam center.

Displacement diagram: Same as previous case.


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Cam profile: Construction is same as previous case, except that the lines

drawn from 1,2,3. are tangential to the offset circle of 10mm dia. as

shown in the fig.

(3) Draw the cam profile for following conditions:

Follower type = roller follower, in-line; lift = 25mm; base circle radius =

20mm; roller radius = 5mm; out stroke with UARM, for 120 cam rotation;

dwell for 60 cam rotation; return stroke with UARM, for 90 cam rotation;

dwell for the remaining period.



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Cam profile: Construct base circle and prime circle (25mm radius). Mark

points 1,2,3..in direction opposite to the direction of cam rotation, on

prime circle. Transfer points a,b,c..l from displacement diagram. At each

of these points a,b,c draw circles of 5mm radius, representing rollers.

Starting from the first point of contact between roller and base circle, draw

a smooth free hand curve, tangential to all successive roller positions. This

forms the required cam profile.

(4) Draw the cam profile for conditions same as in (3), with follower off set

to right of cam center by 5mm and cam rotating counter clockwise.

Displacement diagram: Same as previous case.


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Cam profile: Construction is same as previous case, except that the lines

drawn from 1,2,3. are tangential to the offset circle of 10mm dia. as

shown in the fig.


Follower type = roller follower, off set to the right of cam axis by 18mm; lift

= 35mm;base circle radius = 50mm; roller radius = 14mm; out stroke with

SHM in 0.05sec; dwell for 0.0125sec; return stroke with UARM, during

0.125sec; dwell for the remaining period. During return stroke,

acceleration is 3/5 times retardation.


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Follower type = knife edged follower, in line; lift = 30mm; base circle radius

= 20mm;out stroke with uniform velocity in 1200 of cam rotation; dwell for

600; return stroke with uniform velocity, during 900 of cam rotation; dwell

for the remaining period.



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Follower type = roller follower, off-set to the right by 5mm; lift = 30mm;

base circle radius = 25mm; roller radius = 5mm; out stroke with SHM, for

1200 cam rotation; dwell for 600 cam rotation; return stroke during 1200

cam rotation; first half of return stroke with Uniform velocity and second

half with UARM; dwell for the remaining period.


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(12) A push rod of valve of an IC engine ascends with UARM, along a path

inclined to the vertical at 600. The same descends with SHM. The base

circle diameter of the cam is 50mm and the push rod has a roller of 60mm

diameter, fitted to its end. The axis of the roller and the cam fall on the

same vertical line. The stroke of the follower is 20mm. The angle of action


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for the outstroke and the return stroke is 600 each, interposed by a dwell

period of 600. Draw the profile of the cam.



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Chapter 4Power Transmission

Content4.1 Types of Drives Belt, Chain, Rope, Gear drives & their

comparison.

4.2 Belt Drives - flat belt, V belt & its applications, material for flat

and V-belt, angle of lap, belt length. Slip and creep. Determination of

velocity ratio, ratio of tight side and slack side tension, centrifugal

tension and initial tension, condition for maximum power

transmission (Simple numericals)

4.3 Chain Drives Advantages & Disadvantages, Selection of Chain &

Sprocket wheels, methods of lubrication.

4.4 Gear Drives Spur gear terminology, types of gears and gear

trains, and their selection for different application, train value &

Velocity ratio for compound, reverted and simple epicyclic gear train,

methods of lubrication, Law of gearing.

4.5 Rope Drives Types, applications, advantages & limitations of

Steel ropes.

Introduction

Power transmission is the movement ofenergy from its place of generation

to a location where it is applied to performing useful work.
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Power is defined formally as units ofenergy per unittime. In SI units:

The great majority of mechanical power transmission applicationsinvolve rotating shafts, since rotation is continuous and the shafts /

mountings are cheap relative to other means of power transmission.

Matching a prime- mover to a load thus involves transformation of power

between shafts - usually from a high speed / low torque drive shaft,

through a speed reducer of ratio R 1, to a low speed / high torque load

shaft.

Ideally, for gears and for belts, the speed reduction ratio and thetorque amplification ratio are each equal to the radius ratio, so that the

output power equals the input power and the efficiency is 100%. The speed

ratio across a real pair of gears always equals the ideal ratio because of the

positive drive, however sliding friction results in a torque ratio which is

less than ideal

Belt Drives

Introduction

A belt drive is a method of transferring rotary motion between two shafts.

A belt drive includes one pulley on each shaft and one or more continuous

belts over the two pulleys. The motion of the driving pulley is, generally,

transferred to the driven pulley via the friction between the belt and the

pulley. Synchronous/timing belts have teeth and therefore do not depend

on friction. Belt drives and gear transmissions have a much greater life

expectancy than belt drives. Belt drives also have relatively high

inspection and maintenance demands

Belt Drive advantages
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Easy, flexible equipment design, as tolerances are not important.

Isolation from shock and vibration between driver and driven system.

Driven shaft speed conveniently changed by changing pulley sizes.

Belt drives require no lubrication.

Maintenance is relatively convenient

Very quiet compared to chain drives, and direct spur gear drives

Flat Belts-

Flat Belt transfers torque by friction of the belt over a pulley. Traction

related to angle of contact of belt on pulley. It is susceptible to slip.

Material-Belt made from leather, woven cotton, rubber, and balata

V- Belt-

V-belt drives are essentially short centre drives, V-belt drives are

essentially short centre drives Vee belt Better torque transfer possible

compared to flat belt. Generally arranged with a number of matched vee

belts to transmit power. They are Smooth and reliable. Made from hi-text

woven textiles, polyurethane, etc. Vee Belts Poly-Vee Belt is flat on outside

and Vee Grooved along the inside. It Combines advantages of high traction

of the Vee belt and the use of only one belt

Advantages and Disadvantages of V-belt Drive over Flat Belt Drive

Following are the advantages and disadvantages of the V-belt drive over

flat belt drive:

Advantages

1. The V-belt drive gives compactness due to the small distance between

centres of pulleys.
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2. The drive is positive, because the slip between the belt and the pulley

groove is negligible.

3. Since the V-belts are made endless and there is no joint trouble,

therefore the drive is smooth.

4. It provides longer life, 3 to 5 years.

5. It can be easily installed and removed.

6. The operation of the belt and pulley is quiet.

7. The belts have the ability to cushion the shock when machines are

started.

8. The high velocity ratio (maximum 10) may be obtained.

9. The wedging action of the belt in the groove gives high value of limiting

ratio of tensions. Therefore the power transmitted by V-belts is more than

flat belts for the same coefficient of friction, arc of contact and allowable

tension in the belts.

10. The V-belt may be operated in either direction, with tight side of the

belt at the top or bottom. The centre line may be horizontal, vertical or

inclined.

Disadvantages

1. The V-belt drive cannot be used with large centre distances, because of

larger weight per unit length.

2. The V-belts are not as durable as flat belts.

3. The construction of pulleys for V-belts is more complicated than pulleys

of flat belts.

4. Since the V-belts are subjected to certain amount of creep, therefore

these are not suitable for constant speed applications such as synchronous

machines and timing devices.

5. The belt life is greatly influenced with temperature changes, improper

belt tension and mismatching of belt lengths.

6. The centrifugal tension prevents the use ofV-belts at speeds below 5 m/

s and above 50 m / s

Typical belt drives

Two types of belt drives, an open belt drive, and a crossed belt

drive are shown. In both the drives, a belt is wrapped around the pulleys.

Let us consider the smaller pulley to be the driving pulley. This pulley will


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transmit motion to the belt and the motion of the belt in turn will give a

rotation to the larger driven pulley. In open belt drive system the rotation

of both the pulleys is in the same direction, whereas, for crossed belt drive

system, opposite direction of rotation is observed.

Angle of lap- When the two pulleys of different diameter are connected by

means of an open belt drive, then the angle of contact or lap () at smaller

pulley must be taken into consideration.

Sin =

=

=

=(180-2) *

radian

Length of belt-

D1

- Diameter of the larger pulley

d2

Diameter of the smaller pulley

1- Angle of wrap of the larger pulley

2 Angle of wrap of the smaller pulley

X- Center distance between the two pulleys


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Basic Formulae

1 =

180

+ 2

2 =

180

- 2

Where angle is,

X =

+ 2X +

Similarly length of cross belt drive

X =

+ 2X +

Slip of belt-

It is assumed that there is a firm frictional grip between shaft and

belt. But sometimes this grip is insifficient to carry shaft with it. This may

cause some forward motion iof shaft without carrying belt with it. This may

also cause forward motion of belt without carrying the driven pulley with

it. This is calles slip of belt.

Let S1% = slip between driver and belt

S2% = slip between belt and follower

Velocity of belt passing over the driver pulley /sec

V =

-

*

=

[ ] /100 ..(i)

Velocity of belt passing over follower

V =

-

*

=

[ ] /100 ..(ii)

=

* [ ] /100 *[ ] /100

=

* (1-

-

)

=

* (1-

) where S = S1 + S2

When thickness of the belt is considered.

=

* (1-

) where S = S1 + S2

Creep of Belt-

When the belt passes from the slack side to tight side, a certain

portion of belt extends and it contracts again when belt passes from tight


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side to slack side.Due to these changes of lenghts there is relative motion

between belt and pulley surface. This relative motion is termed as creep.

Presence of friction between pulley and belt causes differentialtension in the belt. This differential tension causes the belt to elongate orcontract and create a relative motion between the belt and the pulleysurface. This relative motion between the belt and the pulley surface iscreated due to the phenomena known as elastic creep.Effect of creep is sligthly reduction in speed.

=

*

Velocity ratio

D1 = diameter of the driver pulley

D2 = diameter of the driven pulley

N1,N2 = Speed of driver and driven pulley repectively in rpm

Length of belt passing over driver pulley in one revolution = D1

Length of the belt passing over driver pulley in n1 revolution = D1 n1

(i)

Similarly Length of the belt passing over driven pulley in n2 revolution =

D2 n2 .(ii)

Since equation I = Equation ii

D1 n1 = D2 n2

Speed ratio, R = n1 / n2 = D2 / D1 1

If thicknesst of the belt is considered

(n1+ t ) / n2+ t= D2 + t / D+ t

Ratio of tight side and slack side tension

The belt drives primarily operate on the friction principle. i.e. the friction

between the belt and the pulley is responsible for transmitting power from


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one pulley to the other. In other words the driving pulley will give a motion

to the belt and the motion of the belt will be transmitted to the driven

pulley. Due to the presence of friction between the pulley and the belt

surfaces, tensions on both the sides of the belt are not equal. So it is

important that one has to identify the higher tension side and the lower

tension side,

It is observed that the slack side of the belt is in the upper

side and the tight side of the belt is in the lower side. The slack side of the

belt, due to self weight, will not be in a straight line but will sag and the

angle of contact will increase. However, the tight side will not sag to that

extent. Hence, the net effect will be an increase of the angle of contact or

angle of wrap. It will be shown later that due to the increase in angle of

contact, the power transmission capacity of the drive system will increase.

On the other hand, if it is other way round, that is, if the slack side is on the

lower side and the tight side is on the upper side, for the same reason as

above, the angle of wrap will decrease and the power transmission capacity

will also decrease. Hence, in case of horizontal drive system the tight side is

on the lower side and the slack side is always on the upper side.

Derivation of relationship between belt tensions

T1, T2 = Tension in tight and slack side respectively

= coefficient of friction between belt and pulley

= angle of lap

R= normal reaction of belt and pulley


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Let us consider small element of belt AB subtending angle

T = tension in slack side

T+ T = Tension in tight side

Considering the element AB under equilibrium and resolving forces

vertically

R T * sin (/2) (T + T) * sin (/2) = 0

R = T * sin (/2) + (T + T) * sin (/2)

As is very small

T * /2 + T * /2 + T*/2 = R

T * /2 + T * /2 = R

Neglecting T*

2T*/2 = R

T* = R ..(i)

Now resolving forces horizontally

(T + T)* cos /2 T* cos /2 = R

T*cos /2 + T* cos/2 T cos /2 = R

T * cos/2 = R.

As is very small

T = R (ii)

From equation I and ii

T = T

T/T =

Integrating above equation

=

[ln T] = []

Ln (T1/T2) = *

T1/T2 =

ii) For V Belts

R = normal Reaction

2 = Angle of groove

R = reaction in the plane grove

= coefficient of friction


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Considering equilibrium and resolving forces vertically and horizontally

Centrifugal tension-

As the belt run at uniform speed over an arc of a circle the centrifugal force

acts on the belt due to its mass. To balance this centrifugal force a tension a

tension is generated equally in tight side and slack side of belt. It is called

centrifugal tension.

M= mass of belt per meter length

R= radius of pulley

V= linear velocity of belt

Fc= centrifugal force

Tc= centrifugal tension

Length of element Mn = r*

Mass of MN = m.r.

Fc = m.r. . V/r

Fc = m * *v .(i)

Now considering equilibrium and resolving forces vertically

Tc*sin(/2) + Tc*sin(/2) = fc

Fc = 2 Tc*sin(/2)

As is very small sin(/2) = /2

2Tc * /2 = m. v .

Tc= m.v

Initial Tension

When belt is mounted on pulley it is provided with some initial tension

Let us determine the magnitude of the initial tension in the belt.


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Tight side elongation (T1

Ti)

Slack side contraction (Ti

T2

)

Where, Tiis the initial belt tension.

Since, belt length remains the same, i.e., the elongation is same as the

contraction,

It is to be noted that with the increase in initial tension power transmission

can be increased. If initial tension is gradually increased then T1

will also

increase and at the same time T 2 will decrease. Thus, if it happens that T 2

is equal to zero, then T1

= 2Ti

and one can achieve maximum power

transmission.

Power transmission of belt drive

Power transmission of a belt drive is expressed as,

P = (T1 T

2)v (i)

Where,P is the power transmission in Watt and v is the belt velocity in m/s.

T1, T2 = Tension in tight side and slack side respectively

T1/T2 =

T2 = T1/.(ii)

Equation I become

P = (T1 - T1/) * v

= T1 (1-1/) * v= T1 * v* c (iii) where C = (1-1/)

We know that Tmax = T1 + Tc

T1 = Tmax Tc

Substituting in equation iii

P = (T max - Tc) * v * C

= (Tmax - mv) * v * C

= (Tm.v - mv) * C


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For maximum power differentiating above equation with respect to

v and equating to zero.

= 0

= 0

T 3m.v = 0

T 3Tc = 0

T= 3 Tc

Numericals -

i) A pump is driven by an electric motor through a open type flat

belt drive. Determine the belt specifications for the following

data.

Motor pulley diameter (dS) = 300 mm, Pump pulley diameter (d

L)

= 600 mm

Coefficient of friction (S) for motor pulley = 0.25

Coefficient of friction (L) for pump pulley = 0.20

Center distance between the pulleys=1000 mm; Rotational speed of the

motor=1440 rpm;

Power transmission = 20kW; density of belt material ()= 1000 kg/m3

;

allowable stress for the belt material () = 2 MPa; thickness of the belt =

5mm.


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3. Find the power transmitted by a belt running over a pulley of 600mm

diameter at 200 rpm. The coefficient of friction between the belt and

pulley is 0.25, angle of lap 160 and maximum tension in belt is 2500 N.

4. In a flat belt drive the initial tension is 2000n. the coefficient of friction

between belt and pulley is 0.3 and the angle of lap for smaller pulley is


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150. The smaller pulley has a radius of 200mm and rotates at 500 rpm.

Find the power in KW transmitted by belt drive.

5. An open belt drive running over two pulleys 240 mm and 600 mm

diameter connects two parallel shafts 3m apart and transmits 4 KW from

the smaller pulley that rotates at 300 rpm. Coefficient of friction between

belt and pulley is 0.3 and safe working tension is 10 n per mm width.

Determine

a) Maximum width of belt

b) Initial tension in belt

c) Length of belt required

5.

Chain Drives

Introduction

Chain drive is a way of transmitting mechanical power from one place

to another. It is often used to convey power to the wheels of a vehicle,

particularly bicycles and motorcycles. It is also used in a wide variety of

machines besides vehicles.

Most often, the power is conveyed by a roller chain, known as the

drive chain or transmission chain, passing over a sprocketgear, with the

teeth of the gear meshing with the holes in the links of the chain. The gear

is turned, and this pulls the chain putting mechanical force into the system.

A chain is a method of transferring rotary motion between two parallel

shafts. The chain drive is positive, efficient and high torques can be

transmitted. The chain is generally made from steel although plastic chains

have been developed

Advantages & Disadvantages,
http://en.wikipedia.org/wiki/Bicyclehttp://en.wikipedia.org/wiki/Motorcyclehttp://en.wikipedia.org/wiki/Roller_chainhttp://en.wikipedia.org/wiki/Sprockethttp://en.wikipedia.org/wiki/Sprockethttp://en.wikipedia.org/wiki/Roller_chainhttp://en.wikipedia.org/wiki/Motorcyclehttp://en.wikipedia.org/wiki/Bicycle


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Power transmission chains can be categorized as roller chain, engineering

steel chain, silent chain, detachable chain, and offset sidebar chain.

Some of the advantages of chain drives over belt drives are:

No slippage between chain and sprocket teeth.

Negligible stretch, allowing chains to carry heavy loads.

Long operating life expectancy because flexure and friction contact occur

between hardened bearing surfaces separated by an oil film.

Operates in hostile environments such as high temperatures, high

moisture or oily areas, dusty, dirty, and corrosive atmospheres, etc.,

especially if high alloy metals and other special materials are used.

Long shelf life because metal chain ordinarily doesnt deteriorate with

age and is unaffected by sun, reasonable ranges of heat, moisture, and oil.

Certain types can be replaced without disturbing other components

mounted on the same shafts as sprockets.

Drawbacks

Noise is usually higher than with belts or gears, but silent chain drives are

relatively quiet.

Chain drives can elongate due to wearing of link and sprocket teeth

contact surfaces.

Chain flexibility is limited to a single plane whereas some belt drives are

not.

Usually limited to somewhat lower-speed applications compared to belts

or gears.

Sprockets usually should be replaced because of wear when worn chain is

replaced. V-belt sheaves exhibit very low wear.

Selection of Chain & Sprocket wheels,

The following data should be taken into consideration while selecting

roller chain drives.

a. Horsepower to be transmitted

b. RPM of the driving and driven sprocket (Speed ratio)

c. Load classification

d. Space limitations if any

e. Driven machine


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f. Source of power

Methods of lubrication

Gear Drives

Spur gear terminology,


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The spur gear terms:

The pitch circle is the circle representing the original cylinder which

transmitted motion by friction, and its diameter the pitch circle diameter.

The center distance of a pair of meshing spur gears is the sum of theirpitch circle radii. One of the advantages of the involute system is that small

variations in the center distance do not affect the correct the correct

working of the gears.

The addendum is the radial height of a tooth above the pitch circle.

The dedendum is the radial depth below the pitch circle.

The clearanceis the difference between the addendum and the dedendum.

The whole depth of a tooth is the sum of the addendum and thededendum.

The working depth of a tooth is the maximum depth that the tooth extends

into the tooth space of a mating gear. It is the sum of the addenda of the

gear.

The addendum circle is that which contains the tops of the teeth and its

diameter is the outside or blank diameter.

The dedendum or root circle is that which contains the bottoms of thetooth spaces and its diameter is the root diameter.

Circular tooth thickness is measured on the tooth around the pitch circle,

that is, it is the length of an arc.

Circular pitch is the distance from a point on one tooth to the

corresponding point on the next tooth, measured around the pitch circle.

The module is the pitch circle diameter divided by the number of teeth.

TheDiametrical pitchis the number of teeth per inch of pitch circlediameter. This is a ratio.

The pitch point is the point of contact between the pitch circles of two gears

in mesh.

The line of action. Contact between the teeth of meshing gears takes place

along a line tangential to the two base circles. This line passes through the

pitch point and is called the line of action.

The pressure angle. The angle between the line of action and the commontangent to the pitch circles at the pitch point is the pressure angle.


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The tooth face is the surface of a tooth above the pitch circle, parallel to the

axis of the gear.

The tooth flank is the tooth surface below the pitch circle, parallel to the

axis of the gear. If any part of the flank extends inside the base circle it

cannot have involute form. It may have ant other form, which does not

interfere with mating teeth, and is usually a straight radial line.

Types of gears

The type of gear depends upon the teeth cut on the cylindrical disc and

their use. When the spiral gears are used to connect parallel shafts, they

are called spur gears and when they are used to connect non-parallel

shafts, they are called spiral gears.

Spur gears

A spur gear is one of the most important ways of tran

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