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A linkageis a kinematic chain in which one of the links is

fixed to the ground which usually is the frame.

A linkage permits relative motion between its links andmay have one or more degree of freedom.

A linkage with zero or negative degree of freedom is a

structure which does not allow any relative motionbetween the links.

2.1. FOURBAR LINKAGE

Fig 2.1 below shows four bar linkage The conventional numbering system is to label the

ground or frame as link 1, & then to number linksclockwise around the mechanism loop

2. LINKAGES

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Link 1: is the frameor ground; generally it is stationary.

Link 2: is the driver; may rotate or oscillate.

Link 3: is the coupleror connecting rodand undergoesgeneral plane motion.

Link 4: is the followeror driven element, may rotate or oscillate depending on the rotary or oscillatory

motion of link 2, and on link dimensions.

These four bar links are joined by four revolute joints.

2

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Four bar linkages are able to produce a variety of non-uniform motion and can transmit large force.

The links of a four-bar mechanism should beproportional in such a way that lookingis avoided.

PositionABO4 shows the case of locking.

For such a position link four can move in any of the twodirections as indicated in the figure.

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2.1.1 Transmission Angle

The angle between the coupler 3 and the output link 4(follower) is called the transmission angle.

The equation for the transmission angle can be derivedas follows. From fig 2.3

cos2cos2 432

4

2

3221

2

2

2

1

2rrrrrrrrz

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From which we obtain the transmission angle.

In general, for good force transmission to the output link,the transmission angle should be in the range of

40o

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2.1.2 Motion of a Four-Bar Mechanism

There are three basic types of motion which a four-bar linkage

can produce. Crank-rocker:- to indicate that link 2 rotates and link 4 oscillates;

Double crank:- to indicate that both the driver and followerrotate;

Double-rocker:- to indicate that both the driver and followeroscillate through certain angles.

To determine whether a four-bar link will operate as one of theabove motion types, Grashoffs law is applied which isstated as follows:

i) If the sum of the length of the longest and shortest links isless than or equal to the sum of the lengths of the other twolinks, then

a) Two different crank rockerswill be formed when the shortestlink is the crank and either of the adjacent links is the fixed link.

http://../animations/fourbar.avihttp://../animations/fourbar.avihttp://../animations/fourbar.avihttp://../animations/fourbar.avihttp://../animations/fourbar.avihttp://../animations/fourbar.avi

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b) a double crankwill be formed when the shortest link is the fixed link;

c) a double rocker will be formed when the link opposite the shortest

link is the fixed link.ii) If the sum of the lengths of the longest and shortest links is

greater than the sum of the lengths of the other two, only adouble-rockermechanism will be formed.

2.1.3 Variation of the Four-Bar Linkage

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2.2 SLIDER-CRANK MECHANISM

Slider-crank mechanismis basically a four-bar mechanism

with three revolute joints, or turning pairs and a prismatic jointor a sliding pair.

In the slider crank mechanism, commonly

- link 1 is the frame, considered to be fixed;- link 2is the crankwhich is the driver (rotating motion);

- link 3is the connecting rod, the link b/n the driver & follower;

- link 4is the sliderwhich is the driven element (reciprocating)

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The slider crank mechanism converts rotary motion intoreciprocating motion and vice-versa.

It is commonly applied in internal combustion engines. During a cycle there are two dead points A and A in which

the crank and the connecting rod are in line.

at the dead positions the crank can move in either direction unlessconstrained by an external force.

In case of an engine the external constraint is provided by the remainingcylinders and a flywheel.

2.2.1 Inversion of the Slider-Crank Mechanism

As many inversions are obtained as the number of links in theoriginal mechanism

N. B. that inversion of a mechanism does not change therelative motion of the link, however, the absolute motion isaltered.

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2.3. THE SCOTCH YOKE

It is widely used as a sine and cosine generator, i.e. it is used

to produce harmonic motion. It is also used to produce desired vibration.

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The displacement of the slider x in moving from A to A is

given by

Substituting =t, the displacement is

The velocity and acceleration of the follower is

)2.2()cos1(cos rrrx

)3.2()cos1( trx

)5.2(coscos

)4.2(sinsin

22

2

2

rtrtd

xda

rtrdt

dxv

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2.4 QUICK-RETURN MECHANISM

Give a quick return-stroke of the followerfor a constant

angular velocity of the driver. The ratio of the crank angle for the working stroke to that

of the return stroke is known as the time-ratio.

The time ratio for quick return mechanisms is always

greater than unity to give a slower cutting stroke and afaster return stroke.

1 strokereturnofangle

strokecuttingofangleratiotime

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2.4.1 Crank-Shaper Mechanism

The figure below shows schematic

representation of six-bar crank-shapermechanism.

Links 1-4 of this mechanism form avariation of the slider-crankmechanism in which the crank is held

fixed.

For the direction of motion indicated inthe figure,

the cutting stroke occures when the

crank rotates from O2Ato O2Athrough angle ,

the idle stroke being when the crankmoves from O2Ato O2Athrough theangle .

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For constant angular speed of the crank, the time ratio Q isgiven by

And for constant angular velocity 2of link 2,

where = 2tw and = 2tr

Length of stroke of the tool holder C is given by

)6.2(r

w

t

t

strokereturnoftime

strokecuttingoftimeQ

)7.2(

Q

90-2

Bsin2O

DB2

BBstrokeoflength

4

)8.2(OO

AOB2Ostrokeoflength

42

24

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2.4.2 Drag Link

Developed by connecting two four bar linkages in series.

For a constant angular velocity of link 2, link 4 will rotate at a non-uniform velocity.

time ratio Q is:

Q

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2.4.3 Whitworth Mechanism

Another variation of the slider crank mechanism in which thecrank is held fixed

Commonly used in shaping and slotting machines

Time ratio Q is:

Q

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2.5 Toggle Mechanisms Simple toggle consists of two links which tend to line-up in a straight

line at one point in their motion.

The mechanical advantage of the simple toggle above is the velocityratio of the input point A to the output point B

Mechanical advantage =

As => 90o, CA & ABcome in to toggle

Used in punch presses, riveting machines, stone crusher, etc

B

A

A

B

v

v

y

x

F

F tan

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2.5.1 Stone Crusher

The stone crusher shown uses twotoggle linkages in series

=> high mechanical advantage.

When links 2 and 3 are in toggle,links 4 & 5 are also in toggle

=> produce high crushing force needed.

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2.6. Straight Line Mechanism

These are mechanisms which can generate straight lines from rotarymotion.

Point on one of the links moves in a straight line with out the need ofguides.

Converts rotary motion into straight line motion.

2.6.1 Watt mechanism Produces an approximate straight line motion

For equal lengths of links 2 & 4, the tracing point P traces anapproximate straight line.

This will happen if AP/PB = O4B/O2A.

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2.7. Parallel Mechanism

Used to produce parallel motions & reproducing motions at differentscale.

Common examples Pantograph

Drafting machine

2.7.1. The Pantograph

Used to enlarge or reduce trajectories to different scales.

Commonly used in cutting tools to duplicate complicated shapes to

desired scales. Links 2, 3, 4, & 5 form a parallelogram

Link 3 is extended to contain point C and point E lies on theintersection of lines O2Cand DB.

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A pen attached at Ereproduces the movement of Cto a reducedscale and vise versa i.e. the motion of Eis parallel to that of C.

To produce this parallel motion the necessary condition to be

satisfied for all positions of Cis

=>for any position of C, triangle O2DEis similar to triangle CBE.

=>

The ratio of the sizes of the figures at Cand Eis

.2

2 constEO

CO

.22

constCB

DO

EC

EO

EO

CO

Eatf igureofsize

Catf igureofsize

2

2

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Another example of parallel mechanism is found indrafting machine.

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2.8. INTERMITTENT MOTION MECHANISM

Converts continuous motion intointermittent motion.

Common examples are the Geneva wheel& ratchet mechanism.

2.8.1. Geneva Mechanism.

Provides intermittent rotary motion.

During one cycle of the crank, the Geneva

wheel rotates through fraction part of arevolution.

The circular segment attached to the cranklocks the wheel against rotation when theroller is not engaged.

Angle is half the angle subtended byadjacent slots.

where n=number of slots.

Let r2=the crank radius, the centerdistance Cis:

n2

360

sin

2rC

http://../animations/geneva.avihttp://../animations/geneva.avi

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Locking-slid Geneva

Ratchet Mechanism

Used to produce intermittent circular motion from an oscillatingor reciprocating member and/or to allow rotational motion in one

direction alone.

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2.9. STEERING GEAR MECHANISM

Used to change the direction of the wheel axle with respect to thechassis which enables motion of an automobile in any desired

direction.

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To avoid skidding or slipping of the wheels sideways, the fronttwo wheels must turn about the same instantaneous center Cwhich lies on the axis of the back wheels.

This avoids undue wear in the tires. This is also the condition for correct steering

this will be satisfied if > . Where: is inner wheel turning angleis outer wheel turning angle.

the condition to be satisfied is obtained as follows.

And the condition for correct steering is obtained to be

this is the fundamental equation for correct steering which, ifsatisfied, eliminates skidding of the front wheel.

bx

OCBOCot

b

ax

OC

AOCot

b

aCotCot

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2.9.1 Ackerman Steering Gear

Consists of a four bar mechanism joined by revolute joints.

The shorter links QRand PSare of equal length and are

connected to the front wheel axles by hinge joints. Links PQand RSare of unequal length.