KINEMATICS

OF MECHANISMS

ApplicationTo select drive and devices for power transmission. To operate and maintain machine and mechanism used in field of Automobile, machine tool, workshop etc.

Procedures Analysis of Mechanism in machine. Velocity and acceleration diagrams, cam profile……etc

Principles Conversion of Kinematic chain to mechanism, Relative Velocity and Acceleration in Mechanism, Law of Inversions and Governor.

Concepts

Mechanism, Inversion, Kinematic link, pair, Kinematic chain,constrain motion, velocity and acceleration in mechanism,displacement diagram, turning moment, torque, vibration,balancing.

Facts Cam, Follower, Belt, Chain, Gear, Flywheel, Governor, brake, clutch, Dynamometers

AIM

TO STUDY VARIOUS TYPES OF KINEMATIC LINKS,PAIRS,CHAINS,

MECHANISM,MACHINE AND STRUCTURE

KINEMATICS

RELEVANCE OF KINEMATIC STUDY

Motion requirements

Design requirements

MOTION STUDY

Study of position, displacement, velocity and acceleration of different elements of mechanism

Types of Motion

Pure rotation: the body possesses one point (center of rotation) that has no motion with respect to the “stationary” frame of reference. All other points move in circular arcs.

Pure translation: all points on the body describe parallel (curvilinear or rectilinear) paths.

Complex motion: a simultaneous combination of rotation and translation.

DESIGN REQUIREMENTSDesign: determination of shape and size

1. Requires knowledge of material

2. Requires knowledge of stress

Requires knowledge of load acting

(i) static load

(ii) dynamic/inertia load

WHAT DO YOU MEAN BY KINEMATIC LINK?

ORLINKOR

ELEMENT

Link:is a rigid body which possesses at lease two nodes which are points for attachment to other links.

nodes

Ternary link

Quaternary linkBinary link

nodes

Machine Element:

Binary link

This link has several machine elements

1. Cylinder 2. Piston 3. Coupler4. Crank5. Pinion6. Flywheel7. Gear8. Inlet cam9. Outlet cam10. Inlet valve11. Outlet valve

Examples of rigid links

EXAMPLE:

SLIDER CRANK MECHANISM CONSIST OF FOUR LINKS:

•FRAME AND GUIDES

• CRANK

•CONNESTING ROD

•SLIDER

SLIDER CRANK MECHANISM:

CONSTRAINED MOTION

One element has got only one definite motion relative to the other

(a) Completely constrained motion

(b) Successfully constrained motion

(c) Incompletely constrained motion

KINEMATIC PAIR

AND CLASSIFICTIO

N

PAIRING ELEMENTS

PAIRING ELEMENTS

KINEMATIC PAIRS ACCORDING TO NATURE OF CONTACT:

LOWER PAIR:LINKS HAVING SURFACE OR AREA

CONTACT

EXAMPLE:I)NUT TURNING ON A SCREW2)SHAFT ROTATING IN A BEARING

1 2

Turning pair

Link 2(guide, slot)

Link 1(slider)

sliding pair

Lower pairs

Prismatic joint Revolute joint Cylindric joint Spherical joint Helical joint Planar joint

Type P joint R joint C joint S joint H joint F joint

DOF 1 1 2 3 1 3

HIGHER PAIR(POINT OR LINE CONTACT BETWEEN LINKS)

EXAMPLE:

• WHEN ROLLLING ON A SURFACE

• CAM AND FOLLOWER

• TOOTHED GEARS

• BALL AND ROLLER BEARINGS

HIGHER PAIRS:

Examples of higher pairsHigher pairs

1 (cam)

2 (follower)

DOF

5 2

Examples of higher pairsHigher pairs

1 (cam)

2 (follower)

KINEMATIC PAIRS ACCORDING TO THE NATURE OF RELATIVE

MOTION:

REVOLUTE PAIR:

PRISMATIC PAIR:

SCREW PAIR:

CYLINDRICAL PAIR:

SPHERICAL PAIR:

PLANAR PAIR:

KINEMATIC PAIRS ACCORDING TO MECHANICAL CONSTRAINT:

CLOSED PAIR: (WHEN THE ELEMENTS OF PAIR ARE

HELD TOGETHER MECHANICALLY)

EXAMPLE ALL THE LOWER PAIRS AND SOME OF

THE HIGHER PAIR

UNCLOSED PAIR: (WHEN TWO LINKS OF PAIR ARE IN

CONTACT EITHER DUE TO FORCE OF GRAVITY OR SOME SPRING ACTION)

EXAMPLECAM AND FOLLOWER

KINEMATIC CHAIN

LOCKED CHAIN OR STRUCTURE

Links connected in such a way that no relative motion is possible.

MECHANISM

Slider crank and four bar mechanisms

Working of slider crank mechanism

Unconstrained kinematic chain

PLANAR MECHANISMSWhen all the links of a mechanism have plane motion, it

is called as a planar mechanism. All the links in a planar mechanism move in planes parallel to the reference plane.

TYPES OF KINEMATIC CHAIN:

• FOUR BAR CHAIN MECHANISM

• SINGLE SLIDER CRANK CHAIN

• DOUBLE SLIDER CRANK CHAIN

MACHINE AND

STRUCTURE

Characteristics of a machine:

1. Composed of man-made elements

2. The elements are arranged to transmit motion

3. Can transmit energy, and fulfill certain work for human being.

Degrees of freedom ormobility of a mechanism

It is the number of inputs (number of independent coordinates) required to describe the configuration or position of all the links of the mechanism, with respect to the fixed link at any given instant.

51

Degrees of Freedom (DOF) or Mobility

• DOF: Number of independent parameters (measurements) needed to uniquely define position of a system in space at any instant of time.

Rigid body in a plane has 3 DOF: x,y,

Rigid body in space has 6 DOF (3 translations & 3 rotations)

GRUBLER’S CRITERIONNumber of degrees of freedom of a

mechanism is given by F = 3(n-1)-2l-h. Where,F = Degrees of freedomn = Number of links in the mechanism.l = Number of lower pairs, which is obtained

by counting the number of joints. If more than two links are joined together at any point, then, one additional lower pair is to be considered for every additional link.

h = Number of higher pairs

Examples - DOF

F = 3(n-1)-2l-hHere, n = 4, l = 4 & h = 0.F = 3(4-1)-2(4) = 1I.e., one input to any one link will

result in definite motion of all the links.

Grubler's Equation

A rigid body is constrained by a lower pair, which allows only rotational or sliding movement. It has one degree of freedom, and the two degrees of freedom are lost.

When a rigid body is constrained by a higher pair, it has two degrees of freedom: translating along the curved surface and turning about the instantaneous contact point.

N - number of kinematic elements including the fixed member.

L - number of lower pairs in the mechanism

H - the number of higher pairs in the mechanism

Then, the resultant number of degrees of freedom in the mechanism (this should be one)

F = 3(N-1) - 2L – H = 1,

which gives

3N - 2L - H = 4

This is known as Grubler’s Equation.

F = 3(n-1)-2l-hHere, n = 5, l = 5 and h = 0.F = 3(5-1)-2(5) = 2I.e., two inputs to any two links are

required to yield definite motions in all the links.

F = 3(n-1)-2l-hHere, n = 6, l = 7 and h = 0.F = 3(6-1)-2(7) = 1I.e., one input to any one link will result in definite motion of all the links.

F = 3(n-1)-2l-hHere, n = 6, l = 7 (at the intersection of 2, 3 and 4, two

lower pairs are to be considered) and h = 0.F = 3(6-1)-2(7) = 1

F = 3(n-1)-2l-hHere, n = 11, l = 15 (two lower pairs at the intersection

of 3, 4, 6; 2, 4, 5; 5, 7, 8; 8, 10, 11) and h = 0.F = 3(11-1)-2(15) = 0

(a)F = 3(n-1)-2l-hHere, n = 4, l = 5 and h = 0.F = 3(4-1)-2(5) = -1I.e., it is a structure

(b) F = 3(n-1)-2l-hHere, n = 3, l = 2 and h = 1.F = 3(3-1)-2(2)-1 = 1

(c) F = 3(n-1)-2l-hHere, n = 3, l = 2 and h = 1.F = 3(3-1)-2(2)-1 = 1

INVERSIONS OF MECHANISM

A mechanism is one in which one of the links of a kinematic chain is fixed. Different mechanisms can be obtained by fixing different links of the same kinematic chain. These are called as inversions of the mechanism.

Where would you see 4-bar mechanisms?

Grashof condition predicts behavior of linkage based only on length of links S=length of shortest link L=length of longest link P,Q=length of two remaining links

If S+L ≤ P+Q the linkage is Grashof :at least one link is capable of making a complete revolution

Otherwise the linkage is non-Grashof : no link is capable of making a complete revolution

The Grashof Condition

For S+L<P+QCrank-rocker if either

link adjacent to shortest is grounded

Double crank if shortest link is grounded

Double rocker if link opposite to shortest is grounded

For S+L>P+Q All inversions will be double rockers

No link can fully rotate

69

For S+L=P+Q (Special case Grashof)

• All inversions will be double cranks or crank rockers

• Linkage can form parallelogram or antiparallelogram

• Often used to keep coupler parallel (drafting machine)

Parallelogram form

Anti parallelogram form

Deltoid form

70

Linkages of more than 4 bars

5-bar 2DOFGeared 5-bar 1DOF

• Provide more complex motion• See Watt’s sixbar and Stephenson’s sixbar

mechanisms in the textbook

Linkages of more than 4 bars

72

More Examples: Front End Loader

73

Drum Brake

EXAMPLE FOR SUPERSTRUCTURE

(link 1) frame(link 2) crank(link 3) coupler (link 4) rocker

Four bar mechanism

1

2

3

4

1

(1:2), (2:3), (3:4), (4:1) - Turning pairs

Link 1 is fixed

INVERSIONS OF FOUR BAR CHAIN

1. Double Crank mechanism: parallel 4 bar linkage which is used in coupled wheels of locomotive)

1. Beam engine (crank- rocker mechanism)

1. Watts indicator diagram (Rocker-rocker)

DOUBLE CRANK MECHANISM

PARALLEL-4 BAR LINKAGE MECHANISM

CRANK-ROCKER MECHANISMDRAG LINK MECHANISM

CRANK-ROCKER MECHANISM

INVERSIONS OF SLIDER CRANK CHAIN

Inversions of Slider-crank Mechanism

Slider crank mechanism (link-1 is fixed)

Main bearing 1(fixed)

Crank 2Connecting rod 3

Piston 4

Cylinder 1 (fixed)

IDC ODC

Stroke length (L=2r)

r

LINK1 IS FIXED

SLOT

CRANK

APPLICATION:RECIPROCATING PUMPS,RECIPROCATINGCOMPRESSORS,OSCILLATING CYLINDER ENGINE

PISTON

CRANK SHAFT

1

23

4

Link 2 is fixed

12

3 4 Inversion 2 (link-2 is fixed)

Whitworth quick-return mechanism or Gnome aircraft engine

Whitworth quick return motion mechanism

APPLICATION:SHAPING AND SLOTTING MACHINE

1

2 3

4

Link 3 is fixed

Inversion 3 (link-3 is fixed)

The oscillating cylinder engine Foot pump

1

2

3

4

Crank and slotted lever quick return motion mechanism

Crank and slotted lever quick return motion mechanism

Application of Crank and slotted lever quick return motion mechanism

1

2

3

4

Inversion 4 (link- 4 is fixed)

Hand-pump

Oscillating cylinder mechanism (connecting rod fixed)

APPLICATION:HAND PUMP

Pendulum pump or bull engine–inversion of slider crank mechanism (slider fixed)

– A simple example (simple crank – slider mechanism)• The structure

• The mathematical model r

l

u

l+r

l-r

x

Rotary engine– (crank fixed)

DOUBLE SLIDER CRANK CHAIN

It is a kinematic chain consisting of two turning pairs and two sliding pairs.

SCOTCH –YOKE MECHANISMTurning pairs –1&2, 2&3; Sliding pairs – 3&4, 4&1

SCOTCH-YOKE MECHANISM

4

1

32

A

B

ELLIPTICAL TRAMMEL

ELLIPTICAL TRAMMEL

11

1

2

4

OLDHAM COUPLING

OLDHAM COUPLING

Quick return motion mechanisms

Drag link mechanism

Whitworth quick return motion mechanism

Crank and slotted lever quick return motion mechanism

DRAG LINK MECHANISM

12

21

ˆ

ˆ

BAB

BAB

urnstrokeTimeforret

wardstrokeTimeforfor

INTERMITTENT MOTION MECHANISMS

Ratchet and pawl mechanism

APPLICATION:CLOCKS AND WATCHES, FEED MECHANISMS,LIFTING JACK

AND COUNTING DEVICES

Application of Ratchet Pawl mechanism

Geneva wheel mechanism

APPLICATION:CLOCKS AND WATCHES,FEEDING OF FIM ROLL IN EARLY MOTION PICTURE

TOGGLE MECHANISM

TOGGLE MECHANISM

APPLICATION:TOGGLE CLAMPS,RIVETTING M/C,SHEET METALPUNCHING,

FORMING, PRESS,STONE CRUSHERS,SWITCHES,CIRCUIT BREAKERS,

Straight line motion mechanisms

..,

.

constACifABconstAE

constbutADAD

ACABAE

AE

AB

AC

AD

ABDAEC

Locus of pt.C will be a straight line, ┴ to AE if, AB*AC

is constant.

Proof:

Peaucellier mechanism

FIG1

SCOTT RUSSEL MECHANISM

PANTOGRAPH

Hooke’s joint

STEERING GEAR MECHANISM

Ackermann steering gear mechanism

CL

SL

Ackermann steering gear mechanism

TEN BAR STEERING

14 BAR STEERING