INTRODUCTION  Theory of Machines may be defined as that branch of Engineering-science, which deals with the study of rela...

1. Department of Mechanical Engineering
JSS Academy of Technical Education, Bangalore-560060
Kinematics of Machines
(Course Code:17ME42)

2. TEXT BOOKS
 Theory of Machines, Rattan S.S, Tata McGraw-Hill Publishing Company.
 Theory of Machines, Sadhu Singh
 Theory of Machines, by R S Khurmi
REFERENCE BOOKS:
 “Theory of Machines & Mechanisms", J.J. Uicker, , G.R. Pennock, J.E. Shigley. OXFORD 3rd Ed.
Further Reference:
 National Programme on Technology Enhanced Learning (NPTEL)
http://nptel.ac.in/courses/112104121/1

3. At the end of the course, students will be able to,
 CO1: Describe the concepts of machines, mechanisms and related terminologies.
 CO2: Identify the mechanisms and predict their motions in mechanical components.
 CO3: Analyze planar mechanism for displacement, velocity and acceleration analytically and
graphically .
 CO4: Analyze various motion transmission elements like gears, gear trains and cams.
 CO5: Utilize analytical, mathematical and graphical aspects of kinematics of machines for
effective design.
Course Outcomes

4. Kinematics of Machines
CHAPTER 1: Introduction to Kinematics of Machines

5. Module 1
Introduction: Definitions Link or element, kinematic pairs, Degrees of freedom, Grubler's
criterion (without derivation), Kinematic chain, Mechanism, Structure, Mobility of Mechanism,
Inversion, Machine.
Kinematic Chains and Inversions: Inversions of Four bar chain; Single slider crank chain
and Double slider crank chain

6. Terminologies
Machine
• Device for transferring and/or transforming motion and force (power) from source
(Input) to the load (output).
or
• A machine is a device which receives energy in some available form and utilizes
to do some work.

7. Classification of Motion
• Continuous rotation motion
• Linear motion / Rectilinear Motion / translatory motion
• Intermittent motion
• Angular Oscillations

8. Basic Definitions
• Link or element
• Kinematic pair
• Kinematic chain
• Mechanism, Machine & Structure
• Degrees of freedom (DOF)
• Grubler’s criterion
• Inversion
Kinematic Chains and Inversions
• Inversions of Four bar chain
• Single slider crank chain and
• Double slider crank chain

9. Basic Definitions
Link/Eleme
nt
Kinematic
Pair/Joint
Kinematic
Chain
Mechanism Machine

10. Kinematic Link or Element
Each part of a machine, which moves relative to some other part, is known as a
kinematic link (or simply link) or element.

11. Types of Links
• Rigid link : Link which does not undergo any deformation while transmitting motion.
• Flexible link: Link which is partly deformed in a manner not to affect the
transmission of motion.
e.g.: belts, ropes, chains and wires
• Fluid link: A link formed by having a fluid in a receptacle and the motion is
transmitted through the fluid by pressure.
e.g.: hydraulic presses, jacks and brakes.

12. Classification of Links
• Unary Link : Link with one Node (bucket of an excavator)
• Binary link : Link connected to other links at two points
• Ternary link: Link connected to other links at three points
• Quaternary link: Link connected to other links at four points

13. Structure
An assembly of a no. of resistant bodies (members) having no relative motion
between them and meant for carrying loads having straining action.
Examples: A railway bridge, a roof truss, machine frames. etc.

14. Difference Between a Machine and a Structure
• The parts of a machine move relative to one another, whereas the members of a
structure do not move relative to one another
• A machine transforms the available energy into some useful work, whereas in a
structure no energy is transformed into useful work
• The links of a machine may transmit both power and motion, while the members
of a structure transmit forces only (load bearing member).

15. Kinematic Pair
The two links or elements of a machine, when in contact with each other, are said to
form a pair, If the relative motion between them is completely or successfully
constrained (i.e. in a definite direction)
Successfully constrained
Completely constrained

16. 1. Completely constrained motion
When the motion between a pair is limited to a definite direction irrespective
of the direction of force applied, then the motion is said to be a completely
constrained motion.
Example:
The motion of a square bar in a square hole
The motion of a shaft with collars at each end in a circular hole
Constrained Motions

17. 2. Incompletely constrained motion
When the motion between a pair can take place in more than one direction,
then it can be an incompletely constrained motion.
E.g.: A circular bar or shaft in a circular hole
It may either rotate or slide in a hole. These both motions have no relationship
with the other (automobile wheel).

18. 3. 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
E.g.: Shaft in a foot-step bearing, I C engine valve, Piston inside an cylinder.

19. Classification of Kinematic Pairs
1. Sliding pair / Prismatic pair (P)
When the two elements of a pair are connected in
such a way that one can only slide relative to the
other, has a completely constrained motion.
E.g.: The piston and cylinder, Cross-head and guides of a
reciprocating steam engine, ram and its guides in shaper, tail
stock on the lathe bed
According to the type of relative motion between the elements
DOF = 1

20. 2. Turning Pair / Revolute Pair (R)
When the two elements of a pair are connected in such a way that one can only
turn or revolve about a fixed axis of another link.
Examples Lathe spindle supported in head stock.
cycle wheels turning over their axles.
DOF = 1

21. 3. Spherical pair (S)
When the two elements of a pair are connected in such a way that one element
turns or swivels about the other fixed element.
Eg: The ball and socket joint.
Attachment of a car mirror.
Pen stand.
DOF = 3

22. 4. Screw pair
When the two elements of a pair are connected in such a way that one element can
turn about the other by screw threads.
Examples:
• The lead screw of a lathe with nut
• Bolt with a nut
DOF = 1

23. 5. Cylindrical pair
If the relative motion between the pairing elements is the combination of turning
and sliding, then it is called as cylindrical pair.
DOF = 2

24. 6. Rolling Pair
When the two elements of a pair are connected in such a way that one rolls over
another fixed link.
E.g.: Ball and roller bearings, railway wheel rolling over a fixed rail
Belt and pulley
DOF = 1

25. According to the type / nature of contact between the elements / links.
Classification of Kinematic Pairs
1. Lower pair
When the two elements of a pair have a surface contact, and the surface of one
element slides or rolls over the surface of the other.
E.g. sliding pairs, turning pairs and screw pairs form lower pairs.

26. According to the type / nature of contact between the elements / links.
Classification of Kinematic Pairs
2. Higher pair
When the two elements of a pair have a line or point contact, and the motion
between the two elements is partly turning and partly sliding.
E.g.: toothed gearing, belt and rope drives, ball and roller bearings and cam and
follower.

27. According to the Mechanical arrangement.
1. Self closed pair
When the two elements of a pair are connected together mechanically in such a
way that only required relative motion occurs.
E.g. Lower pairs are self closed pair.
2. Force - closed pair
When the two elements of a pair are not connected
mechanically but are kept in contact by the action of
external forces, the pair is said to be a force-closed pair.
E.g.: Cam and follower
Classification of Kinematic Pairs

28. Lower pair Higher pair Self closed pair Force - closed pair
• Sliding pairs
• Turning pairs
• Screw pairs
Rolling pair
Kinematic Pairs

29. • Based on the possible motions (Few Important Types only)
Name of Pair Letter Symbol D.O.F
1. Revolute / Turning Pair R 1
2. Prismatic / Sliding Pair P 1
3. Helical / Screw Pair H 1
4. Cylindrical Pair C 2
5. Spherical / Globular Pair S (or) G 3
6. Flat / Planar Pair E 3

31. Kinematic Chain
When the kinematic pairs are coupled in such a way that the last link is joined to
the first link to transmit definite motion (i.e. completely or successfully constrained
motion), it is called a kinematic chain.
Or
Assembly of links (Kinematic link / element) and Kinematic pairs to transmit
required / specified output motion(s) for given input motion(s)

32. Mechanism
• When one of the links of a kinematic chain is fixed, the chain is known as
mechanism.
• It may be used for transmitting or transforming motion
e.g. printing machine, windshield wiper, robot arms
• A mechanism may be regarded as a machine in which each part is reduced
to the simplest form to transmit the required motion.

33. Degrees of Freedom (DOF) / Mobility of a Mechanism
Number of independent coordinates, required to describe / specify the configuration
or position of all the links of the mechanism, with respect to the fixed link at any
given instant.

34. Degrees of freedom (DOF)
In a kinematic pair, depending on the constraints imposed on the motion, the links
may loose some of the six degrees of freedom.

35. Grubler’s Criterion / Equation
Grubler’s mobility equation
M = 3 (n-1) - j1 - j2
M = Mobility or Total no. of DOF
n = Total no. of links in a mechanism
J1 = No. of joints having 1 DOF
J2 = No. of joints having 2 DOF
If, M>0, It gives mechanism with M DOF
M=0, it gives a statically determinate structure
M<0, it gives statically indeterminate structure

36. Find the Degrees of Freedom

37. Find the Degrees of Freedom

38. Inversion 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.

39. Kinematic Chain
When the kinematic pairs are coupled in such a way that the last link is joined to the
first link to transmit definite motion (i.e. completely or successfully constrained
motion).

40. Types of Kinematic Chains
1. Four bar chain or quadric cyclic chain
2. Single slider crank chain
3. Double slider crank chain

41. 1. Four bar chain or quadric chain
• Four bar chain (mechanism) is the simplest and the
basic kinematic chain and consists of four rigid links
• Each of them forms a turning pair at A, B, C and D.
The link that makes a complete revolution is called a crank
• The four links may be of different lengths.
• According to Grashof ’s law for a four bar mechanism, “the sum of the
shortest and longest link lengths should not be greater than the sum of the
remaining two link lengths” if there is to be continuous relative motion between
the two links.

42. 1. Four bar chain or quadric chain
• The shortest link, will make a complete revolution relative to the other three links crank or driver.
In Fig., AD (link 4 ) is a crank.
• link BC (link 2) which makes a partial rotation or oscillates is known as lever or rocker or follower
• link CD (link 3) which connects the crank and lever is called connecting rod or coupler.
• The fixed link AB (link 1) is known as frame of the mechanism.
The mechanism transforms rotary motion into oscillating motion.

44. • In this mechanism, the links AD and BC (having equal length) act as cranks and are connected to
the respective wheels.
• The link CD acts as a coupling rod.
• The link AB is fixed in order to maintain a constant centre to centre distance between them.
Purpose: Transmitting rotary motion from one wheel to the other wheel.

46. Purpose of this mechanism is to convert rotary motion into reciprocating motion.
 When the crank AB rotates about the fixed point A.
 The lever oscillates about another fixed point D.
 The end E of lever is connected to a piston rod which reciprocates in a cylinder.`

47. The four links are : fixed link at A, link AC, link CE and link BFD. Links CE and BFD act as levers.
On displacement of the mechanism, the tracing point E at
the end of the link CE traces out approximately a straight
line.

48. Single Slider Crank Chain & Inversion

49. 2. Single Slider Crank Chain

51. When the crank (link 2) is given a rotary motion, the connecting rod (link 3) oscillates
about a pin pivoted to the fixed link 4 at A. The piston attached to the piston rod (link
1) reciprocates.
Pendulum Pump or Bull Engine

52. Link 3 forming the turning pair is fixed and it corresponds to the connecting rod of a
reciprocating steam engine mechanism.
When the crank (link 2) rotates, the piston attached to piston rod (link 1)
reciprocates and the cylinder (link 4) oscillates about a pin pivoted to the fixed link
at A
A

54. It consists of seven cylinders in one plane all revolve about fixed centre D.
Cylinders form link 1, Crank (link 2) is fixed. When the connecting rod (link4) rotates,
the piston (link 3) reciprocates inside the cylinder.

55. Radial Engine

56. A mechanism used in shaping and slotting
machines, where the metal is cut intermittently.
• Link AC (i.e. link 3) is fixed.
• Crank CB revolves with uniform angular speed
about the fixed center C.
• Sliding block attached to the crank pin at B slides
along the slotted bar AP, thus causing AP to
oscillate about the pivoted point A.
• Short link PR transmits the motion from AP to the
ram which carries the tool and reciprocates along
the line of stroke R1R2

57. • The forward or cutting stroke occurs when the
crank rotates from the position CB1 to CB2 (or
through an angle β) in the clockwise direction.
• The return stroke occurs when the crank rotates
from the position CB2 to CB1 (or through angle
α) in the clockwise direction.

59. Double Slider Crank Chain

60. • This inversion is obtained by fixing the slotted plate (link 4).
• fixed plate or link 4 has two straight grooves cut in it, at right angles to each other.
• link 1 and link 3, are known as sliders and form sliding pairs with link 4.
• link AB (link 2) is a bar which forms turning pair with links 1 and 3.
• When the links 1 and 3 slide along their respective grooves, any point on the link 2 such as P traces
out an ellipse on the surface of link 4

61. Show that AP and BP are the semi-major axis and semi-minor axis of the ellipse.
OX and OY as horizontal and vertical axes
let the link BA is inclined at an angle θ with the horizontal,
Now the co-ordinates of the point P on the link BA will be
This is the equation of an ellipse

62. • This mechanism is used for converting rotary motion into a reciprocating motion.
• The inversion is obtained by fixing either the link 1 or link 3. link 1 is fixed.
• In this mechanism, when the link 2 (crank) rotates about B as centre, the link 4
(frame) reciprocates.
• The fixed link 1 guides the frame.

63. End of Module