2. • Kinematics of Machines
• It deals with the relative motions of different parts of machines and mechanisms
without taking into consideration the Couse of it such as forces, couples, torques,
moments etc
• Thus it is the study from geometric point of view, to know the displacements, velocities
and accelerations of various parts of machines/ mechanisms
Two type study approach are
1. Analysis 2. Synthesis
• Analysis is the study of relative displacements, velocities, accelerations and other
geometric aspects for the existing machines, mechanisms
• Synthesis involve the design and creation of machine/ mechanisms and its various
parts and components as per the desired need
3. Some key words
• Link or element,
• Kinematic pairs
• Kinematic chains
• Mechanism and Machines, Structure, Superstructure,
• Degree of freedom
• Mobility of Mechanism
• Inversions
• Grubbler's criterion
• Kutzbatch criterion
• Grashoff criterion
4. Kinematic Links or Element:
• It is defined as the part of machine which has a relative motion with respect to some
other part of same machine
Or in other words
• All the parts of machine having no relative motion among them can be considered as
same link
A link should have the following two characteristics:
1. It should have relative motion 2. It must be a resistant body
5.
6. • Classification of links:
• As per the number of joints/nodes provided
• Binary, ternary, quaternary
7. • As per strength and nature of links
1. Rigid link/ Resistant link
A rigid link is one which does not undergo any deformation while transmitting motion.
connecting rod, crank, piston etc. of a reciprocating steam engine are example of resistant links
2. Flexible link
A flexible link is one which is partly deformed in a manner not to affect the transmission of
motion. For example, belts, ropes, chains and wires are flexible links and transmit tensile
forces only
3. Fluid link
A fluid link is one which is formed by having a fluid in a receptacle and the motion is
transmitted through the fluid by pressure or compression only, as in the case of
hydraulic presses, jacks and brakes
8.
9. Kinematic Pairs
• When two elements or links are connected together in such a way that their relative
motion is constrained, form a Kinematic pair
Classification of kinematic pairs
• 1. According to the type of relative motion between the links
• (a) Turning pair or Revelatory pair : 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, the pair
is known as turning pair.
A shaft with collars at both ends fitted into a circular hole, the crankshaft in a journal
bearing in an engine, lathe spindle supported in head stock, cycle wheels turning over
their axles etc. are the examples of a turning pair.
•
10. • (b) Sliding pair/ Prismatic pair:
• When the two elements of a pair are connected in such a way that
one can only slide relative to the other, the pair is known as a sliding pair.
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 etc. are the examples of a
sliding pair.
• (c) Rolling pair: When the two elements of a pair are connected in such a way that
one rolls over another fixed link, the pair is known as rolling pair. Ball and roller
bearings are examples of rolling pair.
11. (d) 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, the pair is known as screw pair. The
lead screw of a lathe with nut, and bolt with a nut are examples of a screw pair
(e) Spherical pair: When the two elements of a pair are connected in such a way that one
element (with spherical shape) turns or swivels about the other fixed element, the pair
formed is called a spherical pair. The ball and socket joint, attachment of a car mirror,
pen stand etc., are the examples of a spherical pair
12. • According to the type of contact between the elements
• (a) Lower pair: When the two elements of a pair have a surface contact when
relative motion takes place and the surface of one element slides over the
surface of the other, the pair formed is known as lower pair. Sliding pairs,
turning pairs and screw pairs form lower pairs
• (b) Higher pair: When the two elements of a pair have a line or point contact
when relative motion takes place and the motion between the two elements is
partly turning and partly sliding, then the pair is known as higher pair. A pair of
friction discs, toothed gearing, belt and rope drives, ball and roller bearings and
cam and follower are the examples of higher pairs
13. • According to the type of closure between links
• (a) Self closed pair: When the two elements of a pair are connected together
mechanically in
such a way that only required kind of relative motion occurs, it is then known
as self closed pair. The lower pairs are self closed pair
• (b) 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.
The cam and follower is an example of force closed pair, as it is kept in contact by
the forces exerted by spring and gravity.
14.
15.
16. Kinematic Chains
• When the kinematic pairs are coupled in such a way that the last link is joined to the
first link to transmit/ transform/modify the definite motion (constrained motion), it is
called a kinematic chain
• In other words, a kinematic chain may be defined as a combination of kinematic pairs,
joined in such a way that each link forms a part of two pairs and the relative motion
between the links or elements is constrained
• Constrained Kinematic Chain means that an arrangement of links such that a
movement of one link causes a definite predictable movement of the other links
17. Degree of Freedom
The degrees of freedom (DOF) of a rigid body(link) is defined as the number of
independent relative motions (both translational and rotational ) which it possess
To determine the DOF of this body it is required to know how many distinct ways
the rigid body can be moved
• Maximum Degree of freedom of a link in a space= 6
• Three translational motion(about x,y and z axis)
• Three Rotational motion (about x,y and z axis)
18. • Maximum Degree of freedom of link in a plane= 3
• Two translational motion (about x and y axis)
• One rotational motion in x-y plane about z axis
19. Degree of freedom of pairs and chains
Every pair restrict the motion and only permit the motion for which it is meant
Revolute pair: A rigid body is constrained by a revolute pair which allows only rotational
movement around an axis
Prismatic pair: rigid body is constrained by a prismatic pair which allows only
translational motion
Rolling pair: It has two degrees of freedom: translating along the curved surface and
turning about the instantaneous contact point
20. Degree of Freedom of Linkages, Chains,
Mechanisms and Structure
Mechanisms: 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. engine indicators, typewriter etc. It is also called linkage.
A mechanism with four links is known as simple mechanism, and the
mechanism with more than four links is known as compound mechanism.
Machine: When a mechanism is required to transmit power or to do some
particular type of work, it then becomes a machine. In such cases, the various
links or elements have to be designed to withstand the forces (both static and
kinetic) safely.
21. Degree of freedom of Mechanisms
Gruebler’s Criteria, Kutzback’s Criteria (For Planer mechanisms)
• Let there are n links free in a plane
• Let one link is fixed than number of free link = n-1
• As we know degree of freedom on one link in plane = 3
• Degree of freedom possess by (n-1) free links = 3(n-1)
• Let links are paired
• One pair of one degree of freedom imposes 2 restrain
• Than p1 such type of pair imposes 2p1 restrains
• Than degree of freedom available = 3(n-1)-2p1
• One pair of two degree of freedom imposes 1 restrain
• Than p2 such type of pair imposes p2 restrains
• Than Total degree of freedom available for a mechanism F = 3(n-1)- 2p1 – p2
• Here p1= number of lower pairs(one degree of freedom),
• p2 = number of higher pairs(2 degree of freedom)
22. • Number of joints:
• binary joint: Two links are joined at same connection
• Ternary joint: Three links are joined at same connection
(equivalent to two binary joint)
• Quaternary joint: Four links are joined at same connection
(equivalent to three
binary joint)
27. Four bar mechanisms ( All revolute pairs) Quadratic
chain
A four bar mechanism is a mechanism having four rigid links with one link
fixed, the chain consists of four turning pairs .
The fixed link is referred as the Frame
The link having freedom to make complete rotation is called the driver or
Crank
The link having limited mobility is called the follower or rocker
The link connecting driver to driven is called floating link, coupler link or
connecting rod
28. Grashoff criteria for 4 bar mechanism
• Let the length of shortest link is S
• Length of longest link L
• Length of remaining two links are P and Q
• Ist Observastion:
• The length of any of the link should not be
more then the some of the length of remaining
three links. Otherwise the chain will not form.
• L S+P+Q
29. • Kind I mechanisms (Grashoff Mechanisms)
• S+L P +Q
• Three inversions are possible (crank- rocker, double crank, double
rocker)
• Kind II mechanisms Non Grashoff Mechanisms)
• S+L P +Q
• Only one inversion is possible ( double rocker)
• S +L = P + Q (inderminent motion)
30. Inversions of Kind I Grashoff mechanisms
• Inversions of Mechanism:
• The method of obtaining different mechanisms by fixing different links in a
kinematic chain, is known as inversion of the mechanism.
• The relative motions between the various links will not changed in any manner
through the process of inversion, but their absolute motions (those measured
with respect to the fixed link) may be changed drastically.
31. Inversions of Kind I Grashoff mechanisms
• S+L P +Q
• Ist inversion : let the link adjacent to the shortest link is fixed
• The mechanism will be crank-rocker mechanism
32. Inversions of Kind I Grashoff mechanisms
• S+L P +Q
• IIst inversion : let the Shortest link is fixed
• The mechanism will be crank-crank of double crank mechanism
33. Inversions of Kind I Grashoff mechanisms
S+L P +Q
III rd inversion : Let the opposite to the Shortest link is fixed
The mechanism will be double rocker mechanism
34.
35. Inversions of Kind II (Non Grashoff)
mechanisms
• S+L P +Q
• The mechanism will be double rocker mechanism in
all the cases
44. Single Slider Crank Chain
• A single slider crank chain is a modification of the basic four bar chain. It
consist of one sliding pair and three turning pairs.
• It is, usually, found in reciprocating steam engine mechanism, IC engines,
reciprocating compressors etc
• This type of mechanism converts rotary motion into reciprocating motion
and vice versa.
• Generically it is RRRP chain
45. Inversions of Single Slider Crank Chain
First Inversion (let link number one is fixed )
Link 1 is fixed link 2 is crank and link 4 is slider
And link 3 is connecting rod
Pair between links
1-2 (R), 2-3(R), 3-4(R), 4-1(P)
It is Generically RRRP chain
Applications: IC engines, Reciprocating
compressors
46. Second Inversion
Second Inversion (let link number two is fixed)
Link 2 is fixed,
Link 3 will act like crank and compel link 1 to
rotate
Link 4 is slider and will slide on rotating link
Pair between links
1-2 (R), 2-3(R), 3-4(R), 4-1(P)
It is Generically RRRP chain
Applications: Rotatory engines, Whitworth quick-
return motion mechanisms
47.
48.
49.
50.
51. • In a Whitworth quick return motion mechanism, , the distance
between the fixed centers is 50 mm and the length of the driving
crank is 75 mm. The length of the slotted lever is 150 mm and the
length of the connecting rod is 135 mm. Find the ratio of them time
of cutting stroke to the time of return stroke
• Ans: 2.735
52. Third inversion
Third Inversion (let link number three is fixed)
Link 3 is fixed,
Link 2 will act like crank and compel link 1 to slide with respect with
link 4
Pair between links
1-2 (R), 2-3(R), 3-4(R), 4-1(P)
It is Generically RRRP chain
Applications: Oscillating cylinder engine mechanism, Crank and
slotted lever quick return motion mechanisms
55. • A crank and slotted lever mechanism used in a shaper has a centre
distance of 300 mm between the centre of oscillation of the slotted
lever and the centre of rotation of the crank. The radius of the
crank is 120 mm. Find the ratio of the time of cutting to the time
of return stroke.
• Ans: 1.72
56. • In a crank and slotted lever quick return motion mechanism, the
distance between the fixed centres is 240 mm and the length of the
driving crank is 120 mm. Find the inclination of the slotted bar
with the vertical in the extreme position and the time ratio of
cutting stroke to the return stroke.
• If the length of the slotted bar is 450 mm, find the length of the
stroke if the line of stroke passes through the extreme positions of
the free end of the lever.
• Ans: 30, 2, 450 mm
57. Fourth inversion
• Fourth Inversion (let link number four is
fixed)
• Link 4 is fixed,
• Link 2 will act like crank and compel link 1
to slide with respect with link 4
• Pair between links
• 1-2 (R), 2-3(R), 3-4(R), 4-1(P)
• It is Generically RRRP chain
• Applications: Handpump mechanism
mechanisms
58.
59.
60.
61. Double slider crank mechanisms
• A kinematic chain which consists of two revolute pairs and two
prismatic pairs such that the two pairs of same kind are adjacent is
known as double slider crank chain
62. Inversions of double slider mechanisms
• 1. Elliptical trammels. It is an instrument used for drawing
ellipses. This inversion is obtained by fixing the slotted plate
63. • Scotch yoke mechanism: This mechanism is used for converting
rotary motion into a reciprocating motion
64. • Oldham’s coupling:
An oldham's coupling is used for connecting two parallel shafts
whose axes are at a small distance apart.
The shafts are coupled in such a way that if one shaft rotates, the
other shaft also rotates at the same speed.
This inversion is obtained by fixing the link 2,. The shafts to be
connected have two flanges (link 1 and link 3) rigidly fastened at their
ends by forging.
65.
66.
67.
68. Special Mechanisms
• Exact Straight line motion mechanisms:
• Peaucellier’s Mechanism
• Scott Russel Mechanism
• Approx- Straight line motion mechanism
• Grasshopper
• Watt
• Techebicheff
• Pantograph
