3. Theory of Machines and
Mechanisms
Mechanisms/
Linkages
Parts/ Link
Joints
deals with the determination
of the forces and motions of
links in machines
Subsystems of
machines to facilitate
analysis
Introduction
4. 4
Links: rigid member having nodes, i.e. attachment points
– Binary link: 2 nodes
– Ternary link: 3 nodes
– Quaternary link: 4 nodes
Links & joints
Joint: connection between two links (at their nodes) which allows
motion
Classified by type of contact, number of DOF, type of physical
closure, or number of links joined.
kinematic pair : Joints are also known as kinematic pair
5. 5
Joint Classification
Type of contact:
line/point i.e. higher pair,
area/surface i.e. lower pair
Number of DOF:
full joint=1DOF,
half joint=2DOF
Form closed (closed by geometry) or
Force closed (needs an external force to
keep it closed)
Joint order = number of links-1
6. Full Joint: permits one relative motion between adjacent links. All of these
kinematic pairs are referred to as one degree of freedom(DOF) pairs.
Turning pairs allow relative turning motion between two
links., e.g. bearings, pivots, or pin joints.
Rolling pairs allow relative rolling motion between two links,
e.g. pair of friction wheels For a rolling pair, it is assumed that
there is no slippage between the links.
Sliding pairs allow relative sliding motion between two links,
e.g Piston-Cylinder.
Half Joint: allows two relative motions simultaneously
between the adjacent links and referred to as two degree of
freedom pairs.
Sliding pairs
Turning pairs
Half Joint
Kinematic Pairs
8. Higher Pairs
& Lower Pairs
Lower pairs: A kinematic
pair or joint with
surface/area contact.
9. Higher Pairs & Lower Pairs
Higher pairs: A kinematic pair or joint with point contact or line contact.
10. Mechanism
Mobility: The mobility of a mechanism is defined as the number of
independent parameters required to specify the position of all links of the
mechanism. It also specify the number of input/actuators needed to operate the
mechanism.
kinematic chain : A kinematic chain is an assembly of links formed by
placing kinematic pairs at each of the nodes without specifying the ground
link.
Kinematic chains may be either open type or close type.
Mechanism: It is an assemblage of links and joints with at least one link
grounded and interconnected in a way to provide controlled output motions in
response to supplied input motions.
DOF of a mechanism: The number of independent ways by which a
dynamical system can move without violating any constraint imposed on it. In
other words, the minimum number of independent coordinates, which can
specify the position of the system completely. It is the number of parameters
that determine the state of a physical system.
11. Link classification:
Ground: fixed wrt. reference frame
Crank: pivoted to ground, makes complete revolutions
Rocker: pivoted to ground, has oscillatory motion
Coupler: link has complex motion, not attached to ground
Machine: mechanism designed to do work.
12. A simple machine may also be considered as a single mechanism.
Figure 1.3(b) shows a free body diagram of
the system used to analyze the manual force
required to generate sufficient gripping force.
Figure 1.3(a) The tongs
can be considered either
as a machine or as a
mechanism.
Figure: Inline 4-Cylinder Engine
Figure: IC Engine Demonstration
Machine & Mechanism
14. Mechanisms are widely used in applications where precise relative
movement and transmission of force are required. Motions may be
continuous or intermittent, linear and/or angular.
Mechanisms
Worm_gear Gear-gear Gear-rack
Examples of continuous motion output
Examples of intermittent motion output
Cam –follower
Sewing machine creating a
lockstitch using an
oscillating a boat shuttle
Sewing machine creating a
lockstitch using Allen B
Wilson's rotating hook
Geneva
Mechanism
Cam –follower
Ratchet
Mechanism
15. Every mechanism has one stationary base link. All other links may move
relative to the fixed base link. From the same kinematic chain, an inversion or
a mechanism is obtained by making the originally fixed link into a moving
link and selecting an originally moving link to be the fixed link .
Mechanism Inversion
Figure 1.39 Slider crank mechanism and its three inversions
(a) slider crank mechanism (link 1 fixed),
(b) inversion #1
(link 2 fixed),
(c) inversion #2
(link 3 fixed),
(d) inversion #3
(link 4 fixed).
16. Planar motion is restricted to a plane. For a planar mechanism, the motions of
all of its links must take place either in the same plane or in planes that are
parallel to one another. The slider crank mechanism and four-bar mechanism
are examples of planar mechanisms.
Planar Mechanism
Figure 1.5 Slider crank mechanism
Figure 1.7 Slider crank mechanism with offset
Figure 1.8 Four-bar mechanism
The Gruebler’s equation for the mobility, m,
of a planar mechanism is given as
n= number of links in the mechanism
J1 = number of one degree of freedom pairs or
binary joints
J2= number of two degree of freedom pairs or
higher pair
2
1
2
)
1
(
3 J
J
n
m
17. If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to
assemble / Indeterminate problem .
If, m= 0 , Structure.
If, m>0 i.e. “+ve”, Mechanism.
Mobility
18. 18
Figure 1.36 Examples of mobility.
If, m< 0 i.e. “–ve”, Preloaded Structure, may require force to
assemble.
If, m= 0 , Structure.
If, m>0 i.e. “+ve”, Mechanism.
Mobility
19. n=5; J1=6; J2=0; m=0
n=5; J1=6; J2=0; m=0 but, m=1.
Full Joint,
Pure rolling,
no sliding
In case of pure rolling,
n=3; J1=3; J2=0; m=0
In case of rolling & sliding,
n=3; J1=2; J2=1; m=1
Half Joint,
rolling &
sliding
3
1
1
1
2
4
5
1 1 1
2 4
5
3
2
3
1 1
Mobility Paradoxes
The Gruebler criterion pays no attention to
link sizes or shapes, it can give misleading
results in the face of unique geometric
configurations.
20. Idle Degrees of Freedom
An Idle degree of freedom is one that appears (and is) present but its value has
no effect on the input – output relationships of interest
To identify Idle degrees of freedom, first identify the input and output links
–Then we must determine if a single link or combinations of links can move
without affecting the input/output link positions
–Like a connecting link rotating (about its axis) in a steering mechanism
without changing the relationship between the steering wheel and the front
tires in a vehicle
21. 04/28/2025 ME 3230 Page 21
Note: Pin-in-slot & Cam Contact are half joints
5
2
13
2
)
1
12
(
3
2
)
1
(
3
2
13
12
2
1
2
1
J
J
n
m
J
J
n
Here,
The Structure has five DOF with two Idle DOF’s.
They are the roller and the cam rocker .
mActual = MTheoretical - mIdle
=5-2 = 3
Idle Degrees of Freedom
22. Four-Bar Mechanism-Grashof's Criterion
Four-bar mechanisms may be studied by distinguishing the link lengths as
follows:
s: the length of the shortest link
l: the length of the longest link
p, q: the lengths of the other two links
To assemble the kinematic chain it is necessary that,
l
q
p
s
The type of a four-bar mechanism may be determined using Grashof"s
Criterion,
(i) (ii) (iii)
q
p
l
s
Then, only case (i) offers all three types of a four-bar mechanisms.
q
p
l
s
q
p
l
s
23. Class_I Class_II Class_III
(i) If s is the input link, then
the mechanism is a crank
rocker.
(ii) If s is the base link, then
the mechanism is a drag link.
(iii) If otherwise, then the
mechanism is a rocker-rocker.
Rocker_Rocker Change Point
q
p
l
s
q
p
l
s
q
p
l
s
Four-Bar Mechanism-Grashof's Criterion
24. Figure 1.43 Types of four-bar mechanisms (a) crank rocker, (b) drag link, (c) rocker-rocker.
For S+L<P+Q
Crank-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
25. For S+L>P+Q
All inversions will be double rockers
No link can fully rotate
Figure: Four Bar double rockers
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)
25
Parallelogram form Anti parallelogram form Deltoid form
26. Figure 1.47 Four-bar mechanisms:
crank rocker
Let the lengths of the three moving links are r2= 2.0 cm; r3=4.0
cm; r4=5.0 cm, adjusting the length of the base link we can get the
following inversion of four bar mechanism.
