THEORY OF MACHINES (TOM) MEEG 206
Theory of Machines
(TOM)
Lecture – 01
Basic Concepts
Terms and Definitions
THEORY OF MACHINES (TOM) MEEG 206
• Machines
• Devices used to alter, transmit and direct forces to accomplish a
specific objective.
• transforms the available energy into some useful work
e.g. Chain saw
• Mechanisms
• Mechanical portion of machine containing systems of links
that transmits motion and forces from a power source to
desired output.
THEORY OF MACHINES (TOM) MEEG 206
Kinematic analysis
Kinematic analysis provides insight into
significant design questions:
•What is the significance of the length of the legs
that support the platform?
• Is it necessary for the support legs to cross and be
connected at their midspan, or is it better to arrange
the so that they cross closer to the platform?
• How far must the cylinder extend to raise the
platform 8 in?
Dynamic force analysis of the platform could provide
insight into another set of important design questions:
•What capacity (maximum force) is required of the
hydraulic cylinder?
•Is the platform free of any tendency to tip over?
•What cross-sectional size and material are required of
the support legs so they don’t fail?
Terms and Definitions
THEORY OF MACHINES (TOM) MEEG 206
• Links
• Rigid structure connected with other links to transmit
motion and forces.
• E.g. connecting rod, piston, crankshaft
• Flexible links are ignored during kinematic analysis but
must be included during dynamic force analysis.
• Joints
• Movable connection between links that allows relative
motion between links.
• Primary joints are revolute and sliding joints
THEORY OF MACHINES (TOM) MEEG 206
Types of joints
On the basis of number of links
• Binary joint
• When two links are connected at same joint
• Ternary joint
• When three links are connected at same joint
• Quaternary joint
• When four links are connected at same joint
On the basis of nature of contact
• Higher Pair
• Line or Point Contact (Example: Gear, Cam mesh)
• Lower Pair:
• Area Contact (Example: Slider)
THEORY OF MACHINES (TOM) MEEG 206
Joints
THEORY OF MACHINES (TOM) MEEG 206
Kinematic drawing
• The representation of machine drawing in skeleton
form so that the links and joints that influence the
motion of the mechanism are shown.
Machine drawing
Kinematic drawing
THEORY OF MACHINES (TOM) MEEG 206
Symbols Used in Kinematic Diagrams (Links)
THEORY OF MACHINES (TOM) MEEG 206
Symbols Used in Kinematic Diagrams (Links)
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
Figure 1.9 shows a shear that is used
to cut and trim electronic circuit
board laminates. Draw a kinematic
diagram.
Kinematic Diagram of Machine
THEORY OF MACHINES (TOM) MEEG 206
Figure 1.11 shows a pair of vise grips. Draw a kinematic diagram.
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
THEORY OF MACHINES (TOM) MEEG 206
Kinematic Diagram of Machine
Gruebler’s Equation
THEORY OF MACHINES (TOM) MEEG 206
Degree of Freedom (DOF) = 3(n - 1) - 2jp - jh
Where:
n = Number of Link
jp = Number of Lower Pair Joint
jh = Number of Higher Pair Joint
Note:
Number of DOF implies that the number of driver required to
get desired motion for the mechanism is equal to DOF.
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.14 shows a toggle clamp.
Draw a kinematic diagram, using
the clamping jaw and the handle
as points of interest.
Also compute the degrees of
freedom for the clamp.
n = 4, jp = 4 pins, jh = 0
DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.16 shows a beverage
can crusher used to reduce the
size of cans for easier storage
prior to recycling. Draw a
kinematic diagram, using the end
of the handle as a point of
interest. Also compute the
degrees of freedom for
the device.
n = 4, jp = (3 pins + 1 slider) = 4, jh = 0
DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.18 shows another device that can be
used to shear material. Draw a kinematic
diagram, using the end of the handle and the
cutting edge as points of interest. Also,
compute the degrees of freedom for the shear
press.
n = 3, jp = 2 pins, jh = 1 gear
DOF = 3(n - 1) - 2jp - jh = 3(3 - 1) - 2(2) - 1 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.21 shows an outrigger foot to
stabilize a utility truck. Draw a
kinematic diagram, using the bottom of
the stabilizing foot as a point of interest.
Also compute the degrees of freedom.
n = 4, jp = (3 pins + 1 slider), jh = 0
DOF = 3(n - 1) - 2jp - jh = 3(4 - 1) - 2(4) - 0 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.26 presents a lift table
used to adjust the working height
of different objects. Draw a
kinematic diagram and compute
the degrees of freedom.
n = 6, jp = (5 pins + 2 slider), jh = 0
DOF = 3(n - 1) - 2jp - jh = 3(6 - 1) - 2(7) - 0 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Figure 1.29 shows a mechanical
press used to exert large forces to
insert a small part into a larger one.
Draw a kinematic diagram, using
the end of the handle as a point of
interest. Also compute the degrees
of freedom.
n = 6, jp = (6 pins + 1 slider), jh = 0
DOF = 3(n - 1) - 2jp - jh = 3(6 - 1) - 2(7) - 0 = 1
THEORY OF MACHINES (TOM) MEEG 206
Calculation of DOF
Number of Driver and DOF for desired motion
C
THEORY OF MACHINES (TOM) MEEG 206
C
2
2
3
B
A 1
4
D
C
C' D
1
A
4
1
5
B
D
1
A
DOF = 1
3
Number of Driver required
= 1,
2
To get2'
determined motion3'
B
3
E
D'
4'
4
1
5
E
DOF = 2
Number of Driver required = 2,
To get determined motion
4
THEORY OF MACHINES (TOM) MEEG 206
C
2
3
B
A 1
4
DOF=3(n-1)-2PL-Ph=33
D
-24 =1
THEORY OF MACHINES (TOM) MEEG 206
C
3
2
B
D
1
A
4
1
5
E
DOF=3(n-1)-2PL-Ph=34 -2 5 =2
4
THEORY OF MACHINES (TOM) MEEG 206
3
DOF=3(n-1)-2PL-Ph=32 -
22 -1=1
2
1
THEORY OF MACHINES (TOM) MEEG 206
Conditions for a Mechanism to have a
Determined Motion
Independent parameters are supplied only by
the driving links.
Since the driving links are always connected to
the frame by lower pairs, one driving link
(together with the corresponding lower pair)
will provide only one independent parameter.
Therefore, the condition for a mechanism to
have a determined motion is:
1. DOF1
2. DOF= number of the driving links.
THEORY OF MACHINES (TOM) MEEG 206
DOF=3(n-1)-2PL-Ph
=34 -26=0
DOF=3(n-1)-2PL-Ph
=32 -23=0
2
1
2
1
C
5
3
D
A
3
E
4
If DOF=0, it is a truss (Structure).
F
THEORY OF MACHINES (TOM) MEEG 206
If the number of driving links is greater than the DOF
of the mechanisms, the kinematic chain cannot move
due to the conflict between the input forces. In
extreme cases, the weakest link in the chain may be
broken.
C
2
3
B
A 1
4
C
D
THEORY OF MACHINES (TOM) MEEG 206
If the number of driving links is less than the DOF of
the mechanisms, some driven links will not have
determined motion.
C
3
2
B
DOF=3(n-1)-2PL-Ph=34 -2 5
=2
D
1
A
4
1
5
E
4
THEORY OF MACHINES (TOM) MEEG 206
Compound Hinge (Multiple Joint)
If the axes of two or more
revolutes coincide, they
constitute a compound hinge.
2
11
3
The number of revolutes in a
compound hinge is equal to
one less than the number of
links joined at that hinge.
(a)
(a)
11
22
33
(b)
(b)
THEORY OF MACHINES (TOM) MEEG 206
Note: In many cases, compound hinge does not show
clearly. Attention must be paid to it.
THEORY OF MACHINES (TOM) MEEG 206
Note: In many cases, compound hinge does not
show clearly. Attention must be paid to it.
4
3
1
1
2
2
3
3
3
2
1
2
1
THEORY OF MACHINES (TOM) MEEG 206
All machine elements belonging to the same link in
the kinematic diagram must be connected firmly by
welding symbols.
THEORY OF MACHINES (TOM) MEEG 206
THEORY OF MACHINES (TOM) MEEG 206
Is there compound hinge here?
3
2
1
THEORY OF MACHINES (TOM) MEEG 206
Calculate the DOF of the gear-linkage mechanism. In
the mechanism, there are three moving gears(1, 2, and
3) and two moving rods (4 and 5).
2
4
1
B
A
E
C
F 5
D
6 3
THEORY OF MACHINES (TOM) MEEG 206
C is a compound hinge of links 2, 4, 5
D is a compound hinge of links 3, 5, 6.
DOF=3(n-1)-2PL-Ph=35 -26 -2=1
2
4
1
B
A
E
C
F 5
D
6 3
Passive DOF(Local DOF)
B
THEORY OF MACHINES (TOM) MEEG 206
B
2
1
O
2
1
A
A
O
3
3
DOF=3(n-1)-2PL-Ph=33-23-1=2
DOF=3(n-1)-2PL-Ph=32 -22 -1=1 33-231=2
THEORY OF MACHINES (TOM) MEEG 206
Passive DOF
B
B
2
1
O
2
1
A
A
O
3
3
The DOF which does not change the output motion
of the mechanism is called a passive DOF. Any
passive DOF should be deleted by welding the roller
to the follower before the calculation of the DOF of
mechanism.
THEORY OF MACHINES (TOM) MEEG 206
Passive DOF
DOF=3(n-1)-2PL-Ph=32 -22 -1=1
 33-23-1=2
THEORY OF MACHINES (TOM) MEEG 206
Redundant Constraints (void constraint) have exactly the
same kinematic function as other constraints. Redundant
constraints must not be counted during the calculation.
(1)When two links are connected by more than one
parallel sliding pair, only one sliding pair can be
counted during the
F
calculation, others are
redundant constraints
and must not be counted.
THEORY OF MACHINES (TOM) MEEG 206
For the plate cam with
translating flat-faced
follower, if there are two
parallel sliding pairs
between the follower and
the frame, then one of the
guide-ways is a redundant
constraint
DOF=3(n-1)-2PL-Ph
=32 -22 -1=1
 32-23-1= -1
THEORY OF MACHINES (TOM) MEEG 206
(2) When two links are connected by more than one
revolute pair whose axes coincide,
A
A'
DOF=3(n-1)-2PL-Ph=31 -21=1 31-22= -1
THEORY OF MACHINES (TOM) MEEG 206
(3) When two links are connected by more than one higher
pair whose common normals passing through the points of
contact coincide,
DOF=3(n-1)-2PL-Ph =32 -22-1=1
3 2 -22-2=0
A
1
n
O
3
2
n
B
THEORY OF MACHINES (TOM) MEEG 206
(4) When the distance between two points on two links
remains constant during the motion of the mechanism, adding
one link and two revolutes with their centers at these two
points will create a redundant constraint.
DOF=3(n-1)-2PL-Ph=33 -24=1
34 -26=0
2
B
C
E
1
A
AB
4
CD
3
F
BC
AD
D
BE
AF
1
A
C
2
B
5
4
F
3
D
CD
THEORY OF MACHINES (TOM) MEEG 206
THEORY OF MACHINES (TOM) MEEG 206
Note:
(1) Redundant constraints can improve the rigidity of
a mechanism, improve the force condition in links,
etc. and are widely used.
(2) Deleting the redundant constraints during the
calculation of DOF does not mean that the redundant
constraints should be omitted from real mechanisms.
(3) All redundant constraints require some special
dimensions. Therefore, attention should be paid to
manufacturing accuracy when any redundant
constraint is used.
2
THEORY OF MACHINES (TOM) MEEG 206
B
1
A
C
E
1
3
4
A
F
F
CD
AB EFDOF=3(n-1)-2P
EF CD
AB-24=1.
L-Ph=33
1
A
E
2
B
C
5
F
4
C
5
4
D
E
2
B
3
D
DOF=3(n-1)-2PL-Ph=33 -24=0.
truss!!
It is a
3
D
THEORY OF MACHINES (TOM) MEEG 206
C
1
A
F
B
3
D
4
5
E
2
C
1
A
F
DOF=3(n-1)-2PL-Ph
=33 -24=1
3
D
4
5
E
DOF=3(n-1)-2PL-Ph
=34 -26=0
It is a truss!!
THEORY OF MACHINES (TOM) MEEG 206
Calculate the DOF of the mechanism
shown below.
THEORY OF MACHINES (TOM) MEEG 206
Solution:
(1)The spring should not be counted.
(2) C is a compound hinge of links 2, 3, 4.
(3) The roller has a passive DOF.
(4) There are two parallel sliding pairs E and E
between the frame 8 and the slider 6. One of the
sliding pairs is redundant.
C
2
B
1
3
D
A
E
4
E'
6
G
F
7
O
5
8
THEORY OF MACHINES (TOM) MEEG 206
After the mechanism is redrawn,
DOF=3n-2PL-Ph=37 -29 -1=2. Needs two drivers.
C
2
B
4
3
1
E'
D
A
6
E
8
5
4
3
1
A
7
O
C
2
B
G
F
E
D
G
F
6 7
O
5
8
THEORY OF MACHINES (TOM) MEEG 206
THEORY OF MACHINES (TOM) MEEG 206
Calculate the DOF of the mechanism.
THEORY OF MACHINES (TOM) MEEG 206
(1) C is a compound hinge of links 4, 5, 7.
(2) A is a compound hinge of links 1, 2, 7.
(3) The roller has a passive DOF. After the roller is welded
with the link 6, there is still a revolute pair between links 5 and
6.
DOF=3n-2PL-Ph=36 -28 -1=1
B
1
6
A
2
D
5
C
3
E 4
7
F
THEORY OF MACHINES (TOM) MEEG 206
AB EC FD.
Calculate the DOF of the mechanism.
H
C
D
B
G
E
F
K
A
= This implies parallel and equal
THEORY OF MACHINES (TOM) MEEG 206
Link EC, revolute E, and revolute C create a
redundant constraint. remains 6 moving links.
H
C
D
4
2
B
G
3
5
E
F
K
6
7
1
A
THEORY OF MACHINES (TOM) MEEG 206
D is a compound hinge of links 2, 3, 4.
H
C
D
4
2
B
G
3
5
E
F
K
6
7
1
A
THEORY OF MACHINES (TOM) MEEG 206
The roller has a passive DOF. There is only
one higher pair between links 4 and 5.
H
C
D
4
2
B
G
3
5
E
F
K
6
7
1
A
THEORY OF MACHINES (TOM) MEEG 206
There is only one higher pair between cam 6
and link 5.
H
C
D
4
2
B
G
3
5
E
F
K
6
7
1
A
THEORY OF MACHINES (TOM) MEEG 206
DOF=3n-2PL-Ph=36 -27 -2=2.
Needs two drivers.
H
C
D
4
2
B
G
3
5
E
F
K
6
7
1
A