2/23/2021
SUBJECT: KINEMATICS OF MACHINES
Code: TE2030E
Volume: 3(3-0-1-6)
COURSE DESCRIPTION
This subject provides the knowledge on kinematics of basic
mechanisms commonly found in a system or machine. By the
end of the course students will be able to analyze the
position, velocity, and acceleration of the elements of single
and multiple degree-of-freedom linkages; understand and
have ability to design some common mechanisms in
machines.
COURSE MATERIALS
Textbook: George Henry Martin, Kinematics and Dynamics of
Machines, 1982, McGraw-Hill, Inc. ISBN: 0-07-040657-X
Lecture Notes: Distributed at lectures
Reference books
Robert L. Norton, Design of Machinery: An introduction to the
synthesis and analysis of mechanisms and machines, 1999,
McGraw-Hill Inc., ISBN 0-07-048395-7
John J. Uicker Jr, Gordon R. Pennock, Joseph E. Shigley, Theory of
Machines and Mechanisms, Fifth Edition, 2017, McGraw-Hill Inc.,,
ISBN 9780190264482
Khurmi, R., Theory of Machines, 14th ed, 2005, S. Chand & Co.
Ltd., ISBN 9788121925242
S S Rattan., Theory of Machines, Third Edition 2009, Tata McGrawHill, ISBN 13: 978-0-07-014477-4
1
2/23/2021
Chapter 1. FUNDAMENTAL CONCEPTS
1.1 Kinematics, kinetics and dynamics
- What is mechanics/dynamics/statics/
kinematics/kinetics?
- What is the difference between
kinematics and kinetics?
Chapter 1. FUNDAMENTAL CONCEPTS
Mechanics: concerned with the
state of rest or motion of bodies that are
subjected to the action of forces
Dynamics: treats with the forces acting on
the parts of a machine and the motions
resulting from these forces; deals with the
systems that change with time
Statics: deals with the analysis of stationary
systems which are not changing with time
Kinetics
The study of forces on systems in
motion (the study of forces causing
or resulting from motion)
Kinematics
Study of the relative motion of machine parts
without
regard
to
forces:
position,
displacement, rotation, speed, velocity,
acceleration
2
2/23/2021
Kinematic Variables
Kinetic Variables
•
•
•
•
•
•
•
•
•
•
•
•
•
Time
Position
Displacement & distance
Velocity & speed
Acceleration
….
Force
Presure
Torque
Work
Power
Momentum
…
1.2 Machine and Mechanism
Mechanism
Machine
A device which transforms input
motion to some desirable output
motion/pattern and typically
develops very low forces and
transmits little power
Typically contains mechanisms,
designed to provide significant
forces and transmits significant
power (to do useful work)
Cam and follower
mechanism
Sewing machine
Internal
combustion engine
Adjustable
desk lamp
3
2/23/2021
There is a direct analogy between the terms structure,
mechanism and machine and the three branches of mechanics.
A structure is also a combination of resistant (rigid) bodies
connected by joints, but its purpose is not to do work or to
transform motion. A structure (such as a truss) is intended to
be rigid. It can perhaps be moved from place to place and is
movable in this sense of the word; however, it has no internal
mobility, no relative motions between its various members,
whereas both machines and mechanisms do.
Classification of Mechanisms
Based on the relative motion of the rigid bodies
Planar
mechanism:
all particles describe
plane curves in
space and all these
curves lie in parallel
planes.
Spherical mechanisms:
each link has some point that
remains
stationary as the linkage
moves, the stationary points of all links
lie at a common location; that is, the
locus of each point is a curve contained
in a spherical surface, the spherical
surfaces defined by several arbitrarily
chosen points are all concentric.
Spatial
mechanisms:
No restrictions on
the
relative
motions of the
particles
4
2/23/2021
Based on the degree of freedom (D.O.F) of output only
Constrained Mechanism
• One independent output motion. Output member is
constrained to move in a particular manner only.
• Example: Four-bar mechanism, Slider Crank Mechanism,
Five-bar mechanism with two inputs
Unconstrained mechanism
• Output motion has more than one D.O.F.
• Example: Automobile Differential during turning the
vehicle on a curve, Five-bar mechanism with one input
Terminology
- Link: a machine part or a component of a mechanism, is
assumed to be completely rigid.
The links of a mechanism must be connected together in some
manner in order to transmit motion from the driver (input link) to
the follower (output link).
A link possesses at least two nodes which are points for
attachment to other links.
5
2/23/2021
If two links have no motion relative to one another, they are
considered as a single link (e.g: bearing and cylinder wall).
The link of a mechanism/the part of a machine which is
stationary and which supports the moving members is called the
frame and is designated link 1.
If a link is not completely rigid, e.g bell or chain, that can be
called a flexible link. However, in some cases, that can be replaced
by a rigid link.
A rigid link: deformations are
so small that they can be
neglected in determining the
motions of the various other
links in a machine.
A flexible link
if it is always
in tension
Assuming an
incompressible fluid
6
2/23/2021
A kinematic chain is a system of links, that is, rigid bodies, which
are either joined together or are in contact with one another in a
manner that permits them to move relative to one another.
For example, the crankshaft of an engine forms a
kinematic pair with the bearings which are fixed in a pair,
the connecting rod with the crank forms a second
kinematic pair, the piston with the connecting rod forms a
third pair and the piston with the cylinder forms a fourth
pair. The total combination of these links is a kinematic
chain.
Closed and Open kinematic chains
Closed kinematic chain: every link is
connected to at least two other links,
the chain forms one or more closed
loops.
When one link moves all the other
links will move in a predictable pattern
Open kinematic chain: there is one or
more link which is connected to one
other link.
The distal segment possesses a higher
degree of freedom than the proximal
ones.
7
2/23/2021
1.3 Inversion
Obtaining a different mechanism by making a different link in a
kinematic chain the fixed member
Inversions will have same relative motion between the links but
absolute motion will change.
Example of Slider-crank Inversion
1.4 Pairing
Two bodies in contact constitute a pair.
The connections, joints between the links, are called
kinematic pairs.
Classification of Pairs
Based on the nature of contact between links, kinematic pairs
can be divided into higher pairs and lower pairs:
Lower pairs: surface contact between the pair elements,
such as the pin joint
Higher pairs: line or point contact between the
elemental surfaces, such as the connection between a cam
and its follower, ball and roller bearings
8
2/23/2021
The six lower pairs
Turning/revolute pair
or pin joint
Cylindric pair
Prismatic pair
Globular or Spheric pair
Screw pair or helical pair
Flat pair or planar pair
9
2/23/2021
Note: Spheric pair with pin that is different from spheric pair
due to having 2 degrees of freedom, a rotation about two
coordinate axes
Pin 3
Slot 4
All other joint types are called higher pairs.
E.g: mating gear teeth, a wheel rolling on a rail, a cam contacting its
follower…
If a mechanism has only lower pairs, it is called a linkage
Based on the type of mechanical constraint/How the contact is
maintained:
Self closed pairs: the links in the
pair have direct mechanical
contact, even without the
application of external force (no
external force).
Force closed pair: the links in the
pair are kept in contact by the
application of external forces.
10
2/23/2021
Based on the degree of freedom (D.O.F)
1. Type I / Class I – One D.O.F
2. Type II / Class II – Two D.O.F
3. Type III / Class III – Three D.O.F
4. Type IV / Class IV – Four D.O.F
5. Type V / Class V – Five D.O.F
1.5 Degrees of freedom (DOF)
The system's DOF is equal to the number of independent
parameters (measurements) which are needed to uniquely define
its position in space at any instant of time.
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.
11
2/23/2021
A pencil (as a link) lying on a flat piece of paper requires three parameters
(DOF) to completely define its position:
- Two linear coordinates (x, y) to define the position of anyone point on
the pencil
- One angular coordinate () to define the angle of the pencil with respect
to the axes.
This system of the pencil in a plane then has three DOF
- For a Link: Six in spatial motion, three in planar motion.
- For a Kinematic Pair: Number of independent coordinates/pair
variables to specify the position of one link with another link, or
number of independent relative motions possible between the
links. Maximum five and minimum one in spatial motion.
Maximum two and minimum one in planar motion.
- For a Kinematic Chain/Mechanism – Number of independent
position variables to sketch the configuration with known link
lengths, or number of input motions required to get a
constrained output motion
12
2/23/2021
DETERMINING DEGREE OF FREEDOM
Degree of Freedom in Planar Mechanisms
Gruebler's equation:
In any real mechanism, even if more than one link of the kinematic chain is
grounded, there can be only one ground plane. Thus G is always one, and
Gruebler's equation becomes:
In the above equations, the value of J must reflect the value of all joints in the
mechanism, that means half joints count as ½ because they remove only 1
DOF.
13
2/23/2021
It is less confusing if we use Kutzbach’s modification of
Gruebler's equation in this form:
14
2/23/2021
15
2/23/2021
L = 3; J1= 3; J2 = 0
L = 4; J1= 4; J2= 0
L = 5; J1= 5; J2= 0
M=3 x(3 – 1) – 2 x2 – 1x 0
=6–6–0=0
M = 3x(4 – 1) – 2x4 – 1x0
=9–8–0=1
M =3 x (5 – 1) – 2x5 – 1x0
= 12 – 10 – 0 = 2
M  0: Structure
(M < 0: Preload/Super
Structure
M = 1: Constrained
Mechanism
M > 1:
Unconstrained
Mechanism
L = 2; J1= 2; J2 = 0
M=3 x(2 – 1) – 2 x2 – 1x 0
= 3 – 4 – 0 = -1
16
2/23/2021
It is important to realize that the Kutzbach criterion can give an incorrect
result. In the development of the Kutzbach criterion, no consideration was
given to the lengths of the links or other dimensional properties. Therefore, it
should not be surprising that exceptions to the criterion are found for
particular cases with equal link lengths, parallel links, or other special
geometric features.
For example, Fig. a) represents a structure and that the criterion properly predicts m
= 0. However, if link 5 is arranged as in Fig. b), the result is a double-parallelogram
linkage with a mobility of m = 1.
The actual mobility of m = 1 results only if the parallelogram geometry is achieved.
M=0
M= 1
Examples:
Determine the mobility of the planar mechanism below
a)
b)
c)
17
2/23/2021
Degree of Freedom in Spatial Mechanisms
1.6 Types of Motion
Plane Motion (Planar Motion)
A body has plane motion if all its points move in planes which are parallel to some
reference plane. The reference plane is called the plane of motion.
Plane motion can be one of three types: translation, rotation, or a combination of
translation and rotation.
18
2/23/2021
Pure translation
A body has translation if it moves so that all straight lines in the body move
to parallel positions.
Rectilinear translation is a motion wherein all points of the body move in
straight-line paths.
A translation in which points in the body move along curved paths is called
curvilinear translation.
The piston has rectilinear translation
The connecting link 3 has
curvilinear translation
Pure rotation
All points in a body remain at fixed distances from a line which is
perpendicular to the plane of motion. This line is the axis of rotation, and
points in the body describe circular paths about it.
The crank has a motion of rotation if the
frame of the engine is fixed
19
2/23/2021
Spherical Motion
A point has spherical motion if it moves in three-dimensional space and
remains at a fixed distance from some fixed point. A body has spherical
motion if each point in the body has spherical motion.
In the ball-and-socket joint in the Figure below, if either the socket or rod is held
fixed , the other will move with spherical motion.
Helical Motion
A point which rotates about an axis at a fixed distance and at the same
time moves parallel to the axis describes a helix. A body has helical
motion if each point in the body describes a helix.
The motion of a nut along a
screw is a common example.
20
2/23/2021
Cycle, period and phase of motion
A mechanism completes a cycle of motion when it moves through all
its possible configurations and returns to its starting position.
E.g: The slider-crank mechanism (in the figure below) completes a cycle of
motion as the crank makes one revolution.
The time required for one cycle is the period. The relative positions of
the links at any instant during the cycle of motion for the mechanism
constitute a phase.
E.g: When the crank is in position 1, the mechanism is in one phase of its
motion. When the crank is in position 2, the mechanism is in another
phase.
21