Theory of Machines
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Syllabus and Course Outline
Faculty of Engineering
Department of Mechanical Engineering
EMEC 3302, Theory of Machines
Instructor: Dr. Anwar Abu-Zarifa
Office:
IT Building, Room: I413
Tel: 2821
eMail: aabuzarifa@iugaza.edu.ps
Website: http://site.iugaza.edu.ps/abuzarifa
Office Hrs: see my website
SAT
09:30 – 11:00
Q412
MON
09:30 – 11:00
Q412
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Text Book: R. L. Norton, Design of Machinery “An Introduction to the
Synthesis and Analysis of Mechanisms and Machines”, McGraw Hill
Higher Education; 3rd edition
Reference Books:
 John J. Uicker, Gordon R. Pennock, Joseph E. Shigley, Theory of Machines
and Mechanisms
 R.S. Khurmi, J.K. Gupta,Theory of Machines
 Thomas Bevan, The Theory of Machines
 The Theory of Machines by Robert Ferrier McKay
 Engineering Drawing And Design, Jensen ect., McGraw-Hill Science, 7th
Edition, 2007
 Mechanical Design of Machine Elements and Machines, Collins ect., Wiley,
2 Edition, 2009
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Grading:
Attendance
Design Project
Midterm
Final exam
5%
25%
30%
40%
Course Description:
The course provides students with instruction in the fundamentals of theory of
machines. The Theory of Machines and Mechanisms provides the foundation
for the study of displacements, velocities, accelerations, and static and
dynamic forces required for the proper design of mechanical linkages, cams,
and geared systems.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Course Objectives:
Students combine theory, graphical and analytical skills to understand
the Engineering Design. Upon successful completion of the course,
the student will be able:




To develop the ability to analyze and understand the dynamic
(position, velocity, acceleration, force and torque) characteristics of
mechanisms such as linkages and cams.
To develop the ability to systematically design and optimize
mechanisms to perform a specified task.
To increase the ability of students to effectively present written,
oral, and graphical solutions to design problems.
To increase the ability of students to work cooperatively on teams
in the development of mechanism designs.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Chapter 1
Introduction
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Definitions
The subject Theory of Machines may be defined as that branch of
Engineering-science, which deals with the study of relative motion
between the various parts of a machine, and forces which act on
them. The knowledge of this subject is very essential for an
engineer in designing the various parts of a machine.
Kinematics: The study of motion without regard to forces
More particularly, kinematics is the study of position, displacement,
rotation, speed, velocity, and acceleration.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Kinetics: The study of forces on systems in motion
A mechanism: is a device that transforms motion to some desirable pattern
and typically develops very low forces and transmits little power.
A machine: typically contains mechanisms that are designed to provide
significant forces and transmit significant power.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Application of Kinematics
Any machine or device that moves contains one or more kinematic elements
such
As linkages, … gears…. belts and chains.
Bicycle is a simple example of a kinematic system that contains a chain drive
to provide Torque.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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An Automobile contains many more examples of kin-systems…
the transmission is full of gears….
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Chapter 2
DEGREES OF FREEDOM (MOBILITY)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Degrees of Freedom (DOF) or Mobility
• DOF: Number of independent parameters
(measurements) needed to uniquely define
position of a system in space at any instant of
time.
• A mechanical system’s mobility (M) can be
classified according to the number of degrees
of freedom (DOF).
• DOF is defined with respect to a selected frame
of reference (ground).
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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 Rigid body in a plane has 3 DOF: x,y,z
 Rigid body in 3D-space has 6 DOF, 3 translations & 3
rotations three lengths (x, y, z), plus three angles
(θ, φ, ρ).
 The pencil in these examples represents a rigid body,
or link.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Types of Motion
• Pure rotation: the body possesses one point (center
of rotation) that has no motion with respect to the
“stationary” frame of reference. All other points
move in circular arcs.
• Pure translation: all points on the body describe
parallel (curvilinear or rectilinear) paths.
• Complex motion: a simultaneous combination of
rotation and translation.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Excavator
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Slider-Crank Mechanism
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Links, joints, and kinematic chains
Linkage design:
 Linkages are the basic building blocks of all mechanisms
 All common forms of mechanisms (cams, gears, belts, chains)
are in fact variations on a common theme of linkages.
• Linkages are made up of links and joints.
• Links: rigid member having nodes
• Node: attachment points
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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1. Binary link: 2 nodes
2. Ternary link: 3 nodes
3. Quaternary link: 4 nodes
Joint: connection between two or more links (at their
nodes) which allows motion;
(Joints also called kinematic pairs)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Joint Classification
Joints can be classified in several ways:
1.By the type of contact between the elements, line, point, or
surface.
2.By the number of degrees of freedom allowed at the joint.
3.By the type of physical closure of the joint: either force or
form closed.
4.By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the
number of degrees of freedom that they allow between
the two elements joined.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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A joint with more than one freedom may
also be a higher pair
•
•
•
•
Type of contact: line, point, surface
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
Joint order = number of links-1
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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lower pair to describe joints with surface contact
The six lower pairs
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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The half joint is also called a roll-slide joint
because it allows both rolling and sliding
Form closed (closed by
geometry) or Force closed
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Terminology of Joints
 A joint (also called kinematic pair) is a connection between two or
more links at their nodes, which may allow motion between the links.
 A lower pair is a joint with surface contact; a higher pair is a joint with
point or line contact.
 A full joint has one degree of freedom; a half joint has two degrees
of freedom. Full joints are lower pairs; half-joints are higher pairs and
allow both rotation and translation (roll-slide).
 A form-closed joint is one in which the links are kept together form by
its geometry; a force-closed joint requires some external force to
keep the links together.
 Joint order is the number of links joined minus one (e.g. 1st order
means two links).
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Kinematic chains, mechanisms,
machines, link classification
•
•
•
•
Kinematic chain: links joined together for motion
Mechanism: grounded kinematic chain
Machine: mechanism designed to do work
Link classification:
 Ground: any link or links that are fixed, nonmoving with
respect to the 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
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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crank mechanism
Elements:
0: Ground (Casing, Frame)
1: Rocker
2: Coupler
3: Crank
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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The “Ground” Link
 When studying mechanisms it is very helpful to establish a fixed
reference frame by assigning one of the links as “ground”.
 The motion of all other links are described with respect to the
ground link.
 For example, a fourbar mechanism often looks like a 3-bar
mechanism, where the first “bar” is simply the ground link.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Drawing kinematic Diagrams
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Determining Degrees of Freedom
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Determining Degrees of Freedom
Two unconnected links: 6 DOF
(each link has 3 DOF)
When connected by a full joint: 4 DOF
(each full joint eliminates 2 DOF)
Gruebler’s equation for planar mechanisms: DOF = 3L-2J-3G
Where:
L: number of links
J: number of full joints
G: number of grounded links
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Determining DOF’s
• Gruebler’s equation for planar mechanisms
M= 3L-2J-3G
• Where
M = degree of freedom or mobility
L = number of links
J = number of full joints (half joints count as 0.5)
G = number of grounded links =1
M  3  L  1  2 J
Kutzbach’s modification of Gruebler’s equation
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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The Cylindrical (cylindric) joint - two
degrees of freedom
It permits both angular rotation and an
independent sliding motion (C joint)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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The Spherical (spheric) - Three degree
of freedom
It permits rotational motion about all three
axes, a ball-and-socket joint (S joint)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Example
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Example
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Gruebler’s Equation
Gruebler’s equation can be used to
determine the mobility of planar
mechanisms.
L=2
J=1
G=1
DOF = 1
Link 1
3 DOF
Gruebler’s Equation
DOF
L
J
G
= mobility
= number of links
= number of revolute joints or
prismatic joints
= number of grounded links
DOF (M) = 3*L – 2* J – 3 *G
= 3 (L-1) – 2 * J
1 DOF
Link 2
3 DOF
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Mobility of Vise Grip Pliers
This example applies Gruebler’s equation to
the determine the mobility of a vise grip plier.
1
4
5
1
4
3
3
2
2
Each revolute joint
removes two DOF.
The screw joint
removes two DOF.
L=5
J = 4 (revolute)
J = 1 (screw)
G = 1 (your hand)
DOF = 3*5 - 2*5 - 1*3 = 2
The mobility of the plier is two. Link 3 can be moved relative link1 when
you squeeze your hand and the jaw opening is controlled by rotating
link 5.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Punch Press
Slider-Crank Mechanism
As designated in the figure, there are four
links link 1, link 2, link 3 and link 4. Link 1
acts as a crank. Link 2 acts as
connecting link, link 3 is the slider and
link 4 is ground.
Joint
Number
Formed between links
1
Link 4 and Link 1
2
Link 1 and Link 2
3
Link 2 and Link 3
4
Link 3 and Link 4
Joint type
Revolute
(or Pin)
Revolute
(or Pin)
Revolute
(or Pin)
Translatio
nal or
(Slider)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Mechanisms and Structures
 If DOF > 0, the assembly of links is a mechanism and will
exhibit relative motion
 If DOF = 0, the assembly of links is a structure and no motion
is possible.
 If DOF < 0,then the assembly is a preloaded structure, no
motion is possible, and in general stresses are present.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Paradoxes
• Greubler criterion does not include geometry, so it
can give wrong prediction
• We must use inspection
L=5
J=6
G=1
M=3*5-2*6-3*1=0
E-quintet
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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Rolling cylinders even without slip (The joint between the two wheels can be
postulated to allow no slip, provided that sufficient friction is available) is an
example in which the ground link is exactly the same length as the sum of two
other links.
If no slip occurs, then this is a one-freedom, or full, joint that allows only
relative angular motion (Δθ) between the wheels.
With that assumption, there are 3 links and 3 full joints,
The equation predicts DOF = 0 (L=3,
J1=3), but the mechanism has DOF = 1.
Others paradoxes exist, so
the designer
must not apply the equation
blindly.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
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