THEORY OF MACHINES
AND MECHANISMS
Third Edition
John J. Dicker, Jr.
Professor of Mechanical Engineering
University of Wisconsin-Madison
Gordon R. Pennock
Associate Professor of Mechanical Engineering
Purdue University
Joseph E. Shigley
Late Professor Emeritus of Mechanical Engineering
The University of Michigan
New York
Oxford
OXFORD UNIVERSITY PRESS
2003
Oxford University Press
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Copyright © 2003 by Oxford University Press, Inc.
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Oxford is a registered trademark of Oxford University Press
All rights reserved. No part of this publication may be reproduced,
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electronic, mechanical, photocopying, recording, or otherwise,
without the prior permission of Oxford University Press.
ISBN 0-1 9-5 I 5598-X
Printing number:
9 8 7 6 5 4 3 2 I
Printed in the United States of America
on acid-free paper
This textbook is dedicated to the memory of the third author, the late Joseph E. Shigley,
Professor Emeritus, Mechanical Engineering Department, University of Michigan, Ann
Arbor, on whose previous writings much of this edition is based.
This work is also dedicated to the memory of my father, John J. Uicker, Emeritus Dean
of Engineering, University of Detroit; to my mother, Elizabeth F. Uicker; and to my six
children, Theresa A. Uicker, John J. Uicker Ill, Joseph M. Uicker, Dorothy J. Winger,
Barbara A. Peterson, and Joan E. Uicker.
-John J. Vicker, Jr.
This work is also dedicated first and foremost to my wife, Mollie B., and my son,
Callum R. Pennock. The work is also dedicated to my friend and mentor Dr. An (Andy)
Tzu Yang and my colleagues in the School of Mechanical Engineering, Purdue University, West Lafayette, Indiana.
-Gordon R. Pennock
Contents
PREFACE
XIII
ABOUT THE AUTHORS
XVII
Part 1 KINEMATICS AND MECHANISMS
1 The World of Mechanisms
1
3
1.1
Introduction
1.2
Analysis and Synthesis
3
1.3
The Science of Mechanics
1.4
Terminology, Definitions, and Assumptions
5
10
4
4
1.5
Planar, Spherical, and Spatial Mechanisms
1.6
Mobility
1.7
Classification of Mechanisms
1.8
Kinematic Inversion
1.9
Grashof's Law
II
27
1.10 Mechanical Advantage
Problems
14
26
29
31
2 Position and Displacement
33
2.1
Locus of a Moving Point
33
2.2
Position of a Point
2.3
Position Difference Between Two Points
2.4
Apparent Position of a Point
38
2.5
Absolute Position of a Point
39
36
2.6
The Loop-Closure Equation
2.7
Graphic Position Analysis
2.8
Algebraic Position Analysis
2.9
Complex-Algebra
37
41
45
51
Solutions of Planar Vector Equations
2.10 Complex Polar Algebra
57
2.11 Position Analysis Techniques
60
2.12 The Chace Solutions to Planar Vector Equations
2.13 Coupler-Curve Generation
64
68
2.14 Displacement of a Moving Point
70
2.15 Displacement Difference Between Two Points
71
55
vi
CONTENTS
2.16 Rotation and Translation
72
2.17 Apparent Displacement
74
2.18 Absolute Displacement
75
Problems
3 Velocity
76
79
3.1
Definition of Velocity
3.2
Rotation of a Rigid Body
79
3.3
Velocity Difference Between Points of a Rigid Body
3.4
Graphic Methods; Velocity Polygons
80
82
85
3.5
Apparent Velocity of a Point in a Moving Coordinate System
3.6
Apparent Angular Velocity
3.7
Direct Contact and Rolling Contact
3.8
Systematic Strategy for Velocity Analysis
3.9
Analytic Methods
3.10 Complex-Algebra
92
97
98
99
100
Methods
101
3.11 The Method of Kinematic Coefficients
3.12 The Vector Method
105
116
3.13 Instantaneous Center of Velocity
3.14 The Aronhold-Kennedy
117
Theorem of Three Centers
3.15 Locating Instant Centers of Velocity
120
3.16 Velocity Analysis Using Instant Centers
3.17 The Angular-Velocity-Ratio
119
Theorem
123
126
3.18 Relationships Between First-Order Kinematic Coefficients and Instant Centers
3.19 Freudenstein' s Theorem
129
3.20 Indices of Merit; Mechanical Advantage
3.21 Centrodes
Problems
130
133
135
4 Acceleration
141
4.1
Definition of Acceleration
4.2
Angular Acceleration
4.3
Acceleration Difference Between Points of a Rigid Body
4.4
Acceleration Polygons
4.5
Apparent Acceleration of a Point in a Moving Coordinate System
4.6
Apparent Angular Acceleration
4.7
Direct Contact and Rolling Contact
4.8
Systematic Strategy for Acceleration Analysis
4.9
Analytic Methods
4.10 Complex-Algebra
141
144
144
151
163
168
Methods
169
164
167
155
127
CONTENTS
4.11 The Method of Kinematic Coefficients
4.12 The Chace Solutions
175
4.13 The Instant Center of Acceleration
4.14 The Euler-Savary
171
177
Equation
178
4.15 The Bobillier Constructions
183
4.16 Radius of Curvature of a Point Trajectory Using Kinematic Coefficients
4.17 The Cubic of Stationary Curvature
Problems
188
190
Part 2 DESIGN OF MECHANISMS
5 Carn Design
195
197
5.1
Introduction
197
5.2
Classification of Cams and Followers
5.3
Displacement Diagrams
5.4
Graphical Layout of Cam Profiles
5.5
Kinematic Coefficients of the Follower Motion
5.6
High-Speed Cams
5.7
Standard Cam Motions
198
200
203
207
211
212
5.8
Matching Derivatives of the Displacement Diagrams
5.9
Plate Cam with Reciprocating Flat-Face Follower
5.10 Plate Cam with Reciprocating Roller Follower
Problems
250
6 Spur Gears
252
6.1
Terminology and Definitions
252
6.2
Fundamental Law of Toothed Gearing
6.3
Involute Properties
255
256
6.4
Interchangeable Gears; AGMA Standards
6.5
Fundamentals of Gear-Tooth Action
6.6
The Manufacture of Gear Teeth
6.7
Interference and Undercutting
6.8
Contact Ratio
6.9
Varying the Center Distance
6.10 Involutometry
268
270
271
6.11 Nonstandard Gear Teeth
Problems
262
265
274
282
7 Helical Gears
286
7.1
Parallel-Axis Helical Gears
7.2
Helical Gear Tooth Relations
286
287
259
257
222
225
230
187
vii
viii
CONTENTS
7.3
Helical Gear Tooth Proportions
7.4
Contact of Helical Gear Teeth
7.5
Replacing Spur Gears with Helical Gears
7.6
Herringbone Gears
7.7
Crossed-Axis Helical Gears
Problems
289
290
292
292
295
8 Bevel Gears
297
8.1
Straight-Tooth Bevel Gears
8.2
Tooth Proportions for Bevel Gears
8.3
Crown and Face Gears
8.4
Spiral Bevel Gears
8.5
Hypoid Gears
Problems
9.1
Basics
297
301
302
303
304
305
9 Worms and Worm Gears
Problems
291
306
306
310
10 Mechanism Trains 311
10.1 Parallel-Axis Gear Trains
311
10.2 Examples of Gear Trains
313
10.3 Determining Tooth Numbers
10.4 Epicyclic Gear Trains
314
315
10.5 Bevel Gear Epicyclic Trains
317
10.6 Analysis of Planetary Gear Trains by Formula
10.7 Tabular Analysis of Planetary Gear Trains
10.8 Adders and Differentials
319
323
10.9 All Wheel Drive Train
Problems
317
327
329
11 Synthesisof Linkages 332
11.1 Type, Number, and Dimensional Synthesis
332
11.2 Function Generation, Path Generation, and Body Guidance
11.3 Two-Position Synthesis of Slider-Crank Mechanisms
11.4 Two-Position Synthesis of Crank-and-Rocker
333
333
Mechanisms
334
11.5 Crank-Rocker Mechanisms with Optimum Transmission Angle
11.6 Three-Position Synthesis
338
11.7 Four-Position Synthesis; Point-Precision Reduction
339
. 11.8 Precision Positions; Structural Error; Chebychev Spacing
11.9 The Overlay Method
343
341
335
CONTENTS
11.10 Coupler-Curve Synthesis
344
11.11 Cognate Linkages; The Roberts-Chebychev
11.l2 Bloch's Method of Synthesis
11.I3 Freudenstein's
Equation
350
11.I4 Analytic Synthesis Using Complex Algebra
II.I 6 Intermittent Rotary Motion
356
360
361
366
12 Spatial Mechanisms
368
12.1
Introduction
12.2
Exceptions in the Mobility of Mechanisms
12.3
The Position-Analysis
12.4
Velocity and Acceleration Analyses
12.5
The Eulerian Angles
12.6
The Denavit-Hartenberg
12.7
Transformation-Matrix
12.8
Matrix Velocity and Acceleration Analyses
12.9
Generalized Mechanism Analysis Computer Programs
Problems
368
Problem
369
373
378
384
Parameters
387
Position Analysis
389
392
400
13 Robotics 403
13.1
Introduction
13.2
Topological Arrangements of Robotic Arms
13.3
Forward Kinematics
403
13.4
Inverse Position Analysis
Inverse Velocity and Acceleration Analyses
13.6
Robot Actuator Force Analyses
411
418
421
Part 3 DYNAMICS OF MACHINES
423
14 Static;:Force Analysis 425
14.1
Introduction
14.2
Newton's Laws
404
407
13.5
Problems
348
352
11.15 Synthesis of Dwell Mechanisms
Problems
Theorem
425
427
14.3
Systems of Units
14.4
Applied and Constraint Forces
428
14.5
Free-Body Diagrams
14.6
Conditions for Equilibrium
14.7
Two- and Three-Force Members
14.8
Four-Force Members
429
432
443
433
435
414
397
x
CONTENTS
14.9
Friction-Force Models
445
14.10 Static Force Analysis with Friction
448
14.11 Spur- and Helical-Gear Force Analysis
451
14.12 Straight- Bevel-Gear Force Analysis
14.13 The Method of Virtual Work
Problems
457
461
464
15 Dynamic ForceAnalysis (Planar)
15.1
Introduction
15.2
Centroid and Center of Mass
470
470
470
15.3
Mass Moments and Products of Inertia
15.4
Inertia Forces and D' Alembert's Principle
15.5
The Principle of Superposition
15.6
Planar Rotation About a Fixed Center
15.7
Shaking Forces and Moments
15.8
Complex Algebra Approach
15.9
Equation of Motion
Problems
475
485
489
492
492
502
511
16 Dynamic ForceAnalysis (Spatial)
515
16.1
Introduction
16.2
Measuring Mass Moment of Inertia
16.3
Transformation of Inertia Axes
515
515
519
16.4
Euler's Equations of Motion
16.5
Impulse and Momentum
16.6
Angular Impulse and Angular Momentum
Problems
478
523
527
528
538
17 Vibration Analysis 542
17.1
Differential Equations of Motion
17.2
A Vertical Model
542
17.3
Solution of the Differential Equation
17.4
Step Input Forcing
546
547
551
17.5
Phase-Plane Representation
17.6
Phase-Plane Analysis
553
17.7
Transient Disturbances
17.8
Free Vibration with Viscous Damping
17.9
Damping Obtained by Experiment
555
559
563
565
17.10 Phase-Plane Representation of Damped Vibration
17.11 Response to Periodic Forcing
17.12 Harmonic Forcing
574
571
567
CONTENTS
17.13 Forcing Caused by Unbalance
17.14 Relative Motion
17.15 Isolation
579
580
580
17.16 Rayleigh's Method
583
17.17 First and Second Critical Speeds of a Shaft
17.18 Torsional Systems
Problems
586
592
594
18 Dynamics of Reciprocating Engines 598
18.1
Engine Types
18.2
Indicator Diagrams
598
18.3
Dynamic Analysis-General
18.4
Gas Forces
18.5
Equivalent Masses
18.6
Inertia Forces
603
606
606
609
610
18.7
Bearing Loads in a Single-Cylinder Engine
18.8
Crankshaft Torque
18.9
Engine Shaking Forces
18.10 Computation Hints
Problems
613
616
616
617
620
19 Balancing 621
19.1
Static Unbalance
621
19.2
Equations of Motion
19.3
Static Balancing Machines
19.4
Dynamic Unbalance
19.5
Analysis of Unbalance
622
624
626
627
19.6
Dynamic Balancing
635
19.7
Balancing Machines
638
19.8
Field Balancing with a Programmable Calculator
19.9
Balancing a Single-Cylinder Engine
19.10 Balancing Multicylinder Engines
640
643
647
19.11 Analytical Technique for Balancing Multicylinder Reciprocating Engines
19.12 Balancing Linkages
656
19.13 Balancing of Machines
Problems
663
20 Cam Dynamics
20.1
661
665
Rigid- and Elastic-Body Cam Systems
20.2
Analysis of an Eccentric Cam
20.3
Effect of Sliding Friction
670
666
665
651
xi
xii
CONTENTS
20.4
20.5
Analysis of Disk Cam with Reciprocating Roller Follower
Analysis of Elastic Cam Systems 673
20.6 Unbalance, Spring Surge, and Windup
Problems
676
21 Flywheels
678
21.1
Dynamic Theory
21.2
Integration Technique
678
680
21.3 Multicylinder Engine Torque Summation
Problems
683
22 Governors
22.1
675
682
685
Classification
685
22.2
Centrifugal Governors
22.3
Inertia Governors
686
687
22.4
Mechanical Control Systems
22.5
Standard Input Functions
687
22.6
Solution of Linear Differential Equations
22.7
Analysis of Proportional-Error
689
690
Feedback Systems
695
23 Gyroscopes 699
23.1
Introduction
699
23.2
The Motion of a Gyroscope
23.3
Steady or Regular Precession
23.4 Forced Precession
Problems
711
700
701
704
APPENDIXES
ApPENDIX
A: TABLES
Table 1 Standard SI Prefixes
712
Table 2 Conversion from U.S. Customary Units to SI Units
713
Table 3 Conversion from SI Units to U.S. Customary Units
Table 4 Properties of Areas 714
713
Table 5 Mass Moments ofInertia
Table 6 Involute Function
ApPENDIX
INDEX
B: ANSWERS
725
715
716
TO SELECTED
PROBLEMS
718
671
Preface
This book is intended to cover that field of engineering theory, analysis, design, and
practice that is generally described as mechanisms and kinematics and dynamics of machines. While this text is written primarily for students of engineering, there is much
material that can be of value to practicing engineers. After all, a good engineer knows
that he or she must remain a student throughout their entire professional career.
The continued tremendous growth of knowledge, including the areas of kinematics
and dynamics of machinery, over the past 50 years has resulted in great pressure on the
engineering curricula of many schools for the substitution of "modern" subjects for
those perceived as weaker or outdated. At some schools, depending on the faculty, this
has meant that kinematics and dynamics of machines could only be made available as
an elective topic for specialized study by a small number of students; at others it remained a required subject for all mechanical engineering students. At other schools, it
was required to take on more design emphasis at the expense of depth in analysis. In all,
the times have produced a need for a textbook that satisfies the requirements of new and
changing course structures.
Much of the new knowledge developed over this period exists in a large variety of
technical papers, each couched in its own singular language and nomenclature and each
requiring additional background for its comprehension. The individual contributions
being published might be used to strengthen the engineering courses if first the necessary foundation were provided and a common notation and nomenclature were established. These new developments could then be integrated into existing courses so as to
provide a logical, modern, and comprehensive whole. To provide the background that
will allow such an integration is the purpose of this book.
To develop a broad and basic comprehension, all the methods of analysis and development common to the literature of the field are employed. We have used graphical
methods of analysis and synthesis extensively throughout the book because the authors
are firmly of the opinion that graphical computation provides visual feedback that enhances the student's understanding of the basic nature of and interplay between the
equations involved. Therefore, in this book, graphic methods are presented as one possible solution technique for vector equations defined by the fundamental laws of mechanics, rather than as mysterious graphical "tricks" to be learned by rote and applied
blindly. In addition, although graphic techniques may be lacking in accuracy, they can
be performed quickly and, even though inaccurate, sketches can often provide reasonable estimates of a solution or can be used to check the results of analytic or numeric solution techniques.
We also use conventional methods of vector analysis throughout the book, both
in deriving and presenting the governing equations and in their solution. Raven's methods using complex algebra for the solution of two-dimensional vector equations are
xiii
xiv
PREFACE
presented throughout the book because of their compactness, because they are employed so frequently in the literature, and also because they are so easy to program for
computer evaluation. In the chapters dealing with three-dimensional kinematics and
robotics, we briefly present an introduction to Denavit and Hartenberg's methods using
transformation matrices.
With certain exceptions, we have endeavored to use U.S. Customary units and SI
units in about equal proportions throughout the book.
One of the dilemmas that all writers on the subject of this book have faced is how
to distinguish between the motions of two different points of the same moving body and
the motions of coincident points of two different moving bodies. In other texts it has
been customary to describe both of these as "relative motion"; but because they are two
distinct situations and are described by different equations, this causes the student difficulty in distinguishing between them. We believe that we have greatly relieved this
problem by the introduction of the terms motion difference and apparent motion and two
different notations for the two cases. Thus, for example, the book uses the two terms,
velocity difference and apparent velocity, instead of the term "relative velocity," which
will not be found when speaking rigorously. This approach is introduced beginning with
the concepts of position and displacement, used extensively in the chapter on velocity,
and brought to fulfillment in the chapter on accelerations where the Coriolis component
always arises in, and only in, the apparent acceleration equation.
Another feature, new with the third edition, is the presentation of kinematic coefficients, which are derivatives of various motion variables with respect to the input motion
rather than with respect to tirr.e. The authors believe that these provide several new and
important advantages, among which are the following: (1) They clarify for the student
those parts of a motion problem which are kinematic (geometric) in their nature, and
they clearly separate them from those that are dynamic or speed-dependent. (2) They
help to integrate different types of mechanical systems and their analysis, such as gears,
cams, and linkages, which might not otherwise seem similar.
Access to personal computers and programmable calculators is now commonplace
and is of considerable importance to the material of this book. Yet engineering educators have told us very forcibly that they do not want computer programs included in the
text. They prefer to write their own programs and they expect their students to do so too.
Having programmed almost all the material in the book many times, we also understand
that the book should not become obsolete with changes in computers or programming
languages.
Part 1 of this book is an introduction that deals mostly with theory, with nomenclature, with notation, and with methods of analysis. Serving as an introduction, Chapter 1
also tells what a mechanism is, what a mechanism can do, how mechanisms can be classified, and some of their limitations. Chapters 2, 3, and 4 are concerned totally with
analysis, specifically with kinematic analysis, because they cover position, velocity, and
acceleration analyses, respectively.
Part 2 of the book goes on to show engineering applications involving the selection,
the specification, the design, and the sizing of mechanisms to accomplish specific motion objectives. This part includes chapters on cam systems, gears, gear trains, synthesis
of linkages, spatial mechanisms, and robotics.
Part 3 then adds the dynamics of machines. In a sense this is concerned with the
consequences of the proposed mechanism design specifications. In other words, having
PREFACE
xv
designed a machine by selecting, specifying, and sizing the various components, what
happens during the operation of the machine? What forces are produced? Are there any
unexpected operating results? Will the proposed design be satisfactory in all respects?
In addition, new dynamic devices are presented whose functions cannot be explained o~
understood without dynamic analysis. The third edition includes complete new chapters
on the analysis and design of flywheels, governors, and gyroscopes.
As with all topics and all texts, the subject matter of this book also has limits. Probably the clearest boundary on the coverage in this text is that it is limited to the study of
rigid-body mechanical systems. It does study multibody systems with connections or
constraints between them. However, all elastic effects are assumed to come within the
connections; the shapes of the individual bodies are assumed constant. This assumption
is necessary to allow the separate study of kinematic effects from those of dynamics.
Because each individual body is assumed rigid, it can have no strain; therefore the study
of stress is also outside of the scope of this text. It is hoped, however, that courses using
this text can provide background for the later study of stress, strength, fatigue life,
modes of failure, lubrication, and other aspects important to the proper design of mechanical systems.
John J. Uicker, Jr.
Gordon R. Pennock
About the Authors
John J. Vicker, Jr. is Professor of Mechanical Engineering at the University of
Wisconsin-Madison.
His teaching and research specialties are in solid geometric modeling and the modeling of mechanical motion and their application to computer-aided
design and manufacture; these include the kinematics, dynamics, and simulation of
articulated rigid-body mechanical systems. He was the founder of the Computer-Aided
Engineering Center and served as its director for its initial 10 years of operation.
He received his B.M.E. degree from the University of Detroit and obtained his M.S.
and Ph.D. degrees in mechanical engineering from Northwestern University. Since joining the University of Wisconsin faculty in 1967, he has served on several national committees of ASME and SAE, and he is one of the founding members of the US Council
for the Theory of Machines and Mechanisms and of IFroMM, the international federation. He served for several years as editor-in-chief of the Mechanism and Machine Theory journal of the federation. He is also a registered Mechanical Engineer in the State of
Wisconsin and has served for many years as an active consultant to industry.
As an ASEE Resident Fellow he spent 1972-1973 at Ford Motor Company. He was
also awarded a Fulbright-Hayes Senior Lectureship and became a Visiting Professor to
Cranfield Institute of Technology in England in 1978-1979. He is the pioneering researcher on matrix methods of linkage analysis and was the first to derive the general
dynamic equations of motion for rigid-body articulated mechanical systems. He has
been awarded twice for outstanding teaching, three times for outstanding research publications, and twice for historically significant publications.
Gordon R. Pennock
is Associate Professor of Mechanical Engineering at Purdue
University, West Lafayette, Indiana. His teaching is primarily in the area of mechanisms
and machine design. His research specialties are in theoretical kinematics, and the dynamics of mechanical motion. He has applied his research to robotics, rotary machinery,
and biomechanics; including the kinematics, and dynamics of articulated rigid-body
mechanical systems.
He received his B.Sc. degree (Hons.) from Heriot-Watt University, Edinburgh,
Scotland, his M.Eng.Sc. from the University of New South Wales, Sydney, Australia, and
his Ph.D. degree in mechanical engineering from the University of California, Davis.
Since joining the Purdue University faculty in 1983, he has served on several national
committees and international program committees. He is the Student Section Advisor of
the American Society of Mechanical Engineers (ASME) at Purdue University, Region VI
College Relations Chairman, Senior Representative on the Student Section Committee,
and a member of the Board on Student Affairs. He is an Associate of the Internal Combustion Engine Division, ASME, and served as the Technical Committee Chairman of
Mechanical Design, Internal Combustion Engine Division, from 1993 to 1997.
XVII
~iii
ABOUT THE AUTHORS
He is a Fellow of the American Society of Mechanical Engineers and a Fellow and
a Chartered Engineer with the Institution of Mechanical Engineers (CEng, FIMechE),
United Kingdom. He received the ASME Faculty Advisor of the Year Award, 1998, and
was named the Outstanding Student Section Advisor, Region VI, 2001. The Central Indiana Section recognized him in 1999 by the establishment of the Gordon R. Pennock
Outstanding Student Award to be presented annually to the Senior Student in recognition of academic achievement and outstanding service to the ASME student section at
Purdue University. He received the ASME Dedicated Service Award, 2002, for dedicated voluntary service to the society marked by outstanding performance, demonstrated effective leadership, prolonged and committed service, devotion, enthusiasm,
and faithfulness. He received the SAE Ra]ph R. Teetor Educational Award, 1986, and
the Ferdinand Freudenstein Award at the Fourth National Applied Mechanisms and
Robotics Conference, 1995. He has been at the forefront of many new developments in
mechanical design, primarily in the areas of kinematics and dynamics. He has pub]ished some 80 technical papers and is a regular symposium speaker, workshop presenter, and conference session organizer and chairman.
Joseph E. Shigley
(deceased May ]994) was Professor Emeritus of Mechanical Engineering at the University of Michigan, Fellow in the American Society of Mechanica]
Engineers, received the Mechanisms Committee Award in 1974, the Worcester Reed
Warner medal in ] 977, and the Machine Design Award in 1985. He was author of eight
books, including Mechanical Engineering Design (with Charles R. Mischke) and
Applied Mechanics of Materials. He was Coeditor-in-Chief of the Standard Handbook
of Machine Design. He first wrote Kinematic Analysis of Mechanisms in 1958 and then
wrote Dynamic Analysis of Machines in ]961, and these were published in a single
volume titled Theory of Machines in 1961; these have evolved over the years to become
the current text, Theory of Machines and Mechanisms, now in its third edition.
He was awarded the B.S.M.E. and B.S.E.E. degrees of Purdue University and received his M.S. at the University of Michigan. After severa] years in industry, he devoted
his career to teaching, writing, and service to his profession starting first at Clemson
University and later at the University of Michigan. His textbooks have been widely used
throughout the United States and internationally.
PART 1
Kinematics and
Mechanisms
1
The World of Mechanisms
1.1 INTRODUCTION
The theory of machines and mechanisms is an applied science that is used to understand the
relationships between the geometry and motions of the parts of a machine or mechanism
and the forces that produce these motions. The subject, and therefore this book, divides
itself naturally into three parts. Part 1, which includes Chapters 1 through 4, is concerned
with mechanisms and the kinematics of mechanisms, which is the analysis of their motions.
Part 1 lays the groundwork for Part 2, comprising Chapters 5 through 13, in which we study
the methods of designing mechanisms. Finally, in Part 3, which includes Chapters 14
through 23, we take up the study of kinetics, the time-varying forces in machines and the
resulting dynamic phenomena that must be considered in their design.
The design of a modern machine is often very complex. In the design of a new engine,
for example, the automotive engineer must deal with many interrelated questions. What is
the relationship between the motion of the piston and the motion of the crankshaft? What
will be the sliding velocities and the loads at the lubricated surfaces, and what lubricants
are available for the purpose? How much heat will be generated, and how will the engine
be cooled? What are the synchronization and control requirements, and how wi\I they be
met? What will be the cost to the consumer, both for initial purchase and for continued
operation and maintenance? What materials and manufacturing methods will be used?
What will be the fuel economy, noise, and exhaust emissions; will they meet legal requirements? Although all these and many other important questions must be answered before
the design can be completed, obviously not all can be addressed in a book of this size. Just
as people with diverse skills must be brought together to produce an adequate design, so
too many branches of science must be brought to bear. This book brings together material
that falls into the science of mechanics as it relates to the design of mechanisms and
machines.
3
4
THE WORLD
OF MECHANISMS
1.2 ANALYSIS AND SYNTHESIS
There are two completely different aspects of the study of mechanical systems, design and
analysis. The concept embodied in the word "design" might be more properly Itermed
synthesis, the process of contriving a scheme or a method of accomplishing a given purpose. Design is the process of prescribing the sizes, shapes, material compositions, and
arrangements of parts so that the resulting machine will perform the prescribed task.
Although there are many phases in the design process which can be approached in a
well-ordered, scientific manner, the overall process is by its very nature as much an art as
a science. It calls for imagination, intuition, creativity, judgment, and experience. The role
of science in the design process is merely to provide tools to be used by the designers as
they practice their art.
It is in the process of evaluating the various interacting alternatives that designets find
need for a large collection of mathematical and scientific tools. These tools, when applied
properly, can provide more accurate and more reliable information for use in judging a
design than one can achieve through intuition or estimation. Thus they can be of tremendous help in deciding among alternatives. However, scientific tools cannot make decisions
for designers; they have every right to exert their imagination and creative abilities, even to
the extent of overruling the mathematical predictions.
Probably the largest collection of scientific methods at the designer's disposal fall into
the category called analysis. These are the techniques that allow the designer to critically
examine an already existing or proposed design in order to judge its suitability for the task.
Thus analysis, in itself, is not a creative science but one of evaluation and rating of things
already conceived.
We should always bear in mind that although most of our effort may be spent on analysis, the real goal is synthesis, the design of a machine or system. Analysis is simply a tool.
It is, however, a vital tool and will inevitably be used as one step in the design process.
1.3 THE SCIENCE OF MECHANICS
That branch of scientific analysis that deals with motions, time, and forces is called mechanics
and is made up of two parts, statics and dynamics. Statics deals with the analysis of stationary
systems-that is, those in which time is not a factor-and dynamics deals with systems that
change with time.
As shown in Fig. 1.1, dynamics is also made up of two major disciplines, first recognized as separate entities by Euler in 1775: I
The investigation of the motion of a rigid body may be conveniently separated into
two parts, the one geometrical, the other mechanical. In the first part, the transference
of the body from a given position to any other position must be investigated without
respect to the causes of the motion, and must be represented by analytical formulae,
which will define the position of each point of the body. This investigation will therefore be referable solely to geometry, or rather to stereotomy.
It is clear that by the separation of this part of the question from the other, which
belongs properly to Mechanics, the determination of the motion from dynamical principles will be made much easier than if the two parts were undertaken conjointly.
These two aspects of dynamics were later recognized as the distinct sciences of kinematics (from the Greek word kinema, meaning motion) and kinetics, and they deal with
motion and the forces producing it, respectively.
The initial problem in the design of a mechanical system is therefore understanding its
kinematics. Kinematics is the study of motion, quite apart from the forces which produce
that motion. More particularly, kinematics is the study of position, displacement, rotation,
speed, velocity, and acceleration. The study, say, of planetary or orbital motion is also a
problem in kinematics, but in this book we shall concentrate our attention on kinematic
problems that arise in the design of mechanical systems. Thus, the kinematics of machines
and mechanisms is the focus of the next several chapters of this book. Statics and kinetics,
however, are also vital parts of a complete design analysis, and they are covered as well in
later chapters.
It should be carefully noted in the above quotation that Euler based his separation of
dynamics into kinematics and kinetics on the assumption that they should deal with rigid
bodies. It is this very important assumption that allows the two to be treated separately. For
flexible bodies, the shapes of the bodies themselves, and therefore their motions, depend on
the forces exerted on them. In this situation, the study of force and motion must take place
simultaneously, thus significantly increasing the complexity of the analysis.
Fortunately, although all real machine parts are flexible to some degree, machines are
usually designed from relatively rigid materials, keeping part deflections to a minimum.
Therefore, it is common practice to assume that deflections are negligible and parts are rigid
when analyzing a machine's kinematic performance, and then, after the dynamic analysis
when loads are known, to design the parts so that this assumption is justified.
1.4 TERMINOLOGY,
DEFINITIONS,
AND ASSUMPTIONS
Reuleaux2 defines a machine3 as a "combination of resistant bodies so arranged that by
their means the mechanical forces of nature can be compelled to do work accompanied by
certain determinate motions." He also defines a mechanism as an "assemblage of resistant
bodies. connected by movable joints, to form a closed kinematic chain with one link fixed
and having the purpose of transforming motion."
Some light can be shed on these definitions by contrasting them with the term structure. 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. Indeed, the whole purpose of a machine
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