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Theory of Machines and Mechanisms, Fifth Edition, is an ideal text for the complete study of displacements, velocities, accelerations, and static and dynamic forces required for the proper design of mechanical linkages, cams, and geared systems. The authors present the background, notation, and nomenclature essential for students to understand the various independent technical approaches that exist in the field of mechanisms, kinematics, and dynamics. The fifth edition features streamlined coverage and substantially revised worked examples. This latest edition also includes a greater number of problems, suitable for in-class discussion or homework, at the end of each chapter.


* Offers balanced coverage of all topics by both graphic and analytic methods

* Covers all major analytic approaches

* Provides high-accuracy graphical solutions to exercises, by use of CAD software

* Includes the method of kinematic coefficients and also integrates the coverage of linkages, cams, and geared systems

* An Ancillary Resource Center (ARC) offers an Instructor's Solutions Manual, solutions to 100 of the problems from the text using MatLab, and PowerPoint lecture slides

* A Companion Website includes more than 100 animations of key figures from the text
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Theory of Machines

and Mechanisms

Theory of Machines
and Mechanisms
Fifth Edition

John J. Uicker, Jr.
Professor Emeritus 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 is a department of the University of Oxford.
It furthers the University’s objective of excellence in research,
scholarship, and education by publishing worldwide.
Oxford is a registered trade mark of Oxford University Press
in the UK and certain other countries.

Published in the United States of America by Oxford University Press
198 Madison Avenue, New York, NY 10016, United States of America.

Copyright c© 2017, 2011, 2003 by Oxford University Press; 1995, 1980 by McGraw-Hill

For titles covered by Section 112 of the US Higher Education
Opportunity Act, please visit for the latest
information about pricing and alternate formats.

All rights reserved. No part of this publication may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
without the prior permission in writing of Oxford University Press,
or as expressly permitted by law, by license, or under terms agreed
with the appropriate reproduction rights organization. Inquiries concerning
reproduction outside the scope of the above should be sent to the Rights Department,
Oxford University Press, at the address above.

You must not circulate this work in any other form
and you must impose this same condition on any acquirer.

Library of Congress Cataloging-in-Publication Data

Names: Uicker, John Joseph, author. | Pennock, G. R., author. | Shigley,
Joseph Edward author.

Title: Theory of machines and mechanisms / John J. Uicker, Jr., Professor
Emeritus of Mechanical Engineering, University of Wisconsin–Madison,
Gordon R. Pennock, Associate Professor of Mechanical Engine; ering, Purdue
University, Joseph E. Shigley, Late Professor Emeritus of Mechanical
Engineering, The University of Michigan.

Description: Fifth edition. | New York : Oxford University Press, 2016. |
First-second editions by Joseph E. Shigley. | Includes bibliographical
references and index.

Identifiers: LCCN 2016007605 | ISBN 9780190264482
Subjects: LCSH: Mechanical engineering.
Classification: LCC TJ145 .U33 2016 | DDC 621.8–dc23 LC record available at

9 8 7 6 5 4 3 2 1

Printed by Edwards Brothers Malloy
Printed in the United States of America

This textbook is dedicated to the memory of my parents, John J. Uicker, Emeritus Dean of
Engineering, University of Detroit, Elizabeth F. Uicker, and to my six children, Theresa A.
Zenchenko, John J. Uicker III, Joseph M. Uicker, Dorothy J. Winger, Barbara A. Peterson,
and Joan E. Horne.

—John J. Uicker, 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, the late Dr. An (Andy)
Tzu Yang, and my colleagues in the School of Mechanical Engineering, Purdue University,
West Lafayette, Indiana.

—Gordon R. Pennock

Finally, this text is dedicated to the memory of the late Joseph E. Shigley, Professor
Emeritus, Mechanical Engineering Department, University of Michigan, Ann Arbor.
Although this fifth edition contains significant changes from earlier editions, the text
remains consistent with his previous writings.





1 The World of Mechanisms 3
1.1 Introduction 3

1.2 Analysis and Synthesis 4

1.3 Science of Mechanics 4

1.4 Terminology, Definitions, and Assumptions 6

1.5 Planar, Spheric, and Spatial Mechanisms 10

1.6 Mobility 12

1.7 Characteristics of Mechanisms 17

1.8 Kinematic Inversion 32

1.9 Grashof’s Law 33

1.10 Mechanical Advantage 36

1.11 References 39

Problems 40

2 Position, Posture, and Displacement 48
2.1 Locus of a Moving Point 48

2.2 Position of a Point 51

2.3 Position Difference Between Two Points 53

2.4 Apparent Position of a Point 54

2.5 Absolute Position of a Point 55

2.6 Posture of a Rigid Body 56

2.7 Loop-Closure Equations 57

2.8 Graphic Posture Analysis 62

2.9 Algebraic Posture Analysis 69

2.10 Complex-Algebraic Solutions of Planar Vector Equations 73

2.11 Complex Polar Algebra 74

2.12 Posture Analysis Techniques 78

2.13 Coupler-Curve Generation 86



2.14 Displacement of a Moving Point 89

2.15 Displacement Difference Between Two Points 89

2.16 Translation and Rotation 91

2.17 Apparent Displacement 92

2.18 Absolute Displacement 94

2.19 Apparent Angular Displacement 94

2.20 References 98

Problems 99

3 Velocity 105
3.1 Definition of Velocity 105

3.2 Rotation of a Rigid Body 106

3.3 Velocity Difference Between Points of a Rigid Body 109

3.4 Velocity Polygons; Velocity Images 111

3.5 Apparent Velocity of a Point in a Moving Coordinate System 119

3.6 Apparent Angular Velocity 126

3.7 Direct Contact and Rolling Contact 126

3.8 Systematic Strategy for Velocity Analysis 128

3.9 Algebraic Velocity Analysis 129

3.10 Complex-Algebraic Velocity Analysis 131

3.11 Method of Kinematic Coefficients 135

3.12 Instantaneous Centers of Velocity 145

3.13 Aronhold-Kennedy Theorem of Three Centers 147

3.14 Locating Instantaneous Centers of Velocity 149

3.15 Velocity Analysis Using Instant Centers 153

3.16 Angular-Velocity-Ratio Theorem 156

3.17 Relationships Between First-Order Kinematic Coefficients
and Instant Centers 157

3.18 Freudenstein’s Theorem 160

3.19 Indices of Merit; Mechanical Advantage 162

3.20 Centrodes 164

3.21 References 166

Problems 167

4 Acceleration 180
4.1 Definition of Acceleration 180

4.2 Angular Acceleration 183

4.3 Acceleration Difference Between Points of a Rigid Body 183

4.4 Acceleration Polygons; Acceleration Images 192

4.5 Apparent Acceleration of a Point in a Moving Coordinate System 196


4.6 Apparent Angular Acceleration 205

4.7 Direct Contact and Rolling Contact 206

4.8 Systematic Strategy for Acceleration Analysis 212

4.9 Algebraic Acceleration Analysis 213

4.10 Complex-Algebraic Acceleration Analysis 214

4.11 Method of Kinematic Coefficients 216

4.12 Euler-Savary Equation 225

4.13 Bobillier Constructions 230

4.14 Instantaneous Center of Acceleration 234

4.15 Bresse Circle (or de La Hire Circle) 235

4.16 Radius of Curvature of a Point Trajectory Using Kinematic
Coefficients 239

4.17 Cubic of Stationary Curvature 242

4.18 References 249

Problems 250

5 Multi-Degree-of-Freedom Mechanisms 258
5.1 Introduction 258

5.2 Posture Analysis; Algebraic Solution 262

5.3 Velocity Analysis; Velocity Polygons 263

5.4 Instantaneous Centers of Velocity 265

5.5 First-Order Kinematic Coefficients 268

5.6 Method of Superposition 273

5.7 Acceleration Analysis; Acceleration Polygons 276

5.8 Second-Order Kinematic Coefficients 278

5.9 Path Curvature of a Coupler Point Trajectory 285

5.10 Finite Difference Method 289

5.11 Reference 292

Problems 292


6 Cam Design 297
6.1 Introduction 297

6.2 Classification of Cams and Followers 298

6.3 Displacement Diagrams 300

6.4 Graphic Layout of Cam Profiles 303

6.5 Kinematic Coefficients of Follower 307

6.6 High-Speed Cams 312

6.7 Standard Cam Motions 313


6.8 Matching Derivatives of Displacement Diagrams 323

6.9 Plate Cam with Reciprocating Flat-Face Follower 327

6.10 Plate Cam with Reciprocating Roller Follower 332

6.11 Rigid and Elastic Cam Systems 350

6.12 Dynamics of an Eccentric Cam 351

6.13 Effect of Sliding Friction 355

6.14 Dynamics of Disk Cam with Reciprocating Roller Follower 356

6.15 Dynamics of Elastic Cam Systems 359

6.16 Unbalance, Spring Surge, and Windup 362

6.17 References 363

Problems 363

7 Spur Gears 369
7.1 Terminology and Definitions 369

7.2 Fundamental Law of Toothed Gearing 372

7.3 Involute Properties 373

7.4 Interchangeable Gears; AGMA Standards 375

7.5 Fundamentals of Gear-Tooth Action 376

7.6 Manufacture of Gear Teeth 381

7.7 Interference and Undercutting 384

7.8 Contact Ratio 386

7.9 Varying Center Distance 388

7.10 Involutometry 389

7.11 Nonstandard Gear Teeth 393

7.12 Parallel-Axis Gear Trains 401

7.13 Determining Tooth Numbers 404

7.14 Epicyclic Gear Trains 405

7.15 Analysis of Epicyclic Gear Trains by Formula 407

7.16 Tabular Analysis of Epicyclic Gear Trains 417

7.17 References 421

Problems 421

8 Helical Gears, Bevel Gears, Worms, and Worm Gears 427
8.1 Parallel-Axis Helical Gears 427

8.2 Helical Gear Tooth Relations 428

8.3 Helical Gear Tooth Proportions 430

8.4 Contact of Helical Gear Teeth 431

8.5 Replacing Spur Gears with Helical Gears 432

8.6 Herringbone Gears 433

8.7 Crossed-Axis Helical Gears 434


8.8 Straight-Tooth Bevel Gears 436

8.9 Tooth Proportions for Bevel Gears 440

8.10 Bevel Gear Epicyclic Trains 440

8.11 Crown and Face Gears 443

8.12 Spiral Bevel Gears 443

8.13 Hypoid Gears 445

8.14 Worms and Worm Gears 445

8.15 Summers and Differentials 449

8.16 All-Wheel Drive Train 453

8.17 Note 455

Problems 455

9 Synthesis of Linkages 458
9.1 Type, Number, and Dimensional Synthesis 458

9.2 Function Generation, Path Generation, and Body Guidance 459

9.3 Two Finitely Separated Postures of a Rigid Body (N = 2) 460
9.4 Three Finitely Separated Postures of a Rigid Body (N = 3) 465
9.5 Four Finitely Separated Postures of a Rigid Body (N = 4) 474
9.6 Five Finitely Separated Postures of a Rigid Body (N = 5) 481
9.7 Precision Postures; Structural Error; Chebyshev Spacing 481

9.8 Overlay Method 483

9.9 Coupler-Curve Synthesis 485

9.10 Cognate Linkages; Roberts-Chebyshev Theorem 489

9.11 Freudenstein’s Equation 491

9.12 Analytic Synthesis Using Complex Algebra 495

9.13 Synthesis of Dwell Linkages 499

9.14 Intermittent Rotary Motion 500

9.15 References 504

Problems 504

10 Spatial Mechanisms and Robotics 507
10.1 Introduction 507

10.2 Exceptions to the Mobility Criterion 509

10.3 Spatial Posture-Analysis Problem 513

10.4 Spatial Velocity and Acceleration Analyses 518

10.5 Euler Angles 524

10.6 Denavit-Hartenberg Parameters 528

10.7 Transformation-Matrix Posture Analysis 530

10.8 Matrix Velocity and Acceleration Analyses 533

10.9 Generalized Mechanism Analysis Computer Programs 538


10.10 Introduction to Robotics 541

10.11 Topological Arrangements of Robotic Arms 542

10.12 Forward Kinematics Problem 543

10.13 Inverse Kinematics Problem 550

10.14 Inverse Velocity and Acceleration Analyses 553

10.15 Robot Actuator Force Analysis 558

10.16 References 561

Problems 562


11 Static Force Analysis 569
11.1 Introduction 569

11.2 Newton’s Laws 571

11.3 Systems of Units 571

11.4 Applied and Constraint Forces 573

11.5 Free-Body Diagrams 576

11.6 Conditions for Equilibrium 578

11.7 Two- and Three-Force Members 579

11.8 Four- and More-Force Members 589

11.9 Friction-Force Models 591

11.10 Force Analysis with Friction 594

11.11 Spur- and Helical-Gear Force Analysis 597

11.12 Straight-Tooth Bevel-Gear Force Analysis 604

11.13 Method of Virtual Work 608

11.14 Introduction to Buckling 611

11.15 Euler Column Formula 612

11.16 Critical Unit Load 615

11.17 Critical Unit Load and Slenderness Ratio 618

11.18 Johnson’s Parabolic Equation 619

11.19 References 645

Problems 646

12 Dynamic Force Analysis 658
12.1 Introduction 658

12.2 Centroid and Center of Mass 658

12.3 Mass Moments and Products of Inertia 663

12.4 Inertia Forces and d’Alembert’s Principle 666

12.5 Principle of Superposition 674

12.6 Planar Rotation about a Fixed Center 680


12.7 Shaking Forces and Moments 682

12.8 Complex-Algebraic Approach 683

12.9 Equation of Motion from Power Equation 692

12.10 Measuring Mass Moment of Inertia 702

12.11 Transformation of Inertia Axes 705

12.12 Euler’s Equations of Motion 710

12.13 Impulse and Momentum 714

12.14 Angular Impulse and Angular Momentum 714

12.15 References 724

Problems 725

13 Vibration Analysis 743
13.1 Differential Equations of Motion 743

13.2 A Vertical Model 747

13.3 Solution of the Differential Equation 748

13.4 Step Input Forcing 752

13.5 Phase-Plane Representation 755

13.6 Phase-Plane Analysis 757

13.7 Transient Disturbances 760

13.8 Free Vibration with Viscous Damping 764

13.9 Damping Obtained by Experiment 766

13.10 Phase-Plane Representation of Damped Vibration 768

13.11 Response to Periodic Forcing 772

13.12 Harmonic Forcing 776

13.13 Forcing Caused by Unbalance 780

13.14 Relative Motion 781

13.15 Isolation 782

13.16 Rayleigh’s Method 785

13.17 First and Second Critical Speeds of a Shaft 787

13.18 Torsional Systems 793

13.19 References 795

Problems 796

14 Dynamics of Reciprocating Engines 804
14.1 Engine Types 804

14.2 Indicator Diagrams 811

14.3 Dynamic Analysis—General 814

14.4 Gas Forces 814

14.5 Equivalent Masses 816

14.6 Inertia Forces 818


14.7 Bearing Loads in a Single-Cylinder Engine 821

14.8 Shaking Forces of Engines 824

14.9 Computation Hints 825

Problems 828

15 Balancing 830
15.1 Static Unbalance 830

15.2 Equations of Motion 831

15.3 Static Balancing Machines 834

15.4 Dynamic Unbalance 835

15.5 Analysis of Unbalance 837

15.6 Dynamic Balancing 846

15.7 Dynamic Balancing Machines 848

15.8 Field Balancing with a Programmable Calculator 851

15.9 Balancing a Single-Cylinder Engine 854

15.10 Balancing Multi-Cylinder Engines 858

15.11 Analytic Technique for Balancing Multi-Cylinder Engines 862

15.12 Balancing Linkages 868

15.13 Balancing of Machines 874

15.14 References 875

Problems 875

16 Flywheels, Governors, and Gyroscopes 885
16.1 Dynamic Theory of Flywheels 885

16.2 Integration Technique 887

16.3 Multi-Cylinder Engine Torque Summation 890

16.4 Classification of Governors 890

16.5 Centrifugal Governors 892

16.6 Inertia Governors 893

16.7 Mechanical Control Systems 894

16.8 Standard Input Functions 895

16.9 Solution of Linear Differential Equations 897

16.10 Analysis of Proportional-Error Feedback Systems 901

16.11 Introduction to Gyroscopes 905

16.12 Motion of a Gyroscope 906

16.13 Steady or Regular Precession 908

16.14 Forced Precession 911

16.15 References 917

Problems 917



APPENDIX A: Tables 919

Table 1 Standard SI Prefixes 919

Table 2 Conversion from US Customary Units to SI Units 920

Table 3 Conversion from SI Units to US Customary Units 920

Table 4 Properties of Areas 921

Table 5 Mass Moments of Inertia 922

Table 6 Involute Function 923

APPENDIX B: Answers to Selected Problems 925



The tremendous growth of scientific knowledge over the past 50 years has resulted
in an intense pressure on the engineering curricula of many universities to substitute
“modern” subjects in place of subjects perceived as weaker or outdated. The result
is that, for some, the kinematics and dynamics of machines has remained a critical
component of the curriculum and a requirement for all mechanical engineering students,
while at others, a course on these subjects is only made available as an elective topic for
specialized study by a small number of engineering students. Some schools, depending
largely on the faculty, require a greater emphasis on mechanical design at the expense
of depth of knowledge in analytical techniques. Rapid advances in technology, however,
have produced a need for a textbook that satisfies the requirement of new and changing
course structures.

Much of the new knowledge in the theory of machines and mechanisms currently
exists in a large variety of technical journals and manuscripts, each couched in its
own singular language and nomenclature and each requiring additional background for
clear comprehension. It is possible that the individual published contributions could be
used to strengthen engineering courses if the necessary foundation was provided and
a common notation and nomenclature was established. These new developments could
then be integrated into existing courses to provide a logical, modern, and comprehensive
whole. The purpose of this book is to provide the background that will allow such an

This book is intended to cover that field of engineering theory, analysis, design,
and practice that is generally described as mechanisms or as kinematics and dynamics
of machines. Although this text is written primarily for students of mechanical
engineering, the content can also be of considerable value to practicing engineers
throughout their professional careers.

To develop a broad and basic comprehension, the text presents numerous methods
of analysis and synthesis that are common to the literature of the field. The authors have
included graphic methods of analysis and synthesis extensively throughout the book,
because they are firmly of the opinion that graphic methods provide visual feedback
that enhances the student’s understanding of the basic nature of, and interplay between,
the underlying equations. Therefore, graphic methods are presented as one possible
solution technique, but are always accompanied by vector equations defined by the
fundamental laws of mechanics, rather than as graphic “tricks” to be learned by rote and
applied blindly. In addition, although graphic techniques, performed by hand, may lack
accuracy, they can be performed quickly, and even inaccurate sketches can often provide
reasonable estimates of a solution and can be used to check the results of analytic or
numeric solution techniques.



The authors 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 included because of their compactness, because of the ease of taking
derivatives, because they are employed so frequently in the literature, and because
they are so easy to program for computer evaluation. In the chapter dealing with
three-dimensional kinematics and robotics, the authors present a brief introduction to
Denavit and Hartenberg’s methods using transformation matrices.

Another feature of this text is its focus on the method of kinematic coefficients,
which are derivatives of motion variables with respect to the input position variable(s)
rather than with respect to time. The authors believe that this analytic technique provides
several important advantages, namely: (1) Kinematic coefficients clarify for the student
those parts of a motion problem that are kinematic (geometric) in their nature, and
clearly separate these from the parts that are dynamic or speed dependent. (2) Kinematic
coefficients help to integrate the analysis of different types of mechanical systems, such
as gears, cams, and linkages, which might not otherwise seem similar.

One dilemma that all writers on the subject of this book have faced is how to
distinguish between the motions of different points of the same moving body and the
motions of coincident points of different moving bodies. In other texts, it has been
customary to describe both of these as “relative motion”; however, because they are
two distinctly different situations and are described by different equations, this causes
the student confusion 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 by using different terminology and different notation for the two cases.
Thus, for example, this 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 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 arises in, the
apparent acceleration equation.

Access to personal computers, programmable calculators, and laptop computers
is 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 as well. Having programmed almost all the material in the book
many times, we also understand that the book should not include such programs and
thus become obsolete with changes in computers or programming languages.

The authors have endeavored to use US Customary units and SI units in about
equal proportions throughout the book. However, there are certain exceptions. For
example, in Chapter 14 (Dynamics of Reciprocating Engines), only SI units are
presented, because engines are designed for an international marketplace, even by US
companies. Therefore, they are always rated in kilowatts rather than horsepower, they
have displacements in liters rather than cubic inches, and their cylinder pressures are
measured in kilopascals rather than pounds per square inch.

Part 1 of this book deals mostly with theory, nomenclature, notation, and methods
of analysis. Serving as an introduction, Chapter 1 tells what a mechanism is, what


a mechanism can do, how mechanisms can be classified, and what some of their
limitations are. Chapters 2, 3, and 4 are concerned totally with analysis, specifically with
kinematic analysis, because they cover position, velocity, and acceleration analyses,
respectively, of single-degree-of-freedom planar mechanisms. Chapter 5 expands this
background to include multi-degree-of-freedom planar mechanisms.

Part 2 of the book goes on to demonstrate 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 an introduction to robotics.
Chapter 6 is a study of the geometry, kinematics, proper design of high-speed cam
systems, and now includes material on the dynamics of elastic cam systems. Chapter 7
studies the geometry and kinematics of spur gears, particularly of involute tooth profiles,
their manufacture, and proper tooth meshing, and then studies gear trains, with an
emphasis on epicyclic and differential gear trains. Chapter 8 expands this background to
include helical gears, bevel gears, worms, and worm gears. Chapter 9 is an introduction
to the kinematic synthesis of planar linkages. Chapter 10 is a brief introduction to
the kinematic analysis of spatial mechanisms and robotics, including the forward and
inverse kinematics problems.

Part 3 of the book adds the dynamics of machines. In a sense, this part is concerned
with the consequences of the mechanism design specifications. In other words, having
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? Chapter 11 presents the static force analysis of machines. This chapter
also includes sections focusing on the buckling of two-force members subjected to
axial loads. Chapter 12 studies the planar and spatial aspects of the dynamic force
analysis of machines. Chapter 13 then presents the vibration analysis of mechanical
systems. Chapter 14 is a more detailed study of one particular type of mechanical
system, namely the dynamics of both single- and multi-cylinder reciprocating engines.
Chapter 15 next addresses the static and dynamic balancing of rotating and reciprocating
systems. Finally, Chapter 16 is on the study of the dynamics of flywheels, governors,
and gyroscopes.

As with all texts, the subject matter of this book also has limitations. 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 planar multibody systems with movable
connections or constraints between them. However, all motion effects are assumed to
come within the connections; the shapes of the individual bodies are assumed constant,
except for the dynamics of elastic cam systems. 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, except for buckling of axially
loaded members, the study of stress is also outside 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.

Despite the limitations on the scope of this book, it is still clear that it is not reason-
able to expect that all of the material presented here can be covered in a single-semester


course. As stated above, a variety of methods and applications have been included to
allow the instructor to choose those topics that best fit the course objectives and to still
provide a reference for follow-on courses and help build the student’s library. Yet, many
instructors have asked for suggestions regarding a choice of topics that might fit a 3-hour
per week, 15-week course. Two such outlines follow, as used by two of the authors to
teach such courses at their institutions. It is hoped that these might be used as helpful
guidelines to assist others in making their own parallel choices.

Tentative Schedule I

Kinematics and Dynamics of Machine Systems

Week Topics Sections

1 Introduction to Mechanisms 1.1–1.10

Kutzbach and Grashof Criteria 1.6, 1.9

Advance-to-Return Time Ratio 1.7

Overlay Method of Synthesis 9.8

2 Vector Loop-Closure Equation 2.6, 2.7

Velocity Difference Equation 3.1–3.3

Velocity Polygons; Velocity Images 3.4

3 Apparent Velocity Equation 3.5, 3.6, 3.8

Direct and Rolling Contact Velocity 3.7

4 Instantaneous Centers of Velocity 3.12

Aronhold–Kennedy Theorem of Three Centers 3.13, 3.14

Use of Instant Centers to Find Velocities 3.15, 3.16

5 Exam #1

Acceleration Difference Equation 4.1–4.3

Acceleration Polygons; Acceleration Images 4.4

6 Apparent Acceleration Equation 4.5, 4.6

Coriolis Component of Acceleration

7 Direct and Rolling Contact Acceleration 4.7, 4.8

Review of Velocity and Acceleration Analyses

8 Raven’s Method of Kinematic Analysis 2.10, 3.10, 4.10

Kinematic Coefficients 3.11, 4.11

Computer Methods in Kinematics 10.9


9 Exam #2

Static Forces 11.1–11.6

Two-, Three-, and Four-Force Members 11.7, 11.8

Force Polygons

10 Coulomb Friction Forces in Machines 11.9, 11.10

11 D’Alembert’s Principle 12.1–12.4

Dynamic Forces in Machine Members 12.4, 12.5

12 Introduction to Cam Design 6.1–6.4

Choice of Cam Profiles; Matching Displacement Curves 6.5–6.8

13 First-Order Kinematic Coefficients; Face Width; Pressure


Second-Order Kinematic Coefficients; Pointing and Under-


14 Exam #3

Introduction to Gearing 7.1–7.6

Involute Tooth Geometry; Contact Ratio; Undercutting 7.7–7.9, 7.11

15 Epicyclic and Differential Gear Trains 7.15–7.17


Final Exam

Tentative Schedule II

Machine Design I

Week Topics Sections

1 The World of Mechanisms 1.1–1.6

Measures of Performance (Indices of Merit) 1.10, 3.19

Quick Return Mechanisms 1.7

2 Position Analysis. Vector Loops 2.1–2.7

Newton–Raphson Technique 2.8, 2.11

3 Velocity Analysis 3.1–3.9

First-Order Kinematic Coefficients 3.11

Instant Centers of Zero Velocity 3.12–3.17


4 Rolling Contact, Rack and Pinion, Two Gears 3.10

Acceleration Analysis 4.1–4.4

Second-Order Kinematic Coefficients 4.5–4.11

5 Geometry of a Point Path 4.15

Kinematic Coefficients for Point Path 4.15

Radius and Center of Curvature 4.16

6 Cam Design 6.1–6.4

Lift Curve 6.1–6.4

Exam 1

7 Kinematic Coefficients of the Follower 6.5

Roller Follower 6.10

Flat-Face Follower 6.9

8 Graphic Approach 11.5, 11.6

Two-, Three-, and Four-Force Members 11.7, 11.8

Friction-Force Models 11.9, 11.10

9 Dynamic Force Analysis 12.1–12.3

Force and Moment Equations 12.4–12.6

Static Force Analysis 11.1–11.4

10 Power Equation 12.9

Kinetic, Potential, and Dissipative Energy 12.9

Equivalent Inertia and Equivalent Mass 12.9

11 Equation of Motion 12.9

Critical Speeds of a Shaft 13.17

Exam #2

12 Exact Equation 13.17

Dunkerley and Rayleigh–Ritz Approximations 13.17

Shaking Forces and Moments 14.5

13 Rotating Unbalance 15.3

Discrete Mass System 15.5

Distributed Mass System 15.5


14 Reciprocating Unbalance 15.9

Single-Cylinder Engine 15.9

Multi-Cylinder Engine 15.10

15 Primary Shaking Forces 15.11

Secondary Shaking Forces 15.11

Comparison of Forces 15.9

Final Exam

Supplement packages for this fifth edition have been designed to support both
the student and the instructor in the kinematics and dynamics course. The Companion
Website ( will include a list of any errors discovered in
the text and their corrections. This website also includes over 100 animations of key
figures from the text; these are marked with a symbol in the text. These animations,
created by Zhong Hu of South Dakota State University, are presented in both Working
Model and .avi file formats, and are meant to help students visualize and comprehend
the movement of important mechanisms.

An Ancillary Resource Center site is available for instructors only (registration
is required). A complete solutions manual for all problems is available on that site.
Solutions are also available on that site for 100 problems in the text worked out using
MatLab software, for instructors wishing to incorporate MatLab code into their courses.
Problems marked with a † signify that there is a MatLab-based solution available on that
site; thank you to Bob Williams at Ohio University for his help with those solutions.

The authors wish to thank the reviewers for their very helpful criticisms and

Reviewers of the fourth edition are: Zhuming Bi, Indiana University, Purdue
University Fort Wayne; Mehrdaad Ghorashi, University of Southern Maine; Dominic
M. Halsmer, Oral Roberts University; E. William Jones, Mississippi State University;
Pierre Larochelle, Florida Institute of Technology; John K. Layer, University of
Evansville; Todd Letcher, South Dakota State University; Jizhou Song, University of
Miami; and Michael Uenking, Thomas Nelson Community College.

Reviewers of the third edition were: Efstatios Nikolaidis, University of Toledo; Fred
Choy, University of Akron; Bob Williams, Ohio University; Lubambala Kabengela,
UNC Charlotte; Carol Rubin, Vanderbilt University; Yeau-Jian Liao, Wayne State
University; Chad O’Neal, Louisiana Tech University; Alba Perez-Garcia, Idaho State
University; Zhong Hu, South Dakota State University.

The many instructors and students who have tolerated previous versions of this
book and made their suggestions for its improvement also deserve our continuing

The authors would also like to offer our sincere thanks to Nancy Blaine,
Senior Acquisitions Editor, Engineering; Christine Mahon, Associate Editor; Theresa
Stockton, Production Team Leader; Micheline Frederick, Senior Production Editor;
John Appeldorn, Editorial Assistant; Margaret Wilkinson, copyeditor; Cat Ohala,


proofreader; and Todd Williams, cover designer; Higher Education Group, Oxford
University Press, USA, for their continuing cooperation and assistance in bringing this
edition to completion.

John J. Uicker, Jr.
Gordon R. Pennock

October, 2016

About the Authors

John J. Uicker, Jr. is Professor Emeritus of Mechanical Engineering at the University
of Wisconsin–Madison. He received his B.M.E. degree from the University of Detroit
and his M.S. and Ph.D. degrees in mechanical engineering from Northwestern
University. Since joining the University of Wisconsin faculty in 1967, his teaching
and research specialties have been 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 UW Computer-Aided Engineering
Center and served as its director for its initial 10 years of operation. He has served on
several national committees of the American Society of Mechanical Engineers (ASME)
and the Society of Automotive Engineers (SAE), and he received the Ralph R. Teetor
Educational Award in 1969, the ASMEMechanisms Committee Award in 2004, and the
ASME Fellow Award in 2007. He is one of the founding members of the US Council for
the Theory of Machines and Mechanisms and of IFToMM, the international federation.
He served for several years as editor-in-chief of the federation journal Mechanism and
Machine Theory. He has also been 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 FordMotor Company. He was
also awarded a Fulbright-Hayes Senior Lectureship and became a Visiting Professor
to Cranfield Institute of Technology in Cranfield, England in 1978–1979. He is a
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 machine
design. His research specialties are in theoretical kinematics and the dynamics of
mechanical systems. He has applied his research to robotics, rotary machinery, and
biomechanics, including the kinematics, statics, 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 ASME at Purdue University and a member of the Student Section



Committee. He is a member of the Commission on Standards and Terminology, the
International Federation of the Theory of Machines and Mechanisms. He is also 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. He also served as chairman of the Mechanisms and
Robotics Committee, ASME, from 2008 to 2009.

He is a fellow of the ASME, a fellow of the SAE, and a fellow and chartered
engineer of the Institution of Mechanical Engineers, United Kingdom. He is a senior
member of the Institute of Electrical and Electronics Engineers and a senior member of
the Society of Manufacturing Engineers. He received the ASME Faculty Advisor of the
Year Award in 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 was presented with the Ruth and Joel
Spira Award for outstanding contributions to the School of Mechanical Engineering
and its students in 2003. He received the SAE Ralph R. Teetor Educational Award in
1986, the Ferdinand Freudenstein Award at the Fourth National Applied Mechanisms
and Robotics Conference in 1995, and the A.T. Yang Memorial Award from the Design
Engineering Division of ASME in 2005. He has been at the forefront of many new
developments in mechanical design, primarily in the areas of kinematics and dynamics.
He has published some 100 technical papers and is a regular conference and symposium
speaker, workshop presenter, and conference session organizer and chairman.

Joseph E. Shigley (deceased May 1994) was Professor Emeritus of Mechanical
Engineering at the University of Michigan and a fellow in the ASME. He received the
Mechanisms Committee Award in 1974, the Worcester Reed Warner medal in 1977,
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 wroteDynamic Analysis
of Machines in 1961, and these were published in a single volume titled Theory of
Machines in 1961; they have evolved over the years to become the current text, Theory
of Machines and Mechanisms, now in its fifth 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 several years in industry, he
devoted his career to teaching, writing, and service to his profession, first at Clemson
University and later at the University of Michigan. His textbooks have been widely used
throughout the United States and internationally.

Kinematics and Mechanisms

1 The World of Mechanisms


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 Chaps. 1 through 5, 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 Chaps. 6 through 10, in which we study
methods of designing mechanisms. Finally, in Part 3, which includes Chaps. 11 through
16, 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 are the sliding velocities and the loads at the lubricated surfaces, and what lubricants
are available for this purpose? How much heat is generated, and how is the engine
cooled?What are the synchronization and control requirements, and how are they satisfied?
What is the cost to the consumer, both for initial purchase and for continued operation
and maintenance? What materials and manufacturing methods are used? What are the
fuel economy, noise, and exhaust emissions; do they meet legal requirements? Although
all these and many other important questions must be answered before the design is
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 must many
branches of science be brought together. This book assembles material that falls into the
science of mechanics as it relates to the design of mechanisms and machines.




There are two completely different aspects of the study of mechanical systems: design and
analysis. The concept embodied in the word “design” is more properly termed 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 that 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 designers as they
practice their art.

In the process of evaluating the various interacting alternatives, designers find a need
for a large collection of mathematical and scientific tools. These tools, when applied
properly, provide more accurate and more reliable information for judging a design
than one achieves through intuition or estimation. Thus, the tools are of tremendous
help in deciding among alternatives. However, scientific tools cannot make decisions for
designers; designers have every right to exert their imagination and creative abilities, even
to the extent of overruling the mathematical recommendations.

Probably the largest collection of scientific methods at the designer’s disposal fall
into the category called analysis. These are techniques that allow the designer to critically
examine an already existing, or proposed, design to judge its suitability for the task. Thus,
analysis in itself is not a creative science but one of evaluation and rating things already

We should 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;
however, it is a vital tool and will inevitably be used as one step in the design process.


The 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 1765 [2]:†

The investigation of the motion of a rigid body may be conveniently separated
into two parts, the one, geometrical, and 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

∗ Leonhard Euler (1707–1783).
† Numbers in square brackets refer to references at the end of each chapter.



Kinematics Kinetics

Statics Dynamics

Figure 1.1

investigation will therefore be referable solely to geometry, or rather to stereotomy
[the art of stonecutting, now referred to as descriptive geometry].
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 (cinématique was a term coined by Ampère∗ and derived from the Greek word
kinema, meaning motion) and kinetics and deal with motion and the forces producing the
motion, respectively.

The initial problem in the design of a mechanical system, therefore, is understanding
the kinematics. Kinematics is the study of motion, quite apart from the forces that produce
the motion. In particular, kinematics is the study of position, displacement, rotation,
speed, velocity, acceleration, and jerk. 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 and operation of mechanical systems. Thus,
the kinematics of machines and mechanisms is the focus of the next several chapters of
this book. In addition, statics and kinetics are also vital parts of a complete design analysis,
and they are also covered in later chapters.

It should be carefully noted in the previous quotation that Euler based his separation of
dynamics into kinematics and kinetics on the assumption that they 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 while analyzing a machine’s kinematic performance and then, during dynamic
analysis when loads are sought, to design the parts so that the assumption is justified. A
more detailed discussion of a rigid body compared to a deformable, or flexible, body is
presented in the introduction to static force analysis in Sec. 11.1.

∗ André-Marie Ampère (1775–1836).



Reuleaux∗ defines a machine† 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 the purpose of a structure (such as a truss) is not to do work or to transform motion, but
to be rigid. A truss can perhaps be moved from place to place and is movable in this sense of
the word; however, it has no internalmobility. A structure has no relative motions between
its various links, whereas both machines and mechanisms do. Indeed, the whole purpose of
a machine or mechanism is to utilize these relative internal motions in transmitting power
or transforming motion.

A machine is an arrangement of parts for doing work, a device for applying power or
changing the direction of motion. It differs from a mechanism in its purpose. In a machine,
terms such as force, torque, work, and power describe the predominant concepts. In a
mechanism, though it may transmit power or force, the predominant idea in the mind of
the designer is one of achieving a desired motion. There is a direct analogy between the
terms structure, mechanism, and machine and the branches of mechanics illustrated in
Fig. 1.1. The term “structure” is to statics as the term “mechanism” is to kinematics and as
the term “machine” is to kinetics.

We use the word link to designate a machine part or a component of a mechanism.
As discussed in the previous section, a link is assumed to be completely rigid. Machine
components that do not fit this assumption of rigidity, such as springs, usually have no
effect on the kinematics of a device but do play a role in supplying forces. Such parts or
components are not called links; they are usually ignored during kinematic analysis, and
their force effects are introduced during force analysis (see the analysis of buckling in Secs.
11.14–11.18). Sometimes, as with a belt or chain, a machine part may possess one-way
rigidity; such a body can be considered a link when in tension but not under compression.

The links of a mechanism must be connected in some manner in order to transmit
motion from the driver, or input, to the driven, or follower, or output. The connections, the
joints between the links, are called kinematic pairs (or simply pairs), because each joint
consists of a pair of mating surfaces, two elements, one mating surface or element being
a part of each of the joined links. Thus, we can also define a link as the rigid connection
between two or more joint elements.

Stated explicitly, the assumption of rigidity is that there can be no relative motion (no
change in distance) between two arbitrarily chosen points on the same link. In particular,

∗ Much of the material of this section is based on definitions originally set down by Franz Reuleaux
(1829–1905), a German kinematician whose work marked the beginning of a systematic treatment
of kinematics [7].
† There appears to be no agreement at all on the proper definition of a machine. In a footnote
Reuleaux gives 17 definitions, and his translator gives 7 more and discusses the whole problem
in detail [7].


the relative positions of joint elements on any given link do not change no matter what loads
are applied. In other words, the purpose of a link is to hold a constant spatial relationship
between its joint elements.

As a result of the assumption of rigidity, many of the intricate details of the actual part
shapes are unimportant when studying the kinematics of a machine or mechanism. For this
reason, it is common practice to draw highly simplified schematic diagrams that contain
important features of the shape of each link, such as the relative locations of joint elements,
but that completely subdue the real geometry of the manufactured part. The slider-crank
linkage of the internal combustion engine, for example, can be simplified for purposes of
analysis to the schematic diagram illustrated later in Fig. 1.3b. Such simplified schematics
are a great help since they eliminate confusing factors that do not affect the analysis; such
diagrams are used extensively throughout this text. However, these schematics also have
the drawback of bearing little resemblance to physical hardware. As a result they may give
the impression that they represent only academic constructs rather than real machinery. We
should continually bear in mind that these simplified diagrams are intended to carry only
the minimum necessary information so as not to confuse the issue with unimportant detail
(for kinematic purposes) or complexity of the true machine parts.

When several links are connected together by joints, they are said to form a kinematic
chain. Links containing only two joint elements are called binary links, those having
three joint elements are called ternary links, those having four joint elements are called
quaternary links, and so on. If every link in a chain is connected to at least two other links,
the chain forms one or more closed loops and is called a closed kinematic chain; if not, the
chain is referred to as open. If a chain consists entirely of binary links, it is a simple-closed
chain. Compound-closed chains, however, include other than binary links and thus form
more than a single closed loop.

Recalling Reuleaux’s definition of a mechanism, we see that it is necessary to have a
closed kinematic chain with one link fixed. When we say that one link is fixed, we mean
that it is chosen as the frame of reference for all other links; that is, the motions of all
points on the links of the mechanism are measured with respect to the fixed link. This
link, in a practical machine, usually takes the form of a stationary platform or base (or
a housing rigidly attached to such a base) and is commonly referred to as the ground,
frame, or base link.∗ The question of whether this reference frame is truly stationary (in
the sense of being an inertial reference frame) is immaterial in the study of kinematics, but
becomes important in the investigation of kinetics, where forces are considered. In either
case, once a frame link is designated (and other conditions are met), the kinematic chain
becomes a mechanism and, as the driver is moved through various positions, all other links
have well-defined motions with respect to the chosen frame of reference. We use the term
kinematic chain to specify a particular arrangement of links and joints when it is not clear
which link is to be treated as the frame. When the frame link is specified, the kinematic
chain is called a mechanism.

For a mechanism to be useful, the motions between links cannot be completely
arbitrary; they too must be constrained to produce the proper relative motions—those
chosen by the designer for the particular task to be performed. These desired relative

∗ In this text, the ground, frame, or base of the mechanism is commonly numbered 1.


motions are achieved by proper choice of the number of links and the kinds of joints used
to connect them. Thus we are led to the concept that, in addition to the distances between
successive joints, the nature of the joints themselves and the relative motions they permit
are essential in determining the kinematics of a mechanism. For this reason, it is important
to look more closely at the nature of joints in general terms, and in particular at several of
the more common types.

The controlling factors that determine the relative motions allowed by a given joint are
the shapes of the mating surfaces or elements. Each type of joint has its own characteristic
shapes for the elements, and each allows a given type of motion, which is determined by
the possible ways in which these elemental surfaces can move with respect to each other.
For example, the pin joint in Fig. 1.2a, has cylindric elements, and, assuming that the links
cannot slide axially, these surfaces permit only relative rotational motion. Thus a pin joint
allows the two connected links to experience relative rotation about the pin center. So, too,
other joints each have their own characteristic element shapes and relative motions. These
shapes restrict the totally arbitrary motion of two unconnected links to some prescribed
type of relative motion and form constraining conditions (constraints) on the mechanism’s

It should be pointed out that the element shapes may often be subtly disguised and
difficult to recognize. For example, a pin joint might include a needle bearing, so that
two mating surfaces, as such, are not distinguishable. Nevertheless, if the motions of the
individual rollers are not of interest, the motions allowed by the joints are equivalent, and
the joints are of the same generic type. Thus the criterion for distinguishing different joint
types is the relative motions they permit and not necessarily the shapes of the elements,
though these may provide vital clues. The diameter of the pin used (or other dimensional
data) is also of no more importance than the exact sizes and shapes of the connected
links. As stated previously, the kinematic function of a link is to hold a fixed geometric
relationship between the joint elements. Similarly, the only kinematic function of a joint,
or pair, is to determine the relative motion between the connected links. All other features
are determined for other reasons and are unimportant in the study of kinematics.

When a kinematic problem is formulated, it is necessary to recognize the type of
relative motion permitted in each of the joints and to assign to it some variable parameter(s)
for measuring or calculating the motion. There will be as many of these parameters as there
are degrees of freedom of the joint in question, and they are referred to as joint variables.
Thus, the joint variable of a pinned joint will be a single angle measured between reference
lines fixed in the adjacent links, while a spheric joint will have three joint variables (all
angles) to specify its three-dimensional rotation.

Reuleaux separated kinematic pairs into two categories: namely, higher pairs and
lower pairs, with the latter category consisting of the six prescribed types to be discussed
next. He distinguished between the categories by noting that lower pairs, such as the
pin joint, have surface contact between the joint elements, while higher pairs, such as
the connection between a cam and its follower, have line or point contact between the
elemental surfaces. This criterion, however, can be misleading (as noted in the case of a
needle bearing). We should rather look for distinguishing features in the relative motion(s)
that the joint allows between the connected links.

Lower pairs consist of the six prescribed types shown in Fig. 1.2.









(d )


�s �u



(e) (f )




Figure 1.2 (a) Revolute; (b) prism; (c) screw; (d) cylinder; (e) sphere; ( f ) flat pairs.

The names and the symbols (Hartenberg and Denavit [4]) that are commonly
employed for the six lower pairs are presented in Table 1.1. The table also includes the
number of degrees of freedom and the joint variables that are associated with each lower

The revolute or turning pair, R (Fig. 1.2a), permits only relative rotation and is often
referred to as a pin joint. This joint has one degree of freedom.

The prism or prismatic pair, P (Fig. 1.2b), permits only relative sliding motion and
therefore is often called a sliding joint. This joint also has one degree of freedom.

The screw or helical pair, H (Fig. 1.2c), permits both rotation and sliding motion.
However, it only has one degree of freedom, since the rotation and sliding motions are
related by the helix angle of the thread. Thus, the joint variable may be chosen as either
�s or �θ , but not both. Note that the helical pair reduces to a revolute if the helix angle is
made zero, and to a prism if the helix angle is made 90◦.

Table 1.1 Lower Pairs

Pair Symbol Pair Variable Degrees of Freedom Relative Motion

Revolute R �θ 1 Circular

Prism P �s 1 Rectilinear

Screw H �θ or �s 1 Helical

Cylinder C �θ and �s 2 Cylindric

Sphere S �θ , �φ, �ψ 3 Spheric

Flat F �x, �y, �θ 3 Planar


The cylinder or cylindric pair, C (Fig. 1.2d), permits both rotation and an independent
sliding motion. Thus, the cylindric pair has two degrees of freedom.

The sphere or globular pair, S (Fig. 1.2e), is a ball-and-socket joint. It has three
degrees of freedom, sometimes taken as rotations about each of the coordinate axes.

The flat or planar pair, sometimes called an ebene pair (German), F (Fig. 1.2f ), is
seldom found in mechanisms in its undisguised form, except at a support point. It has three
degrees of freedom, that is, two translations and a rotation.

All other joint types are called higher pairs. Examples include mating gear teeth, a
wheel rolling and/or sliding on a rail, a ball rolling on a flat surface, and a cam contacting
its follower. Since an unlimited variety of higher pairs exist, a systematic accounting of
them is not a realistic objective. We shall treat each separately as it arises.

Among the higher pairs is a subcategory known as wrapping pairs. Examples are the
connections between a belt and a pulley, a chain and a sprocket, or a rope and a drum. In
each case, one of the links has only one-way rigidity.

The treatment of various joint types, whether lower or higher pairs, includes another
important limiting assumption. Throughout the book, we assume that the actual joint, as
manufactured, can be reasonably represented by a mathematical abstraction having perfect
geometry. That is, when a real machine joint is assumed to be a spheric joint, for example,
it is also assumed that there is no “play” or clearance between the joint elements and that
any deviation from spheric geometry of the elements is negligible. When a pin joint is
treated as a revolute, it is assumed that no axial motion takes place; if it is necessary to
study the small axial motions resulting from clearances between real elements, the joint
must be treated as cylindric, thus allowing the axial motion.

The term “mechanism,” as defined earlier, can refer to a wide variety of devices,
including both higher and lower pairs. A more limited term, however, refers to those
mechanisms having only lower pairs; such a mechanism is commonly called a linkage.
A linkage, then, is connected only by the lower pairs shown in Fig. 1.2.


Mechanisms may be categorized in several different ways to emphasize their similarities
and differences. One such grouping divides mechanisms into planar, spheric, and spatial
categories. All three groups have many things in common; the criterion that distinguishes
the groups, however, is to be found in the characteristics of the motions of the

A planar mechanism is one in which all particles describe planar curves in space, and
all these curves lie in parallel planes; that is, the loci of all points are planar curves parallel
to a single common plane. This characteristic makes it possible to represent the locus of any
chosen point of a planar mechanism in its true size and shape in a single drawing or figure.
The motion transformation of any such mechanism is called coplanar. The planar four-bar
linkage, the slider-crank linkage, the plate cam-and-follower mechanism, and meshing
gears are familiar examples of planar mechanisms.

Planar mechanisms utilizing only lower pairs are called planar linkages; they include
only revolute and prismatic joints. Although the planar pair might theoretically be included
in a planar linkage, this would impose no constraint on the motion. Planar motion also


requires that all revolute axes be normal to the plane of motion, and that all prismatic joint
axes be parallel to the plane.

As already pointed out, it is possible to observe the motions of all particles of a planar
mechanism in true size and shape from a single direction. In other words, all motions can
be represented graphically in a single view. Thus, graphic techniques are well suited to their
analysis, and this background is beneficial to the student once mastered. Since spheric and
spatial mechanisms do not have this special geometry, visualization becomes more difficult
and more powerful techniques must be used for their study.

A spheric mechanism is one in which each moving link has a point that remains
stationary as the mechanism moves. Also, arbitrary points fixed in each moving link travel
on spheric surfaces; the spheric surfaces must all be concentric. Therefore, the motions of
all these points can be completely described by their radial projections (or shadows) on the
surface of a sphere with a properly chosen center. Note that the only lower pairs (Table 1.1)
that allow spheric motion are the revolute pair and the spheric pair. In a spheric linkage,
the axes of all revolute pairs must intersect at a single point. In addition, a spheric pair
center must be concentric with this point, and, then, it would not produce any constraint on
the motions of the other links. Therefore, a spheric linkage must consist of only revolute
pairs, and the axes of all such pairs must intersect at a single point. A familiar example
of a spheric mechanism is the Hooke universal joint (also referred to as the Cardan joint)
shown in Fig. 1.21b.

Spatial mechanisms include no restrictions on the relative motions of the links.
For example, a mechanism that contains a screw joint (Fig. 1.2c) must be a spatial
mechanism, since the relative motion within a screw joint is helical. An example of a spatial
mechanism is the differential screw shown in Fig. 1.11. Because of the more complex
motion characteristics of spatial mechanisms, and since these motions can not be analyzed
graphically from a single viewing direction, more powerful techniques are required for
their analysis. Such techniques are introduced in Chap. 10 for a detailed study of spatial
mechanisms and robotics.

Since the majority of mechanisms in modern machinery are planar, one might
question the need to study these complex mathematical techniques. However, even though
the simpler graphic techniques for planar mechanisms may have been mastered, an
understanding of the more complex techniques is of value for the following reasons:

1. They provide new, alternative methods that can solve problems in a different way.
Thus, they provide a means for checking results. Certain problems by their nature
may also be more amenable to one method than another.

2. Methods that are analytic in nature are better suited to solution by a calculator or
a digital computer than by graphic techniques.

3. One reason why planar mechanisms are so common is that good methods for the
analysis of spatial mechanisms have not been available until relatively recently.
Without these methods, the design and application of spatial mechanisms has been
hindered, even though they may be inherently better suited to certain applications.

4. We will discover that spatial mechanisms are, in fact, much more common in
practice than their formal description indicates.


Consider the planar four-bar linkage (Fig. 1.3c), which has four links connected
by four revolute pairs whose axes are parallel. This “parallelism” is a mathematical
hypothesis; it is not a reality. The axes, as produced in a machine shop—in any machine
shop, no matter how precise the machining—are only approximately parallel. If the axes
are far out of parallel, there is binding in no uncertain terms, and the linkage moves only
because the “rigid” links flex and twist, producing loads in the bearings. If the axes are
nearly parallel, the linkage operates because of looseness of the running fits of the bearings
or flexibility of the links. A common way of compensating for small nonparallelism
is to connect the links with self-aligning bearings, actually spheric joints allowing
three-dimensional rotation. Such a “planar” linkage is thus a low-grade spatial linkage.

Thus, the overwhelmingly large category of planar mechanisms and the category of
spheric mechanisms are special cases, or subsets, of the all-inclusive category of spatial
mechanisms. They occur as a consequence of the special orientations of their joint axes.


One of the first concerns in either the design or the analysis of a mechanism is the
number of degrees of freedom, also called the mobility of the device. The mobility∗ of
a mechanism is the number of input parameters (usually joint variables) that must be
controlled independently to bring the device into a particular posture. Ignoring, for the
moment, certain exceptions to be mentioned later, it is possible to determine the mobility
of a mechanism directly from a count of the number of links and the number and types of
joints comprising the mechanism.

To develop this relationship, consider that—before they are connected together—each
link of a planar mechanism has three degrees of freedom when moving with planar motion
relative to the fixed link. Not counting the fixed link, therefore, an n-link planar mechanism
has 3(n− 1) degrees of freedom before any of the joints are connected. Connecting two of
the links by a joint that has one degree of freedom, such as a revolute, has the effect of
providing two constraints between the connected links. If the two links are connected by a
two-degree-of-freedom joint, it provides one constraint. When the constraints for all joints
are subtracted from the total degrees of freedom of the unconnected links, we find the
resulting mobility of the assembled mechanism.

If we denote the number of single-degree-of-freedom joints as j1 and the number
of two-degree-of-freedom joints as j2, then the resulting mobility, m, of a planar n-link
mechanism is given by

m= 3(n− 1)− 2j1 − j2. (1.1)
Written in this form, Eq. (1.1) is called the Kutzbach criterion for the mobility of a planar
mechanism [8]. Its application is illustrated for several simple examples in Fig. 1.3.

∗ The German literature distinguishes between movability and mobility. Movability includes the six
degrees of freedom of the device as a whole, as though the ground link were not fixed, and thus
applies to a kinematic chain. Mobility neglects these degrees of freedom and considers only the
internal relative motions, thus applying to a mechanism. The English literature seldom recognizes
this distinction, and the terms are used somewhat interchangeably.


n = 3,  j1 = 3,
j2 = 0, m = 0


n = 4,  j1 = 4,
j2 = 0, m = 1


n = 4,  j1 = 4,
j2 = 0, m = 1


n = 5,  j1 = 5,
j2 = 0, m = 2

(d )

Figure 1.3 Applications of the Kutzbach criterion.

If the Kutzbach criterion yields m > 0, the mechanism has m degrees of freedom.
If m = 1, the mechanism can be driven by a single input motion to produce constrained
(uniquely defined) motion. Two examples are the slider-crank linkage and the four-bar
linkage, shown in Figs. 1.3b and 1.3c, respectively. If m = 2, then two separate input
motions are necessary to produce constrained motion for the mechanism; such a case is
the five-bar linkage shown in Fig. 1.3d.

If the Kutzbach criterion yields m= 0, as in Figs. 1.3a and 1.4a, motion is impossible
and the mechanism forms a structure.

If the criterion yields m < 0, then there are redundant constraints in the chain and it
forms a statically indeterminate structure. An example is illustrated in Fig. 1.4b. Note in
the examples of Fig. 1.4 that when three links are joined by a single pin, such a connection
is treated as two separate but concentric joints; two j1 joints must be counted.

Figure 1.5 shows two examples of the Kutzbach criterion applied to mechanisms with
two-degree-of-freedom joints—that is, j2 joints. Particular attention should be paid to the
contact (joint) between the wheel and the fixed link in Fig. 1.5b. Here it is assumed that
slipping is possible between the two links. If this contact prevents slipping, the joint would
be counted as a one-degree-of-freedom joint—that is, a j1 joint—because only one relative
motion would then be possible between the links. Recall that, in this case, the mechanism
is generally referred to as a “linkage.”

It is important to realize that the Kutzbach criterion can give an incorrect result. For
example, note that Fig. 1.6a represents a structure and that the criterion properly predicts
m= 0. However, if link 5 is arranged as in Fig. 1.6b, the result is a double-parallelogram
linkage with a mobility of m = 1, even though Eq. (1.1) indicates that it is a structure.
The actual mobility of m = 1 results only if the parallelogram geometry is achieved. In
the development of the Kutzbach criterion, no consideration was given to the lengths of the

n = 5, j1 = 6,
j2 = 0, m = 0


n = 6, j1 = 8,
j2 = 0, m = –1


Figure 1.4 Applications of the Kutzbach criterion to structures.


n = 3, j1 = 2,
j2 = 1, m = 1


n = 4, j1 = 3,
j2 = 1, m = 2



Figure 1.5

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.

Although there are exceptions, the Kutzbach criterion remains useful, because it is so
easily applied during mechanism design. To avoid exceptions, it would be necessary to
include all the dimensional properties of the mechanism. The resulting criterion would be
very complex and would be useless at the early stages of design when dimensions may not
be known.

An earlier mobility criterion, named after Grübler [3], applies to a planar linkage
where the overall mobility is m = 1. Substituting j2 = 0 and m = 1 into Eq. (1.1) and
rearranging, we find that Grübler’s criterion for planar linkages can be written as

3n− 2j1 − 4= 0. (1.2)
Rearranging this equation, the number of links is

n= 2j1 + 4

. (1.3)

From this equation, we see that a planar linkage with a mobility of m= 1 cannot have an
odd number of links. Also, the simplest possible linkage with all binary links has n= j1 = 4.
This explains one reason why the slider-crank linkage (Fig. 1.3b) and the four-bar linkage
(Fig. 1.3c) appear so commonly in machines.

n = 5, j1 = 6,
j2 = 0, m = 0


5 42


n = 5, j1 = 6,
j2 = 0, m = 1


5 42


Figure 1.6


The Kutzbach criterion, Eq. (1.1), and the Grübler criterion, Eq. (1.2), were derived for
the case of planar mechanisms and linkages, respectively. If similar criteria are developed
for spatial mechanisms and linkages, which is the subject of Chap. 10, we must recall that
each unconnected link has six degrees of freedom and each single-degree-of-freedom joint
provides five constraints, each two-degree-of-freedom joint provides four constraints, and
so on. Similar arguments then lead to the Kutzbach criterion for spatial mechanisms,

m= 6(n− 1)− 5j1 − 4j2 − 3j3 − 2j4 − j5,

and the Grübler criterion for spatial linkages,

6n− 5j1 − 7= 0. (1.4)

Therefore, the simplest form of a spatial linkage∗ with a mobility of m= 1 is n= j1 = 7.


Determine the mobility of the planar mechanism shown in Fig. 1.7a.






1 (b)






Figure 1.7 Planar mechanism.


The link numbers and the joint types for the mechanism are shown in Fig. 1.7b. The
number of links is n = 5, the number of lower pairs is j1 = 5, and the number of higher
pairs is j2 = 1. Substituting these values into the Kutzbach criterion, Eq. (1.1), the mobility
of the mechanism is

m= 3(5− 1)− 2(5)− 1(1)= 1. Ans.
Note that the Kutzbach criterion gives the correct answer for the mobility of this
mechanism; that is, a single input motion is required to give a unique output motion.

∗ Note that all planar linkages are exceptions to the spatial mobility criterion. They have the special
geometric characteristics that all revolute axes are parallel and perpendicular to the plane of motion
and that all prismatic axes lie in the plane of motion.


For example, rotation of link 2 could be used as the input and rotation of link 5 could be
used as the output.


For the mechanism shown in Fig. 1.8a, determine: (a) the number of lower pairs (j1 joints)
and the number of higher pairs (j2 joints); and (b) the mobility of the mechanism using the
Kutzbach criterion. Treating rolling contact to mean rolling with no slipping. Does this
criterion provide the correct answer for the mobility of this mechanism? Briefly explain
why or why not.

Figure 1.8 Planar mechanism.


(a) The links and the joint types of the mechanism are labeled in Fig. 1.8b. The
number of links is n= 7, the number of lower pairs is j1 = 9, and the number of
higher pairs is

j2 = 1. Ans.
(b) Substituting these values into the Kutzbach criterion, Eq. (1.1), the mobility of

the mechanism is

m= 3(7− 1)− 2(9)− 1(1)= −1. Ans.
However, this answer is not corrrect; that is, the Kutzbach criterion does not give
the correct mobility for this mechanism. The mobility of this mechanism is, in
fact, m= 1; that is, a single input motion gives a unique output motion.

Reasoning: Links 3 and 4 are superfluous to the constraints of the mechanism. If links
3 and 4 were removed, the motion of the remaining links would be unaffected. With links
3 and 4 removed, the mobility of the mechanism using the Kutzbach criterion is m = 1.
Note that if links 3 and 4 were attached with no special conditions—that is, not pinned at
their centers, for example—then the mechanism would indeed be locked and the answer
m= −1 would be correct.



For the mechanism shown in Fig. 1.9a, determine: (a) the number of lower pairs and
the number of higher pairs; and (b) the mobility of the mechanism predicted by the
Kutzbach criterion. Does this criterion provide the correct answer for this mechanism?
Briefly explain why or why not.



















Figure 1.9 Planar mechanism.


(a) The links and the joints of the mechanism are labeled as shown in Fig. 1.9b. The
number of links is n= 5, the number of lower pairs is j1 = 5, and the number of
higher pairs is

j2 = 1. Ans.
(b) Substituting these values into the Kutzbach criterion, Eq. (1.1), the mobility of

the mechanism is

m= 3(5− 1)− 2(5)− 1(1)= 1. Ans.
For this mechanism, the mobility is indeed 1, which indicates that the Kutzbach
criterion gives the correct answer for this mechanism.

For a mechanism, or a linkage, with a mobility of m= 1, the input or driving link will, in
general, be numbered as 2 in this text.


An ideal system for the classification of mechanisms would be a system that allows
a designer to enter the system with a set of specifications and leave with one or
more mechanisms that satisfy those specifications. Although history∗ demonstrates that

∗ For an excellent short history of the kinematics of mechanisms, see [4, Chap. 1].


3 1



2 3



Figure 1.10 (a) Bistable mechanism; (b) true toggle mechanism.

many attempts have been made, few have been particularly successful in devising a
satisfactory classification system. In view of the fact that the purpose of a mechanism
is the transformation of motion, we will follow Torfason’s lead [9] and classify
mechanisms according to the type of motion transformation. In total, Torfason displays 262
mechanisms, each of which can have a variety of dimensions. His categories are as follows:

Snap-Action Mechanisms Snap-action, toggle, or flip-flop mechanisms are used for
switches, clamps, or fasteners. Torfason also includes spring clips and circuit breakers.
Fig. 1.10 shows examples of bistable and true toggle mechanisms.

Linear Actuators Linear actuators include stationary screws with traveling nuts,
stationary nuts with traveling screws, and single-acting and double-acting hydraulic and
pneumatic cylinders.

Fine Adjustments Fine adjustments may be obtained with screws, including differen-
tial screws, worm gearing, wedges, levers, levers in series, and various motion-adjusting
mechanisms. For the differential screw shown in Fig. 1.11, you should be able to determine
that the translation of the carriage resulting from one turn of the handle is 0.0069 in to the
left (see Prob. 1.17).

Carriage Frame



Figure 1.11 Differential


Clamping Mechanisms Typical clamping mechanisms are the C-clamp, the wood
worker’s screw clamp, cam-actuated and lever-actuated clamps, vises, presses (such as the
toggle press shown in Fig. 1.10b), collets, and stamp mills.

Locational Devices Torfason [9] shows 15 locational mechanisms. These are usually
self-centering and locate either axially or angularly using springs and detents.

Ratchets and Escapements There are many different forms of ratchets and escape-
ments, some quite clever. They are used in locks, jacks, clockwork, and other applications
requiring some form of intermittent motion. Figure 1.12 shows four typical applications.

The ratchet in Fig. 1.12a allows only one direction of rotation of wheel 2. Pawl 3 is
held against the wheel by gravity or by a spring. A similar arrangement is used for lifting
jacks, which then employ a toothed rack for rectilinear motion.

Figure 1.12b is an escapement used for rotary adjustments.
Graham’s escapement shown in Fig. 1.12c is used to regulate the movement of

clockwork. Anchor 3 drives a pendulum whose oscillating motion is caused by the two
clicks engaging wheel 2. One is a push click, the other is a pull click. The lifting and
engaging of each click caused by oscillation of the pendulum results in a wheel motion







(a) (b)

(c) (d)




Figure 1.12 Ratchets and escapements.


(a) 3










Figure 1.13 Indexing

that, at the same time, presses each respective click and adds a gentle force to the motion
of the pendulum.

The escapement shown in Fig. 1.12d has a control wheel, 2, that may rotate
continuously to allow wheel 3 to be driven (by another source) in either direction.

Indexing Mechanisms The indexer shown in Fig. 1.13a uses standard gear teeth; for
light loads, pins can be used in the input wheel 2 with corresponding slots in wheel 3, but
neither form should be used if the shaft inertias are large.

Figure 1.13b shows a Geneva-wheel, sometimes called a “Maltese-cross,” indexer.
Three or more slots may be used in the driven link, 2, which can be attached to, or geared
to, the output to be indexed. High speeds and large inertias may cause problems with this

Toothless ratchet 5 in Fig. 1.13c is driven by the oscillating crank, 2, of variable throw.
Note the similarity of this indexing mechanism to the ratchet of Fig. 1.12a.

Torfason [9] lists nine different indexing mechanisms, and many variations are

Swinging or Rocking Mechanisms The category of swinging or rocking mechanisms
is often termed oscillators; in each case, the output rocks or swings through an angle that
is generally less than 360◦. However, the output shaft can be geared to a successor shaft to
produce a larger angle of oscillation.

Figure 1.14a is a mechanism consisting of a rotating crank 2 and a coupler 3 containing
a rack, which meshes with output gear 4 to produce the oscillating motion.












(d )



Figure 1.14 Oscillating mechanisms.

In Fig. 1.14b, crank 2 drives link 3, which slides on output link 4, producing a rocking
motion. This mechanism is described as a quick-return mechanism, because crank 2 rotates
through a larger angle on the forward stroke of link 4 than on the return stroke.

Figure 1.14c is a four-bar linkage called the crank-rocker linkage (Sec. 1.9). Crank 2
drives rocker 4 through the coupler 3. Of course, link 1 is the frame. The characteristics of
the rocking motion depend on the dimensions of the links and the placement of the frame

Figure 1.14d shows a cam-and-follower mechanism, in which the rotating link 2,
called the cam, drives link 3, called the follower, in a rocking motion. An endless variety
of cam-and-follower mechanisms are possible, many of which are discussed in Chap. 6.
In each case, the cam can be designed to produce an output motion with the desired

Reciprocating Mechanisms Repeating straight-line motion is commonly obtained
using either a pneumatic or hydraulic cylinder, a stationary screw with a traveling nut,
a rectilinear drive using a reversible motor or reversing gears, or a cam-and-follower








C1 C2








2 3








Figure 1.15 Reciprocating linkages.

mechanism. A variety of typical linkages for obtaining reciprocating motion are shown
in Figs. 1.15 and 1.16 [5].

The offset slider-crank linkage shown in Fig. 1.15a has kinematic characteristics that
differ from the in-line (or on-center) slider-crank, shown in Fig. 1.3b. If the length of
connecting rod 3 is long compared to the length of crank 2, then the resulting motion is
nearly harmonic. Exact harmonic motion can be obtained from link 4 of the Scotch-yoke
linkage shown in Fig. 1.15b.

The six-bar linkage shown in Fig. 1.15c is often called the shaper linkage, after the
name of the machine tool in which it is used. Note that it is obtained from Fig. 1.14b by
adding coupler 5 and slider 6. The stroke of the slider has a quick-return characteristic.

Figure 1.15d shows another version of the shaper linkage, which is termed the
Whitworth quick-return linkage. The linkage is presented in an upside-down posture to
illustrate its similarity to Fig. 1.15c.

Another example of a six-bar linkage is the Wanzer needle-bar linkage [5] shown in
Fig. 1.16.

Figure 1.17a shows a six-bar linkage derived from the crank-rocker linkage of
Fig. 1.14c by expanding coupler 3 and adding coupler 5 and slider 6. Coupler point C
should be located to produce the desired motion characteristic for slider 6.

A crank-driven toggle linkage is shown in Fig. 1.17b. With this linkage, a high
mechanical advantage is obtained at one end of the stroke of slider 6. (For a detailed
discussion of the mechanical advantage of a mechanism, see Secs. 1.10 and 3.20).






Figure 1.16 Wanzer needle-bar linkage.
(Richard Mott Wanzer, 1812–1900).












Figure 1.17 Additional six-bar reciprocating linkages.

In many applications, mechanisms are used to perform repetitive operations, such as
pushing parts along an assembly line, clamping parts together while they are welded, or
folding cardboard boxes in an automated packaging machine. In such applications it is
often desirable to use a constant-speed motor; this leads us to a discussion of Grashof’s
law in Sec. 1.9. In addition, however, we should give some consideration to the power and
timing requirements.

In such repetitive operations, there is usually a part of the cycle when the mechanism
is under load, called the advance or working stroke, and a part of the cycle, called the return
stroke, when the mechanism is not working but simply returning to repeat the operation.
For example, consider the offset slider-crank linkage shown in Fig. 1.15a. Work may be
required to overcome the load, F, while the piston moves to the right from position C1 to
position C2 but not during its return to position C1, since the load may have been removed.
In such situations, in order to keep the power requirement of the motor to a minimum and
to avoid wasting valuable time, it is desirable to design a mechanism so that the piston
moves much faster through the return stroke than it does during the advance (or working)
stroke—that is, to use a higher portion of the cycle time for doing work than for returning.


A measure of the suitability of a mechanism from this viewpoint, called the
advance-to-return ratio is defined as

Q= cycle fraction for advance stroke
cycle fraction for return stroke

. (a)

A mechanism for which the value of Q is high is more desirable for such repetitive
operations than one in which the value of Q is lower. Certainly, any such operation would
call for a mechanism for which Q is greater than unity. Because of this, mechanisms with
Q greater than unity are called quick-return mechanisms.

As shown in Fig. 1.15a, the first step is to determine the two crank postures, AB1
and AB2, that mark the beginning and the end of the working stroke. Next, noting the
direction of rotation of the crank, we determine the crank angle α traveled through during
the advance stroke and the remaining crank angle β of the return stroke. Then,

cycle fraction for advance stroke= α

, (b)


cycle fraction for return stroke= β

. (c)

Finally, substituting Eqs. (b) and (c) into Eq. (a), the advance-to-return ratio can be
written as

Q= α
. (1.5)

Note that the advance-to-return ratio depends only on geometry (that is, on changes in
the crank position); this ratio does not depend on the amount of work being done or on the
speed of the driving motor. It is a kinematic property of the mechanism itself. Therefore,
this ratio can be used for either design or analysis totally by graphic constructions. The
following two examples illustrate applications in design.


The rocker of a crank-rocker four-bar linkage is required to have a length of 4 in and swing
through a total angle of 45◦. Also, the advance-to-return ratio of the linkage is required to
be 2.0. Determine a suitable set of link lengths for the remaining three links.


Equation (1.5) requires

Q= α

= 2.0, (1)


α = 180◦ +φ (2)

β = 180◦ −φ. (3)


Substituting Eqs. (2) and (3) into Eq. (1) allows us to solve for

φ = 60◦, α = 240◦, and β = 120◦.
Now, referring to Fig. 1.18, we apply the following graphic procedure:

(a) Draw the rocker (r4 = 4.0 in) to a suitable scale in its two extreme postures; that
is, show the swing angle of the rocker of 45◦. Label the ground pivot O4, and
label pin B in the two positions B1 and B2.

(b) Through point B1, draw an arbitrary line (labeled the X-line). Through B2, draw
a line parallel to the X-line.

(c) Measure the angle φ = 60◦ counterclockwise from the X-line through point B1.
The intersection of this line with the line parallel to the X-line is the required
position of the input crank pivot O2.

(d) The length O2O4 of the ground link can be measured from the drawing—that is

r1 = 1.50 in. Ans.
(e) The lengths of crank r2 and the coupler r3 can be determined from the


O2B1 = r3 + r2 = 3.50 in and O2B2 = r3 − r2 = 2.50 in.
That is,

r2 = 0.5(O2B1 −O2B2)= 0.50 in and r3 = 0.5(O2B2 +O2B1)= 3.00 in.

The solution for the synthesized four-bar linkage is shown in Fig. 1.18.


B1 B2


f = 60º









Figure 1.18 Synthesized
four-bar linkage.


In the two limiting positions, B1 and B2, the output is momentarily stopped, and, for
this reason, these two postures of the linkage are referred to as dead-center postures (see
Prob. 1.35). Also, note that this problem is an example of two-posture synthesis. For a
detailed discussion on the general problems of two-, three-, and four-posture synthesis, see
Chap. 9.


Determine a suitable set of link lengths for a slider-crank linkage such that the stroke is
2.50 in and the advance-to-return ratio is 1.4.


Equation (1.5) requires

Q= α

= 1.40, (1)


α = 180◦ +φ (2)

β = 180◦ −φ. (3)
Substituting Eqs. (2) and (3) into Eq. (1) allows us to solve for

φ = 30◦, α = 210◦, and β = 150◦.
Now, referring to Fig. 1.19, we apply the following graphic procedure:

(a) Draw the stroke (shown horizontal) of 2.50 in of the slider-crank linkage to a
suitable scale. Label pin B in its two extreme positions B1 and B2 .

(b) Through point B2, draw an arbitrary line (labeled the X-line). Through point B1,
draw a line parallel to the X-line.

(c) Measure the angle φ = 30◦ clockwise from the X-line. The intersection of this
line with the line parallel to the X-line is the ground pivot O2.

(d) The length of the ground link—that is, the offset or eccentricity (the perpendicular
distance from the ground pivot O2 to the line of travel of the slider)—can be
measured from the drawing. That is,

r1 = 2.17 in. Ans.
(e) The lengths of crank r2 and coupler r3 can be determined from the measurements

O2B1 = r3 + r2 = 4.33 in and O2B2 = r3 − r2 = 2.50 in.
That is,

r2 = 0.5(O2B1 −O2B2)= 0.92 in and r3 = 0.5(O2B2 +O2B1)= 3.42 in.




r1 r3


B2 B1




f = 30�




Figure 1.19 Synthesized slider-crank linkage.

The solution of the synthesized slider-crank linkage is shown in Fig. 1.19.

Note that there is a proper and an improper direction of rotation for the input of such
a device. If the direction of crank rotation were reversed in the example of Fig. 1.19, the
roles of α and β would also be reversed, and the advance-to-return ratio would be less than
1.0. Thus, the motor must rotate clockwise for this mechanism to have the quick-return

Other mechanisms with quick-return characteristics are the shaper linkage, shown in
Fig. 1.15c, and theWhitworth linkage, shown in Fig. 1.15d. The synthesis of a quick-return
mechanism, as well as mechanisms with other properties, is discused in detail in Chap. 9.

Reversing Mechanisms When a mechanism capable of delivering output rotation in
either direction is desired, some form of reversing mechanism is required. Many such
devices make use of a two-way clutch that connects the output shaft to either of two
driveshafts turning in opposite directions. This method is used in both gear and belt drives
and does not require that the drive be stopped to change direction. Gear-shift devices, as in
automotive transmissions, are also in common use.

Couplings and Connectors Couplings and connectors are used to transmit motion
between coaxial, parallel, intersecting, or skewed shafts. Gears of one kind or another can
be used for any of these situations. These are discussed in Chaps. 7 and 8.

Flat belts can be used to transmit motion between parallel shafts. They can also be used
between intersecting or skewed shafts if guide pulleys are used, as shown in Fig. 1.20a.
When parallel shafts are involved, the belts can be open or crossed, depending on the
direction of rotation desired.

Figure 1.20b shows a drag-link (also referred to as a double-crank) four-bar linkage
used to transmit rotary motion between parallel shafts. Here crank 2 is the driver and crank


(a) Pulley



Shaft 1

Shaft 2






Figure 1.20 Two-shaft coupling mechanisms.

4 is the output. This is a very interesting linkage; you should try to construct one using
cardboard strips and thumbtacks for joints to observe its motion. Can you devise a working
model that allows complete rotation of both links 2 and 4 (see Prob. 1.14)?

The Reuleaux coupling, shown in Fig. 1.21a, for intersecting shafts is recommended
only for light loads. The Hooke joint, shown in Fig. 1.21b, is also used for intersecting
shafts. However, this joint can withstand heavy loads and is commonly used with a
driveshaft in rear-wheel-drive vehicles. It is customary to use two of these joints in series
for connecting parallel shafts.

Sliding Connectors Sliding connectors are used when one slider (the input) is to drive
another slider (the output). The usual problem is that the two sliders operate in the same
plane but in different directions. The possible solutions are:









Figure 1.21 Coupling mechanisms for intersecting shafts.


1. A rigid link pivoted at each end to a slider.
2. A belt or chain connecting the two sliders with the use of a guide pulley or

3. Rack gear teeth cut on each slider and the connection completed using one or

more gears.
4. A flexible cable connector.

Stop, Pause, and Hesitation Mechanisms In an automotive engine a valve must
open, remain open for a short period of time, and then close. A conveyor line may need
to halt for an interval of time while an operation is being performed and then continue
its motion. Many similar requirements occur in the design of machines. Torfason [9]
classifies these as stop-and-dwell, stop-and-return, stop-and-advance, and so on. Such
requirements can often be met using cam-and-follower mechanisms (Chap. 6), indexing
mechanisms, including those of Fig. 1.13, ratchets, linkages at the limits of their motion,
and gear-and-clutch mechanisms.

The six-bar linkage of Fig. 1.22 is a clever method to obtain a rocking motion (of
link 6) containing a dwell. This linkage, an extension of the four-bar linkage, consisting of
frame 1, crank 2, coupler 3, and rocker 4, can be designed such that point C on the coupler
generates the curve shown by dashed lines. A portion of this curve will then fit closely to a
circular arc whose radius is equal to the length of link 5—that is, distance DC. Thus, when
point C traverses this portion of the coupler curve, link 6, the output rocker, is stationary.

Curve Generators The connecting rod, or coupler, of a planar four-bar linkage may be
imagined as an infinite plane extending in all directions but pin-connected to the input and
output cranks. Then, during motion of the linkage, any point attached to the plane of the
coupler generates a path with respect to the fixed link; this path is called a coupler curve.
Two of these paths, namely those generated by the pin connectors of the coupler, are true
circles with centers at the two fixed pivots. However, other points can be found that trace
much more complex curves.












Figure 1.22 Six-bar
stop-and-dwell linkage.


Figure 1.23 A set of coupler curves [6].

One of the best sources of coupler curves for the four-bar linkage is the Hrones and
Nelson atlas [6]. This book consists of a set of 11 in × 17 in drawings containing over
7,000 coupler curves of crank-rocker linkages. Figure 1.23 is a reproduction of a typical
page of this atlas (by permission of the publishers). In each case, the crank has unit length,
and the lengths of the remaining links vary from page to page to produce the different
combinations. On each page a number of different coupler points are chosen, and their
coupler curves are shown. This atlas of coupler curves has proven to be invaluable to the
designer who needs a linkage to generate a curve with specified characteristics.

The algebraic equation of a four-bar linkage coupler curve is, in general, a sixth-order
polynomial [1]; thus, it is possible to find coupler curves with a wide variety of shapes and
many interesting features. Some coupler curves have sections that are nearly straight line
segments; others have almost exact circular arc segments; still others have one or more
cusps or cross over themselves like a figure eight. Therefore, it is often not necessary to
use a mechanism with a large number of links to obtain a complex motion of a coupler

Yet the complexity of the coupler-curve equation is also a hinderance; it means that
hand-calculation methods can become very cumbersome. Thus, over the years, many
mechanisms have been designed by strictly intuitive procedures and proven with cardboard
models, without the use of kinematic principles or procedures. Until quite recently,


those techniques that did offer a rational approach have been graphic, avoiding tedious
computations. Finally, with the availability of digital computers, and particularaly with
computer graphics, useful design methods are now emerging that can deal directly with
the complex calculations required without burdening the designer with the computational
drudgery (Sec. 10.9 has details on some of these).

One of the more curious and interesting facts about the coupler-curve equation is that
the same curve can always be generated by three different four-bar linkages. These are
called cognate linkages, and the theory is developed in Sec. 9.10.

Straight-Line Generators In the late 17th century, before the development of the
milling machine, it was extremely difficult to machine straight, flat surfaces. For this
reason, good prismatic joints with close clearances were not available. During that era,
much thought was given to the problem of attaining a straight line as a part of the coupler
curve of a linkage having only revolute connections. Probably the best-known result of this
search is the straight-line mechanism developed by Watt for guiding the piston of early
steam engines. Figure 1.24a shows a four-bar linkage, known as Watt’s linkage, which
generates an approximate straight line as a part of its coupler curve. Although the coupler
point (tracing point P) does not generate an exact straight line, a good approximation is
achieved over a considerable distance of travel.

Another four-bar linkage in which the tracing point P generates an approximate
straight-line coupler-curve segment is Roberts’ linkage (Fig. 1.24b). The dashed lines
BP and CP in the figure indicate that the linkage is defined by forming three congruent
isosceles triangles; thus, BC= AP= PD= AD/2.













2 4









(d )











Figure 1.24 (a) Watt’s linkage; (b) Roberts’ linkage; (c) Chebychev linkage; (d) Peaucillier inversor.


3 2
4 A



Figure 1.25 AB=AP=O2A.



2 5

O2 O3





Figure 1.26 Pantograph linkage.

The tracing point P (the midpoint of coupler link BC) of the Chebychev linkage shown
in Fig. 1.24c also generates an approximate straight line. The linkage forms a 3:4:5 triangle
when link 4 is in the vertical posture, as shown by the dashed lines; thus, DB′ = 3 units,
AD= 4 units, and AB′ = 5 units. Note that AB= DC,DC′ = 5 units, and tracing point P′
is the midpoint of dashed line B′C′. Also, note that DP′C forms another 3:4:5 triangle, and
hence the line containing P and P′ is parallel to the ground link AD.

A linkage that generates an exact straight line is the Peaucillier inversor shown in
Fig. 1.24d. The conditions describing its geometry are that BC = BP = EC = EP and
AB= AE such that, by symmetry, points A, C, and P always lie on a straight line passing
through A. Under these conditions AC ·AP= k, a constant, and the curves generated by C
and P are said to be inverses of each other. If we place the other fixed pivot D such that
AD = CD, then point C must trace a circular arc while point P follows an exact straight
line. Another interesting property is that if AD is not equal to CD, then point P traces a
true circular arc of very large radius. This was the first straight-line generator, and it was
important in the development of the steam engine.

Figure 1.25 shows another linkage that generates exact straight-line motion: the
Scott-Russell linkage. However, note that it employs a slider.

The pantograph shown in Fig. 1.26 is used to trace figures at a larger or smaller size. If,
for example, point P traces a map, then a pen atQwill draw a similar map at a smaller scale.
The dimensions O2A, AC, CB, and BO3 must conform to an equal-sided parallelogram.

Torfason [9] also include