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THEORY OF MACHINES AND MECHANISMS Third Edition

John J. Dicker, Jr. Professor of Mechanical Engineering University of Wisconsin-Madison

Gordon R. Pennock Associate Professor of Mechanical Engineering Purdue University

Joseph E. Shigley Late Professor Emeritus of Mechanical Engineering The University of Michigan

New York

Oxford

OXFORD UNIVERSITY PRESS 2003

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Oxford University Press Oxford New York Auckland Bangkok Buenos Aires Cape Town Chennai Dar es Salaam Delhi Hong Kong Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Sao Paulo Shanghai Taipei Tokyo Toronto

Copyright © 2003 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York, 10016 www.mechassis.com Oxford is a registered trademark of Oxford University Press 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, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press.

ISBN 0-1 9-5 I 5598-X

Printing number:

9 8 7 6 5 4 3 2 I

Printed in the United States of America on acid-free paper

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This textbook is dedicated to the memory of the third author, the late Joseph E. Shigley, Professor Emeritus, Mechanical Engineering Department, University of Michigan, Ann Arbor, on whose previous writings much of this edition is based.

This work is also dedicated to the memory of my father, John J. Uicker, Emeritus Dean of Engineering, University of Detroit; to my mother, Elizabeth F. Uicker; and to my six children, Theresa A. Uicker, John J. Uicker Ill, Joseph M. Uicker, Dorothy J. Winger, Barbara A. Peterson, and Joan E. Uicker.

-John J. Vicker, Jr. This work is also dedicated first and foremost to my wife, Mollie B., and my son, Callum R. Pennock. The work is also dedicated to my friend and mentor Dr. An (Andy) Tzu Yang and my colleagues in the School of Mechanical Engineering, Purdue University, West Lafayette, Indiana.

-Gordon R. Pennock

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Contents

PREFACE

XIII

ABOUT THE AUTHORS

XVII

Part 1 KINEMATICS AND MECHANISMS 1 The World of Mechanisms 1.1

Introduction

1.2

Analysis and Synthesis

1 3

3 4

1.3

The Science of Mechanics

1.4

Terminology, Definitions, and Assumptions

5

1.5

Planar, Spherical, and Spatial Mechanisms

10

1.6

Mobility

1.7

Classification of Mechanisms

1.8

Kinematic Inversion

1.9

Grashof's Law

II 14

26

27

1.10 Mechanical Advantage Problems

4

29

31

2 Position and Displacement

33

2.1

Locus of a Moving Point

33

2.2

Position of a Point

2.3

Position Difference Between Two Points

2.4

Apparent Position of a Point

38

2.5

Absolute Position of a Point

39

2.6

The Loop-Closure Equation

2.7

Graphic Position Analysis

2.8

Algebraic Position Analysis

2.9

Complex-Algebra

36 37

41 45 51

Solutions of Planar Vector Equations

2.10 Complex Polar Algebra

57

2.11 Position Analysis Techniques

60

2.12 The Chace Solutions to Planar Vector Equations 2.13 Coupler-Curve Generation

2.14 Displacement of a Moving Point

70

2.15 Displacement Difference Between Two Points

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64

68 71

55

vi

CONTENTS

2.16 Rotation and Translation

72

2.17 Apparent Displacement

74

2.18 Absolute Displacement

75

Problems

3 Velocity

76

79

3.1

Definition of Velocity

3.2

Rotation of a Rigid Body

3.3

Velocity Difference Between Points of a Rigid Body

3.4

Graphic Methods; Velocity Polygons

3.5

Apparent Velocity of a Point in a Moving Coordinate System

3.6

Apparent Angular Velocity

3.7

Direct Contact and Rolling Contact

3.8

Systematic Strategy for Velocity Analysis

3.9

Analytic Methods

3.10 Complex-Algebra

79 80 82

85 92

97 98 99

100 Methods

101

3.11 The Method of Kinematic Coefficients 3.12 The Vector Method

105

116

3.13 Instantaneous Center of Velocity 3.14 The Aronhold-Kennedy

117

Theorem of Three Centers

3.15 Locating Instant Centers of Velocity

120

3.16 Velocity Analysis Using Instant Centers 3.17 The Angular-Velocity-Ratio

119

Theorem

123 126

3.18 Relationships Between First-Order Kinematic Coefficients and Instant Centers 3.19 Freudenstein' s Theorem

129

3.20 Indices of Merit; Mechanical Advantage 3.21 Centrodes Problems

130

133

135

4 Acceleration

141

4.1

Definition of Acceleration

4.2

Angular Acceleration

4.3

Acceleration Difference Between Points of a Rigid Body

4.4

Acceleration Polygons

4.5

Apparent Acceleration of a Point in a Moving Coordinate System

4.6

Apparent Angular Acceleration

4.7

Direct Contact and Rolling Contact

4.8

Systematic Strategy for Acceleration Analysis

4.9

Analytic Methods

4.10 Complex-Algebra

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141

144 144

151 163

168 Methods

169

164 167

155

127

CONTENTS 4.11 The Method of Kinematic Coefficients 4.12 The Chace Solutions

175

4.13 The Instant Center of Acceleration 4.14 The Euler-Savary

171

177

Equation

178

4.15 The Bobillier Constructions

183

4.16 Radius of Curvature of a Point Trajectory Using Kinematic Coefficients 4.17 The Cubic of Stationary Curvature Problems

188

190

Part 2 DESIGN OF MECHANISMS 5 Carn Design

195

197

5.1

Introduction

197

5.2

Classification of Cams and Followers

5.3

Displacement Diagrams

5.4

Graphical Layout of Cam Profiles

5.5

Kinematic Coefficients of the Follower Motion

5.6

High-Speed Cams

5.7

Standard Cam Motions

5.8

Matching Derivatives of the Displacement Diagrams

5.9

Plate Cam with Reciprocating Flat-Face Follower

198

200 203

211 212

5.10 Plate Cam with Reciprocating Roller Follower Problems

250

6 Spur Gears

252

6.1

Terminology and Definitions

6.2

Fundamental Law of Toothed Gearing

6.3

Involute Properties

252 255

256

6.4

Interchangeable Gears; AGMA Standards

6.5

Fundamentals of Gear-Tooth Action

6.6

The Manufacture of Gear Teeth

6.7

Interference and Undercutting

6.8

Contact Ratio

6.9

Varying the Center Distance

6.10 Involutometry

262 265

268 270

271

6.11 Nonstandard Gear Teeth Problems

274

282

7 Helical Gears

286

7.1

Parallel-Axis Helical Gears

7.2

Helical Gear Tooth Relations

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207

286 287

259

257

222 225

230

187

vii

viii

CONTENTS

7.3

Helical Gear Tooth Proportions

7.4

Contact of Helical Gear Teeth

7.5

Replacing Spur Gears with Helical Gears

7.6

Herringbone Gears

7.7

Crossed-Axis Helical Gears

Problems

289 290

292 292

295

8 Bevel Gears

297

8.1

Straight-Tooth Bevel Gears

8.2

Tooth Proportions for Bevel Gears

8.3

Crown and Face Gears

8.4

Spiral Bevel Gears

8.5

Hypoid Gears

Problems

Basics

Problems

297 301

302

303

304

305

9 Worms and Worm Gears 9.1

291

306

306 310

10 Mechanism Trains 311 10.1 Parallel-Axis Gear Trains

311

10.2 Examples of Gear Trains

313

10.3 Determining Tooth Numbers 10.4 Epicyclic Gear Trains

314

315

10.5 Bevel Gear Epicyclic Trains

317

10.6 Analysis of Planetary Gear Trains by Formula 10.7 Tabular Analysis of Planetary Gear Trains 10.8 Adders and Differentials

319

323

10.9 All Wheel Drive Train Problems

317

327

329

11 Synthesisof Linkages 332 11.1 Type, Number, and Dimensional Synthesis

332

11.2 Function Generation, Path Generation, and Body Guidance 11.3 Two-Position Synthesis of Slider-Crank Mechanisms 11.4 Two-Position Synthesis of Crank-and-Rocker

333

333

Mechanisms

334

11.5 Crank-Rocker Mechanisms with Optimum Transmission Angle 11.6 Three-Position Synthesis

338

11.7 Four-Position Synthesis; Point-Precision Reduction

339

. 11.8 Precision Positions; Structural Error; Chebychev Spacing 11.9 The Overlay Method

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343

341

335

CONTENTS 11.10 Coupler-Curve Synthesis

344

11.11 Cognate Linkages; The Roberts-Chebychev 11.l2 Bloch's Method of Synthesis 11.I3 Freudenstein's

Equation

350

11.15 Synthesis of Dwell Mechanisms II.I 6 Intermittent Rotary Motion

356

360

361

366

12 Spatial Mechanisms

368

12.1

Introduction

12.2

Exceptions in the Mobility of Mechanisms

12.3

The Position-Analysis

12.4

Velocity and Acceleration Analyses

12.5

The Eulerian Angles

12.6

The Denavit-Hartenberg

12.7

Transformation-Matrix

12.8

Matrix Velocity and Acceleration Analyses

12.9

Generalized Mechanism Analysis Computer Programs

Problems

368 Problem

369

373 378

384 Parameters

387

Position Analysis

389 392

400

13 Robotics 403 13.1

Introduction

13.2

Topological Arrangements of Robotic Arms

13.3

Forward Kinematics

13.4

Inverse Position Analysis

13.5

Inverse Velocity and Acceleration Analyses

13.6

Robot Actuator Force Analyses

Problems

403

411 418

421

423

14 Static;:Force Analysis 425 14.1

Introduction

14.2

Newton's Laws

404

407

Part 3 DYNAMICS OF MACHINES

425 427

14.3

Systems of Units

14.4

Applied and Constraint Forces

428

14.5

Free-Body Diagrams

14.6

Conditions for Equilibrium

14.7

Two- and Three-Force Members

14.8

Four-Force Members

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348

352

11.I4 Analytic Synthesis Using Complex Algebra

Problems

Theorem

429

432

443

433 435

414

397

x

CONTENTS

14.9

Friction-Force Models

445

14.10 Static Force Analysis with Friction

448

14.11 Spur- and Helical-Gear Force Analysis

451

14.12 Straight- Bevel-Gear Force Analysis 14.13 The Method of Virtual Work Problems

457

461

464

15 Dynamic ForceAnalysis (Planar)

470

15.1

Introduction

15.2

Centroid and Center of Mass

470

15.3

Mass Moments and Products of Inertia

15.4

Inertia Forces and D' Alembert's Principle

15.5

The Principle of Superposition

15.6

Planar Rotation About a Fixed Center

15.7

Shaking Forces and Moments

15.8

Complex Algebra Approach

15.9

Equation of Motion

Problems

470 475

485 489

492 492

502

511

16 Dynamic ForceAnalysis (Spatial)

515

16.1

Introduction

16.2

Measuring Mass Moment of Inertia

16.3

Transformation of Inertia Axes

515 515

519

16.4

Euler's Equations of Motion

16.5

Impulse and Momentum

16.6

Angular Impulse and Angular Momentum

Problems

478

523

527 528

538

17 Vibration Analysis 542 17.1

Differential Equations of Motion

17.2

A Vertical Model

542

17.3

Solution of the Differential Equation

17.4

Step Input Forcing

17.5

Phase-Plane Representation

17.6

Phase-Plane Analysis

17.7

Transient Disturbances

17.8

Free Vibration with Viscous Damping

17.9

Damping Obtained by Experiment

546 547

551 553

555 559 563

565

17.10 Phase-Plane Representation of Damped Vibration 17.11 Response to Periodic Forcing 17.12 Harmonic Forcing

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574

571

567

CONTENTS 17.13 Forcing Caused by Unbalance 17.14 Relative Motion 17.15 Isolation

579

580

580

17.16 Rayleigh's Method

583

17.17 First and Second Critical Speeds of a Shaft 17.18 Torsional Systems Problems

586

592

594

18 Dynamics of Reciprocating Engines 598 18.1

Engine Types

18.2

Indicator Diagrams

598

18.3

Dynamic Analysis-General

18.4

Gas Forces

18.5

Equivalent Masses

18.6

Inertia Forces

18.7

Bearing Loads in a Single-Cylinder Engine

18.8

Crankshaft Torque

18.9

Engine Shaking Forces

603

609

610 613

616

18.10 Computation Hints Problems

606

606

616

617

620

19 Balancing 621 19.1

Static Unbalance

621

19.2

Equations of Motion

19.3

Static Balancing Machines

622 624

19.4

Dynamic Unbalance

19.5

Analysis of Unbalance

626

19.6

Dynamic Balancing

635

19.7

Balancing Machines

638

19.8

Field Balancing with a Programmable Calculator

19.9

Balancing a Single-Cylinder Engine

627

19.10 Balancing Multicylinder Engines

640

643 647

19.11 Analytical Technique for Balancing Multicylinder Reciprocating Engines 19.12 Balancing Linkages

656

19.13 Balancing of Machines Problems

661

663

20 Cam Dynamics

665

20.1

Rigid- and Elastic-Body Cam Systems

20.2

Analysis of an Eccentric Cam

20.3

Effect of Sliding Friction

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670

666

665

651

xi

xii

CONTENTS 20.4 20.5

Analysis of Disk Cam with Reciprocating Roller Follower Analysis of Elastic Cam Systems 673

20.6 Unbalance, Spring Surge, and Windup Problems 676

21 Flywheels

678

21.1

Dynamic Theory

21.2

Integration Technique

678 680

21.3 Multicylinder Engine Torque Summation Problems 683

22 Governors

675

682

685

22.1

Classification

685

22.2

Centrifugal Governors

22.3

Inertia Governors

686

687

22.4

Mechanical Control Systems

22.5

Standard Input Functions

687

22.6

Solution of Linear Differential Equations

22.7

Analysis of Proportional-Error

689 690

Feedback Systems

695

23 Gyroscopes 699 23.1

Introduction

699

23.2

The Motion of a Gyroscope

23.3

Steady or Regular Precession

23.4 Forced Precession Problems 711

700 701

704

APPENDIXES ApPENDIX

A: TABLES

Table 1 Standard SI Prefixes

712

Table 2 Conversion from U.S. Customary Units to SI Units

713

Table 3 Conversion from SI Units to U.S. Customary Units Table 4 Properties of Areas 714

713

Table 5 Mass Moments ofInertia Table 6 Involute Function ApPENDIX

INDEX

B: ANSWERS 725

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715

716

TO SELECTED

PROBLEMS

718

671

Preface

This book is intended to cover that field of engineering theory, analysis, design, and practice that is generally described as mechanisms and kinematics and dynamics of machines. While this text is written primarily for students of engineering, there is much material that can be of value to practicing engineers. After all, a good engineer knows that he or she must remain a student throughout their entire professional career. The continued tremendous growth of knowledge, including the areas of kinematics and dynamics of machinery, over the past 50 years has resulted in great pressure on the engineering curricula of many schools for the substitution of "modern" subjects for those perceived as weaker or outdated. At some schools, depending on the faculty, this has meant that kinematics and dynamics of machines could only be made available as an elective topic for specialized study by a small number of students; at others it remained a required subject for all mechanical engineering students. At other schools, it was required to take on more design emphasis at the expense of depth in analysis. In all, the times have produced a need for a textbook that satisfies the requirements of new and changing course structures. Much of the new knowledge developed over this period exists in a large variety of technical papers, each couched in its own singular language and nomenclature and each requiring additional background for its comprehension. The individual contributions being published might be used to strengthen the engineering courses if first the necessary foundation were provided and a common notation and nomenclature were established. These new developments could then be integrated into existing courses so as to provide a logical, modern, and comprehensive whole. To provide the background that will allow such an integration is the purpose of this book. To develop a broad and basic comprehension, all the methods of analysis and development common to the literature of the field are employed. We have used graphical methods of analysis and synthesis extensively throughout the book because the authors are firmly of the opinion that graphical computation provides visual feedback that enhances the student's understanding of the basic nature of and interplay between the equations involved. Therefore, in this book, graphic methods are presented as one possible solution technique for vector equations defined by the fundamental laws of mechanics, rather than as mysterious graphical "tricks" to be learned by rote and applied blindly. In addition, although graphic techniques may be lacking in accuracy, they can be performed quickly and, even though inaccurate, sketches can often provide reasonable estimates of a solution or can be used to check the results of analytic or numeric solution techniques. We also use conventional methods of vector analysis throughout the book, both in deriving and presenting the governing equations and in their solution. Raven's methods using complex algebra for the solution of two-dimensional vector equations are xiii

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xiv

PREFACE

presented throughout the book because of their compactness, because they are employed so frequently in the literature, and also because they are so easy to program for computer evaluation. In the chapters dealing with three-dimensional kinematics and robotics, we briefly present an introduction to Denavit and Hartenberg's methods using transformation matrices. With certain exceptions, we have endeavored to use U.S. Customary units and SI units in about equal proportions throughout the book. One of the dilemmas that all writers on the subject of this book have faced is how to distinguish between the motions of two different points of the same moving body and the motions of coincident points of two different moving bodies. In other texts it has been customary to describe both of these as "relative motion"; but because they are two distinct situations and are described by different equations, this causes the student difficulty in distinguishing between them. We believe that we have greatly relieved this problem by the introduction of the terms motion difference and apparent motion and two different notations for the two cases. Thus, for example, the book uses the two terms, velocity difference and apparent velocity, instead of the term "relative velocity," which will not be found when speaking rigorously. This approach is introduced beginning with the concepts of position and displacement, used extensively in the chapter on velocity, and brought to fulfillment in the chapter on accelerations where the Coriolis component always arises in, and only in, the apparent acceleration equation. Another feature, new with the third edition, is the presentation of kinematic coefficients, which are derivatives of various motion variables with respect to the input motion rather than with respect to tirr.e. The authors believe that these provide several new and important advantages, among which are the following: (1) They clarify for the student those parts of a motion problem which are kinematic (geometric) in their nature, and they clearly separate them from those that are dynamic or speed-dependent. (2) They help to integrate different types of mechanical systems and their analysis, such as gears, cams, and linkages, which might not otherwise seem similar. Access to personal computers and programmable calculators is now commonplace and is of considerable importance to the material of this book. Yet engineering educators have told us very forcibly that they do not want computer programs included in the text. They prefer to write their own programs and they expect their students to do so too. Having programmed almost all the material in the book many times, we also understand that the book should not become obsolete with changes in computers or programming languages. Part 1 of this book is an introduction that deals mostly with theory, with nomenclature, with notation, and with methods of analysis. Serving as an introduction, Chapter 1 also tells what a mechanism is, what a mechanism can do, how mechanisms can be classified, and some of their limitations. Chapters 2, 3, and 4 are concerned totally with analysis, specifically with kinematic analysis, because they cover position, velocity, and acceleration analyses, respectively. Part 2 of the book goes on to show engineering applications involving the selection, the specification, the design, and the sizing of mechanisms to accomplish specific motion objectives. This part includes chapters on cam systems, gears, gear trains, synthesis of linkages, spatial mechanisms, and robotics. Part 3 then adds the dynamics of machines. In a sense this is concerned with the consequences of the proposed mechanism design specifications. In other words, having

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PREFACE

xv

designed a machine by selecting, specifying, and sizing the various components, what happens during the operation of the machine? What forces are produced? Are there any unexpected operating results? Will the proposed design be satisfactory in all respects? In addition, new dynamic devices are presented whose functions cannot be explained o~ understood without dynamic analysis. The third edition includes complete new chapters on the analysis and design of flywheels, governors, and gyroscopes. As with all topics and all texts, the subject matter of this book also has limits. Probably the clearest boundary on the coverage in this text is that it is limited to the study of rigid-body mechanical systems. It does study multibody systems with connections or constraints between them. However, all elastic effects are assumed to come within the connections; the shapes of the individual bodies are assumed constant. This assumption is necessary to allow the separate study of kinematic effects from those of dynamics. Because each individual body is assumed rigid, it can have no strain; therefore the study of stress is also outside of the scope of this text. It is hoped, however, that courses using this text can provide background for the later study of stress, strength, fatigue life, modes of failure, lubrication, and other aspects important to the proper design of mechanical systems. John J. Uicker, Jr. Gordon R. Pennock

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About the Authors

John J. Vicker, Jr. is Professor of Mechanical Engineering at the University of Wisconsin-Madison. His teaching and research specialties are in solid geometric modeling and the modeling of mechanical motion and their application to computer-aided design and manufacture; these include the kinematics, dynamics, and simulation of articulated rigid-body mechanical systems. He was the founder of the Computer-Aided Engineering Center and served as its director for its initial 10 years of operation. He received his B.M.E. degree from the University of Detroit and obtained his M.S. and Ph.D. degrees in mechanical engineering from Northwestern University. Since joining the University of Wisconsin faculty in 1967, he has served on several national committees of ASME and SAE, and he is one of the founding members of the US Council for the Theory of Machines and Mechanisms and of IFroMM, the international federation. He served for several years as editor-in-chief of the Mechanism and Machine Theory journal of the federation. He is also a registered Mechanical Engineer in the State of Wisconsin and has served for many years as an active consultant to industry. As an ASEE Resident Fellow he spent 1972-1973 at Ford Motor Company. He was also awarded a Fulbright-Hayes Senior Lectureship and became a Visiting Professor to Cranfield Institute of Technology in England in 1978-1979. He is the pioneering researcher on matrix methods of linkage analysis and was the first to derive the general dynamic equations of motion for rigid-body articulated mechanical systems. He has been awarded twice for outstanding teaching, three times for outstanding research publications, and twice for historically significant publications. Gordon R. Pennock is Associate Professor of Mechanical Engineering at Purdue University, West Lafayette, Indiana. His teaching is primarily in the area of mechanisms and machine design. His research specialties are in theoretical kinematics, and the dynamics of mechanical motion. He has applied his research to robotics, rotary machinery, and biomechanics; including the kinematics, and dynamics of articulated rigid-body mechanical systems. He received his B.Sc. degree (Hons.) from Heriot-Watt University, Edinburgh, Scotland, his M.Eng.Sc. from the University of New South Wales, Sydney, Australia, and his Ph.D. degree in mechanical engineering from the University of California, Davis. Since joining the Purdue University faculty in 1983, he has served on several national committees and international program committees. He is the Student Section Advisor of the American Society of Mechanical Engineers (ASME) at Purdue University, Region VI College Relations Chairman, Senior Representative on the Student Section Committee, and a member of the Board on Student Affairs. He is an Associate of the Internal Combustion Engine Division, ASME, and served as the Technical Committee Chairman of Mechanical Design, Internal Combustion Engine Division, from 1993 to 1997. XVII

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~iii

ABOUT THE AUTHORS

He is a Fellow of the American Society of Mechanical Engineers and a Fellow and a Chartered Engineer with the Institution of Mechanical Engineers (CEng, FIMechE), United Kingdom. He received the ASME Faculty Advisor of the Year Award, 1998, and was named the Outstanding Student Section Advisor, Region VI, 2001. The Central Indiana Section recognized him in 1999 by the establishment of the Gordon R. Pennock Outstanding Student Award to be presented annually to the Senior Student in recognition of academic achievement and outstanding service to the ASME student section at Purdue University. He received the ASME Dedicated Service Award, 2002, for dedicated voluntary service to the society marked by outstanding performance, demonstrated effective leadership, prolonged and committed service, devotion, enthusiasm, and faithfulness. He received the SAE Ra]ph R. Teetor Educational Award, 1986, and the Ferdinand Freudenstein Award at the Fourth National Applied Mechanisms and Robotics Conference, 1995. He has been at the forefront of many new developments in mechanical design, primarily in the areas of kinematics and dynamics. He has pub]ished some 80 technical papers and is a regular symposium speaker, workshop presenter, and conference session organizer and chairman. Joseph E. Shigley (deceased May ]994) was Professor Emeritus of Mechanical Engineering at the University of Michigan, Fellow in the American Society of Mechanica] Engineers, received the Mechanisms Committee Award in 1974, the Worcester Reed Warner medal in ] 977, and the Machine Design Award in 1985. He was author of eight books, including Mechanical Engineering Design (with Charles R. Mischke) and Applied Mechanics of Materials. He was Coeditor-in-Chief of the Standard Handbook of Machine Design. He first wrote Kinematic Analysis of Mechanisms in 1958 and then wrote Dynamic Analysis of Machines in ]961, and these were published in a single volume titled Theory of Machines in 1961; these have evolved over the years to become the current text, Theory of Machines and Mechanisms, now in its third edition. He was awarded the B.S.M.E. and B.S.E.E. degrees of Purdue University and received his M.S. at the University of Michigan. After severa] years in industry, he devoted his career to teaching, writing, and service to his profession starting first at Clemson University and later at the University of Michigan. His textbooks have been widely used throughout the United States and internationally.

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PART 1 Kinematics and Mechanisms

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1

The World of Mechanisms

1.1 INTRODUCTION The theory of machines and mechanisms is an applied science that is used to understand the relationships between the geometry and motions of the parts of a machine or mechanism and the forces that produce these motions. The subject, and therefore this book, divides itself naturally into three parts. Part 1, which includes Chapters 1 through 4, is concerned with mechanisms and the kinematics of mechanisms, which is the analysis of their motions. Part 1 lays the groundwork for Part 2, comprising Chapters 5 through 13, in which we study the methods of designing mechanisms. Finally, in Part 3, which includes Chapters 14 through 23, we take up the study of kinetics, the time-varying forces in machines and the resulting dynamic phenomena that must be considered in their design. The design of a modern machine is often very complex. In the design of a new engine, for example, the automotive engineer must deal with many interrelated questions. What is the relationship between the motion of the piston and the motion of the crankshaft? What will be the sliding velocities and the loads at the lubricated surfaces, and what lubricants are available for the purpose? How much heat will be generated, and how will the engine be cooled? What are the synchronization and control requirements, and how wi\I they be met? What will be the cost to the consumer, both for initial purchase and for continued operation and maintenance? What materials and manufacturing methods will be used? What will be the fuel economy, noise, and exhaust emissions; will they meet legal requirements? Although all these and many other important questions must be answered before the design can be completed, obviously not all can be addressed in a book of this size. Just as people with diverse skills must be brought together to produce an adequate design, so too many branches of science must be brought to bear. This book brings together material that falls into the science of mechanics as it relates to the design of mechanisms and machines. 3

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4

THE WORLD

OF MECHANISMS

1.2 ANALYSIS AND SYNTHESIS There are two completely different aspects of the study of mechanical systems, design and analysis. The concept embodied in the word "design" might be more properly Itermed synthesis, the process of contriving a scheme or a method of accomplishing a given purpose. Design is the process of prescribing the sizes, shapes, material compositions, and arrangements of parts so that the resulting machine will perform the prescribed task. Although there are many phases in the design process which can be approached in a well-ordered, scientific manner, the overall process is by its very nature as much an art as a science. It calls for imagination, intuition, creativity, judgment, and experience. The role of science in the design process is merely to provide tools to be used by the designers as they practice their art. It is in the process of evaluating the various interacting alternatives that designets find need for a large collection of mathematical and scientific tools. These tools, when applied properly, can provide more accurate and more reliable information for use in judging a design than one can achieve through intuition or estimation. Thus they can be of tremendous help in deciding among alternatives. However, scientific tools cannot make decisions for designers; they have every right to exert their imagination and creative abilities, even to the extent of overruling the mathematical predictions. Probably the largest collection of scientific methods at the designer's disposal fall into the category called analysis. These are the techniques that allow the designer to critically examine an already existing or proposed design in order to judge its suitability for the task. Thus analysis, in itself, is not a creative science but one of evaluation and rating of things already conceived. We should always bear in mind that although most of our effort may be spent on analysis, the real goal is synthesis, the design of a machine or system. Analysis is simply a tool. It is, however, a vital tool and will inevitably be used as one step in the design process.

1.3 THE SCIENCE OF MECHANICS That branch of scientific analysis that deals with motions, time, and forces is called mechanics and is made up of two parts, statics and dynamics. Statics deals with the analysis of stationary systems-that is, those in which time is not a factor-and dynamics deals with systems that change with time. As shown in Fig. 1.1, dynamics is also made up of two major disciplines, first recognized as separate entities by Euler in 1775: I The investigation of the motion of a rigid body may be conveniently separated into two parts, the one geometrical, the other mechanical. In the first part, the transference of the body from a given position to any other position must be investigated without respect to the causes of the motion, and must be represented by analytical formulae, which will define the position of each point of the body. This investigation will therefore be referable solely to geometry, or rather to stereotomy. It is clear that by the separation of this part of the question from the other, which belongs properly to Mechanics, the determination of the motion from dynamical principles will be made much easier than if the two parts were undertaken conjointly.

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These two aspects of dynamics were later recognized as the distinct sciences of kinematics (from the Greek word kinema, meaning motion) and kinetics, and they deal with motion and the forces producing it, respectively. The initial problem in the design of a mechanical system is therefore understanding its kinematics. Kinematics is the study of motion, quite apart from the forces which produce that motion. More particularly, kinematics is the study of position, displacement, rotation, speed, velocity, and acceleration. The study, say, of planetary or orbital motion is also a problem in kinematics, but in this book we shall concentrate our attention on kinematic problems that arise in the design of mechanical systems. Thus, the kinematics of machines and mechanisms is the focus of the next several chapters of this book. Statics and kinetics, however, are also vital parts of a complete design analysis, and they are covered as well in later chapters. It should be carefully noted in the above quotation that Euler based his separation of dynamics into kinematics and kinetics on the assumption that they should deal with rigid bodies. It is this very important assumption that allows the two to be treated separately. For flexible bodies, the shapes of the bodies themselves, and therefore their motions, depend on the forces exerted on them. In this situation, the study of force and motion must take place simultaneously, thus significantly increasing the complexity of the analysis. Fortunately, although all real machine parts are flexible to some degree, machines are usually designed from relatively rigid materials, keeping part deflections to a minimum. Therefore, it is common practice to assume that deflections are negligible and parts are rigid when analyzing a machine's kinematic performance, and then, after the dynamic analysis when loads are known, to design the parts so that this assumption is justified.

1.4 TERMINOLOGY,

DEFINITIONS,

AND ASSUMPTIONS

Reuleaux2 defines a machine3 as a "combination of resistant bodies so arranged that by their means the mechanical forces of nature can be compelled to do work accompanied by certain determinate motions." He also defines a mechanism as an "assemblage of resistant bodies. connected by movable joints, to form a closed kinematic chain with one link fixed and having the purpose of transforming motion." Some light can be shed on these definitions by contrasting them with the term structure. A structure is also a combination of resistant (rigid) bodies connected by joints, but its purpose is not to do work or to transform motion. A structure (such as a truss) is intended to be rigid. It can perhaps be moved from place to place and is movable in this sense of the word; however, it has no internal mobility, no relative motions between its various members, whereas both machines and mechanisms do. Indeed, the whole purpose of a machine

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THE WORLD OF MECHANISMS 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 qr changing its direction. 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 three branches of mechanics shown in Fig. 1.1. The term "structure" is to statics as the term "mechanism" is to kinematics as the term "machine" is to kinetics. We shall 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 playa role in supplying forces. Such members are not called links; they are usually ignored during kinematic analysis, and their force effects are introduced during dynamic analysis. Sometimes, as with a belt or chain, a machine member may possess one-way rigidity; such a member would be considered a link when in tension but not under compression. The links of a mechanism must be connected together in some manner in order to transmit motion from the driver, or input link, to the follower, or output link. These connections, joints between the links, are called kinematic pairs (or just pairs), because each joint consists of a pair of mating surfaces, or two elements, with 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 elements of different kinematic pairs. Stated explicitly, the assumption of rigidity is that there can be no relative motion (change in distance) between two arbitrarily chosen points on the same link. In particular, the relative positions of pairing elements on any given link do not change. In other words, the purpose of a link is to hold constant spatial relationships between the elements of its pairs. 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, which contain important features of the shape of each link, such as the relative locations of pair elements, but which completely subdue the real geometry of the manufactured parts. The slider-crank mechanism of the internal combustion engine, for example, can be simplified to the schematic diagram shown later in Fig. 1.3b for purposes of analysis. Such simplified schematics are a great help because 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 always bear in mind that these simplified diagrams are intended to carry only the minimum necessary information so as not to confuse the issue with all the unimportant detail (for kinematic purposes) or complexity of the true machine parts. When several links are movably connected together by joints, they are said to form a kinematic chain. Links containing only two pair element connections are called binary links; those having three are called ternary links, and so on. If every link in the chain is connected to at least two other links, the chain forms one or more closed loops and is called

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a closed kinematic chain; if not, the chain is referred to as open. When no distinction is made, the chain is assumed to be closed. If the chain consists entirely of binary links, it is simple-closed; compound-closed chains, however, include other than binary links