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