Principles ofEngineering Mechanics

Volume 2

Dynamics - The Analysis of Motion

MATHEMATICAL CONCEPTSAND METHODSIN SCIENCE AND ENGINEERING

Series Editor: Angelo MieleMechanical Engineering and Mathematical SciencesRice University

Latest volumesin this series:

PRINCIPLES OF ENGINEERING MECHANICS:VOLUME 2. DYNAMICS-THE ANALYSIS OF MOTION· MillardF. Beatty. ir.

CONSTRAINED OPTIMIZATION AND IMAGE SPACE ANALYSIS:VOLUME 1. SEPARATION OF SETSAND OPTIMALITYCONDITIONS • Franco Giannessi

ADVANCED DESIGN PROBLEMS IN AEROSPACE ENGINEERING:VOLUME 1. ADVANCED AEROSPACE SYSTEMS • Editors AngeloMieleand Aida Frediani

UNIFIED PLASTICITY FORENGINEERING APPLICATIONS· Sol R. Bodner

THEORY ANDAPPLICATIONS OF PARTIAL DIFFERENTIALEQUATIONS • PieroBassanini and Alan R. Elcrat

NONLINEAR EFFECTS IN FLUIDS ANDSOLIDS • Editors MichaelM. Carrolland MichaelA. Hayes

Principles ofEngineering Mechanics

Volume 2

Dynamics - The Analysis of Motion

Millard F. Beatty, Jr.Professor Emeritus, Department ofEngineering Mechanics

University ofNebraska-Lincoln. Lincoln. NE

~ Springer

Millard F. Beatty, Jr., PhDProfessor Emeritus, Department of Engineerin g Mechanic sUniversity of Nebraska-LincolnLincoln, NE 68588-0526E-mail: mbeatty2 @unl.edu

Library of Congress Control Number: 20059350 I0

ISBN-lO: 0-387-23704-6ISBN-13: 9780387237046

Printed on acid-free paper.

© 2006 Springer Science+Business Media, Inc.All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science- Business Media, Inc., 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Usein connection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and simliar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Printed in the United States of America.

9 8 7 6 5 4 3 2 I

springeronline.com

To my wife and best friend

NADINE CHUMLEY BEATTY,

to our children

LAURA,ANNandSCOT~

and to our grandchildren

BOONE, BRITTANY, HUNTER,

TANNER and TREVOR

Preface

This volume, which begins with Chapter 5, is a vector treatment of the principlesof mechanics written primarily for advanced undergraduate and first year graduatestudents specializing in engineering. A substantial part of the material, however,exclusive of obvious advanced topics , has been tailored to provide a first course forsophomore or junior level undergraduates having two years of university mathe­matics through differential equations, which may be taken concurrently. The bookalso serves as a sound, self-study review for advanced students specializing inmechanics, engineering science, engineering physics , and applied mathematics. Itis prerequisite, of course, that the reader be familiar with vector and elementarymatrix methods, the primitive concepts of mechanics, and the basic kinematicalequations developed in Volume I, which comprises the first four chapters of thistwo volume series. In addition, it is recommended that advanced students be fa­miliar with the elements of tensor algebra and finite rigid body rotations treatedthere in Chapter 3.

The arrangement of the subject into the separate parts kinematics and dy­namics has always seemed to me the best didactic approach, and it still does .Unfortunately, the luxury of a slow-paced two semester course on these subjects isseldom afforded . Therefore, in a concentrated, fast-paced single semester course,especially for a diverse group of undergraduates, a level and variety of topics maybe chosen for presentation to address student needs and to satisfy curriculum ob­jectives. But a great deal of the subject matter also can be assigned first to readingand later discussed briefly in lectures . Assigned reading of my discourse on thefoundation principles of classical mechanics, for example, might offer an oppor­tunity for an interesting dialogue among students and the instructor. At the bottomline, however, the classroom usage of this material must be assessed by the teacherin keeping with the class level, the course objectives, the curriculum schedule, andthe desired emphasis.

Naturally, the presentation is influenced by my personal interests and back­ground in mechanical engineering, engineering science, and mechanics. Conse­quently, my approach definitely is somewhat more sophisticated and mathemat­ical than is commonly found in traditional textbooks on engineering mechanics.

vii

viii Preface

In keeping with this approach, the prerequisite mathematics, principally that ofthe seventeenth and eighteenth centuries, is used without apology. Nevertheless,aware that many readers may not have mastered these prerequisite materials, Ihave exercised care to reinforce the essential tools both directly within the text andindirectly in the illustrations, exercises and problems selected for study. Elementsof the theory of linear differential equations of the second order, for example,are presented; hyperbolic functions are introduced and illustrated graphically; theanalytic geometry of conic sections is reviewed; and advanced students will find athoughtful discussion on the subject of elliptic functions and integrals applied inseveral examples. These and other topics are illustrated in many physical applica­tions throughout the book; and several exercises and problems expand on relateddetails. Instances where material may be omitted without loss of continuity areclearly indicated.

This presentation of dynamics focuses steadily on illustrating the predictivevalue of the methods and principles of mechanics in describing and explaining themotion and physical behavior of particles and rigid bodies under various kinds offorces and torques . The emphasis throughout this book is principally analytical,not computational, though many numerical examples and problems are provided toexpose the relevant aspects of an example or to illustrate certain theoretical details.Examples have been selected for their interest and instructive value and to help thestudent achieve understanding of the various concepts, principles, and analyticalmethods. In some instances, experimental results, factual situations, and applica­tions that confirm analytical predictions are described. I hope the historical remarksand several relevant short stories add interest and reality to the subject. Numerousassignment problems, ranging from easy, straightforward extensions or reinforce­ments of the subject matter to more difficult problems that challenge the creativeskills of better students, are provided at the end of each chapter. To assist the stu­dent's study of dynamics, answers to the odd-numbered problems are provided atthe back of the book.

It is axiomatic that physical intuition or insight cannot be taught. On the otherhand , competence in mathematical and physical reasoning may be developed sothat these special human qualities may be intelligently cultivated through study ofphysical applications that mirror the world around us and through practice of therational process of reasoning from first principles. With these attributes in mind,one objective of this volume is to help the engineering student develop confi­dence in transforming problems into appropriate mathematical language that maybe manipulated to derive substantive and useful physical conclusions or specificnumerical results. I intend that this treatment should present a penetrating andinteresting treatise on the elements of classical mechanics and their applicationsto engineering problems; therefore, this text is designed to deepen and broadenthe student's understanding and to develop his or her mastery of the fundamentals .However, to reap a harvest from the seeds sown here, it is important that the studentwork through many of the examples and problems provided for study. When teach­ing this course over the years , I have usually assigned six problems per week, two

Preface ix

per class period , roughly ninety per semester. If the assignment problems chosenrange from easy to difficult, I don't think it is wise to assign more. The mere under­standing that one may apply theoretical concepts and formula s to solve a particularproblem is not equivalent to possession of the knowledge and skills required toproduce its solution. These talents grow only from experience in dealing repeat­edly with these matters . My view of the importance of solving many problems isexpressed further at the beginning of the problem set for Chapter 5, the first chapterof this volume. The attitude emphasi zed there is echoed throughout this text. It ismy hope that these books on kinematics and dynamics may provide engineeringstudents and others with solid mathematical and mechanical foundation s for fu­ture advanced study of topics in mechanical design analysis, advanced kinematic sof mechani sms and analytical dynamics, mechanical vibrations and control s, andareas of continuum mechanic s of solids and fluids.

The Contents of This Volume

Volume 2: Dynamics, concerns the analysi s of motion based on classicalfoundation princip les of mechanic s due to Newton and Euler, later cast in a gener­alized energy based formulation by Lagrange. A fresh development of a classicalsubject like this is seldom seen. I believe, however, that the reader will find withinthese pages many fresh developments, beginning with my discussion of the fun­damental classical princip les of mechanic s and concluding with the presentationof the elegant analytical formulation of the Lagrange equations of motion . And inbetween, the reader will find here a refreshingly different, consistent , logical, andgradual development and careful application of all of the classical principles ofmechanic s for a particle , a system of particles, a rigid body, actually any number ofparticle s and rigid bodies, subjected to various kinds of forces and torques. Someunusual, hopefully interesting things to look for are sketched below.

Chapter 5: The Foundation Principles of Classical Mechanic s, presents adetailed study and fresh discussion of Newton 's laws of motion, Newton 's theoryof gravitation , the role of inertial reference frames, and Newton 's second law ofmotion relativeto the moving Earth frame . Further, two fascinating applicationsof Coulomb's laws of friction having engineering significance are presented. Thechapter concludes with an advanced topic borrowed from continuum mechanics ,here applied to deduce the general law of mutual interaction between two particles .This interesting topic has not previously appeared in any textbook.

The kinematical equations characterizing the velocity and acceleration of aparticle in terms of rectangular, intrinsic, cylindrical, and spherical coordinates areapplied in Chapter 6: Dynamics of a Particle , in the formulation of the equationsof motion in a variety of special applications. Besides gravitational and frictionalforces, additional electromagnetic, elastic spring, and drag force laws are intro­duced and illustrated in some atypical examples that include the mechanics of

x Preface

an ink jet printer and Millikan 's famous oil drop experiment. Of course, familiarelementary problems, sometimes having a novel twist, appear here as well. Theelements of projectile motion with examples that include drag force effects andthe fanciful exploits of Percy Panther and Arnold Aardvark surely will attract at­tention. Several additional distinctive examples include a nonlinear oscillator, thefinite nonlinear radial oscillation of a particle of an incompressible rubber tube,and problems of motion relative to the moving Earth frame that include Reich'sexperiment on the deflection of falling pellets and Foucault's experiment demon­strating the rotation of the Earth . The projectile deflection problem is solved; and,with this in mind, the historic tragic battle of the Falkland Islands is described.

The first integral principles of mechanics for a particle-impulse-momentum,torque-impulse, work-energy, and the conservation laws of momentum, momentof momentum, and energy-are developed in Chapter 7: Momentum, Work, andEnergy. Along with several typical applications, some advanced topics treated thereinclude an introduction to elliptic functions and integrals in the problem of thefinite amplitude oscillations of a pendulum. The isochronous cycloidal pendulumand Huygens's clock, and orbital motion and Kepler's laws, are other interestingexamples.

The classical principles of mechanics and first integrals for a particle areextended to study in Chapter 8: Dynamics of a System of Particles. The importanceof the center of mass of the system, first mentioned in Chapter 5, is now evident.The two body problem correcting Kepler's third law is a distinctive illustration.In preparation for later work on rigid bodies, the moment of momentum principlewith respect to a moving reference point, first derived in Chapter 6 for a particle,and the reduction to its simplest form, is carefully described . Several examples ofsimple systems of particles are demonstrated.

The elements of tensor algebra, including the Cartesian tensor transformationlaw, introduced in Volume 1, Chapter 3, are exploited in Chapter 9: The Momentof Inertia Tensor. Here the inertia tensor and its general properties are studied andapplied to complex structured (composite) bodies . The center of mass concept fora simple rigid body, first studied in Chapter 5, is extended to composite bodies aswell. The tensorial form of the parallel axis theorem is proved, and its interpretationis illustrated in examples. The principal (extremal) values of the inertia tensor andtheir directions are derived by the method of Lagrange multipliers; and their easygeometrical interpretation in terms of Cauchy's momental ellipsoid is described.These topics playa key role in the general theory of rigid body motion . Someadvanced analytical topics on the principal values of general symmetric tensors andthe multiplicity of principal values and corresponding principal axes are exploredat the end.

Chapter 10: Dynamics of a Rigid Body, the most difficult of the subjectstreated in this book, focuses on Euler's generalization of Newton's principles ofmechanics, now extended to characterize the equations of motion for all bodies­solids, fluids, gasses-but herein restricted principally to the motions of rigid solidbodies. Euler's laws are set down separately and discussed in detail, somewhat

Preface xi

parallel to my earlier discussion of Newton's principles; but there are sharp dif­ferences . First, the principle of determinism includes both forces and torques, andthe third law of mutual interaction of both forces and torques follows as a theoremfrom Euler's first and second laws. The remaining developments, however, arefairly standard leading to Euler's nonlinear vector differential equation of motionfor a rigid body in terms of its moment of inertia tensor. The key role of the princi­pal body axes in the simplification of Euler's scalar equations is demonstrated. Theusual general first integral principles on impulse-momentum, torque-impulse, con­servation of momentum and moment of momentum, the work-energy and energyconservation principles for a rigid body are obtained. It is worth noting, however,that two forms of Euler's second law of motion with respect to a moving referencepoint, seldom discussed in other works, are derived. Their reduction to the simplestcommon case is thoroughly discussed, and several typical sorts of applications arestudied. Some advanced examples describe the effect of the Earth 's rotation incontrolling the motion of a gyrocompass, the stability of the torque-free rotationof a spinning rigid body, the sliding, slipping and rolling motion of a billiard ball ,and the motion of a general rigid lineal body in a Stokes medium.

A careful, thorough introduction to Lagrange's equations of motion for holo­nomic dynamical systems is provided in Chapter II : Introduction to AdvancedDynamics. The general form of Lagrange 's equations, an energy based formu­lation of the equations of motion, is first motivated in their derivation for aholonomic system consisting of a single particle. The principle of virtual work re­lates the generalized forces in these equations to the physical forces that act on theparticle. The work-energy principle, Lagrange's equations for conservative sclero­nomic systems, and their first integral principle of conservation of energy follow.Precisely all of the same equations are shown to hold for any system of discrete ma­terial points having any number of degrees of freedom. For conservative systems,recognition of so-called ignorable coordinates in the Lagrangian energy functionleads immediately to easy first integrals of Lagrange's equations. Generalized mo­menta are introduced and Lagrange's equations for each generalized coordinateare recast in a form having the appearance of Newton 's second law in which theLagrange forces consist of the generalized forces and certain pseudoforces of thesort encountered differently in earlier studies. Generalized first integral principlesof conservation of momenta and the generalized impulse-momentum principlefollow. All of the results are illustrated in several examples together with the phys­ical interpretation of the terms. Lagrange 's equations for both conservative andnonconservative general dynamical systems are then deduced from Hamilton 'svariational principle of least action . As a consequence, Lagrange's equations areimmediately applicable to study the motion of rigid bodies ; in fact, any dynamicalsystem consisting of any number of particles and rigid bodies. It is only necessaryto define the Lagrangian energy function, however complex, and to identify thecorresponding generalized forces. Accordingly, the Lagrange equations are ap­plied to formulate the equations of motion for several such physical systems . Thetheory of small vibrations for nondegenerate, multidegree of freedom systems is

xii Preface

derived and demonstrated in an example. Finally, Lagrange's equations are modi­fied to account for the effect of Stokes type damping in a general formulation of theRayleigh dissipation funct ion, and the result is illustrated in some concluding ex­amples that include an application to the theory of small vibrations with damping .This relatively brief introducti on to advanced dynamics presents the reader withtools adequate to pursue more advanced, abstract studies of analytical dynamics.

In every chapter throughout this work the reader will find a great many his­torical remarks, annotation s, and references on the several topics treated here. Ihope these little stories, digressions, and annotation s will add life and bring tothe subject a reality not seen in most books of this kind. Countle ss examples andproblems are provided as aids to understanding and visualizing the details of thetheory; and, in most instances, a physical interpretation is presented or perhapssome unusual effect is identified. This sort of discovery and prediction derivedfrom the principles of mechanics and leading to results that most often are not in­tuitively evident, drive the purpose of the analysis . It is important that the studentcultivate the habit of attempting to interpret the physical phenomena or unusualeffects that emerge from problem analysi s. Ample opportunity for the reader to doso is provided within the problem s provided at the end of each chapter. So far as Iam aware, the treatment of problem s provided herein usually is quite different andmore thorough than found in other comparable sources known to me. Nevertheles s,by consulting the listed references or their own favorite books, both the teacherand the student should be able to supplement the many examples and problems tomeet their special needs.

In writing this book, I have appealed over many years to numerou s sources.While I do not necessarily subscribe to their approach to mechanics, several ofthe references on classical and engineering mechanics that I found particularlyhelpful are cited at the end of each chapter, usually with annotations to describethe substance of the work and to identify specific topics that may be consulted forcollateral study. The presentation has also benefited greatly from the very manyhistorical papers and books referenced there . I hope the acknowledgments arecomplete. It is impossible, of course, to be precise in citing my specific use ofevery source, and I apologize if I may have overlooked or forgotten a particularreference.

Acknowledgment

lowe much to Mr. Joe Haas at the University of Kentucky during the 1980s,for his good humor, untiring patience, and painstaking care in the tedious task ofadding typeset lettering to my numerous ink drawings. I came to appreciate thiseven more when years later I did this myself. I am also grateful to Ms. MarjorieBisbee and Mr. James Pester at the University of Nebraska-Lincoln in the 1990s,and I thank Mr. Haas as well, for applying their artistic talents and computer skills

Preface xiii

to draw from my pencil sketches some of the illustrations that appear here. I hopethey are not offended by my modifications and addition of shading to all of theillustrations.

The students that took my courses in dynamics at the University of Delaware,the University of Kentucky, and the University of Nebraska-Lincoln deserve myspecial thanks for the opportunity to learn from them as well. It was especiallyexciting to find in grading some assignment papers, an essential but usually dulland tiresome task, that from time to time a student would discover an interestingsolution method that I had not thought of. In this regard, I thank my former graduatestudent at Nebraska, Professor Alex Elfas-Ziifiiga at the Instituto Technol6gicoy de Estudios Superiores de Monterrey, Mexico; he read an early draft of themanuscript in the 1990s and worked through countless problems in confirmationof my solutions. As a consequence of such student efforts over many years, forwhich I am most grateful, virtually all of the problems in this book have been testedand their solutions confirmed . Any errors that may yet appear in the answers at theback of this volume are mine.

The joy of work with graduate students in areas of finite elasticity, relatedfunded research interests, and classroom, profess ional, and administrative obli­gations as Chairman of the Department of Engineering Mechanics at Nebraska,constantly took precedence over my efforts to complete this volume started so longago. When at last the manuscript was finished in the spring of 2004, I sought thecriticism and comment of several respected colleagues who kindly accepted theburden to read all or a portion of the manuscript. Their thoughtful and constructivecritiques led to many improvements that served to enhance the quality of the book.I am grateful for their input and for their unsolicited supportive complimentarycomments. Such errors as may remain, however, are my sole responsibility. Itis a pleasure to thank Professor Eveline Baesu, University of Nebraska-Lincoln,and Professor David Steigmann, University of California, Berkeley; they providedhelpful remark s and useful suggestions on various parts of the manuscript. I amindebted to Professor Emeritus Oscar W. Dillon, University of Kentucky; his con­structive critical comments, helpful discussions, and thoughtful recommendationsand remarks covering a great many aspects of the manuscript have been invaluable .Finally, I am indeed most grateful to Professor Emeritus Michael A. Hayes, Univer­sity College, Dublin, for his thoughtful, painstaking review of both the manuscriptand the proof; he exposed in both a great many typographical slips and providednumerous beneficial suggestions and invaluable comments from which the finalversion of this book has greatly benefited throughout.

Millard F. Beatty, Jr.Lexington, Kentucky

Contents

Note to the reader: This book, which begins with Chapter 5, is the second of a series of twovolumes on the Principles of Engineering Mechanics. Chapters 1 through 4 and Appendices A

and B appear in Volume J: Kinematics-The Geometry ofMotion, which includes the Answers

to Selected Problems for that volume and a separate Index on its contents. Volume2 includesChapters 5 through 11, AppendicesC and D, Answers to Selected Problemsfor this volume, and

a separate Index on its contents.

5. The Foundation Principles of Classical Mechanics5.1. Introduction 35.2. Mass, Momentum, and the Center of Mass 5

5.2.1. Mass and Momentum of a Particle 6

5.2.2. Mass, Momentum, and Center of Mass of a System of Particles . .. . . . . . . 65.2.3. Mass, Momentum, and Center of Mass of a Rigid Body ... ... .. .. ... .... 10

5.3. Moment of a Vector About a Point. . .. .. . .. . .. . . .. . .. . . .. . . . .. . . . .. . . . . . . . . . . . . . . 155.3.1. Reference Point TransformationRule 165.3.2. Moment of a Force About a Point 165.3.3. Equipollent Force Systems . .. . .. ... . .. . ... .... ... ... ... ..... ... . .. ... . .. . 18

5.4. Moment of Momentum .. . . .. . .. . . . . . .. . . . . . . . ... . . . . .. . . . . . .. . . . . . . . . . . . . . .. . . . . . 205.4.1. Moment of Momentum of a Particle . 215.4.2. Moment of Momentum of a System of Particles. ... . .. . .. . . . . . .. . .. .... . 215.4.3. Moment of Momentum of a Body 22

5.5. Newton's Laws of Motion .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 245.5.1. The Material Universeand Forces... . . . . . .. . .. . . . . . .. . . . . . . . . . . . . . . . . . . . . 255.5.2. The First Law of Motion. ... . .. . .. ... . .. ... . .. . .. . . .. ... . ... ... .. . .. . .. ... 265.5.3. The Second Law of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . .. . .. . . . 275.5.4. The Law of Mutual Action 30

5.6. Newton's Law of Gravitation 31

5.6.1. The Gravitational Field of a Particle . ... ... .... .. . .. . . ... . . . . . . . . . . . . . . . .. 34

5.6.2. The Gravitational Field of a System of Particles 35

5.6.3. The Gravitational Field of a Body 365.7. Some Applications of Newton's Theory of Gravitation . . ... ... ... ... ... .. ... ... . 38

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xvi Contents

5.8. Gravitational Attraction by an Ideal Planet.. .. ... . ... . ... ... .. . .. .. ... . .. . . . .. 42

5.9. Gravitational Force on an Object Near an Ideal Planet . . . . . . . . . . . . . . . . .. . . . . . . 445.10. Weight of a Body and its Center of Gravity 45

5.10.1. The Local Acceleration of Gravity-An Estimate 46

5.10.2. Center of Gravity of a Body 46

5.11. Coulomb 's Laws of Friction 50

5.1I.l . Contact Force between Bodies 51

5.11.2. Governing Principles of Sliding Friction 53

5.1 1.3. Equilibrium of a Block on an Inclined Plane 55

5.11.4 . Motion of a Block on an Inclined Plane. . . ... .. . . . .. . . .. . . . . . . . .. . . . . 58

5.12. Applications of Coulomb 's Laws 59

5.12.1. The Sliding Can Experiment 60

5.12.2. DamntheTorpedoes! 645.13. What is the Inertial Frame? 68

5.14. The Second Law of Motion in a Noninertial Frame 69

5.15. Newton' s Law in the Noninertial Earth Frame 725.16. The Apparent Gravitational Field Strength of the Earth 73

5.17. Newton 's Law in the Earth Frame 77

5.18. Frame Indifference and the Law of Mutual Internal Action 78

5.18.1. Change of Reference Frame 79

5.18.2. The Principle of Material Frame Indifference 80

5.18.3. The Law of Mutual Internal Action 81

References 83Appendix: Measure Units in Mechanics 86

Problems 87

6. Dynamics of a Particle

6.1. Introduction 95

6.2. Component Forms of the Newton-Euler Law 96

6.3. Some Introductory Examples and Additional Concepts . . . . . . . . . . . . . . . . ... . . . . 97

6.3.1. Some Applications in a Rectangular Cartesian Reference

Frame 97

6.3.2. Intrinsic Equation of Motion for a Relativistic Particle 102

6.3.3. Electric and Magnetic Forces on a Charged Particle .. . .... ... . ... . ... . 104

6.3.4. Fun at the Amusement Park ...... . . ... .... . . . . . . .. . . . . . .. . . . . . . . . . . . . .. 107

6.3.5. Formulation of the Particle Dynamics Problem 108

6.4. Analysis of Motion for Time Dependent and Constant Forces 109

6.4.1. Motion under a Time Varying Force 109

6.4.2. Motion under a Constant Force 110

6.5. Motion under Velocity Dependent Force 118

6.5.1. Stokes 's Law of Resistance . . 119

6.5.2. Formulation of the Resistance Problem 119

6.6. An Important Differential Equation 126

6.6.1. General Solution of the Homogeneous Equation 126

Contents xvii

6.6.2. Particular Solution of the General Equation 127

6.6.3. Summary of the General Solution 128

6.6.4. Physical Character of the Solution . .. . . . . .. . . .. . . . . . . . . .. . . . . .. . ... .. . . 128

6.7. The Simple Harmonic Oscillator 132

6.7.1. Hooke's Law of Linear Elasticity.. ... ..... ... ... ... . ... 132

6.7.2. The Linear Spring-Mass System 133

6.7.3. The Simple Pendulum 137

6.7.4. The Common Mathematical Model in Review . . . .. . . . . 139

6.8. Motion of a Charged Particle in an Electromagnetic Field . . . . . . . . . . . . . . .. .. . . 141

6.9. Motion of a Slider Block in a Rotating Reference Frame . . . ... . .. . ...... ... .. 143

6.9.1. Uncontrolled Motion of a Slider Block 143

6.9.2. Controlled Motion and Instability of a Slider Block. ... ... . .. . .. . . .. . . 145

6.10. The Moment of Momentum Principle 148

6.10.1. Application to the Simple Pendulum Problem 149

6.10.2. The Moment of Momentum Principle for a Moving Point 149

6.11. Free Vibrations with Viscous Damping 151

6.11. I. The Equation of Motion for Damped, Free Vibrations 152

6.11.2. Analysis of the Damped, Free Vibrational Motion 153

6.11.3. Summary of Solutions for the Damped, Free Vibrational Motion .... 157

6.12. Steady, Forced Vibrations with and without Damping 158

6.12.1. Undamped Forced Vibrational Motion 160

6.12.2. Steady-State Vibrational Response of a Damped System 165

6.12.3. Force Transmissibility in a Damped System.. 168

6.13. Motion under a General Nonlinear Force f(x , i) 169

6.13.1. Special Class of Nonlinear Equations of Motion 170

6.13.2. Radial Oscillations of an Incompressible Rubber Tube 1716.14. Infinitesimal Stability of the Relative Equilibrium States of a System 172

6.14.1. Stability of the Equilibrium Positions of a Pendulum 176

6.14.2. Application to Linear Oscillators 1776.15. Equations of Motion Relative to the Earth 1776.16. Free Fall Relative to the Earth-An Exact Solution 179

6.16. I. Free Fall Deflection Analysis 180

6.16.2. Reich's Experiment 181

6.17. Foucault' s Pendulum 181

6.17. I. General Formulation of the Problem 184

6.17.2. Equations for Small Amplitude Oscillations 185

6.17.3. Solution of the Small Amplitude Equations 186

6.17.4. Physical Interpretation of the Solution .... ... ... ... .......... ........ 187

6.18. Relative Motion under a Constant Force 189

6.18.1. First Order Vector Solution for Projectile Motion 190

6.18.2. The Battle of the Falkland Islands 192

6.18.3. Concluding Remarks 195

References 196

Problems 198

xviii Contents

7. Momentum, Work, and Energy

7.1. Introduction 22I7.2. The Impulse-Momentum Principle 221

7.2.1. Instantaneous Impulse and Momentum 223

7.2.2. Linear Momentum in an Instantaneous Impulse . ... . .. . ... .. .. . . . . . .. . 2267.3. The Torque-Impul se Principle . .. . . . . .. . . .. . . . . . . . . . .. . . .. . . .. . . . . . . . . . .. . . .. . . 227

7.3.1. Instantaneous Torque-Impul se 228

7.3.2. Moment of Momentum in an Instantaneous Torque-Impulse . .. . . . . . . 228

7.4. Work and Conservative Force . . ... . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . .. . . . . . . . . . 229

7.4.1. Work Done by a Constant Force .. . .. .. .. .. 2337.4.2. Work Done by the Coulomb Frictional Force . . . . . . . . . . . . . . . . . . . . . . . . . . 2347.4.3. Work Done by a Linear Force .. 234

7.5. The Work-Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 235

7.6. Potential Energy . . .. . . .. ... . .. . . . . . .. . .. . ... . .. .. .. . ... . . .. . .. . .. . . . . .. . .. ... . . 2407.6.1. Theorem on Conservative Force 241

7.6.2. The Needle in the Haystack . . .. . . . . . . ...... .... . .. . . ...... . ... . . . . . .. . . 244

7.7. Some Basic Potential Energy Functions . . . . .. . . . . . . . . .. . . . ... . . . . . . . . .. .. ... . . 246

7.7.1. Gravitational Potential Energy .. .. 246

7.7.2. Elastic Potential Energy of a Spring . .. . . . . . . . . . . . . . . . . ... . . . . . ... ... . . 248

7.8. General Conservation Principles 248

7.8.1. A General Conservation Theorem . :.... .... ...... .. .... .. .. .. .. .. .... . 2487.8.2. The Principle of Conservation of Linear Momentum 2497.8.3. The Principle of Conservation of Moment of Momentum . . . . . . . . . . . . . 2507.8.4. The Energy Principle . .. . ... .. . .. . ..... ... ... ... . . ... ... .. ... ...... . . .. . 252

7.9. Some Further Applications of the Fundamental Principles .. 2587.10. The Simple Pendulum Revisited: The Exact Solution 264

7.10.1. The Energy Equation and the Rod Tension 2657.10.2. The Finite Pendulum Motion and Its Period .. .. .. .. 2657.10.3. Introduction to Jacobian Elliptic Functions 267

7.10.4. The Pendulum Motion in Terms of an Elliptic Function 2697.10.5. The Small Amplitude Motion 269

7.10.6. Nonoscillatory Motion of the Pendulum 2717.11. The Isochronous Pendulum 272

7.11.1. Equation of Motion on an Arbitrary Concave Path 2737.11.2. Small Oscillations on a Shallow, Concave Curve 2747.11.3. Finite Amplitude Oscillations on a Cycloid 2747.11.4. Uniqueness of the Isochronal, Cycloidal Pendulum . . . . . . . . . . . .. . . .. . 2767.11.5. Huygens's Isochronous Clock .. .. .... .. .. .. .. .. .. .. . .... .. .. .. .. .... . 277

7.12. Orbital Motion and Kepler's Laws 2787.12.1. Equation of the Path 2787.12.2. Geometry of the Orbit and Kepler's First Law 2797.12.3. Kepler's Third Law. . 281

References 284Problems 285

Contents xix

8. Dynamics of a System of Particles

8.1. Introduction 30I8.2. Equation of Motion for the Center of Mass 30I8.3. The Two Body Problem 303

8.4. The Impulse-Momentum Equation 3058.5. Moment of Momentum of a System of Particles 3078.6. The Moment of Momentum Principle 310

8.6.1. Moment of Momentum Principle for a Moving Reference Point 3118.6.2. Second Form of the Moment of Momentum Principle for

a Moving Point .. .. .. .. . 3138.6.3. Summary: The Moment of Momentum Principle for a System

of Particles 3148.7. Kinetic Energy of a System of Particles 317

8.8. Work-Energy Equations for a System of Particles 318

8.9. The Principle of Conservation of Energy. . .. . ... .. . . . . . . .. . .. . .. . ... ... . ... . .. 323

8.9.1. External Potential Energy... 3248.9.2. Internal Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . ... ... ... ... . . .. . 3258.9.3. Total Energy of the System 3278.9.4. The General Energy Principle for a System of Particles . . . . . . . . ... .... 328

8.10. An Application of the General Energy Principle 3358.11. An Application of the Energy and Impulse-Momentum Principles 3368.12. Motion of a Chain on a Smooth Curve by the Energy Method 3378.13. Law of Restitution 339References 342

Problems 343

9. The Moment of Inertia Tensor9.1. Introduction ... ... . . . .. ... . . . . . . . . . . . . . . . . .. . .. . ... . .. . . . . . . . . .. . . . . . . . . . . . . . . . . 3559.2. The Center of Mass of a Complex Structured Rigid Body 3559.3. The Moment of Inertia Tensor . .. . . . . . . . . . . . . . . . . . . . . . .. ... .... . . . . .. . . . . . . . . . . 3589.4. Rectangular Cartesian Components of the Inertia Tensor . . . . ... . .. .... .. . .. . . 358

9.4.1. Moments of Inertia for a Lamina 3599.4.2. Moment of Inertia About an Arbitrary Axis and the Radius

of Gyration 3609.4.3. Moment of Inertia Properties of Symmetric Bodies . ... ... ... ... . . . . . . 361

9.5. Moments of Inertia for Some Special Homogeneous Bodies 363

9.5.1. Moment of Inertia Tensor for a Rectangular Body 364

9.5.2. Moment of Inertia of a Circular Cylindrical Body . . . . . ... ... ... ... .. . . 3659.5.3. Moment of Inertia Tensor for a Sphere 367

9.6. The Moment of Inertia of a Complex Structured Body 367

9.7. The Parallel Axis Theorem . . . . . . . . . .. . ... ... .. . . . . .. . . . . .. . . . . .. ... ... . .. . .. . . 3719.8. Moment of Inertia Tensor Transformation Law 3769.9. Extremal Properties of the Moment of Inertia Tensor . . . . . . . . . . . . . . .. . .. . .. . .. 378

9.9.1. Introduction to the Principal Values Problem . . . . . . . . .. . .. . ... . .. . .. . .. 378

xx Contents

9.9.2. The Method of Lagrange Multipliers 379

9.9.3. Principal Values and Directions for the Inertia Tensor. . ... ... .... . . . . . 3829.9.4. Cauchy's Inertia Ellipsoid 387

9.10. Loose Ends and Generalities for Symmetric Tensors 3909.10.1. Summary of the Principal Values Problem 3919.10.2. Reality of the Principal Valuesof a Symmetric Tensor.. ...... ....... 391

9.10.3. Orthogonality of Principal Directions 392

9.10.4. Multiplicity of Principal Values 392

References 396Problems ; . .. . . .. . . . . . .. . . . . . . .. . .. . . .. .. . . .. . 396

10. Dynamicsof a RigidBody10.1. Introduction 40510.2. Euler' s Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

10.2.1. Forces and Torques 40910.2.2. The Principle of Determinism 41110.2.3. Euler's Laws 412

10.3. The Law of Mutual Action .. .. .. .. . .. . .. . .. .. .. .. .. .. .. . .. .. .. . .. .. .. .. .. . .. . 41410.4. Euler's First Principle and Motion of the Center of Mass . .. . .. .. .. .. .. . .. . .. 41810.5. Moment of Momentum Transformation Equations 422

10.5.1. Point Transformation Rule for the Moment of Momentum 422

10.5.2. VelocityTransformation Rule for the Moment of Momentum 423

10.6. The Second Law of Motion for a Moving Reference Point 42610.6.1. The First Form of Euler's Law for a Moving Point 42610.6.2. The Second Form of Euler's Law for a Moving Point 427

10.7. Euler's Second Principle of Motion and Related First Integrals. . ... ........ 42910.8. Moment of Momentum of a Rigid Body 43110.9. Euler's Equations of Motion for a Rigid Body .. . .. . .. .. .. . .... .... .... .. .... 433

10.10. Equilibrium of a Rigid Body. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43410.11. Rotation about a Fixed Body Axis and Plane Motion 435

10.11.1. Rotation about a Fixed Body Axis 436

10.11.2. Plane Motion of a Rigid Body 438

10.11.3. Dynamic Balance of a Rotor 438

10.12. Further Applications of Euler's Laws .. .. ... ........... ........ ... ... .. .. . .. . 439

10.12.1. Application to a Belt Driven Machine 44010.12.2. Application to a Flywheel .. .. . .. . .. .. ... . .. .... .... ... ........ 441

10.12.3. Application to the Plane Motion of a Rigid Link 442

10.12.4. Analysis of Dynamical Bearing Reaction Forces 444

10.12.5. Bearing Reaction Torque and Stability of Relative Equilibrium 44710.13. The Gyrocompass 450

10.13.1. General Formulation of the Problem.. ... ............... ... . .. .. .. . 45110.13.2. Small Angular Displacement Solution . . . . . . . . . . . .. . . . . . ... . .. . ... . 45210.13.3. Small Angular Motion with Viscous Damping 453

10.14. Stability of the Torque-free, Steady Rotation About a Principal Axis 45410.14.1. An Illustration and an Experiment .. 455

Contents xxi

10.14.2. Discussion of the General Torque-Free Steady Rotation

of a Rigid Body 456

10.15. Motion of a Billiard Ball 45610.15.1. Motion on a Smooth Surface 457

10.15.2. Motion on a Rough Surface 45710.16. Kinetic Energy of a Body 460

10.16.1. Kinetic Energy Relative to the Center of Mass 46010.16.2. Kinetic Energy Relative to an Arbitrary Point 46110.16.3. Kinetic Energy of a Rigid Body .. .. .. . .. .. .. . .. . .. . 461

10.17. Torque-Free Rotation of an Axisymmetric Body . .. . . . .. ... .. . ... . . . . .. ... .. 46310.18. Mechanical Power and Work 465

10.19. The Work-Energy Principle .... . ........... . .. . ... . .. . . .. . ..... . .. . ......... . 466

10.20. Potential Energy 46910.21. The General Energy Principle 471

10.22. Motion of Lineal Bodies Subject to a Stokes Retarding Force .. ... . ... . . .. . . 473

10.22.1. Universal Motions for Equipollent Systems . . . . . . . . . . . . . . . . . . . . . . . 47410.22.2. Universal Motions for Workless Force Systems . . . . .. . . . . . . . . . . . . . 475

10.22.3. Universal Motions of Rotating Rigid Bodies 47610.22.4. General Motion with Additional Forces 477

References . . . .. .. . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . 478Problems 480

11. Introduction to Advanced Dynamics11.1. Introduction 495

11.2. Generalized Coordinates, Degrees of Freedom, and Constraints 496

11.2.1. Holonomic Constraints 49711.2.2. Nonholonomic Constraints 498

11.3. Lagrange's Equations of Motion for a Particle . .. .. ....... . .. .. .. .. .. ..... ... 50011.3.1. Two Useful Identities 50011 .3.2. First Fundamental Form of the Lagrange Equations of Motion . . . . . SOl

11.4. The Generalized Forces and Virtual Work 50211.5. The Work-Energy Principle for Scleronomic Systems 50511.6. Conservative Scleronomic Dynamical Systems 50711.7. Lagrange's Equations for General Conservative Systems . 50711.8. Second Fundamental Form of Lagrange's Equations 509

11.9. Lagrange's Equations for a System of Particles.. ...... ..... .... .. ... .. . .. .. . 512

11.10. First Integrals of the Lagrange Equations 517

11.10.1. Ignorable Coordinates: An Easy First Integral of

Lagrange's Equations 518

11.10.2. Principle of Conservation of Generalized Momentum 519

11.10.3. The Principle of Conservation of Energy for Scleronomic

Systems Revisited . . . . . . . . .. . .. . . . . . . . . .. . .. . . . . . . . . . . . .. . .. . . . . .. . . 52111.10.4. The General Energy Principle 52211.10.5. The Generalized Impulse-Momentum Principle 523

11.11. Hamilton's Principle .. . .. ..... .. . .. .. . .. .. .... ... .. .. .... .. . . .. . .. . .. . . .. . .. .. 526

xxii Contents

l l .Ll .I. Hamilton's Principle for a Conservative, Holonomic System 527

11.11.2. Hamilton's Principle for a Nonconservative, HolonomicSystem 531

11.12. Additional Applications of the Lagrange Equations 53411.12.1. A Problem in Engineering Design Analysis 534

11.12.2. Rotating Simple Pendulum 53611.12.3. The Gyrocompass with Torsional Damping Revisited . .. . . .. ... ... 538

11.12.4. Motion of a Symmetrical Top about a Fixed Point 53911.13. Introduction to the Theory of Vibrations . . . . . .... ... ...... . . .. .... .. . .. . .. ... 547

11.13.1. Small Oscillations of a Simple Pendulum Revisited 548

11.13.2. The Theory of Small Vibrations . .. . . . . . .. . .. . . . . . . . . . . . . .. . ...... . 549

11.14. Dissipative Dynamical Systems of the Stokes Type 556

11.15. Closure 558

References 560

Problems ......... ......................... ........................ ................... 561

Appendix C. Internal Potential Energy of a System of Particles 567

Appendix D. Properties of Homogeneous Rigid Bodies

D.1. Nomenclature 571D.2. A Word of Caution . . .. ... . .. . . . . . .. . . . . . . . . .. . . . .. . . . . .. . .. . . . . . . . .. .. . . . . . .. . .. 571

D.3. Table of Properties 572

Answers to Selected Problems 577

Index 585

Volume 2

DynamicsThe Analysis of Motion

Mechanics is the paradise of the mathematical sciences, because here wecome to the fruits of mathematics.

Leonardo da VinciThe Notebooks