Table Of ContentVA RIATIONAL METHODS AND COMPLEMENTARY FORMULATIONS
INDYNAMICS
SOLID MECHANICS AND ITS APPLICATIONS
Volume 31
Series Editor: G.M.L. GLADWELL
Solid Mechanics Division, Faculty ofE ngineering
University ofWaterloo
Waterloo, Ontario, Canada N2L 3Gl
Aims and Scope of the Series
The fundamental questions arising in mechanics are: Why?, How?, and How much?
The aim of this series is to provide lucid accounts written by authoritative research
ers giving vision and insight in answering these questions on the subject of
mechanics as it relates to solids.
The scope of the series covers the entire spectrum of solid mechanics. Thus it
includes the foundation of mechanics; variational formulations; computational
mechanics; statics, kinematics and dynamics of rigid and elastic bodies; vibrations
of solids and structures; dynamical systems and chaos; the theories of elasticity,
plasticity and viscoelasticity; composite materials; rods, beams, shells and
membranes; structural control and stability; soils, rocks and geomechanics;
fracture; tribology; experimental mechanics; biomechanics and machine design.
The median level of presentation is the first year graduate student. Some texts are
monographs defining the current state of the field; others are accessible to final
year undergraduates; but essentially the emphasis is on readability and clarity.
For a list ofrelated mechanics titles, seefinal pages.
Variational Methods
and Complementary
Formulations
in Dynamics
by
B. TABARROK
Department of Mechanical Engineering,
University of Victoria,
Victoria, Canada
and
F.P.J. RIMROTT
Department of Mechanical Engineering,
University of Toronto,
Toronto, Canada
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-4422-8 ISBN 978-94-015-8259-9 (eBook)
DOI 10.1007/978-94-015-8259-9
Printed on acid-free paper
AII Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1994
Softcover reprint of the hardcover 1s t edition 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Dedicated to
Carolyn and Doreen
Table of Contents
Preface ................................................................................................................................. xi
Chapter I -FUNDAMENTALS ....................................................................................... .
A Historical Note ............................................................................................................... .
1.1 Review of Basic Concepts .......................................................................................... 2
1.2 Generalized Displacement Variables ......................................................................... 14
1.3 Degrees of Freedom aud Constraints ....................................................................... 18
Holonomic Constraints ................................................................................................ 18
Reduced Transformation Equations ............................................................................ 21
Nonholonomic Constraints .......................................................................................... 22
1.4 Virtual Displacements and Virtual Work ................................................................ 26
Rheonomic Constraints ..... ....... ........ ........ ........ ......... ... ........ ..... ... .... ................ ........ ... 29
Generalized Forces .... .... ........ ..... ........... ........ ................ ........ ....... ..... ............. .... ......... 30
1.5 Work Function and Potential Energy ....................................................................... 33
Other Forms of Potential Energy . ... .... ............ ..... ... ......... ... .... ................. ....... ..... ... .... 39
1.6 Stability of Static Equilibrium .................................................................................. 41
1.7 A Second Look at the Expression for Increment of Work ..................................... 49
1.8 Energies and Complementary Energies ........ ........ ........ .... .... ........ .... ........ ............ .... 52
Potential Energy aud Complementary Potential Energy ............................................. 52
Kinetic Energy aud Complementary Kinetic Energy ..... ... ........ ..... .................. ..... ...... 54
1.9 Kinetic Energy and Complementary Kinetic Energy in Terms of
Generalized Displacements .... .... ............ ........ ........ .... .... .... ............ .... ........ ............ .... 58
Suggested Reading ......... ........... ..... ....... .... ......... ................ ... ................ ................ ..... ... ..... 64
Chapter II - DIFFERENTIAL VA RIATIONAL FORMULA TIONS ........................... 67
2.1 D' Alembert's Principle ............................................................................................... 67
2.2 Complementary Form of D' Alembert's Principle ................................................... 72
2.3 Conservation of Energy ......... ... .... ........................ .... .... ..... ................... ........ ........ ..... 78
2.4 Gauss' Principle of Least Constraint ....................................................................... 81
2.5 Configuration Space and Hertz' Principle of Least Curvature .............................. 84
2.6 Lagrange's Equations ................................................................................................. 88
2.7 Invariance of Lagrange's Equations ......................................................................... 94
viii
2.8 Complementary Form of Lagrange's Equations ..................................................... 97
2.9 Dissipation Forces ....................................................................................................... 105
2.10 Limitations of the Complementary Formulation ................................................... 113
Suggested Reading ........ ................................ ............ ............ ................ .................... ......... 117
m -
Chapter INTEGRAL VARIATIONAL FORMULATIONS .................................. 119
3.1 Hamilton's Principle ................................................................................................... 119
3.2 Complementary Form of Hamilton's Principle ....................................................... 123
3.3 Lagrange's Equations from Hamilton's Principle ............................ ....................... 126
3.4 Conservation of Energy ............................................................................................. 127
3.5 Change of Energy Over a Period of Time ............................................................... 135
3.6 Constraint Equations and Lagrange Multipliers .................... ................................. 138
Holonomic Constraints ................................................................................................ 138
Nonholonomic Constraints .......................................................................................... 142
3.7 Ignorable Coordinates ................................................................................................ 150
3.8 Hamilton's Canonical Equations ............................................................................... 157
3.9 Complementary Canonical Equations ...................................................................... 164
3.10 Elimination of Ignorable Coordinates ...................................................................... 168
3.11 Phase Space and State Space ............................................................ ........................ 170
3.12 Hydrodynamic Analogy ........................................................................ ........ ............. 174
3.13 Principle of Least Action ..... ... .... ..... ... .... .... .... .... .... ........ .... .... .... ........ ....... ..... ........... 177
3.14 Jacobian Form of the Principle of Least Action ..................................................... 185
3.1S Some Generalizations of the Principle of Least Action ........ ..... ... ........ .... .............. 188
Suggested Reading ............................................................................................................. 192
Chapter IV -CANONICAL TRANSFORMATIONS AND THE HAMILTON-
JACOBI EQUATION ........................................................................................................ 193
Introduction ........................................................................................................................ 193
4.1 Canonical Transformations ....................................................................................... 193
4.2 Infinitesimal Canonical Transformations .... .... ........ .... ... ..... ........ ................ .... .... ..... 199
4.3 Hamilton-Jacobi Equation ................................ .................... ........................ ............. 200
4.4 Hamilton's Characteristic Function .......................................................................... 208
4.5 Hamilton's Optico-Mechanical Analogy ................................................................... 209
Suggested Reading ............................................................................................................. 213
ix
Chapter V - RIGID BODY DYNAMICS ........................................................................ 215
Introduction ........................................................................................................................ 215
5.1 Rotating Coordinate System ...................................................................................... 215
5.2 Momentum Velocity Relation .................................................................................... 219
5.3 Rotational Kinetic Energy and Kinetic Coenergy .... .... .... .... .... .... .... .... ....... ..... ....... 226
5.4 Euler Equations ........................ ............. ........................... ........ ................ ..... ............. 227
5.5 Euler Angles ................................................ ............................ ............................. ....... 231
5.6 Components of 00, Hand M in Different Coordinate Systems .............................. 236
5.7 Angular Momentum, Kinetic Energy and Coenergy in Terms of Euler
Angles ........................................................................................................................... 239
5.8 Lagrange's Equations ................................................................................................. 244
5.9 Other Generalised Variables for Describing Angular Positions ............................. 249
Suggested Reading .... .... ... ..... ........ ........ .... .... .... ........ .... .... .... ........ .... .... ............ .... .... ......... 250
Chapter VI - SPECIAL APPLICATIONS ...................................................................... 251
Introduction ......... ... ................................................................. ... .... ............................. ....... 251
6.1 Approximate Expressions for T*, V, D, and '6W/ ................................................ 251
6.2 Free Undamped Oscillations ............ .... ..... ... .... .... .... ........ .... ............ .... .... .... ........ ...... 256
Orthogonality of Modes of Vibration .................................. .... ................ ................. 260
6.3 Modal Matrix and Decoupling of Equations of Motion ........ ........ .... .... .... ........ ...... 265
6.4 Forced Undamped Oscillations .... ........ ........ .... .... .... ........ ........ ........ .... ........ ........ ...... 270
6.5 Free Damped Oscillations .... ........ .... .... ........ .... .... .... ........ .... .... ........ ........ .... .............. 274
6.6 Forced Damped Oscillations ........ .... .... ........ .... ........ ........ ........ ............ .... .... ....... ....... 276
6.7 Small Oscillations about Steady Motion ................................................................... 277
Taylor Series Expension ofRouthian ....................................................................... 279
6.8 Solution of the Linear Equations of Motion .. ............... ....... ........................ ............ 283
6.9 System Response to a Step Impulse .......................................................................... 286
Lagrangian Method ........ .... ........ ........ .... .... ........ .... .... .... ........ .... ............ .... ........ .... ..... 286
Constraint Consideration ......... ....... .... ................ ......... .................... ........... ..... ....... ..... 290
Energy Change .... .... .... ........ .... .... ..... ... .... .... ............ .... .... ........ .... ........ ........ .... .... .... .... 293
Bertrand's Theorem .................................................................................................... 296
Kelvin's Theorem ...................................................................................................... . 298
Carnot's Theorem ...................................................................................................... . 300
6.10 System Response to a Step Displacement ............................................................... 302
Analogue of Bertrand's Theorem ............................................................................... 305
x
Ana10gue of Kelvin's Theorem ...................................................................... 306
Analogue of Carnot's Theorem ...................................................................... 306
Suggested Reading ................ ................ .................... ........ .................... ................. 307
Appendix A -THE CALCULUS OF VARIA TIONS ........................................ 309
Introduction ........................................................................................................... 309
A.I Fundions and Functionals .................................... ................ ............ ........... 309
A.2 Review of Extremum Values of Functions .................... ............ ........ ......... 311
A.3 Stationary Values of Definite Integrals ....................................................... 317
A.4 A Note about Weak and Strong Variations ............................................... 323
A.5 An Alternative Expression for a Single Euler-Lagrange Equation ........... 323
A.6 The Brachystochrone Problem ............................ ........................ ................ 324
A.7 Path-independent Functionals ........ .... .... ..... ........... .... ........ .... ........ ... ..... ...... 327
A.8 Several Dependent Functions ................................................ ............ ........ ... 329
A.9 Variational Notation ......................................... ,. ..... .... .... ........ .... .... .... .... ..... 332
A.IO Constraint Equations ................................................ .................... .............. 335
Lagrange Multipliers .. ,. .... .... .... .... .... .... .... .... .... .... .... ..... ... .... .... .... ..... ....... .... ... 337
Algebraic and Differential Equation Constraints . .... ............ .... ... ......... ... .... ..... 341
A.II Variable End Points ................................................ ~.................................. 343
Suggested Reading ............................ ................................................ ........ ........ ..... 348
Appendix B -DEVELOPMENTS IN MECHANICS -SOME
mSTORICAL PERSPECTIVES ........................................................................ 349
Author Index ......................................................................................................... 361
Subject Index .... ............ .................................................................... ........ ........ ..... 363
Preface
Not many disciplines can c1aim the richness of creative ideas that make up the
subject of analytical mechanics. This is not surprising since the beginnings of analyti
cal mechanics mark also the beginnings of the theoretical treatment of other physical
sciences, and contributors to analytical mechanics have been many, inc1uding the most
brilliant mathematicians and theoreticians in the history of mankind.
As the foundation for theoretical physics and the associated branches of the
engineering sciences, an adequate command of analytical mechanics is an essential tool
for any engineer, physicist, and mathematician active in dynamics. A fascinating dis
cipline, analytical mechanics is not only indispensable for the solution of certain
mechanics problems but also contributes so effectively towards a fundamental under
standing of the subject of mechanics and its applications.
In analytical mechanics the fundamental laws are expressed in terms of work
done and energy exchanged. The extensive use of mathematics is a consequence of
the fact that in analytical mechanics problems can be expressed by variational State
ments, thus giving rise to the employment of variational methods. Further it can be
shown that the independent variables may be either displacements or impulses, thus
providing in principle the possibility of two complementary formulations, i.e. a dis
placement formulation and an impulse formulation, for each problem. This duality is
an important characteristic of mechanics problems and is given special emphasis in the
present book.
The book begins by discussing fundamental concepts such as coordinate transfor
mation, degrees of freedom, virtual displacement and virtual work, holonomic and
nonholonomic constraints, work, potential energy, duality, complementary potential
energy, kinetic energy, complementary kinetic energy, and generalized variables, alI in
the fust chapter.
Readers not familiar with variational methods are now directed to Appendix A,
which is devoted to the calculus of variations, and introduces concepts such as func
tions and functionals, extrema, the Euler-Lagrange equations, extrema of functionals
subject to some constraints, and functionals with variable end points.
In the second chapter variational methods are used to develop extremum formula
tions of differential type for problems in dynamics. It begins with D' Alembert's prin
ciple, and then introduces Gauss' principle of least constraint, Hertz' principle of least
curvature in configuration space, and ends with Lagrange's equation, in dual represen
tation, i.e. in displacement and impulse variables.
The third chapter presents variational methods in integral form. Duality, e.g. for
mulation in conventional and complementary form, is again emphasized, and
Hamilton' s principle is discussed, as is the significance of Lagrange multipliers, and
the concept of ignorable coordinates.
The fourth chapter deals with canonical transformations, the fifth is devoted to the
applications of variational methods in gyrodynamics and the sixth chapter illustrates
some of the foregoing theory in special applications.
The text is intended for senior undergraduate or for graduate students, as weB as
for engineers, physicists, and mathematicians engaged in work in the multifarious
branches of theoretical and applied mechanics.
xi
