Table Of ContentHorst Leipholz
LEIPHOLZ,STABIL.
THEORY
Second Edition
Stability Theory
Stability Theory
2nd Edition
An Introduction to the Stability of
Dynamic Systems and Rigid Bodies
Horst Leipholz
University of Waterloo
Waterloo, Ontario, Canada
SPRINGER F ACHMEDIEN WIESBADEN GMBH
Copyright © 1987 by Springer Fachmedien Wiesbaden
Originally published by B. G. Teubner, Stuttgart in 1987
All rights reserved.
No part of th1s book may be reproduced by any means, or
transmitted, or translated into a machine language
without the written permission of the publisher.
CIP-Kurztitelaufnahme der Deutschen Bibliothek:
Leipholz, Horst:
Stability theory: an introd. to the
stability of dynam systems and rigid H.
bodies/Horst Leipholz. - 2nd ed. -
Stuttgart: Teubner; Chichester; New York;
Brisbane; Toronto; Singapore: Wiley, 1987.
ISBN 978-3-519-02105-6 ISBN 978-3-663-10648-7 (eBook)
DOI 10.1007/978-3-663-10648-7
Library of Congress Cataloging-in-Publication Data:
Leipholz, H. H. E. (Horst H. E.), 1919-
Stability theory.
Translation of: Stabilitiitstheorie.
Includes index.
I. Stability. I. Title
QA87l.L47313 1987 531 86-15695
British Library Cataloguing in Publication Data:
Le1pholz, H.
Stability theory :an introduction to the
stability of dynamic systems and ngid
bodies. -2nd ed.
I. System analysis 2. Stability
I. Title
003 QA402
Table of Contents
Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v11
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
PART I. FUNDAMENTALS ................................. .
1.1. Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . 3
1.1.1. Historical Development of the Stability Concept . 3
1.1.2. Common Characteristics of Modern Stability
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. Sensitivity Equations and Variational Equations . . . . . . 21
Exercises 1.1 and 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3. Linear Variational Equations . . . . . . . . . . . . . . . . . . . . . . 24
1. 3 .I. Linear Variational Equations with Constant
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.2. Linear Variational Equations with Variable
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.3.3. Linear Variational Equations with Periodic
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Exercises 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
1.4. Investigations in Phase Space . . . . . . . . . . . . . . . . . . . . . . 61
1.4.1. Stationary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
1.4.2. Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Exercises 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1.5. The Direct Method of Lyapunov . . . . . . . . . . . . . . . . . . . 77
1.5.1. The Stability Definition of Lyapunov . . . . . . . . . . . 77
1.5.2. Lyapunov Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 78
1.5.3. Stability and Instability Theorems of Lyapunov . . 79
Exercises I. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
1.6. The Energy Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
1.6.1. Conservative, Scleronomic Systems . . . . . . . . . . . . . 89
v
Vl
1.6.2. Non-autonomous, Non-conservative Systems . . . . 93
Exercises 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
1. 7. Mathematical Approximation Methods . . . . . . . . . . . . . . 106
1. 7 .1. Isocline Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
1.7.2. Methods of Small Oscillations . . . . . . . . . . . . . . . . . 107
1.7.3. Variation of Constants . . . . . . . . . . . . . . . . . . . . . . . 107
1.7.4. Perturbation Method . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.7.5. The Methods of Galerkin and Ritz . . . . . . . . . . . . . 120
1.7.6. The Harmonic Balance Method . . . . . . . . . . . . . . . . 135
Exercises 1. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
PART II. APPLICATION OF STABILITY THEORY TO
SELECTED PROBLEMS OF MECHANICS . . . . . . . . . . . . . . . . . . . . . 143
2.1. Celestial Mechanics Problems . . . . . . . . . . . . . . . . . . . . . . 145
2.1.1. Stability of Central Force Motion . . . . . . . . . . . . . . 145
2.1.2. Stability Investigations of the Three-body Problem 149
Exercises 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.2. Problems of the Mechanics of Rigid Bodies and of
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
2.2.1. Stability of Vibrations . . . . . . . . . . . . . . . . . . . . . . . . 159
2.2.2. Stability of Control Systems . . . . . . . . . . . . . . . . . . . 167
2.2.3. Stability of the Gyroscope . . . . . . . . . . . . . . . . . . . . . 174
2.2.4. Stability of Aircraft, Rockets, Satellites . . . . . . . . . 180
Exercises 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
2.3. Problems of Elastomechanics . . . . . . . . . . . . . . . . . . . . . . 196
2.3.1. Stability Criteria of Elastostatics . . . . . . . . . . . . . . . 197
2.3.2. Application of General Methods of Elastic Systems
with Time-independent Parameters and Forces . . . 200
2.3.3. Conservative Stability Problems and the Application
of the Classical Energy Method . . . . . . . . . . . . . . . . 203
2.3.4. Outline of an Algebraic Theory . . . . . . . . . . . . . . . . 214
2.3.5. Examples for the Algebraic Approach to
Elastostatic Stability Problems . . . . . . . . . . . . . . . . . 249
2.3.6. Nonconservative Stability Problems and the
Application of an Extended Version of the Energy
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
2. 3. 7. Lyapunov Stability of Continuous Elastic Systems 305
2.3.8. Problems of Elastokinetics . . . . . . . . . . . . . . . . . . . . 337
Exercises 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
Preface to the First Edition
There have been great advances in theory of stability in recent decades due to
the requirements of control theory and flight mechanics, for example. We need
only mention the theory of A. M. Lyapunov. A number of specialists have
given a very mathematical and abstract description of the Lyapunov stability
theory which resulted in a 'stability theory of motion' applicable to the kinetics
of rigid bodies and systems. The stability theory of elastomechanics was
developed independently. However, there have been a number of important
developments in recent years, also with respect to this theory, dealing with the
following problems: The concept of the 'follower forces', non-conservative
loads, respectively, has been introduced in aeroelasticity. A number of prob
lems in elastic kinetics that involve pulsating loads or periodically varying
parameters has led to new stability questions. So-called 'kinetic' methods have
become necessary in elastomechnics in order to determine the stability bound
aries. An evaluation of the stability criteria of elastostatics, which have been
assumed to be generally valid, has shown that they can only be applied to a
limited number of problems under special assumptions. The transition from
stability to instability is a kinetic process in elastomechanics. Therefore, the
most general and most certain method of determining stability is the kinetic
stability criterion even if in special cases the classical stability criteria of
elastostatics may remain valid. This will be discussed in detail in Section 2. 3.
On this basis, it is possible to go beyond the original framework of the
theory and treat stability problems of kinetics of rigid bodies and systems as
well as of elastomechanics using a single, unified theory. One of the primary
purposes of the author was to show the common traits of stability problems
which occur in all branches of mechanics. The stability problems of hydro
mechanics and aeromechanics were omitted, because they are discussed in
separate lectures, outside of technical mechanics lectures in our schools.
However, the interested and knowledgeable reader may readily apply the
methods for determining stability also to stability problems of continuum
mechanics of liquid and gaseous media.
Stability theory is primarily expressed in mathematical terms. For purposes
of brevity, it was assumed that the reader is familiar with certain basic
Vll
viii
mathematical methods. In order to ensure that the reader is able to pick up
the requisite mathematical knowledge, several references are given in an ap
pendix. For example, in the section on mathematical approximation methods
used to solve non-self-adjoint or non-linear problems, the Galerkin method
was discussed in detail.
The Einstein summation convention was used in most cases, unless this was
not possible or not appropriate. The usual index notation was used in vector
and tensor calculus.
I would like to express particular appreciation to Professor Dr. K. Magnus,
Stuttgart, who inspired me to write the book and who aided me with many
valuable suggestions. I am indebted to my former assistant, Mr. Dipl.-Ing.
H. J. Hoffmann, for his critical remarks. Dipl. Math. W. Hauger made the
drawings and performed a number of necessary calculations. Mrs. L. Dot
tinger and Miss G. Walter worked with great dedication in completing the
manuscript.
Preface to the Second Edition
Over the past decades, interest in new stability problems in elastomechanics
and a redesigned theory of stability for that area have made a second edition
of the book desirable and even necessary. This opportunity has been used by
the author to revise and expand certain parts of the book intensively, so as
to take essential developments into account which have led to a remarkable
reformulation of the stability theory of elastomechanics.
Thanks are due to B. G. Teubner-Verlag, Stuttgart, for making the new
edition possible. Also, I am indebted to Mrs. G. Moseley for preparing the
typescript of the book in its new form in a competent and dedicated manner.
IX
PART 1
Fundamentals
