The present book deals with the issues of stability of Motion which most often are encountered in the analysis of scientific and technical problems. There are many comprehensive monographs on the theory of stability of motion, with each one devoted to a separate complicated issue of the theory. The main advantage of this book, however, is its simple yet simultaneous rigorous presentation of the concepts of the theory, which often are presented in the context of applied problems with detailed examples demonstrating effective methods of solving practical problems.

Preface to the English Edition

v

(2)

Preface to the Third Russian Edition

vii

(2)

From the Editors

ix

(2)

About the Author and the Editors

xi

Introduction

1

(4)

1 Formulation of the Problem

5

(20)

1.1 Basic Definitions

5

(5)

1.2 Equations of Perturbed Motion

10

(4)

1.3 Examples of Derivation of Equations of a Perturbed Motion

14

(5)

1.4 Problems

19

(6)

2 The Direct Liapunov Method. Autonomous Systems

25

(50)

2.1 Liapunov Functions. Sylvester's Criterion

25

(7)

2.2 Liapunov's Theorem of Motion Stability

32

(2)

2.3 Theorems of Asymptotic Stability

34

(9)

2.4 Motion Instability Theorems

43

(4)

2.5 Methods of Obtaining Liapunov Functions

47

(4)

2.6 Application of Liapunov's Theorem

51

(10)

2.7 Application of Stability Theorems

61

(11)

2.8 Problems

72

(3)

3 Stability of Equilibrium States and Stationary Motions of Conservative Systems

75

(28)

3.1 Lagrange's Theorem

75

(4)

3.2 Invertibility of Lagrange's Theorem

79

(1)

3.3 Cyclic Coordinates. The Routh Transform

80

(3)

3.4 Stationary Motion and Its Stability Conditions

83

(2)

3.5 Examples

85

(9)

3.6 Problems

94

(9)

4 Stability in First Approximation

103

(30)

4.1 Formulation of the Problem

103

(1)

4.2 Preliminary Remarks

104

(3)

4.3 Main Theorems of Stability in First Approximation

107

(4)

4.4 Hurwitz's Criterion

111

(4)

4.5 Examples

115

(14)

4.6 Problems

129

(4)

5 Stability of Linear Autonomous Systems

133

(26)

5.1 Introduction

133

(1)

5.2 Matrices and Basic Matrix Operations

134

(7)

5.3 Elementary Divisors

141

(9)

5.4 Autonomous Linear Systems

150

(7)

5.5 Problems

157

(2)

6 The Effect of Force Type on Stability of Motion

159

(62)

6.1 Introduction

159

(1)

6.2 Classification of Forces

160

(11)

6.3 Formulation of the Problem

171

(3)

6.4 The Stability Coefficients

174

(2)

6.5 The Effect of Gyroscopic and Dissipative Forces

176

(5)

6.6 Application of the Thomson-Tait-Chetaev Theorems

181

(8)

6.7 Stability Under Gyroscopic and Dissipative Forces

189

(8)

6.8 The Effect of Nonconservative Positional Forces

197

(10)

6.9 Stability in Systems with Nonconservative Forces

207

(11)

6.10 Problems

218

(3)

7 The Stability of Nonautonomous Systems

221

(92)

7.1 Liapunov Functions and Sylvester Criterion

221

(5)

7.2 The Main Theorems of the Direct Method

226

(3)

7.3 Examples of Constructing Liapunov Functions

229

(3)

7.4 System with Nonlinear Stiffness

232

(5)

7.5 Systems with Periodic Coefficients

237

(7)

7.6 Stability of Solutions of Mathieu-Hill Equations

244

(12)

7.7 Examples of Stability Analysis

256

(7)

7.8 Problems

263

(2)

8 Application of the Direct Method of Liapunov to the Investigation of Automatic Control Systems

313

8.1 Introduction

265

(1)

8.2 Differential Equations of Perturbed Motion of Automatic Control Systems

266

(3)

8.3 Canonical Equations of Perturbed Motion of Control Systems

269

(4)

8.4 Constructing Liapunov Functions

273

(6)

8.5 Conditions of Absolute Stability

279

(10)

9 The Frequency Method of Stability Analysis

289

(18)

9.1 Introduction

289

(1)

9.2 Transfer Functions and Frequency Characteristics

290

(4)

9.3 The Nyquist Stability Criterion for a Linear System

294

(1)

9.4 Stability of Continuously Nonlinear Systems

295

(3)

9.5 Examples

298

(6)

9.6 Problems

304

(3)

References

307

(10)

Index

317

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Introduction to the Theory of Stability

9780387947617

0387947612

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