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Introduction to the Theory of Stability

Many books on stability theory of motion have been published in various lan­ guages, including English. Most of these are comprehensive monographs, with each one devoted to a separate complicated issue of the theory. Generally, the examples included in such books are very interesting from the point...

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Bibliographic Details
Main Author: Merkin, David R.
Other Authors: Afagh, F.F. (Editor), Smirnov, A.L. (Editor)
Format: eBook
Language:English
Published: New York, NY Springer New York 1997, 1997
Edition:1st ed. 1997
Series:Texts in Applied Mathematics
Subjects:
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 1 Formulation of the Problem
  • 1.1 Basic Definitions
  • 1.2 Equations of Perturbed Motion
  • 1.3 Examples of Derivation of Equations of a Perturbed Motion
  • 1.4 Problems
  • 2 The Direct Liapunov Method. Autonomous Systems
  • 2.1 Liapunov Functions. Sylvester’s Criterion
  • 2.2 Liapunov’s Theorem of Motion Stability
  • 2.3 Theorems of Asymptotic Stability
  • 2.4 Motion Instability Theorems
  • 2.5 Methods of Obtaining Liapunov Functions
  • 2.6 Application of Liapunov’s Theorem
  • 2.7 Application of Stability Theorems
  • 2.8 Problems
  • 3 Stability of Equilibrium States and Stationary Motions of Conservative Systems
  • 3.1 Lagrange’s Theorem
  • 3.2 Invertibility of Lagrange’s Theorem
  • 3.3 Cyclic Coordinates. The Routh Transform
  • 3.4 Stationary Motion and Its Stability Conditions
  • 3.5 Examples
  • 3.6 Problems
  • 4 Stability in First Approximation
  • 4.1 Formulation of the Problem
  • 4.2 Preliminary Remarks
  • 4.3 Main Theorems of Stability in First Approximation
  • 7.6 Stability of Solutions of Mathieu-Hill Equations
  • 7.7 Examples of Stability Analysis
  • 7.8 Problems
  • 8 Application of the Direct Method of Liapunov to the Investigation of Automatic Control Systems
  • 8.1 Introduction
  • 8.2 Differential Equations of Perturbed Motion of Automatic Control Systems
  • 8.3 Canonical Equations of Perturbed Motion of Control Systems
  • 8.4 Constructing Liapunov Functions
  • 8.5 Conditions of Absolute Stability
  • 9 The Frequency Method of Stability Analysis
  • 9.1 Introduction
  • 9.2 Transfer Functions and Frequency Characteristics
  • 9.3 The Nyquist Stability Criterion for a Linear System
  • 9.4 Stability of Continuously Nonlinear Systems
  • 9.5 Examples
  • 9.6 Problems
  • References
  • 4.4 Hurwitz’s Criterion
  • 4.5 Examples.-4.6 Problems
  • 5 Stability of Linear Autonomous Systems
  • 5.1 Introduction
  • 5.2 Matrices and Basic Matrix Operations
  • 5.3 Elementary Divisors
  • 5.4 Autonomous Linear Systems
  • 5.5 Problems
  • 6 The Effect of Force Type on Stability of Motion
  • 6.1 Introduction
  • 6.2 Classification of Forces
  • 6.3 Formulation of the Problem
  • 6.4 The Stability Coefficients
  • 6.5 The Effect of Gyroscopic and Dissipative Forces
  • 6.6 Application of the Thomson-Tait-Chetaev Theorems
  • 6.7 Stability Under Gyroscopic and Dissipative Forces
  • 6.8 The Effect of Nonconservative Positional Forces
  • 6.9 Stability in Systems with Nonconservative Forces
  • 6.10 Problems
  • 7 The Stability of Nonautonomous Systems
  • 7.1 Liapunov Functions and Sylvester Criterion
  • 7.2 The Main Theorems of the Direct Method
  • 7.3 Examples of Constructing Liapunov Functions
  • 7.4 System with Nonlinear Stiffness
  • 7.5 Systems with Periodic Coefficients

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