Introduction to the Theory of Stability

Introduction to the Theory of Stability


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Product Details

ISBN-13: 9781461284772
Publisher: Springer New York
Publication date: 09/17/2011
Series: Texts in Applied Mathematics , #24
Edition description: 1997
Pages: 320
Product dimensions: 6.10(w) x 9.25(h) x 0.03(d)

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.- 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.- 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.

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