Lambda-Matrices and Vibrating Systems

This text covers several aspects and solutions of the problems of linear vibrating systems with a finite number of degrees of freedom. It offers a detailed account of the part of the theory of matrices necessary for efficient problem-solving, beginning with the first four chapters' focus on developing the necesary tools in matrix theory. The following chapters present numerical procedures for the relevant matrix formulations and the relevant theory of differential equations. The book is directed toward a wide audience of applied mathematicans, scientists, and engineers who are interested in these problems from either practical or theoretical points of view. Although mathematically sound, the treatment involves readers in a minimum of mathematical abstraction. A familiarity and facility with matrix theory is assumed, along with a knowlege of elementary calculus, including the rudiments of the theory of functions of a complex variable. Unabridged republication of the edition published by Pergamon Press, Oxford, 1966. Preface. Bibliographical Notes. References. Index.

Preface to the Dover Edition

xi

Preface

xvii

A Sketch of Some Matrix Theory

1

(22)

Definitions

1

(2)

Column and Row Vectors

3

(1)

Square Matrices

4

(3)

Linear Dependence, Rank, and Degeneracy

7

(1)

Special Kinds of Matrices

8

(2)

Matrices Dependent on a Scalar Parameter; Latent Roots and Vectors

10

(1)

Eigenvalues and Vectors

11

(3)

Equivalent Matrices and Similar Matrices

14

(4)

The Jordan Canonical Form

18

(2)

Bounds for Eigenvalues

20

(3)

Regular Pencils of Matrices and Eigenvalue Problems

23

(19)

Introduction

23

(1)

Orthogonality Properties of the Latent Vectors

24

(3)

The Inverse of a Simple Matrix Pencil

27

(1)

Application to the Eigenvalue Problem

28

(5)

The Constituent Matrices

33

(2)

Conditions for a Regular Pencil to be Simple

35

(3)

Geometric Implications of the Jordan Canonical Form

38

(1)

The Rayleigh Quotient

39

(1)

Simple Matrix Pencils with Latent Vectors in Common

40

(2)

Lambda-Matrices, I

42

(14)

Introduction

42

(1)

A Canonical Form for Regular λ-Matrices

43

(2)

Elementary Divisors

45

(2)

Division of Square λ-Matrices

47

(2)

The Cayley-Hamilton Theorem

49

(1)

Decomposition of λ-Matrices

50

(3)

Matrix Polynomials with a Matrix Argument

53

(3)

Lambda-Matrices, II

56

(19)

Introduction

56

(1)

An Associated Matrix Pencil

56

(3)

The Inverse of a Simple λ-Matrix in Spectral Form

59

(5)

Properties of the Latent Vectors

64

(3)

The Inverse of a Simple λ-Matrix in Terms of its Adjoint

67

(1)

Lambda-matrices of the Second Degree

68

(3)

A Generalization of the Rayleigh Quotient

71

(2)

Derivatives of Multiple Eigenvalues

73

(2)

Some Numerical Methods for Lambda-Matrices

75

(25)

Introduction

75

(2)

A Rayleigh Quotient Iterative Process

77

(2)

Numerical Example for the RQ Algorithm

79

(2)

The Newton-Raphson Method

81

(1)

Methods Using the Trace Theorem

82

(4)

Iteration of Rational Functions

86

(3)

Behavior at Infinity

89

(1)

A Comparison of Algorithms

90

(2)

Algorithms for a Stability Problem

92

(3)

Illustration of the Stability Algorithms

95

(5)

Appendix to Chapter 5

98

(2)

Ordinary Differential Equations with Constant Coefficients

100

(16)

Introduction

100

(1)

General Solutions

101

(7)

The Particular Integral when f(t) is Exponential

108

(1)

One-point Boundary Conditions

109

(2)

The Laplace Transform Method

111

(3)

Second Order Differential Equations

114

(2)

The Theory of Vibrating Systems

116

(27)

Introduction

116

(1)

Equations of Motion

117

(5)

Solutions under the Action of Conservative Restoring Forces Only

122

(2)

The Inhomogeneous Case

124

(1)

Solutions Including the Effects of Viscous Internal Forces

125

(5)

Overdamped Systems

130

(5)

Gyroscopic Systems

135

(2)

Sinusoidal Motion with Hysteretic Damping

137

(1)

Solutions for Some Non-conservative Systems

138

(2)

Some Properties of the Latent Vectors

140

(3)

On the Theory of Resonance Testing

143

(15)

Introduction

143

(1)

The Method of Stationary Phase

144

(4)

Properties of the Proper Numbers and Vectors

148

(4)

Determination of the Natural Frequencies

152

(1)

Determination of the Natural Modes

153

(5)

Appendix to Chapter 8

156

(2)

Further Results for Systems with Damping

158

(26)

Preliminaries

158

(2)

Global Bounds for the Latent Roots when B is Symmetric

160

(2)

The Use of Theorems on Bounds for Eigenvalues

162

(6)

Preliminary Remarks on Perturbation Theory

168

(3)

The Classical Perturbation Technique for Light Damping

171

(3)

The Case of Coincident Undamped Natural Frequencies

174

(4)

The Case of Neighboring Undamped Natural Frequencies

178

(6)

Bibliographical Notes

184

(3)

References

187

(4)

Index

191

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Lambda-Matrices and Vibrating Systems

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