Lambda-Matrices and Vibrating Systems

by Peter Lancaster, Mathematics

Overview

Features aspects and solutions of problems of linear vibrating systems with a finite number of degrees of freedom. Starts with development of necessary tools in matrix theory, followed by numerical procedures for relevant matrix formulations and relevant theory of differential equations. Minimum of mathematical abstraction; assumes a familiarity with matrix theory, elementary calculus. 1966 edition.

Product Details

ISBN-13:	9780486425467
Publisher:	Dover Publications
Publication date:	12/20/2002
Series:	Dover Books on Mathematics
Edition description:	DOVER
Pages:	224
Product dimensions:	5.50(w) x 8.56(h) x 0.47(d)

Table of Contents

PREFACE TO THE DOVER EDITION
PREFACE
CHAPTER 1. A SKETCH OF SOME MATRIX THEORY
1.1 Definitions
1.2 Column and Row Vectors
1.3 Square Matrices
1.4 "Linear Dependence, Rank, and Degeneracy"
1.5 Special Kinds of Matrices
1.6 Matrices Dependent on a Scalar Parameter; Latent Roots and Vectors
1.7 Eigenvalues and Vectors
1.8 Equivalent Matrices and Similar Matrices
1.9 The Jordan Canonical Form
1.10 Bounds for Eigenvalues
CHAPTER 2. REGULAR PENCILS OF MATRICES AND EIGENVALUE PROBLEMS
2.1 Introduction
2.2 Orthogonality Properties of the Latent Vectors
2.3 The Inverse of a Simple Matrix Pencil
2.4 Application to the Eigenvalue Problem
2.5 The Constituent Matrices
2.6 Conditions for a Regular Pencil to be Simple
2.7 Geometric Implications of the Jordan Canonical Form
2.8 The Rayleigh Quotient
2.9 Simple Matrix Pencils with Latent Vectors in Common
"CHAPTER 3. LAMBDA-MATRICES, I"
3.1 Introduction
3.2 A Canonical Form for Regular ?-Matrices
3.3 Elementary Divisors
3.4 Division of Square ?-Matrices
3.5 The Cayley-Hamilton Theory
3.6 Decomposition of ?-Matrices
3.7 Matrix Polynomials with a Matrix Argument
"CHAPTER 4. LAMBDA-MATRICES, II"
4.1 Introduction
4.2 An Associated Matrix Pencil
4.3 The Inverse of a Simple ?-Matrix in Spectral Form
4.4 Properties of the Latent Vectors
4.5 The Inverse of a Simple ?-Matrix in Terms of its Adjoint
4.6 Lambda-matrices of the Second Degree
4.7 A Generalization of the Rayleigh Quotient
4.8 Derivatives of Multiple Eigenvalues
CHAPTER 5. SOME NUMBERICAL METHODS FOR LAMBDA-MATRICES
5.1 Introduction
5.2 A Rayleigh Quotient Iterative Process
5.3 Numerical Example for the RQ Algorithm
5.4 The Newton-Raphson Method
5.5 Methods Using the Trace Theorem
5.6 Iteration of Rational Functions
5.7 Behavior at Infinity
5.8 A Comparison of Algorithms
5.9 Algorithms for a Stability Problem
5.10 Illustration of the Stability Algorithms
  APPENDIX to Chapter 5
CHAPTER 6. ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS
6.1 Introduction
6.2 General Solutions
6.3 The Particular Integral with f(t) is Exponential
6.4 One-point Boundary Conditions
6.5 The Laplace Transform Method
6.6 Second Order Differential Equations
CHAPTER 7. THE THEORY OF VIBRATING SYSTEMS
7.1 Introduction
7.2 Equations of Motion
7.3 Solutions under the Action of Conservative Restoring Forces Only
7.4 The Inhomogeneous Case
7.5 Solutions Including the Effects of Viscous Internal Forces
7.6 Overdamped Systems
7.7 Gyroscopic Systems
7.8 Sinusoidal Motion with Hysteretic Damping
7.9 Solutions for Some Non-conservative Systems
7.10 Some Properties of the Latent Vectors
CHAPTER 8. ON THE THEORY OF RESONANCE TESTING
8.1 Introduction
8.2 The Method of Stationary Phase
8.3 Properties of the Proper Numbers and Vectors
8.4 Determination of the Natural Frequencies
8.5 Determination of the Natural Modes
  APPENDIX to Chapter 8
CHAPTER 9. FURTHER RESULTS FOR SYSTEMS WITH DAMPING
9.1 Preliminaries
9.2 Global Bounds for the Latent Roots when B is Symmetric
9.3 The Use of Theorems on Bounds for Eigenvalues
9.4 Preliminary Remarks on Perturbation Theory
9.5 The Classical Perturbation Technique for Light Damping
9.6 The Case of Coincident Undamped Natural Frequencies
9.7 The Case of Neighboring Undamped Natural Frequencies
BIBLIOGRAPHICAL NOTES
REFERENCES
INDEX