Lambda-Matrices and Vibrating Systems

by Peter Lancaster

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:	9780486153261
Publisher:	Dover Publications
Publication date:	11/30/2011
Sold by:	Barnes & Noble
Format:	eBook
Pages:	208
File size:	33 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface to the Dover Edition	xi
	Preface	xvii
Chapter 1.	A Sketch of Some Matrix Theory	1
1.1	Definitions	1
1.2	Column and Row Vectors	3
1.3	Square Matrices	4
1.4	Linear Dependence, Rank, and Degeneracy	7
1.5	Special Kinds of Matrices	8
1.6	Matrices Dependent on a Scalar Parameter; Latent Roots and Vectors	10
1.7	Eigenvalues and Vectors	11
1.8	Equivalent Matrices and Similar Matrices	14
1.9	The Jordan Canonical Form	18
1.10	Bounds for Eigenvalues	20
Chapter 2.	Regular Pencils of Matrices and Eigenvalue Problems	23
2.1	Introduction	23
2.2	Orthogonality Properties of the Latent Vectors	24
2.3	The Inverse of a Simple Matrix Pencil	27
2.4	Application to the Eigenvalue Problem	28
2.5	The Constituent Matrices	33
2.6	Conditions for a Regular Pencil to be Simple	35
2.7	Geometric Implications of the Jordan Canonical Form	38
2.8	The Rayleigh Quotient	39
2.9	Simple Matrix Pencils with Latent Vectors in Common	40
Chapter 3.	Lambda-Matrices, I	42
3.1	Introduction	42
3.2	A Canonical Form for Regular [lambda]-Matrices	43
3.3	Elementary Divisors	45
3.4	Division of Square [lambda]-Matrices	47
3.5	The Cayley-Hamilton Theorem	49
3.6	Decomposition of [lambda]-Matrices	50
3.7	Matrix Polynomials with a Matrix Argument	53
Chapter 4.	Lambda-Matrices, II	56
4.1	Introduction	56
4.2	An Associated Matrix Pencil	56
4.3	The Inverse of a Simple [lambda]-Matrix in Spectral Form	59
4.4	Properties of the Latent Vectors	64
4.5	The Inverse of a Simple [lambda]-Matrix in Terms of its Adjoint	67
4.6	Lambda-matrices of the Second Degree	68
4.7	A Generalization of the Rayleigh Quotient	71
4.8	Derivatives of Multiple Eigenvalues	73
Chapter 5.	Some Numerical Methods for Lambda-matrices	75
5.1	Introduction	75
5.2	A Rayleigh Quotient Iterative Process	77
5.3	Numerical Example for the RQ Algorithm	79
5.4	The Newton-Raphson Method	81
5.5	Methods Using the Trace Theorem	82
5.6	Iteration of Rational Functions	86
5.7	Behavior at Infinity	89
5.8	A Comparison of Algorithms	90
5.9	Algorithms for a Stability Problem	92
5.10	Illustration of the Stability Algorithms	95
	Appendix to Chapter 5	98
Chapter 6.	Ordinary Differential Equations with Constant Coefficients	100
6.1	Introduction	100
6.2	General Solutions	101
6.3	The Particular Integral when f(t) is Exponential	108
6.4	One-point Boundary Conditions	109
6.5	The Laplace Transform Method	111
6.6	Second Order Differential Equations	114
Chapter 7.	The Theory of Vibrating Systems	116
7.1	Introduction	116
7.2	Equations of Motion	117
7.3	Solutions under the Action of Conservative Restoring Forces Only	122
7.4	The Inhomogeneous Case	124
7.5	Solutions Including the Effects of Viscous Internal Forces	125
7.6	Overdamped Systems	130
7.7	Gyroscopic Systems	135
7.8	Sinusoidal Motion with Hysteretic Damping	137
7.9	Solutions for Some Non-conservative Systems	138
7.10	Some Properties of the Latent Vectors	140
Chapter 8.	On the Theory of Resonance Testing	143
8.1	Introduction	143
8.2	The Method of Stationary Phase	144
8.3	Properties of the Proper Numbers and Vectors	148
8.4	Determination of the Natural Frequencies	152
8.5	Determination of the Natural Modes	153
	Appendix to Chapter 8	156
Chapter 9.	Further Results for Systems with Damping	158
9.1	Preliminaries	158
9.2	Global Bounds for the Latent Roots when B is Symmetric	160
9.3	The Use of Theorems on Bounds for Eigenvalues	162
9.4	Preliminary Remarks on Perturbation Theory	168
9.5	The Classical Perturbation Technique for Light Damping	171
9.6	The Case of Coincident Undamped Natural Frequencies	174
9.7	The Case of Neighboring Undamped Natural Frequencies	178
	Bibliographical Notes	184
	References	187
	Index	191