Lambda-Matrices and Vibrating Systems

by Lancaster, Peter

• ISBN13: 9780486425467
• ISBN10: 0486425460
• Format: Paperback
• Copyright: 2002-12-20
• Publisher: Dover Publications

Summary

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.

Table of Contents

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