Lambda-Matrices and Vibrating Systems: (Dover Books on Mathematics)

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 a focus on developing the necessary tools in matrix theory in the first four chapters. The following chapters present numerical procedures for the relevant matrix formulations and the relevant theory of differential equations.Directed toward a wide audience of applied mathematicians, scientists, and engineers, this book has much to offer all those interested in problem-solving from both practical and theoretical points of view. The mathematically sound treatment involves readers in a minimum of mathematical abstraction; it assumes a familiarity and facility with matrix theory, along with a knowledge of elementary calculus (including the rudiments of the theory of functions of a complex variable).Those already engaged in the practical analysis of vibrating systems have the option of proceeding directly to the more applications-oriented material, starting with Chapter 7; however, this comprehensive treatment offers ample background in the early chapters for less experienced readers. New Preface to the Dover Edition. Errata List. Preface. Bibliographical Notes. References. Index.

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