Description
DOVER PUBLICATIONS Lambda-Matrices And Vibrating Systems by PETER LANCASTER
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. PREFACE TO THE DOVER EDITIONPREFACECHAPTER 1. A SKETCH OF SOME MATRIX THEORY1.1 Definitions1.2 Column and Row Vectors1.3 Square Matrices1.4 "Linear Dependence, Rank, and Degeneracy"1.5 Special Kinds of Matrices1.6 Matrices Dependent on a Scalar Parameter; Latent Roots and Vectors1.7 Eigenvalues and Vectors1.8 Equivalent Matrices and Similar Matrices1.9 The Jordan Canonical Form1.10 Bounds for EigenvaluesCHAPTER 2. REGULAR PENCILS OF MATRICES AND EIGENVALUE PROBLEMS2.1 Introduction2.2 Orthogonality Properties of the Latent Vectors2.3 The Inverse of a Simple Matrix Pencil2.4 Application to the Eigenvalue Problem2.5 The Constituent Matrices2.6 Conditions for a Regular Pencil to be Simple2.7 Geometric Implications of the Jordan Canonical Form2.8 The Rayleigh Quotient2.9 Simple Matrix Pencils with Latent Vectors in Common"CHAPTER 3. LAMBDA-MATRICES, I"3.1 Introduction3.2 A Canonical Form for Regular ?-Matrices3.3 Elementary Divisors3.4 Division of Square ?-Matrices3.5 The Cayley-Hamilton Theory3.6 Decomposition of ?-Matrices3.7 Matrix Polynomials with a Matrix Argument"CHAPTER 4. LAMBDA-MATRICES, II"4.1 Introduction4.2 An Associated Matrix Pencil4.3 The Inverse of a Simple ?-Matrix in Spectral Form4.4 Properties of the Latent Vectors4.5 The Inverse of a Simple ?-Matrix in Terms of its Adjoint4.6 Lambda-matrices of the Second Degree4.7 A Generalization of the Rayleigh Quotient4.8 Derivatives of Multiple EigenvaluesCHAPTER 5. SOME NUMBERICAL METHODS FOR LAMBDA-MATRICES5.1 Introduction5.2 A Rayleigh Quotient Iterative Process5.3 Numerical Example for the RQ Algorithm5.4 The Newton-Raphson Method5.5 Methods Using the Trace Theorem5.6 Iteration of Rational Functions5.7 Behavior at Infinity5.8 A Comparison of Algorithms5.9 Algorithms for a Stability Problem5.10 Illustration of the Stability Algorithms APPENDIX to Chapter 5CHAPTER 6. ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS6.1 Introduction6.2 General Solutions6.3 The Particular Integral with f(t) is Exponential6.4 One-point Boundary Conditions6.5 The Laplace Transform Method6.6 Second Order Differential EquationsCHAPTER 7. THE THEORY OF VIBRATING SYSTEMS7.1 Introduction7.2 Equations of Motion7.3 Solutions under the Action of Conservative Restoring Forces Only7.4 The Inhomogeneous Case7.5 Solutions Including the Effects of Viscous Internal Forces7.6 Overdamped Systems7.7 Gyroscopic Systems7.8 Sinusoidal Motion with Hysteretic Damping7.9 Solutions for Some Non-conservative Systems7.10 Some Properties of the Latent VectorsCHAPTER 8. ON THE THEORY OF RESONANCE TESTING8.1 Introduction8.2 The Method of Stationary Phase8.3 Properties of the Proper Numbers and Vectors8.4 Determination of the Natural Frequencies8.5 Determination of the Natural Modes APPENDIX to Chapter 8CHAPTER 9. FURTHER RESULTS FOR SYSTEMS WITH DAMPING9.1 Preliminaries9.2 Global Bounds for the Latent Roots when B is Symmetric9.3 The Use of Theorems on Bounds for Eigenvalues9.4 Preliminary Remarks on Perturbation Theory9.5 The Classical Perturbation Technique for Light Damping9.6 The Case of Coincident Undamped Natural Frequencies9.7 The Case of Neighboring Undamped Natural FrequenciesBIBLIOGRAPHICAL NOTESREFERENCESINDEX