Lancaster / Sneddon / Stark Lambda-Matrices and Vibrating Systems

1. Auflage 2014
ISBN: 978-1-4831-5096-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 310 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Lambda-Matrices and Vibrating Systems presents aspects and solutions to problems concerned with linear vibrating systems with a finite degrees of freedom and the theory of matrices. The book discusses some parts of the theory of matrices that will account for the solutions of the problems. The text starts with an outline of matrix theory, and some theorems are proved. The Jordan canonical form is also applied to understand the structure of square matrices. Classical theorems are discussed further by applying the Jordan canonical form, the Rayleigh quotient, and simple matrix pencils with latent vectors in common. The book then expounds on Lambda matrices and on some numerical methods for Lambda matrices. These methods explain developments of known approximations and rates of convergence. The text then addresses general solutions for simultaneous ordinary differential equations with constant coefficients. The results of some of the studies are then applied to the theory of vibration by applying the Lagrange method for formulating equations of motion, after the formula establishing the energies and dissipation functions are completed. The book describes the theory of resonance testing using the stationary phase method, where the test is carried out by applying certain forces to the structure being studied, and the amplitude of response in the structure is measured. The book also discusses other difficult problems. The text can be used by physicists, engineers, mathematicians, and designers of industrial equipment that incorporates motion in the design.

Autoren/Hrsg.

Lancaster, Peter

Sneddon, I. N.

Stark, M.

Kahane, J. P.

Inhaltsverzeichnis

1;Front Cover;1
2;Lambda-Matrices and Vibrating Systems;4
3;Copyright Page;5
4;Table of Contents;8
5;Dedication;6
6;PREFACE;12
7;CHAPTER 1. A SKETCH OF SOME MATRIX THEORY;16
7.1;1.1 DEFINITIONS;16
7.2;1.2 COLUMN AND ROW VECTORS;18
7.3;1.3 SQUARE MATRICES;19
7.4;1.4 LINEAR DEPENDENCE, RANK, AND DEGENERACY;22
7.5;1.5 SPECIAL FINDS OF MATRICES;23
7.6;1.6 MATRICES DEPENDENT ON A SCALAR PARAMETER; LATENT ROOTS AND VECTORS;25
7.7;1.7 EIGENVALIIES AND VECTORS;26
7.8;1.8 EQUIVALENT MATRICES AND SIMILAR MATRICES;29
7.9;1.9 THE JORDAN CANONICAL FORM;33
7.10;1.10 BOUNDS FOR EIGENVALUES;35
8;CHAPTER 2. REGULAR PENCILS OF MATRICES AND EIGENVALUE PROBLEMS;38
8.1;2.1 INTRODUCTION;38
8.2;2.2 ORTHOGONALITY PROPERTIES OF THE LATENT VECTORS;39
8.3;2.3 THE INVERSE OF A SIMPLE MATRIX PENCIL;42
8.4;2.4 APPLICATION TO THE EIGENVALUE PROBLEM;43
8.5;2.5 THE CONSTITUENT MATRICES;48
8.6;2.6 CONDITIONS FOR A REGULAR PENCIL TO BE SIMPLE;50
8.7;2.7 GEOMETRIC IMPLICATIONS OF THE JORDAN CANONICAL FORM;53
8.8;2.8 THE RAYLEIGH QUOTIENT;54
8.9;2.9 SIMPLE MATRIX PENCILS WITH LATENT VECTORS IN COMMON;55
9;CHAPTER 3. LAMBDA-MATRICES, I;57
9.1;3.1 INTRODUOTION;57
9.2;3.2 A CANONICAL FORM FOR REGULAR .-MATRICES;58
9.3;3.3 ELEMENTARY DIVISORS;60
9.4;3.4 DIVISION OF SQUARE .-MATRICES;62
9.5;3.5 THE CAYLEY-HAMILTON THEOREM;64
9.6;3.6 DECOMPOSITION OF .-MATRICES;65
9.7;3.7 MATRIX POLYNOMIALS WITH A MATRIX ARGUMENT;68
10;CHAPTER 4. LAMBDA-MATRICES, II;71
10.1;4.1 INTRODUCTION;71
10.2;4.2 AN ASSOCIATED MATRIX PENCIL;71
10.3;4.3 THE INVERSE OF A SIMPLE .-MATRIX IN SPECTRAL FORM;74
10.4;4.4 PROPERTIES OF THE LATENT VECTORS;79
10.5;4.5 THE INVSRRSE OF A SIMPLE .-MATRIX IN TERMS OF ITS ADJOINT;82
10.6;4.6 LAMBDA-MATRICES OF TIIE SECOND DEGREE;83
10.7;4.7 A GENERALIZATION OF THE RAYLEIGH QUOTIENT;86
10.8;4.8 DERIVATIVES OF MULTIPLE EIGENVALUES;88
11;CHAPTER 5. SOME NUMERICAL METHODS FOR LAMBDA-MATRICES;90
11.1;5.1 INTRODUCTION;90
11.2;5.2 A RAYLEIGH QUOTIENT ITERATIVE PROCESS;92
11.3;5.3 NUMERICAL EXAMPLE FOR THE RQ ALGORITHM;94
11.4;5.4 THE NEWTON-RAPHSON METHOD;96
11.5;5.5 METHODS USING THE TRACE THEOREM;97
11.6;5.6 ITERATION OF RATIONAL FUNCTIONS;101
11.7;5.7 BEHAVIOR AT INFINITY;104
11.8;5.8 A COMPARISON OF ALGORITHMS;105
11.9;5.9 ALGORITHMS FOR A STABILITY PROBLEM;107
11.10;5.10. ILLUSTRATION OF THE STABILITY ALGORITHMS;110
11.11;APPENDIX TO CHAPTER 5;113
12;CHAPTER. 6 ORDINARY DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS;115
12.1;6.1 INTRODIICTION;115
12.2;6.2 GENERAL SOLUTIONS;116
12.3;6.3 THE PARTICULAR INTEGRAL WHEN f(t) IS EXPONENTIAL;123
12.4;6.4 ONE-POINT BOUNDARY CONDITIONS;124
12.5;6.5 THE LAPLACE TRANSFORM METHOD;126
12.6;6.6 SECOND ORDER DIFFERENTIAL EQUATIONS;129
13;CHAPTER 7. THE THEORY OF VIBRATING SYSTEMS;131
13.1;7.1 INTRODUCTION;131
13.2;7.2 EQUATIONS OF MOTION;132
13.3;7.3 SOLUTIONS UNDER THE ACTION OF CONSERVATIVE RESTORING FORCES ONLY;137
13.4;7.4 THE INHOMOGENEOUS CASE;139
13.5;7.5 SOLUTIONS INCLUDING THS EFFECTS OF VISCOUS INTERNAL FORCES;140
13.6;7.6 OVERDAMPED SYSTEMS;145
13.7;7.7 GYROSCOPIC SYSTEMS;150
13.8;7.8 SINUSOIDAL MOTION WITH HYSTERETIC DAMPING;152
13.9;7.9 SOLUTIONS FOR SOME NON-CONSERVATIVE SYSTEMS;153
13.10;7.10 SOME PROPERTIES OF THE LATENT VECTORS;155
14;CHAPTER 8. ON THE THEORY OF RESONANCE TESTING;158
14.1;8.1 INTRODUCTION;158
14.2;8.2 THE METHOD OF STATIONARY PHASE;159
14.3;8.3 PROPERTIES OF THE PROPER NUMBERS AND VECTORS;163
14.4;8.4 DETERMINATION OF THE NATURAL FREQUENCIES;167
14.5;8.5 DETERMINATION OF THE NATURAL MODES;168
14.6;APPENDIX TO CHAPTER 8;171
15;CHAPTER 9. FURTHER RESULTS FOR SYSTEMS WITH DAMPING;173
15.1;9.1 PRELIMINARIES;173
15.2;9.2 GLOBAL BOUNDS FOR THE LATENT ROOTS WHEN BIS SYMMETRIC;175
15.3;9.3 THE USE OF THEOREMS ON BOUNDS FOR EIGENVALUES;177
15.4;9.4 PRELIMINARY REMARKS ON PSRTDRBATION THEORY;183
15.5;9.5 THE CLASSICAL PERTURBATION TECHNIQUE FOR LIGHT DAMPING;186
15.6;9.6 THE CASE OF COINCIDENT UNDAMPED NATURAL FREQUENCIES;189
15.7;9.7 THE CASE OF NEIGHBORING UNDAMPED NATURAL FREQUENCIES;193
16;BIBLIOGRAPHICAL NOTES;199
17;REFERENCES;202
18;INDEX;206
19;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;209