# Lambda-Matrices and Vibrating Systems

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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.

## Table of Contents

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 λ-Matrices

3.3 Elementary Divisors

3.4 Division of Square λ-Matrices

3.5 The Cayley-Hamilton Theorem

3.6 Decomposition of λ-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 λ-Matrix in Spectral Form

4.4 Properties of the Latent Vectors

4.5 The Inverse of a Simple λ-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 Numerical 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 when 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

Other Titles Published in this Series

## Product details

- No. of pages: 310
- Language: English
- Copyright: © Pergamon 1966
- Published: January 1, 1966
- Imprint: Pergamon
- eBook ISBN: 9781483150963

## About the Author

### Peter Lancaster

#### Affiliations and Expertise

Dept. of Mathematics and Statistics, University of Calgary

## About the Editors

### I. N. Sneddon

### M. Stark

### J. P. Kahane

## Ratings and Reviews

### Latest reviews

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**Michael S.**Mon Jan 17 2022**Classic but still interesting**Classic but still interesting