A matrix perturbation theory in structural dynamic design is presented in this book．The theory covers a broad spectrum of subjects，the perturbation methods of the distinctei genvalues and repeated/close eigenvalues，the perturbation methods of the complexmodes of systems with real unsymmetric matrices，the perturbation methods of thedefective/near defective systems，random eigenproblem and the interval eigenproblem for the uncertain structures．The contents synthesized the most recent research results in the structural dynamiCS．Numerical examples are provided to illustrate the applications of the theory in this book．

A matrix perturbation theory in structural dynamic design is presented in this book．The theory covers a broad spectrum of subjects，the perturbation methods of the distinctei genvalues and repeated/close eigenvalues，the perturbation methods of the complexmodes of systems with real unsymmetric matrices，the perturbation methods of thedefective/near defective systems，random eigenproblem and the interval eigenproblem for the uncertain structures．The contents synthesized the most recent research results in the structural dynamiCS．Numerical examples are provided to illustrate the applications of the theory in this book．

This book is recommended to graduates，engineers and scientist of mechanical，civil，aerospace，ocean and vehicle engineering．

Preface

Chapter 1 Finite Element Method for Vibration Analysis of Structures

 1.1 Introduction

 1.2 The Hamilton Variational Principle for Discrete Svstems

 1.3 Finite Element Method for Structural Vibration nalvsis

 1.4 The Mechanics Characteristic Matrices of E1ements

1.4.1 Consistent Mass Matrix of a Rod E1ement

1.4.2 Consistent Mass Matrix of a Beam E1ement

1.4.3 Plate Element Vibrating in the P1ane

1.4.4 Plate Element in Bending Vibration

1.4.5 Lumped Mass Modal

 1.5 Vibration Eigenproblem of Structures

 1.6 Orthogonality of Modal Vectors

 1.7 The Rayleigh—Ritz Analysis

 1.8 The Response to Harmonic Excitation

 1.9 Response to Arbitrary Excitation

 1.10 Direct Integration Methods for Vibration Enuations

1.10.1 The Central Difference Method

1.10.2 The Wilson Method

1.10.3 The Newmark Method

 1.11 Drect Integration Approximation and Load Operators in Modal Uoordinate System

1.11.1 The Central Difference Method

1.11.2 The Wilson Method

1.11.3 The Newmark Method

Chapter 2 Matrix Perturbation Theory for Distinct Eigenvalues

 2.2 Matrix Perturbation for Distinct Eigenvalues

2.2.1 The 1st Order Perturbation

2.2.2 The 2nd Order Perturbation

2.2.3 Computing for the Expansion Coefficients cl and c2

2.2.4 Numerical Examples

 2.3 The Improvement for Matrix Perturbation

2.3.1 The William BBickford Method

2.3.2 The Mixed Method of Matrix Perturbation and Rayleigh's Quotient

2.3.3 Numerical Example

 2.4 High Accurate Modal Superposition for Derivatives of Modal Vlectors

2.4.1 The BPWang Method

2.4.2 High Accurate Modal Superposition

2.4.3 Numerical Example

 2.5 Mixed Basis Superposition for Eigenvector Perturbation

2.5.1 Constructing for Mixed—Basis

2.5.2 The 1st Order Perturbation Using Mixed—Basis Expansion

2.5.3 The 2nd Order Perturbation Using Mixed—Basis Expansion

2.5.4 Numerical Example

 2.6 Eigenvector Derivatives for Free—nee Structures

2.6.1 The Theory Analysis

2.6.2 Effect ofEigenvalue Shift on the Convergent Speed

2.6.3 Numerical Example

 2.7 Extracting Modal Parameters of Free—Free Structures from Modes of Constrained Structures Using Matrix Perturbation

 2.8 Determination of Frequencies and Modes of Free—Free Structures Using Experimental Data for the Constrained Structures

2.8.1 Generalized Stiffness.Mass.and the Response to Harmonic Excitation for Free—Free Structures

2.8.2 Przemieniecki’s Method (Method 1)

2.8.3 Chen—Liu Method (Method 2)

2.8.4 Zhang—Zerva Method (Method 3)

2.8.5 Further Improvement on Zhang-Zerva Method(Method 4)

2.8.6 Numerical Example

 2.9 Response Analysis to Harmonic Excitation Using High Accurate Modal Superposition

2.9.1 High Accurate Modal Superposition(HAMS)

2.9.2 Numerical Examples

2.9.3 Extension of High Accurate Modal Superposition

 2.10 Sensitivity Analysis of Response Using High Accurate Modal Superposition

Chapter 3 Matrix Perturbation Theory for Multiple Eigenvalues

 3.1 Introduction

 3.2 Matrix Perturbation for Multiple Eigenvalues

3.2.1 Basic Equations

3.2.2 Computing for the 1st Order Perturbation of Eigenvalues

3.2.3 Computing for the 1st Order Perturbation of Eigenvectors

 3.3 Approximate Modal Superposition for the 1st Order Perturbation of Eigenvectors of Repeated Eigenvalues

 3.4 High Accurate Modal Superposition for the 1st Order Perturbation of Eigenvectors of Repeated Eigenvalues

 3.5 Exact Method for Computing Eigenvector Derivatives of repeated Eigenvalues

3.5.1 Theoretical Background

3.5.2 A New Method for Computing v

3.5.3 Numerical Example

 3.6 Hu'S Method for Computing the 1st Order Perturbation of Eigenvectors

3.6.1 Hu'S Small Parameter Method

3.6.2 Improved Hu'S Method

Chapter 4 Matrix Perturbation Theory for Close Eigenvalues.

 4.1 Introduction

 4.2 Behavior of Modes of Close Eigenvalues

 4.3 Identification of Modes of Close Eigenvalues

 4.4 Matrix Perturbation for Close Eigenvalues

4.4.1 Preliminary Considerations

4.4.2 Spectral Decomposition of Matrices K and M

4.4.3 Matrix Perturbation for Close Eigenvalues

 4.5 Numerical Example

 4.6 Derivatives of Modes for Close Eigenvalues

Chapter 5 Matrix Perturbation Theory for Complex Modes

 5.1 IntrOduction

 5.2 Basia Equations

 5.3 Matrix Perturbation for Distinct Eigenvalues

5.3.1 Basic Equations of Matrix Perturbation for Complex Modes

5.3.2 The ist Order Perturbation

5.3.3 The 2nd Order Perturbation

5.3.4 Computing for Coefficients C1，Dl，C2 and D2

 5.4 High Accurate Modal Superposition for Eigenvector Derivatives

5.4.1 Improved Modal Superposition

5.4.2 High Accurate Modal Superposition

5.4.3 Numerical Example

 5.5 Matrix Perturbation for Repeated Eigenvalues of Nondefective Systems

5.5.1 Basic Equations

5.5.2 The 1st Order Perturbation of Eigenvalues

5.5.3 The 1st Order Perturbation of Eigenvectors

 5.6 Matrix Perturbation for Close Eigenvalues

5.6.1 Spectral Decomposition of Matrices A and B

5.6.2 Matrix Perturbation for Close Eigenvalues

Chapter 6 Matrix Perturbation Theory for Linear Vibration Defective Systems

 6.1 Introduction

 6.2 Generalized Modal Theory of Defective Systems

 6.3 Singular Value Decomposition(SVD)and Eigensolutions

 6.4 The SVD Method for Modal Analysis of Defective Systems

6.4.1 Rank Analysis for Identification of Defectiveness

6.4.2 SVD Method for Identification of Defectiveness and Modal Analysis

 6.5 Invariant Subspace Recursive Method for Computing the Generalized Modes

6.5.1 Invariant Subspace Recursive Relationship

6.5.2 SVD and Reductive Method for Computing the Orthogonal Basis of Invariant Subspace

6.5.3 Numerical Example

 6.6 Matrix Perturbation for Defective Systems

6.6.1 The Puiseux Expansion for Eigensolutions of Defective Systems

6.6.2 Improved perturbation for Defective Eigenvalues

6.6.3 Numerical Examples

 6.7 Matrix Perturbation for Generalized Eigenproblem of Defective Systems

6.7.1 Perturbation of Defective Eigenvalues

6.7.2 Improved Perturbation for Defective Eigenvalues

6.7.3 Numerical Example

Chapter 7 Matrix Perturbation Theory for Near Defective Systems

 7.1 Introduction

 7.2 Relationship Between Repeated and Close Eigenvalues and Its Identification

7.2.1 Relationship Between Repeated and Close Eigenvalues

7.2.2 Identification for Repeated Eigenvalues

7.2.3 Identification for Close Eigenvalues

 7.3 Matrix Perturbation for Near Defective Systems

7.3.1 Matrix Perturbation for Standard Eigenproblem of Near Defective Systems

7.3.2 Matrix Perturbation for Generalized Eigenproblem of Near Defective Systems

 7.4 Numerical Example

Chapter 8 Random Eigenvalue Analysis of Structures with Random Parameters

 8.1 Introduction

 8.2 Random Finite Element Method for Random Eigenvalue Analysis

 8.3 Random Perturbation for Random Eigenvatue Analysis

 8.4 Statistical Properties of Random Eigensolutions

 8.5 Examples

Chapter 9 Matrix Perturbation Theory for Interval Eigenproblem

 9.1 Introduction

 9.2 ElementS of Interval Mathematics

9.2.1 Interval Algorithm

9.2.2 Interval Vector and Matrix

9.2.3 Interval Extension

 9.3 Interval Eigenproblem

 9.4 The Deif's Method for Interval Eigenvalue Analysis

 9.5 Generalized Deif's Method

 9.6 Matrix Perturbation for Interval Eigenvalue Analysis Based on the Deif's Method

9.6.1 Application of Matrix Perturbation to Interval Eigenvalues

9.6.2 Numerical Example

 9.7 Matrix Perturbation for Interval Eigenproblem

9.7.1 Interval Perturbation Formulation

9.7.2 Numerical Example

References