33.1 Introduction

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In modern engineering problems, the dynamic design of structures is becoming increasingly important.

In order to achieve an optimal design, we repeatedly have to modify the structural parameters and solve

the generalized eigenvalue problem. The iterative vibration analysis can be very tedious for large and

complex structures. Therefore, it is necessary to seek a fast computation method for sensitivity analysis

and reanalysis. The matrix perturbation method is an extremely useful tool for this purpose.

The matrix perturbation method is concerned with how the natural frequencies and modal vectors

change if small modifications are imposed on the parameters of structures. Engineering problems often

involve many small modifications in the structural parameters, such as material property variations,

manufacturing errors, iterative design of structural parameters, design sensitivity analysis, random

eigenvalue analysis, robustness analysis of control systems, and so on.

In this chapter, it is assumed that the reader has an undergraduate knowledge in vibration theory

(see Chapter 1 to Chapter 5, and Chapter 14) and a working knowledge in the finite element method

(see Chapter 9).

The contents of the chapter include the basic preliminaries: vibration equations of the finite element

model, eigenvalue problem, modal vectors, orthogonality conditions, modal expansion theorem, and

the power series expansion of eigensolutions. The chapter also covers such topics as: the perturbation

method for distinct eigenvalues and corresponding eigenvectors; sensitivities of eigenvalues and

eigenvectors; the high-accuracy modal superposition method for eigenvector derivatives; eigenvector

derivatives for free – free structures; perturbation method for systems with repeated eigenvalues and

close eigenvalues; and perturbation method of the complex modes of systems with real unsymmetric

matrices.

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