Random matrix eigenvalue problems in structural dynamics

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Abstract
Natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems. Considering that the system parameters are known only probabilistically, we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems. Current methods to deal with such problems are dominated by mean‐centred perturbation‐based methods. Here two new approaches are proposed. The first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is ‘best’ in some sense. The second approach is based on an asymptotic approximation of multidimensional integrals. A closed‐form expression is derived for a generalrth‐order moment of the eigenvalues. Two approaches are presented to obtain the probability density functions of the eigenvalues. The first is based on the maximum entropy method and the second is based on a chi‐square distribution. Both approaches result in simple closed‐form expressions which can be easily calculated. The proposed methods are applied to two problems and the analytical results are compared with Monte Carlo simulations. It is expected that the ‘small randomness’ assumption usually employed in mean‐centred‐perturbation‐based methods can be relaxed considerably using these methods. Copyright © 2006 John Wiley & Sons, Ltd.