Random matrix eigenvalue problems in structural dynamics

S Adhikari, MI Friswell

Research output: Contribution to journalArticle (Academic Journal)peer-review

79 Citations (Scopus)

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-centered 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' is some sense. The second approach is based on an asymptotic approximation of multidimensional integrals. A closed-form expression is derived for a general rth 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 unusually employed in mean-centered perturbation based methods can be relaxed considerably using these methods.
Translated title of the contributionRandom matrix eigenvalue problems in structural dynamics
Original languageEnglish
Pages (from-to)562 - 591
Number of pages30
JournalInternational Journal for Numerical Methods in Engineering
Volume69(3)
DOIs
Publication statusPublished - Jan 2007

Bibliographical note

Publisher: Wiley InterScience

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