INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2007; 69:562–591 Published online 16 June 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1781 Random matrix eigenvalue problems in structural dynamics S. Adhikari ∗, † and M. I. Friswell Department of Aerospace Engineering, University of Bristol, Bristol BS8 1TR, U.K. SUMMARY 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 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 usually employed in mean-centred-perturbation-based methods can be relaxed considerably using these methods. Copyright 2006 John Wiley & Sons, Ltd. Received 2 August 2005; Revised 7 April 2006; Accepted 19 April 2006 KEY WORDS: random eigenvalue problems; asymptotic analysis; statistical distributions; random matrices; linear stochastic systems 1. INTRODUCTION Characterization of the natural frequencies and mode shapes play a key role in the analysis and design of engineering dynamic systems. The determination of natural frequency and mode shapes require the solution of an eigenvalue problem. Eigenvalue problems also arise in the context of the ∗ Correspondence to: S. Adhikari, Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, U.K. † E-mail: s.adhikari@bristol.ac.uk Contract/grant sponsor: Engineering and Physical Sciences Research Council (EPSRC); contract/grant number: GR/T03369/01 Contract/grant sponsor: Royal Society Copyright 2006 John Wiley & Sons, Ltd.