Distribution of Eigenvalues of Linear Stochastic Systems

Dynamic characteristics of linear structural systems are governed by the natural frequencies and the mode-shapes. In this paper the statistical properties of the eigenvalues of linear dynamic systems are considered. It is assumed that the mass and the stiffness matrices are smooth, continuous and at least twice differentiable functions of a random parameter vector. The random parameter vector is assumed to be standard Gaussian or can be transformed to standard Gaussian. Two approaches are proposed to obtain moments, cumulants and probability density functions of the eigenval-ues. The first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is best is some sense. This optimal point is obtained by using the concepts borrowed from structural reliability analysis. The second approach is based on asymptotic analysis. Moments of the eigenvalues are obtained by asymptotic expansion of a mul-tidimensional integral involving the joint probability density function of the random variables. Based on theses methods, two simple expressions of the probability density functions of the eigenvalues are derived. A numerical example is given to compare the proposed methods with Monte Carlo simulations.