Random Eigenvalue Problems in Structural Dynamics

Dynamic characteristics of linear structural systems are governed by the natural frequencies and the mode-shapes. In this paper moments and probability density functions of the eigenvalues of linear stochastic dynamic systems are considered. It is assumed that the mass and the stiffness matrices are smooth and at least twice differentiable functions of a random parameter vector. The random parameter vector is assumed to be non-Gaussian in general. Current methods to solve such problems are dominated by perturbation based methods. Here a new approach based on an asymptotic approximation of the multidimensional integrals is proposed. A closedform 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 fitting of a chi-square random variable. Both approaches result in simple closedform expressions which can be easily calculated. The proposed methods are applied to a three degrees-of-freedom spring-mass system and the results are compared with Monte Carlo simulations. Two different cases, namely (a) when all eigenvalues are well separated, and (b) when some eigenvalues are closely spaced, are considered to illustrate some inherent properties of the methodologies proposed in the paper.