Hybrid perturbation-Polynomial Chaos approaches to the random algebraic eigenvalue problem

The analysis of structures is affected by uncertainty in the structure’s material properties, geometric parameters, boundary conditions and applied loads. These uncertainties can be modelled by random variables and random fields. Amongst the various problems affected by uncertainty, the random eigenvalue problem is specially important when analyzing the dynamic behavior or the buckling of a structure. The methods that stand out in dealing with the random eigenvalue problem are the perturbation method and methods based on Monte Carlo Simulation. In the past few years, methods based on Polynomial Chaos (PC) have been developed for this problem, where each eigenvalue and eigenvector are represented by a PC expansion. In this paper four variants of a method hybridizing perturbation and PC expansion approaches are proposed and compared. The methods use Rayleigh quotient, the power method, the inverse power method and the eigenvalue equation. PC expansions of eigenvalues and eigenvectors are obtained with the proposed methods. The new methods are applied to the problem of an Euler Bernoulli beam and a thin plate with stochastic properties.