An approximate solution scheme for the algebraic random eigenvalue problem

A reduced basis formulation is presented for the efficient solution of large-scale algebraic random eigenvalue problems. This formulation aims to improve the accuracy of the first order perturbation method, and also allow the efficient computation of higher order statistical moments of the eigenparameters. In the present method, the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for Ritz analysis of the governing random eigenvalue problem. This leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest. Since, only two basis vectors are used to represent each eigenvector, explicit expressions for the random eigenvalues and eigenvectors can readily be derived. This enables the statistics of the random eigenparameters and the forced response to be efficiently computed. Numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system. It is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method.