Dynamic analysis of stochastic structural systems using frequency adaptive spectral functions

A novel Galerkin subspace projection scheme for linear structural dynamic systems with stochastic coefficients is developed in this paper. The fundamental idea is to solve a discretized stochastic system in the frequency domain by projecting the solution on a reduced subspace of eigenvectors of the deterministic operator weighted by a set of frequency dependent stochastic functions, termed as the spectral functions. These spectral functions are rational functions of the input random variables and a study of the different orders of spectral functions are presented. A set of undetermined Galerkin coefficients are utilized to orthogonalize the residual to the reduced eigenvector space in the mean sense. The complex system response is represented explicitly with these Galerkin coefficients in conjunction with the modal basis and the associated stochastic spectral functions. The statistical moments of the solution are evaluated at all frequencies and the solution accuracy is verified in terms of a relative error norm. Two examples involving a beam and a plate with stochastic parameters subjected to harmonic excitations have been studied. The results are compared with the direct Monte-Carlo simulation, the classical Neumann expansion technique and the polynomial chaos method for different orders stochastic functions and varying degrees of variability of input randomness.