Extended Wittrick–Williams algorithm for eigenvalue solution of stochastic dynamic stiffness method

Abstract

This paper proposes an efficient and reliable eigenvalue solution technique for analytical stochastic dynamic stiffness (SDS) formulations of beam built-up structures with parametric uncertainties. The SDS formulations are developed based on frequency-dependent shape functions in conjunction with both random-variable and random-field structural parameters. The overall numerical framework is aimed towards representing the broadband dynamics of structures using very few degrees of freedom. This paper proposes a novel approach combining the Wittrick–Williams(WW) algorithm, the Newton iteration method and numerical perturbation method to extract eigensolutions from SDS formulations. First, the eigenvalues and eigenvectors of the deterministic DS formulations are computed by the WW algorithm and the corresponding mode finding technique, which are used as the initial solution. Then, a numerical perturbation technique based on the inverse iteration and homotopy method is proposed to update the eigenvectors and eigenvalues. The robustness and efficiency of the proposed method are guaranteed through several technique arrangements. Through numerical examples, the proposed method is demonstrated to be robust within the whole frequency range. This method provides an efficient and reliable tool for stochastic analysis of eigenvalue problems relevant to free vibration and buckling analysis of built-up structures.