Approximate analysis of higher cumulants for multi-degree-of-freedom random vibration

Abstract

A computationally efficient methodology is presented for calculating the higher-order cumulants of the response of a linear system subjected to stationary or non- stationary stochastic excitation. The technique of state space analysis is used to derive, from the original linear system, a new system of ordinary differential equations governing the evolution of cumulants of various response variables. Complex modal analysis is employed to uncouple the new system and obtain the approximate cumulants in a reduced space of modal coordinates. An approximate methodology is developed to obtain simplified analytical solutions for some of the contributing modes by neglecting secondary dynamical effects in the corresponding modal equations. Valuable insight into the important factors affecting the reliability of such approximations is provided. The derivation and storage of the large matrix describing the system of cumulants is avoided. Instead, one uses only the eigenvalues and eigenvectors of the matrix describing the system of cumulants and of its transpose, and in the approximate analysis only the first few of these eigensolutions are usually needed. These eigensolutions are efficiently obtained from the eigensolutions of two much smaller matrices. The approximate methodology reduces drastically the computer storage and the computational effort required to solve for the response cumulants of any order. The performance and accuracy of the methododology are illustrated by examples.