Generalized Linear Random Vibration Analysis Using Auto-Covariance Orthogonal Decomposition

Abstract

Application of a stationary Gaussian random process to describe a nondeterministic forcing function of a linear vibrating system is well studied and documented. Two algorithms, Karhunen-Loeve expansion method, and collocation technique for nonstationary and non-Gaussian forcing processes (narrow- or broadband, but not white noise) for linear dynamic systems are developed here. In the Karhunen-Loeve expansion method, the forcing random process autocovariance is decomposed using the well-known Karhunen-Loeve expansion. In the Karhunen-Loeve expansion, Galerkin projection (the weighted-residual method) and collocation technique (discretized covariance matrix) are used to get the eigenvalues and the eigenfunctions/eigenvectors of the autocovariance function, numerically. The steady-state and the transient response of a single-degree-of-freedom system for an exponential autocovariance (Gaussian random process) is obtained using three methods: 1) analytical, 2) semi-analytical and 3) numerical. In the semi-analytical method, the eigenvalues and the eigenfunctions of the exponential autocovariance function are obtained analytically by solving the Fredholm integral equation of the second kind and those definitions of the eigenfunctions and the eigenvalues are used to obtain the numerical response of the single-degree-of-freedom system. In the case of the steady-state response of the single-degree-of-freedom system, the convergence of the standard deviation of the response is shown to be a function of the number of Karhunen-Loeve expansion terms used in the expansion of the autocovariance of the forcing function. The transient analysis of the same single-degree-of-freedom system is carried out using an exponentially modulated nonstationary process. Comparison of the proposed methods with respect to the analytical solutions are presented for both the stationary and the nonstationary Gaussian excitations. The steady-state responses as well as the transient responses for non-Gaussian random processes (uniform, triangular, and beta) for the same single-degree-of-freedom system are also presented.