Generalized Linear Random Vibration Analysis Using Autocovariance Orthogonal Decomposition

Abstract

Application of a stationary Gaussian Random process to describe a non-deterministic forcing function of a linear vibrating system is well studied and documented. Two al-gorithms: 1) KL Expansion Method, and 2) Collocation Technique for non-stationary and non-Gaussian forcing processes (narrow or broad band, but not a white noise) for lin-ear dynamic systems are developed here. In the KL Expansion Method, the forcing ran-dom process auto-covariance is decomposed using the well-known Karhunen-Loeve(KL) expansion. In the KL expansion, Galerkin projection (the weighted-residual method) and Collocation Technique (discretized covariance matrix) are used to get the eigenvalues and the eigenvectors of the auto-covariance function, numerically. The steady-state and the transient response of a single-degree-of-freedom (SDOF) system for an exponential auto-covariance (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 eigenvectors of the exponential auto-covariance function are obtained analytically by solving the Fredholm Integral equation of the 2nd kind and those analytical definitions of the eigenvectors and the eigenvalues are used to obtain the numerical response of the SDOF

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