An Approximate Approach for Nonlinear System Evolutionary Response Spectrum Determination via Wavelets

Abstract

A novel harmonic wavelet-based statistical linearization approach is proposed for determining the evolutionary power spectrum (EPS) of the response of nonlinear oscillators subject to stochastic excitation. Specifically, first a mathematically rigorous wavelet-based representation of non-stationary stochastic processes is presented. Next, a representation of the process corresponding to a specific scale and translation level is derived. This procedure leads to an EPS estimation approach which is applicable for estimating not only separable but non-separable in time and frequency EPS as well. Next, focusing on the case of the stochastic response of a nonlinear system and relying on the orthogonality properties of the developed representation an excitation-response EPS relationship is derived via statistical linearization. The approach involves the concept of assigning optimal and response dependent equivalent stiffness and damping elements corresponding to the specific frequency and time bands. This leads to an iterative determination of the EPS of the oscillator response. Pertinent Monte Carlo simulations demonstrate the reliability and versatility of the approach.