Harmonic wavelets based statistical linearization for response evolutionary power spectrum determination

A novel harmonic wavelets 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. Several numerical results are presented in this context. Next, focusing on the case of the stochastic response of a linear system and relying on the orthogonality properties of the developed representation an excitation-response EPS relationship is derived. It is further shown that the excitation-response EPS relationship is valid even for linear time-variant (LTV) systems since the approach possesses inherently the element of time-dependence. Further, an extension via statistical linearization of the input-output EPS relationship for the case of a nonlinear system is developed. 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 system response. Pertinent Monte Carlo simulations demonstrate the reliability and versatility of the approach.