Stochastic joint time–frequency response analysis of nonlinear structural systems

A novel approximate analytical approach for determining the response evolutionary power spectrum (EPS) of nonlinear/hysteretic structural systems subject to stochastic excitation is developed. Specifically, relying on the theory of locally stationary processes and utilizing a recently proposed representation of non-stationary stochastic processes via wavelets, a versatile formula for determining the nonlinear system response EPS is derived; this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets. Further, the nonlinear system non-stationary response amplitude probability density function (PDF), which is required as input for the developed approach, is determined either by utilizing a numerical path integral scheme, or by employing a time-dependent Rayleigh PDF approximation technique. A significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non-separable in time and frequency EPS as well. The hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in the numerical examples section. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the approach.