Probability Density and Excursions of Structural Response to Imperfectly Periodic Excitation

Abstract

A single-degree-of-freedom system under periodic excitation with random phase modulation is considered. Probability density functions (PDF) of the response are obtained numerically using the path integration method. Basic results are presented in the form of expected number of excursions over a given displacement level. They clearly illustrate the transformation of the response PDF with increasing excitation/system bandwidth ratio from one corresponding to the sinusoid with random phase at small values to asymptotically Gaussian PDF at high values of the above ratio. While this qualitative trend is known from previous analysis of the response excess factor by the method of moments, the present quantitative results may be of direct use for reliability predictions. An analytical study is also made for reduced stochastic differential equations of motion, as obtained by stochastic averaging, by the method of moments resulting in a simple explicit expression for the mean square amplitude.