On the background and biresonant components of the random response of single degree-of-freedom systems under non-Gaussian random loading

The variance of the response of a single degree-of-freedom system subjected to a low frequency excitation is usually decomposed into its background and resonant contributions. In this paper we aim at the formulation of such a decomposition for the third statistical moment, which should in principle sidestep the heavy double integration of the bispectrum of the response. In large finite element models, the estimation of the bispectrum of the loading for a given frequency (ω1,ω2)(ω1,ω2), which is a necessary stage towards estimation of the response, is the most expensive computational task. We therefore formulate the problem with the underlying constraint that the number of estimations of the bispectrum of the force be kept as small as possible. Invoking the perturbation theory in the context of the computation of integrals, we propose to decompose the third moment into background and biresonant components with expressions that are not trivially adapted from the decomposition of the variance. Thanks to the proposed method, the double integration of the bispectrum is avoided and represents accurately the response of lightly to moderately damped structures under a low-frequency loading. The formal expression for the biresonant component still requires an integration that should preferably be avoided. In the second part of the paper, we then investigate the practical implementation of the proposed formula and study the possible application of local, global or hybrid numerical approximations of that remaining integral, so as to further increase the computational efficiency. Finally two numerical experiments illustrate the prospect of the proposed method: the third statistical moment of the response is accurately computed with less than 20 estimates of the bispectrum of the loading, whereas an advanced numerical procedure for the double integration would require a mesh of probably more than (a) thousand(s) points.