A multiple timescale approach of bispectral correlation

Abstract

The paper develops an approximate semi-analytical solution for the computation of the third statistical cross-moments of modal responses in a stochastic dynamic analysis. These moments would require heavy twofold numerical integration in a general context but are drastically simplified in the proposed formulation by taking advantage of the assumed distinctness between the low characteristic frequency of the loading and the natural frequencies of the structure. This condition is typically respected and acknowledged in wind engineering where the buffeting analysis of large structures hinges on the Background/Resonant decomposition. As such, the proposed formulation extends to third statistical order the existing developments for the estimation of the modal variances and covariances. It allows the third order spectral analysis of large structures to be conducted within a reasonable amount of time. It also reveals the existence of three main components to the response: background, bi-resonant and tri-resonant. The latter one is specific to this very own problem and is shown to be important when the sum of two natural frequencies is equal to a third one, although the structural behavior is linear. Mathematics highlight this and other findings which are then illustrated on a minimum working example, easily reproducible by readers. Overall, it clearly demonstrates the benefits of the proposed decomposition in terms of both behavioral comprehension and computational consumption.