Effects of non-Gaussian random excitation on displacement and velocity responses of a linear system by using bispectrum and cross-bispectrum

Abstract

We consider a linear system under non-Gaussian random excitation modeled by the Cai and Lin's model and investigate the effects of the non-Gaussianity of the excitation on the displacement and velocity responses based on the bispectrum and cross-bispectrum. Bispectrum is one of the higher-order spectra which are generalization of power spectrum. Just as the power spectrum is the distribution of the 2nd-order moment of a stationary stochastic process in the frequency domain, the bispectrum is the distribution of its 3rd-order moment in the frequency domain. Cross-bispectrum is generalization of cross-spectrum, and the cross-bispectrum between the excitation and the response provides the input-output relation of the non-Gaussianity in the frequency domain. The Cai and Lin's model for the excitation can represent a wide variety of non-Gaussian processes observed in various engineering problems. In this paper, we derive the bispectrum of the Cai and Lin's model and show the properties of the non-Gaussianity of the model in the frequency domain. Using the derived bispectrum, we obtain the cross-bispectrum between the excitation and the response, and it reveals that how the effect of the non-Gaussianity of the excitation on the response changes depending on a bandwidth parameter α of the Cai and Lin's model. It is shown that when α is small, the non-Gaussianity of the excitation concentrated on the DC component is mostly transferred to the DC component of the displacement response, but is hardly transferred to the velocity response.On the other hand, when α is large, the power of the resonant frequency component is dominant in the response, but the resonant frequency component hardly transfers or amplifies the non-Gaussianity of the excitation. These findings can only be obtained from the bispectrum and cross-bispectrum, and cannot from the power spectrum.