Response Characteristics of a Linear System under Non-Gaussian Random Excitation using Polyspectra

Abstract

We consider a linear system under non-Gaussian excitation and give an insight into the effect of the non-Gaussianity of the excitation on the response based on the bispectrum. A bispectrum is defined as a double Fourier transform of the 3rd-order cumulant and regarded as an extension of a traditional power spectral density. The bispectrum of the response can be written with the bispectrum of the excitation and the frequency response function of the system. Furthermore, by integrating the bispectum of the response, we can obtain the 3rd-order moments of the displacement and velocity responses. By using these properties, we can calculate the 3rd-order moments of the displacement and velocity responses from the bispectrum of the excitation. The non-Gaussianity of the response is evaluated with skewness, which is the 3rd-order moment standardized by the variance to the power of 1.5. We adopt the Cai and Lin’s model as a non-Gaussian excitation. The model includes a bandwidth parameter α. We derive the bispectrum of the Cai and Lin’s model, and calculate the skewness of the displacement and velocity responses for a variety of α. When α is small, the skewness of the displacement response is closed to the skewness of the excitation, and as α gets larger, the skewness of the displacement response gets close to 0. The skewness of the velocity response is always close to 0 independent of α. We discuss these relationships between α and the skewness of the response based on the shapes of the bispectra of the excitation and response.