Application of the Wiener series to the frequency response estimation of a structural system excited through a bilinear device

Analytical or numerical methods, such as Finite Element Analysis (FEA), can be used to estimate the dynamic response of a system excited with a random input. However, in order to achieve reliable results, the computational cost is usually expensive. Furthermore, at high frequencies, real systems tend to have statistical responses rather than deterministic, and energy techniques, such as the Statistical Energy Analysis (SEA), can be used to estimate the averaged response over an ensemble of nominally identical components. Additionally, it is known that the response is dependent on the linearity of the path through which the force exerts on the system, and the analytical solution of a nonlinear equation of motion might not be always available. In this work, the Wiener series have been used to derive analytical expressions of the first and second order contributions to the response of a nonlinear structural system in the frequency domain. A flat thin plate excited by Gaussian white noise through a bilinear spring has been used as a case study. Following an SEA approach, the structural system has been modelled as a dissipative mechanism. The first order Wiener kernel is found to be equivalent to the transfer function of a linear system, and can be used to compute the first order contribution of the response; whereas the second order Wiener kernel is used to estimate the contribution of second order terms, which become significant at higher degrees of nonlinearity. The reconstruction of the total power spectral density of the response of the system, by adding the two contributions estimated by the Wiener theory, is in good agreement with data generated by simulation