Approximate analytical methods for stationary response probability density of a SDOF system with a nonlinear spring under non-Gaussian random excitation

Abstract

Approximate analytical methods for finding the stationary response probability density of a SDOF system with a wide class of nonlinear springs under non-Gaussian random excitation are proposed. The non-Gaussian excitation is a stationary stochastic process prescribed by an arbitrary probability density with finite variance and a power spectral density with a bandwidth parameter. In order to obtain the response probability distribution of the system, two approximate methods are developed by theoretically extending the Monte Carlo simulation observations on the properties of the response distribution obtained in the previous study. One of the methods is applied when the bandwidth of the excitation power spectrum is narrower than that of the primary resonance peak of the system. In this case, the system becomes quasi-static, and its equation of motion reduces to the equilibrium equation for the stiffness and excitation terms. This equilibrium equation is used to derive the approximate solution of the response distribution. The only requirement of this method is that the nonlinear spring of the system is continuous and monotonically increasing. The other method is proposed for the case of the wide excitation bandwidth compared to the system bandwidth. In this method, the exact solution of the response distribution under Gaussian white noise is utilized. The value of the white noise spectral density parameter in the solution is determined by the equivalent linearization. The second method is applicable to systems with arbitrary integrable nonlinear springs. In numerical examples, we consider three nonlinear systems whose spring characteristics are quite different. Besides, two types of non-Gaussian excitation probability densities with differences in shape are used. It is demonstrated that the proposed methods can reproduce the unique shape of the response probability density caused by the spring nonlinearity and the excitation non-Gaussianity. From the results, the range of the bandwidth ratio between the excitation and the system where the response distribution is accurately obtainable is also examined for each method.