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Based on Volterra series theory, a Volterra-pseudo excitation method (Volterra-PEM) is developed to compute the response power spectral density (PSD) for nonlinear systems under random excitation. Firstly, within the framework of Volterra series theory, an improved pseudo excitation method (PEM) is established for multi-dimensional power spectral density (MPSD) analysis of nonlinear systems. As a generalized PEM for linear random vibration analysis, the Volterra-PEM is used to analyse the response MPSD, which also has a very concise expression. Secondly, in the case of computation difficulties when multi-dimensional integrating from MPSD to PSD, the computational efficiency is improved by converting the multi-dimensional integral into a matrix operation. Finally, as numerical examples, the Volterra-PEM is used to estimate the response PSD for stationary random vibration of a nonlinear spring-damped oscillator and a non-ideal boundary beam with geometrical nonlinearity. Compared with Monte Carlo simulation (MCS), the results show that the proposed method can predict the response PSD for nonlinear systems quickly and accurately under appropriate excitation intensities.

A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.