Abstract
The objective of this study is to analyze the transient response of a system with a nonlinear spring under Gaussian white noise excitation using a complex fractional moment (CFM). The CFM is defined by extending an order of a classical integer-order moment to a complex number, and is related to the Mellin transform of a probability density function. In order to find the transient probability density function of the response, first, we linearize an equation of motion by introducing the effective natural frequency, which is the functions of the response amplitude. Secondly, the Fokker-Planck equation for the response amplitude is derived based on the stochastic averaging procedure. Thirdly, the governing equations of the response CFMs are obtained by applying the Mellin transform to the Fokker-Planck equation. The governing equations are easily solved since they are coupled linear ordinary differential equations. Finally, the inverse Mellin transform of the response CFMs yields the probability density function of the response. In numerical examples, we analyze a Duffing oscillator with different two values of a nonlinear parameter and the results are compared with the results of Monte Carlo simulation. Except for a few cases, the present solutions are in good agreement in the wide range including the tail region. We also consider the relation between the accuracy of the analytical result and the choice of the analytical parameters.
