A new fractional moment equation method for the response prediction of nonlinear stochastic systems

Moment equation method is commonly used in the analysis of nonlinear stochastic systems. However, in some cases, this method may lead to poor accuracy, or even invalid results, because only limited number of moments of the response is available. In order to overcome this limitation, this paper develops a new fractional moment equation methodology. The new method involves the derivation of fractional moment equation of nonlinear systems and the approximation of probability density function (PDF) of response by means of a novel fractional moment closure scheme. Benefitting from the valuable features of fractional moments, i.e., few number of fractional moments contain large amount of statistical information about the variable, the developed method achieves more accurate PDF estimation of the response when compared with the present methods, especially for nonlinear systems with multiple equilibria. The effectiveness of the new method is finally demonstrated by a Duffing and a bistable Duffing oscillator that is subjected to Gaussian white noise. The results are compared with the Gaussian closure approximation and exact solution.