The Approximate Nonstationary Probability Densities For Responses Of Nonlinear Stochastic Systems Subject To Gaussian White Noise Excitations

The nonstationary probability densities for responses of Nonlinear Stochastic Systems subject to Gaussian white noise excitations are interesting and difficult topics in the field of stochastic dynamical systems. The approximate nonstationary responses of a single-degree-of-freedom(SDOF) nonlinear stochasic system with fractional derivative and the coupled two-degree-of-freedom(TDOF) nonlinear stochastic system are investigated in the present thesis.The first part is mainly about the nonstationary probability densities of SDOF stochastic system with fractional derivative which is of great significance in engineering field. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method, and the results from analytical solution are compared to with numerical simulation of the original system.In the second part of the thesis, the approximate nonstationary probability densities for responses of two Duffing-Van der Pol oscillators with coupled nonlinear damping subject to Gaussian white noise excitations are investigated. Firstly, the stochastic averaging method based on generalized harmonic functions is applied. Then, the nonstationary probability density for amplitude response is approximately expressed as a series expansion in terms of a set of Laguerre orthogonal basis functions with time-dependent coefficients. Finally, the approximate solution of the nonstationary probability density for amplitude response is derived with the application of the Galerkin method. The results obtained from proposed procedures are compared with those obtained by Monte Carlo simulation of the original systems, and it is shown that the proposed method is of high precision and applicability. It is concluded that the method in the thesis can be applied to obtain nonstationary probability densities of nonlinear Stochastic Systems subject to Gaussian white noise excitations.