Stationary response analysis for a stochastic Duffing oscillator comprising fractional derivative element

Stationary response is investigated for a Duffing oscillator comprising fractional derivative elements excited by Gaussian white noise in the present paper. Firstly, harmonic balance technique is adopted to form a statistically equivalent linear system. Then, stochastic averaging is applied to the system to obtain a Markovian approximation of the response amplitude, and the associated Fokker-Planck equation and its stationary solution are derived. Furthermore, in virtue of Laplace transform, the transfer function of the equivalent linear system with amplitude-dependent coefficients is derived and it gives the conditional power spectral density, after weighted by the stationary probability density function, estimations of the power spectral density for the response and related statistics are derived. Numerical simulations verify the reliability of the proposed procedure, even for strongly nonlinear oscillators with properties like spectrum broadening and multimodal pattern.