Abstract
First, the restoring force modeled by a fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique. Then, an averaged It? stochastic differential equation for the amplitude is derived by using the stochastic averaging method based on the generalized harmonic function, and the Fokker-Planck-Kolmogorov (FPK) equation associated with the averaged It? equation is solved to obtain the stationary probability densities of amplitude. Finally, the approximate analytical solutions are validated by those from the Monte Carlo simulation of original system.
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