Response of fractional oscillators with viscoelastic term under random excitation

A system with fractional damping and a viscoelastic term subject to narrow-band noise is considered in this paper. Based on the revisit of the Lindstedt-Poincaré (LP) and multiple scales method, we present a new procedure to obtain the second-order approximate analytical solution, and then the frequency-amplitude response equations in the deterministic case and the first- and second-order steady-state moments in the stochastic case are derived theoretically. Numerical simulation is applied to verify the effectiveness of the proposed method, which shows good agreement with the analytical results. Specially, we find that the new method is valid for strongly nonlinear systems. In addition, the influences of fractional order and the viscoelastic parameter on the system are explored, and the results indicate that the steady-state amplitude will increase at a fixed point with the increase of fractional order or viscoelastic parameter. At last, stochastic jump is investigated via the received Fokker-Planck-Kolmogorov (FPK) equation to compute the stationary solution of probability density functions with its shape changing from one peak to two peaks with the increase of noise intensity, and the phenomena of stochastic jump is consistent with the solution of frequency-amplitude response equations.