Response statistics of strongly nonlinear system to random narrowband excitation

A technique coupling with the parameter transformation method and the multiple scales method is presented for determining the primary resonance response of strongly nonlinear Duffing–Rayleigh oscillator subject to random narrowband excitation. By introducing a new expansion parameter α=α(ε,u0)α=α(ε,u0), the multiple scales method is adapted to determine the equations describing the modulation of response amplitude and phase. The effect of the random excitation on the stable periodic response is analyzed as a perturbation. By the moment method steady-state mean square response is obtained and its local stability is checked by Routh–Hurwitz criterion. Theoretical analyses and numerical calculations show that when the intensity of random excitation increases, the steady-state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady-state solutions. The results obtained for strongly nonlinear oscillator complement previous results in the literature for weakly nonlinear case.