Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations

A stochastic averaging procedure of strongly non-linear oscillators subject to external and (or) parametric excitations of both harmonic and white-noise forces is developed by using the so-called generalized harmonic functions. The procedure is applied to a Duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises. The averaged Fokker-Planck-Kolmogrov equation is solved by using the path integration method. Based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration, the stochastic jump of the Duffing oscillator under combined harmonic and white-noise excitations and its bifurcation as the system parameters (frequency ratio, strength of the non-linearity, amplitude of harmonic excitation and intensity of white noise) change are examined for the first time. (C) 2000 Academic Press.