Response and stability of strongly non-linear oscillators under wide-band random excitation

A new stochastic averaging procedure for single-degree-of-freedom strongly non-linear oscillators with lightly linear and (or) non-linear dampings sub ject to weakly external and (or) parametric excitations of wide-band random processes is developed by using the so-called generalized harmonic functio ns. The procedure is applied to predict the response of Duffing-van der Pol oscillator under both external and parametric excitations of wide-band sta tionary random processes. The analytical stationary probability density is verified by digital simulation and the factors affecting the accuracy of th e procedure are analyzed. The proposed procedure is also applied to study t he asymptotic stability in probability and stochastic Hopf bifurcation of D uffing-van der Pol oscillator under parametric excitations of wide-band sta tionary random processes in both stiffness and damping terms. The stability conditions and bifurcation parameter are simply determined by examining th e asymptotic behaviors of averaged square-root of total energy and averaged total energy, respectively, at its boundaries. It is shown that the stabil ity analysis using linearized equation is correct only if the linear stiffn ess term does not vanish. (C) 2001 Elsevier Science Ltd. All rights reserve d.