Response of a Deterministic and Stochastically Excited Nonlinear Oscillator

When compared to independent harmonic or stochastic excitation, there exist relatively few methods to model the response of nonlinear systems to a combination of deterministic and stochastic vibration despite the likelihood of encountering such vibrations in realistic applications. This paper uses the Duffing oscillator to illustrate how the joint probability density function (JPDF) of the displacement and velocity responds to this form of excitation. Monte-Carlo simulations were performed to generate the JPDF which was observed, in general, to spread around the trajectory that would be observed if only deterministic excitation was present. A new method is then presented to compute the JPDF of the response of a deterministic and stochastically excited system. The form of the JPDF representing the response is proposed and substituted into the Fokker-Planck equation that governs the probability density. A number of weighted integrals of the resulting equation are taken, generating equations governing the non-stationary response of the JPDF. This method has been compared to Monte-Carlo and path integral results and shows accurate and rapid solutions. Due to the governing equations becoming ill-conditioned, the method fails to achieve a solution as the shape of the JPDF required by the dynamics becomes too complex.