Numerical methods for calculating the response of a deterministic and stochastically excited Duffing oscillator

When compared to independent harmonic or stochastic excitation, there exist relatively few methods to model the response of non-linear systems to a combination of deterministic and stochastic vibration despite the likelihood of harmonic oscillations containing noise 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. In the deterministic chaotic case, the JPDF is known to be a diffuse chaotic attractor when noise is present. This paper assesses the ability of a useful class of methods, global weighted residual methods, to produce the geometrically complex JPDF responses produced from harmonic and white noise excitation. A technique using a JPDF in the form of a Gram-Charlier type C series was found to produce accurate results, although the method fails due to ill-conditioning as the shape of the JPDF required by the dynamics becomes too complex