On the Stochastic Response of a Fractionally-damped Duffing Oscillator

A numerical method is presented to compute the response of a viscoelastic Duffing oscillatorwith fractional derivative damping, subjected to a stochastic input. The key ideainvolves an appropriate discretization of the fractional derivative, based on a preliminarychange of variable, that allows to approximate the original system by an equivalent systemwith additional degrees of freedom, the number of which depends on the discretization ofthe fractional derivative. Unlike the original system that, due to the presence of the fractionalderivative, is governed by non-ordinary differential equations, the equivalent systemis governed by ordinary differential equations that can be readily handled by standard integrationmethods such as the Runge–Kutta method. In this manner, a significant reductionof computational effort is achieved with respect to the classical solution methods, wherethe fractional derivative is reverted to a Grunwald–Letnikov series expansion and numericalintegration methods are applied in incremental form. The method applies for fractionaldamping of arbitrary order a (0 < a < 1) and yields very satisfactory results. With respect toits applications, it is worth remarking that the method may be considered for evaluatingthe dynamic response of a structural system under stochastic excitations such as earthquakeand wind, or of a motorcycle equipped with viscoelastic devices on a stochastic roadground profile.