On the stochastic response of a fractionally-damped Duffing oscillator

A numerical method is presented to compute the response of a viscoelastic Duffing oscillator
with fractional derivative damping, subjected to a stochastic input. The key idea
involves an appropriate discretization of the fractional derivative, based on a preliminary
change of variable, that allows to approximate the original system by an equivalent system
with additional degrees of freedom, the number of which depends on the discretization of
the fractional derivative. Unlike the original system that, due to the presence of the fractional
derivative, is governed by non-ordinary differential equations, the equivalent system
is governed by ordinary differential equations that can be readily handled by standard integration
methods such as the Runge–Kutta method. In this manner, a significant reduction
of computational effort is achieved with respect to the classical solution methods, where
the fractional derivative is reverted to a Grunwald–Letnikov series expansion and numerical
integration methods are applied in incremental form. The method applies for fractional
damping of arbitrary order α (0 < α < 1) and yields very satisfactory results. With respect to
its applications, it is worth remarking that the method may be considered for evaluating
the dynamic response of a structural system under stochastic excitations such as earthquake
and wind, or of a motorcycle equipped with viscoelastic devices on a stochastic road
ground profile