Fractional oscillators with fractional damping in the presence of external forces

Abstract

We investigate how a fractional oscillator reacts to external forces when the dynamic law includes a damping term that is explicitly fractional. The corresponding three-parametric fractional differential equation (in the Caputo sense) admits exact solution. Two of the fractional parameters are associated with the intrinsic dissipation mechanism that produces continuous dissipation of energy. The third fractional parameter characterizes the damping term against which the oscillator is driven by the external force. When external time-dependent forces are included, the fractional dynamic law is associated with the Newton-Scott-Blair model of viscoelastic materials. The Laplace transform of the external force defines the profile of the solution; specific examples include the absence of external forces as well as constant, sinusoidal and stepped external forces. With a constant driving force, the system reaches a fixed position after a transient period. With a sinusoidal driving force, the system exhibits persistent oscillatory behavior. These results suggest that the external driving force determines the long-term behavior of the fractionally damped oscillator.