Forced vibrations of a nonlinear oscillator with weak fractional damping

This article deals with force-driven vibrations of nonlinear mechanical oscillators whose constitutive equations involve fractional derivatives, defined as fractional powers of the conventional time-derivative operator.This definition of fractional derivatives enables one to analyze approximately the vibratory regimes of the oscillator.The assumption of small fractional derivative terms allows one to use the method of multiple time scales, whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and disturbing force terms can be carried out.The relationship between the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when the orders of the small fractional derivative term and of the cubic nonlinearity appearing in the oscillator's constitutive equation coincide.