Averaging for Oscillations With Light Fractional Order Damping

Abstract

We apply the method of averaging to study small, possibly nonlinear perturbations to a harmonic oscillator, particularly including fractional derivative damping. The averaging procedure here corresponds to the general or aperiodic case, and requires evaluation of Fresnel type integrals. We also describe a conceptually simple numerical scheme which can be used to verify the accuracy of the averaged equations. Systems studied are a linear, fractionally damped oscillator, a van der Pol type oscillator with a limit cycle, and a weakly nonlinear system forced sinusoidally near resonance. The averaged dynamics closely matches the full numerical solution in each case. It is discussed, in light of the averaged equations, why fractional damping is not equivalent to some simple effective traditional damping.