Oscillators with a power-form restoring force and fractional derivative damping: Application of averaging

In this paper free oscillators with a power-form restoring force and with a fractional derivative damping term are considered. An analytical approach based on the averaging method is adjusted to derive analytical expressions for the amplitude and phase of oscillations. Effects of the fractional-order derivative on the amplitude and frequency of oscillations are discussed in several examples, including a generalized van der Pol oscillator, purely nonlinear oscillators and a linear oscillator.

► Oscillators with power-form restoring forces and fractional derivative damping are studied.
► An averaging method is developed to obtained solution for free vibrations.
► General equations are derived for multi and single-term restoring forces.
► Effects of the fractional-order derivative on some vibration properties are discussed.