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In this work, we consider a fractional nonlinear vibration system of Duffing type with harmonic excitation by using the fractional derivative operator  -∞ −1D α  t  and the averaging method. We derive the steady-state periodic response and the amplitude-frequency and phase-frequency relations. Jumping phenomena caused by the nonlinear term and resonance peaks are displayed, which is similar to the integer-order case. It is possible that a minimum of the amplitude exists before the resonance appears for some values of the modelling parameters, which is a feature for the fractional case. The effects of the parameters in the fractional derivative term on the amplitude-frequency curve are discussed.

Response of a fractional nonlinear system to harmonic excitation by the averaging method