Primary resonance of Duffing oscillator with fractional-order derivative

► The approximately analytical solution and the stability condition are presented. ► Effects of the fractional-order parameters are denoted by equivalent coefficients. ► The finding is different from the existing results on fractional-order derivative. ► Effects of the fractional-order parameters on system dynamics are illustrated. In this paper the primary resonance of Duffing oscillator with fractional-order derivative is researched by the averaging method. At first the approximately analytical solution and the amplitude–frequency equation are obtained. Additionally, the effect of the fractional-order derivative on the system dynamics is analyzed, and it is found that the fractional-order derivative could affect not only the viscous damping, but also the linear stiffness, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. This conclusion is remarkably different from the existing research results about nonlinear system with fractional-order derivative. Moreover, the comparisons of the amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two parameters of the fractional-order derivative, i.e. the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated, which are different from the traditional integer-order Duffing oscillator.