Calculating Parametric Oscillation of Third-Order Nonlinear System With Dynamic Friction and Fractional Damping

Abstract

In this paper, the parametric resonance of third-order parametric nonlinear system with dynamic friction and fractional damping is investigated using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the resonance solution is then obtained by means of Lyapunov theory. Additionally, the effect of the fractional derivative on the system dynamics is analyzed. 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.