Stability and bifurcation analysis of single-degree-of-freedom linear vibro-impact system with fractional-order derivative

The stability and bifurcation of the single-degree-of-freedom linear vibro-impact system with fractional-order derivative are investigated, where the one-sided impact model under external harmonic excitation is considered. The approximate analytical solutions of the transient and steady state of the vibro-impact system are obtained by the averaging method, and the general solution of the system is obtained by the superposition method. The approximate analytical solution and numerical solution of the system are compared, and they are in good agreement with each other, which proves the accuracy of the approximate solution. Based on the approximate analytical solution, the stability of periodic motion of the vibro-impact system is studied by changing it to the fixed point on the mapping plane with the help of Poincare mapping. With the change of the fractional order, the frequency and amplitude of the external excitation, the bifurcation behaviors of the system are analyzed in detail. As the results show, saddle-node bifurcation, grazing bifurcation, period-doubling bifurcation and chaotic motion are found in the system. (C) 2019 Elsevier Ltd. All rights reserved.