Vibration control of primary and subharmonic simultaneous resonance of nonlinear system with fractional-order Bingham model

The passive vibration control of primary and subharmonic simultaneous resonance for the Duffing-type nonlinear system under the base excitation and external excitation by using magnetorheological (MR) fluid damper is studied, where the fractional-order derivative Bingham model of MR fluid damper is considered. The approximate analytical solution of the system is obtained by using the incremental averaging method. On the basis of obtaining the primary resonance of the system under base excitation by the averaging method, the subharmonic resonance solution of the system is obtained by taking the subharmonic resonance of the system under base excitation and external excitation as an increment, so as to obtain the approximate analytical solution of the simultaneous resonance of primary and subharmonic resonance. And the amplitude–frequency equation and phase–frequency equation of the steady-state solutions for the primary and subharmonic resonance of the system are derived respectively. According to the approximate analytical solutions, the stability conditions of the steady-state solution of the primary resonance and subharmonic simultaneous resonance are obtained by Lyapunov method. Compared with the numerical solution, the correctness and accuracy of the analytical solution of the primary resonance and subharmonic simultaneous resonance are demonstrated. The influence of system parameters on the resonant response of the system is analyzed in detail, with emphasis on the resonance bifurcation behavior of the system. The analysis results show that the damping passive vibration reduction control has obvious vibration suppression effect on the primary and subharmonic simultaneous resonance of the Duffing-type nonlinear system. Under certain conditions, there are 9 branches in the steady-state amplitude–frequency response of the primary and subharmonic simultaneous resonance of the Duffing-type system, of which 5 branches are stable solutions.