时变参数下非线性扭振系统的Hopf分岔研究

Study on Hopf bifurcation due to time-varying parameter in nonlinear torsional vibration system

建立了一类含时变刚度和非线性阻尼的两自由度非线性扭振系统动力学方程,利用多尺度方法推导出了系统的平均方程。根据Hopf分岔理论分析了系统稳定性,给出了系统发生Hopf分岔的充要条件及系统周期运动稳定性的判别方法,分析了主共振情况下超临界Hopf分岔和亚临界Hopf分岔对系统振荡的影响。最后通过数值仿真验证了结论的正确性,对确保该类扭振系统的稳定运行有一定指导意义。

The nonlinear dynamic equation of torsional vibration system with two degrees of freedom is established including the time-varying stiffness and nonlinear damping. The average equation of system is deduced with the aid of multi-scale time method. Besides, according to Hopfbifurcation theory, the stability of system is analyzed, the necessary and sufficient condition of Hopf bifurcation and the judgment method of periodic motion＇s stability are given, and the influence ofsupercritical and subcritical Hopf bifurcation on vibration of system is analyzed under the condition of primary resonance. Last, the numerical simulation verifies the results. A significant contribution of this study is to ensure smooth running of this kind of torsional vibration system.