二自由度碰振准哈密顿系统亚谐轨道分析

Subharmonic orbits analysis for a 2-DOF vibro-impact quasi-Hamiltonian system

研究了具有立方非线性项和外部激励项的二自由度非线性碰振系统的动力学特性。运用摄动方法推导出了碰振系统的局部亚谐Melnikov函数,并应用该Melnikov函数和数值方法确定了二自由度碰振系统稳定周期运动的存在条件。系统以频率ω和激励力f等为分岔参数,仿真结果表明:系统的碰振运动经历了稳定的单碰和双碰周期运动,然后进入混沌状态,从而验证了Melnikov方法的有效性。此外,适当控制参数取值可以避免系统出现多周期和复杂的混沌运动,实现系统的稳态运动。

The dynamic characteristics of a 2-DOF vibro-impact system with a cubic non-linear item under external excitation were investigated here. Firstly,the system 'slocal subharmonic Melnikov function was derived adopting the perturbation method. Then,Melnikov function and numerical methods were applied to determine the existence conditions of the system's stable periodic motions. When the frequency ω and the excited force f were taken as bifurcation parameters,the simulation results showed that the system performs a stable single-impact periodic motion and a double-impact periodic one,then enters a chaotic state,the validity of Melnikov method is verified;besides,appropriately controlling parameter values can avoid the system to have multi-period motion and complex chaotic one,and realize the stable motion of the system.