van der Pol型自激单摆的张弛振荡特性

含van der Pol型自激项的单摆系统是典型的自激机械系统,本文研究了该系统的张弛振荡特性.首先通过引入新的时间尺度和变量,把原系统表示成标准的快慢系统.然后基于几何奇异摄动理论,求得系统的慢变流形及其结构,从而证明了张弛振荡解的存在性,并进一步求得了张弛振荡解及其周期的近似表达式.理论结果表明,发生张弛振荡时,单摆快速通过其平衡位置,而在远离平衡位置的一段区域上停留较长时间,且存在两个分界点把快速运动和慢速运动分隔开.数值算例证明了理论分析的正确性.

Relaxation oscillation of pendulum with van der pol type self-excition

The pendulum system with van der Pol type self-excitation is one typical self-excited mechanic system, and this paper studied the relaxation oscillation of the pendulum system. Firstly, the new time-scale and variable were introduced, so the pendulum system was described with standard slow-fast system. And then, on the basis of the geometric singular perturbation theory, the slow manifold and its structure of the system were obtained, and the relaxation oscillation was proved. Moreover, the expression of the relaxation oscillation and its period were obtained approximately. The analytical results indicate that, when the system undergoes relaxation oscillation, the pendulum passes through equilibrium position rapidly, and stays in the regions far away from the equilibrium position for a long time, and there are two break points which separate the fast dynamics from the slow dynamics. Numerical studies validate the analytical results.