行星齿轮传动系的周期运动及其稳定性

Coexisting periodic solutions and their stability of a nonlinear planetary gear train

研究了间隙行星齿轮非线性传动系的周期轨道及其稳定性。针对PNF方法在求解非线性动力系统时存在的两点缺陷,即研究对象必须光滑和迭代初始点要求距离周期轨道足够近,提出了改进措施使之能够适应间隙行星齿轮传动系统的周期轨道的求解以及判稳工作。改进后的PNF方法对算例的计算结果和直接数值积分结果的吻合证明了改进措施的有效性。采用改进后的PNF方法研究了行星齿轮系统在一组给定参数下共存的周期运动,并判断了各共存周期运动的稳定性;通过延续判断不同转速下系统周期轨道的稳定性,研究了行星齿轮传动系统的周期运动状态随无量纲转速的分岔特性。结果发现,行星齿轮非线性传动系统在某些参数组合下可以共存几个稳定或者是不稳定的周期轨道;转速的变化可以使行星齿轮系统通过倍周期分岔的形式最终通往混沌。

Coexisting Periodic solutions and their stability of a nonlinear torsional vibration model for a planetary gear train with gear backlashes are studied by using the method of PNF. Some adaptations are made for the PNF method so as to make it suit- able for the non-smooth and nonlinear planetary gear train. The coexisting periodic solutions of the system under certain dimen- sionless rotational speed are studied by using the method of modified PNF,with the stability of each periodic solution investiga ted at the same time. The bifurcation characteristics of the periodic trajectories are studied as well by using PNF at different speeds. The results show that a nonlinear planetary gear train with certain parameters may have several coexisting stable or un- stable periodic traiectories Moreover, the motion of the system could also evolve into chaos by way of period doubling bifurca tion as the speed increases.