单自由度齿轮传动系统安全盆侵蚀与分岔

Erosion and bifurcation of the safe basin for a single-degree-of-freedom spur gear system

齿轮啮合过程中的振动幅值超过一定限度,会造成系统稳定性的破坏,影响其安全特性。为研究齿轮传动系统的安全特性,以单自由度直齿圆柱齿轮传动系统为研究对象,数值计算了随着综合传递误差波动系数和时变刚度波动幅值增大时系统在考察区域内的安全盆,分析了系统的安全盆侵蚀与分岔过程。结合系统相图、Poincaré映射图分析了系统在考察区域内的运动转迁过程,并计算了系统的最大Lyapunov指数图(TLE)及分岔点的Floquet乘子;借助多初值分岔图进一步分析了系统安全盆侵蚀与分岔现象及演变规律。结果表明:一定范围内随着参数的增大,系统不同运动的吸引域演变导致安全盆的侵蚀,系统的安全性逐渐变差;在考察区域内,系统运动分岔和周期跳跃引起安全盆的分岔;周期跳跃使原运动部分初值的安全盆产生新的安全盆,另一部分初值的安全盆保持不变,而倍化分岔和鞍结分岔则使原安全盆全部分岔为新的安全盆。研究结果可为直齿圆柱齿轮系统的参数设计和选择提供指导,为保证齿轮传动系统的安全服役提供理论依据。

In order to study the vibration-boundedness and the safety characteristics of gear transmission systems,a single-degree-of-freedom spur gear transmission system was taken as a studied object.The safety basin of the system in the inspection area was analyzed with the variation of the fluctuant coefficient of comprehension-transmission error and the fluctuant amplitude of time-varying stiffness.The erosion and bifurcation process of the safe basin were analyzed.Based on the system phase diagram and Poincarémap,the motion bifurcations process of the system in the investigation area was analyzed,and the top Lyapunov exponent(TLE)and the Floquet multiplier at the bifurcation point were calculated,and the evolution of the system safety basin erosion and bifurcation was further studied by means of the multi-initial bifurcation diagram.The results show that,with the increase of the parameters in a certain range,the evolution of different motions of the system leads to the erosion of safe basin,and the safety of the system becomes worse;in the investigation area,the bifurcation of the safety basin is caused by the bifurcation and periodic jumping of the system,and the periodic jumping will transform a part of the original safe basin into a new safe basin,while the other part of the original safety basin remains unchanged,and the doubling bifurcation and saddle knot bifurcation makes the original whole safety basin become a new safe basin.The results provide a guidance for the design and selection of spur gear system parameters,and provide a theoretical basis for ensuring the safety service of gear transmission systems.