考虑齿面冲击及摩擦的单级齿轮系统动力学建模及分析

Dynamic modeling and analysis of single-stage gear system considering tooth surface impact and friction

为研究齿轮啮合过程中齿面冲击对系统运动状态的影响,建立了包含单齿齿面啮合、脱啮和齿背啮合三种冲击状态的单级齿轮系统离散动力学模型,模型中考虑了摩擦、齿侧间隙、时变啮合刚度、综合传递误差等因素。基于三种冲击状态,定义了三种不同的Poincaré映射,利用变步长四阶Runge-Kutta法对系统动力学模型进行数值求解,得到了系统进入啮合时啮合力变化图,结合系统分岔图、最大Lyapunov指数谱、相图和随时间变化的动态啮合力图,分析啮合频率和齿侧间隙对系统啮合力的影响以及啮合力对系统动力学特性的影响。研究发现随着啮合频率和齿侧间隙的变化,齿轮系统齿面啮合力和齿背啮合力的复杂变化对系统运动状态有较大的影响。系统啮合力的周期或混沌突变可能会引起系统运动状态发生改变,也可能不会引起系统运动状态发生改变。系统在较高频率或小间隙的情况下出现齿背冲击现象,导致系统振动特性不稳定。此研究结果是建立考虑重合度大于1及载荷分配的齿轮系统动力学模型的基础。

In order to study the effect of the tooth surface impact in the process of gear meshing on the motion state of the system,a single stage discrete dynamical model was established by considering three kinds of impact state wherein the single gear tooth surface meshing,tooth detachment and tooth back-meshing.The model was established by considering the friction,the backlash,the timevarying mesh stiffness,the transmission error and other comprehensive factors.The system dynamic model was numerically solved by employing variable step size Runge-Kutta integration method.The meshing force variation diagrams of the system were obtained according to the definition of three different Poincaré mapping based on the three kinds of impact state.The influence of meshing frequency and backlash on the meshing force of the system and the influence of meshing force on the dynamic characteristics of the system were analyzed by combining with the system bifurcation diagram,the Top Lyapunov Exponent,phase diagram and Poincarémap.The results show that the complex changes of tooth surface meshing force and back-meshing force of gear system have a great influence on the motion state of the system with the change of meshing frequency and backlash.Periodic or chaotic changes in the meshing force of the system may cause changes in the motion state of the system,or may not cause changes in the motion state of the system.In the case of higher frequency or small clearance,the system has tooth back-meshing impact phenomenon,which leads to the instability of the system vibration characteristics.The study is a basis for establishing a gear system model which considers the contact ratio greater than one and the load distribution.