含磨损故障的齿轮传动系统非线性动力学特性

为揭示磨损故障对于齿轮传动系统非线性动态特性的影响，利用Archard和Weber-Banaschek公式分别计算了齿面动态累积磨损量和磨损齿轮对的时变啮合刚度。建立含有非线性齿侧间隙、内部误差激励和含磨损故障的时变啮合刚度的三自由度齿轮传动系统平移-扭转耦合动力学方程。采用变步长Gill积分方法对动力学模型进行了数值仿真分析，以系统的激励频率为分岔参数，计算系统的对应的分岔图；引入GRAM-SCHMIDT方法对系统的Jacobi矩阵进行正交化处理，计算系统的李雅普诺夫指数谱，同时结合Poincaré映射图和功率谱验证了李雅普诺夫指数谱和分岔图计算结果的正确性。通过研究发现了系统内部存在的丰富非线性现象，包括倍周期分岔途径、阵发性途径和多种拟周期通过锁相进入混沌的现象；在系统经由拟周期进入混沌的过程中发现了交替出现的拟周期与锁相现象以及拟周期运动时功率谱分量存在的Farey序列现象。研究结果表明含有磨损故障的齿轮传动系统具有非常复杂的动力学特性，而系统由周期运动进入混沌运动的途径也是丰富多样的。

The Nonlinear Dynamics Analysis of Gear Transmission System with Wear Fault

The Achard and Weber-Banaschek formula have been employed to calculate the dynamic surface wear and the mesh stiffness’s of worn surface gear pair respectively in this research. The three-degree-of-freedom translational-rotational coupled nonlinear dynamic equations of gear transmission system，including the worn teeth pair’s time-varying mesh stiffness, piece-wise backlash and internal error excitation, have been presented in this paper to do in-depth investigation of the surface wear’s effect on gear transmission’s nonlinear vibration characteristics. Vary step GILL integration method is employed to perform numerical simulation of the dynamic model. The internal excitation frequency is selected as the parameter to calculate the bifurcation diagram. The orthonormalization treatment of the system’s Jacobie matrix is carried out by utilizing GRAM-SCHMIDT method in order to obtain the Lyapunov exponents. The Poincaré section and power spectrum are used to verify the results of bifurcation and Lyapunov exponents of the system under some special parameter settings. Under the influence system’s strong nonlinearities, a rich variety of bifurcation phenomena have been illustrated in this paper. The classic periodic-doubling routes to chaos, intermittent routes to chaos and abundant different quasi-periodic routes have been revealed by bifurcation diagram and Lyapunov exponents. One ordinary periodic-doubling route and two particular quasi-periodic routes have been demonstrated in detail with the aid of Poincaré maps plotted in the phase plane. Alternant quasi-period and phase-locking have been observed in the system’s quasi-route to chaos. In addition, it has been observed that the frequencies of quasi-periodic motion satisfy familiar Farey sequence. All the results indicate that the dynamic characteristics of gear transmission with wear fault is very complex and the system’s routes to chaos are abundant and diverse.