为研究非线性因素对直齿轮纯扭转模型弹性动力学的影响，根据达朗贝尔原理，采用微位移思想，建立4自由度直齿轮纯扭转动力学模型.首先，通过Runge-Kutta法结合时间离散法求解系统在齿轮啮合误差、时变啮合刚度和啮合间隙等非线性因素影响下的时变方程，得到系统振动的响应以及对应频谱图.其次，结合非线性特性分析理论，通过改变系统的啮合刚度、激励频率和齿侧间隙，获得系统动态响应的全局分岔图和最大Lyapunov指数.然后，综合运用时间历程曲线、相空间轨线、Poincáre映射图和功率谱，定量和定性分析系统的周期运动、拟周期运动和混沌运动.最后，考虑非线性因素耦合作用对系统的影响，为齿轮传动系统的非线性特性分析提供一套较为完整的方法.
In order to study the influence of nonlinear factors on the elastodynamics of spur gear’s pure torsional model, a pure torsional dynamic model of four-degree-of-freedom spur gear is established by using the principle of D’Alembert and the method of micro-displacement. Firstly, a time-varying equation set is solved by Runge-Kutta and time-dispersion methods with the consideration of the nonlinear factors such as gear meshing error, time-varying meshing stiffness and meshing gap, and the dynamic response and the corresponding spectrum of the system are obtained. Next, combined with the nonlinear analysis theory, the global bifurcation diagram and the maximum Lyapunov exponent are obtained by changing the meshing stiffness, the excitation frequency and the tooth side clearance of the system. Then, the periodic motion, quasi-periodic motion and chaos of the system are analyzed from the quantitative and qualitative point of view respectively by using the time history plot, phase space trajectory, Poincáre map and power spectrum. Finally, the coupling influence of nonlinear factors on the system are considered, which provides a complete method for the nonlinear characteristic analysis of gear transmission system.
［1］LIU J K, CHEN F X, CHEN Y M. Bifurcation ana-lysis of aeroelastic systems with hysteresis by incremental harmonic balance method ［J］. Applied Mathematics and Computation, 2012, 219(5): 2398-2411.
［2］姚红良, 王重阳, 王帆, 等. 多频激励局部非线性系统响应求解的降维增量谐波平衡法［J］. 振动工程学报, 2015, 28(5): 741-747.
YAO Hongliang, WANG Chongyang, WANG Fan, et al. The dimension-reductive incremental harmonic balance method for solving the response of local nonlinear dynamic system with multi-frequency excitation ［J］. Journal of Vibration Engineering, 2015, 28(5): 741-747.
［3］刘振皓.车辆复合行星传动系统动力学特性研究［D］.湖北: 武汉大学, 2012.
LIU Zhenhao. Research on dynamic characteristics of vehicle compound planetary gear train sets ［D］. Hubei: Wuhan University, 2012.
［4］刘江楠, 苏典, 龚佑发.基于MATLAB的齿轮传动系统振动与分岔理论分析［J］. 机械传动, 2016, 40(2): 19-22.
LIU Jiangnan, SU Dian, GONG Youfa. Analysis of the bifurcation and theory vibration of the gear transmission system based on the MATLAB ［J］. Journal of Mechanical Transmission, 2016, 40(2): 19-22.
［5］ERITENEL T, PARKER R G. An investigation of tooth mesh nonlinearity and partial contact loss in gear pairs using a lumped-parameter model ［J］. Mechanism and Machine Theory, 2012, 56: 28-51.
［6］ZHANG W, DING Q. Torsion vibration and parametric instability analysis of a spur gear system with time-varying and square nonlinearities ［J］. International Journal of Applied Mechanics, 2014, 6(1): 1450007.
［7］DEL RINCON A F, VIADERO F, IGLESIAS M A, et al. A model for the study of meshing stiffness in spur gear transmissions［J］. Mechanism and Machine Theory, 2013, 61: 30-58.
［8］方禹鑫, 丁千, 张微.多齿侧间隙传动系统非线性特性研究［J］.振动与冲击, 2016, 35(23): 29-34.
FANG Yuxin, DING Qian, ZHANG Wei. Non-linear dynamic features of a steering gear system with backlashes ［J］. Journal of Vibration and Shock, 2016, 35(23): 29-34.
［9］COOLEY C G, PARKER R G. A review of planetary and epicyclic gear dynamics and vibrations research［J］. Applied Mechanics Reviews, 2014, 66(4): 040804.
［10］韩清凯, 于海, 孙伟.机械振动系统的现代动态设计与分析［M］.北京: 科学出版社, 2010.
HAN Qingkai, YU Hai, SUN Wei. Modern dynamic design and analysis of mechanical vibration system ［M］.Beijing: Science Press, 2010.
