机械

为探究非线性因素对单级齿轮传动系统的运动影响，综合考虑齿侧间隙、激励频率等非线性因素对系统的震动影响，考虑齿轮系统的纵向振动位移，采用集中质量法建立纵向位移和扭转位移耦合的非线性动力学模型，并建立相关动力学方程。运用ode45函数对齿轮系统状态方程进行数值求解，结合系统的最大李雅普诺夫指数图分析随激励频率和齿侧间隙变化的分岔和混沌特性，得到以下结论：激励频率在[2.2, 3.1]区间变化时，随着激励频率的增加，系统由单周期运动激变为混沌运动，最后又经逆倍化分岔回到单周期运动，对应的李雅普诺夫指数呈现负-正-负的变化；齿侧间隙在[0.06, 0.075]区间变化时，随着齿侧间隙的增加，系统由单周期运动激变为混沌运动，经倍化分岔进入4周期运动，最后变为带有短暂周期窗口的不稳定混沌运动，对应的李雅普诺夫指数出现正负交替，在不稳定的混沌区域更是出现频繁正负交替的现象。以上分析结果可为齿轮激励频率的选取、齿侧间隙的设计提供理论依据。

In order to explore the influence of nonlinear factors on the motion of the single-stage gear transmission system, the influence of nonlinear factors such as backlash and excitation frequency on the vibration of the system is comprehensively investigated, and by taking into account the longitudinal vibration displacement of the gear system, the nonlinear dynamic model of longitudinal displacement and torsional displacement coupling is established through the concentrated mass method, and the relevant dynamic equations are derived. The ode45 function is adopted to numerically solve the equation of state of the gear system. Combined with the maximum Lyapunov exponential diagram of the system, the bifurcation and chaotic characteristics with the excitation frequency and tooth clearance are analyzed, and the following conclusions are obtained: when the excitation frequency varies within [2.2, 3.1], with the increase of the excitation frequency, the system changes from single-periodic motion to chaotic motion, and finally returns to single-period motion through inverse doubling, and the corresponding Lyapunov exponent shows a negative-positive-negative change pattern; when the backlash varies within [0.06, 0.075], with the increase of tooth clearance, the system changes from a single-period motion to a chaotic motion, evolves into a four-period motion through doubling bifurcation, and finally becomes an unstable chaotic motion with a short periodic window, and the corresponding Lyapunov exponent alternates between positive and negative values, especially frequent in the unstable chaotic region. The analysis results can provide a theoretical basis for the selection of gear excitation frequency and the design of backlash.