基于弹性圆柱壳的柔性齿轮副建模及动力学分析

Modelling and Dynamic Analysis of Flexible Gear Pairs Based on Elastic Cylindrical Shell Theory

基于弹性圆柱壳理论,建立柔性齿轮副等效简化模型。首先采用具有时变刚度的弹簧模拟齿轮交替啮合过程,其中时变啮合刚度通过解析模型得到;同时采用径向和切向支撑弹簧模拟轴和轴承的柔性。接着根据能量原理,推导柔性齿轮副系统的动能、应变能、啮合弹簧势能以及支撑弹簧势能。随后基于微分求积有限法(DQFEM)对能量表达式进行离散,分别使用拉格朗日方程和Newmark迭代来求解柔性齿轮副模型的模态和动态响应。通过收敛性分析,研究模型的数值稳定性,再与ANSYS的有限元模型进行模态特性的对比,验证模型的正确性。然后分析柔性齿轮副模型的固有频率随转速的变化规律,并给出齿轮啮合模型自然频率的坎贝尔图。最后,通过与基于齿轮体刚性假设的扭振模型进行对比,从时域和频域两方面阐述齿轮柔性对系统动力学特性带来的影响。

Based on the elastic cylindrical shell theory,the simplified equivalent model of the flexible gear pair is established.The spring with time-varying stiffness obtained through an analytical model is used to simulate the alternative meshing process of the gear.And the radial and tangential support springs are used to simulate the flexibilities of the shaft and the bearings.According to the energy principle,the kinetic energy,strain energy of the flexible gear pair system,and the potential energy of meshing spring and supporting spring are derived.The energy expression is discretized by the differential quadrature finite element method(DQFEM),and the modal and dynamic response of the flexible gear pair model are solved by the Lagrange equation and Newmark iteration.Through the convergence analysis,the numerical stability of the presented model is studied.Then,the modal characteristics of the presented model are compared with those of the finite element model in ANSYS to verify the reliability and accuracy of the presented model.The natural frequency variation law of the flexible gear pair model with the rotational speed is analyzed,and the Campbell diagram of the natural frequency of the gear meshing model is given.Finally,by comparing with the torsional vibration model based on the rigidity assumption of the gear body,the influence of gear flexibility on the system dynamic characteristics is explained from the viewpoint of time domain and frequency domain.