# Accurate modeling of time-varying meshing stiffness of spur gear pair based on geometry and potential energy method

In mechanical transmission system, gear transmission system is the most widely used in power and motion transmission devices.Time-varying meshing stiffness (TVMS) is one of the important parameter variables in gear system dynamics research. The internal excitation load generated by TVMS directly affects the dynamic performance response indexes of gear transmission system such as vibration and noise.Accurate, efficient and time-efficient simulation and calculation of TVMS has become an important subject for many scholars to study gear system dynamics in recent years.

At present, the common research methods for TVMS are finite element method (FEM), analytical method (AM), analytical-finite element hybrid method (AM-FE) and experimental method.Due to the efficiency and timeliness of analytical method, it is the primary method to study and calculate TVMS.Yang et al. 189-196 simplified the gear teeth as a cantilever beam model on the base circle, considered the Hertzian contact stiffness, bending stiffness and axial compression stiffness, put forward for the first time an analytical method for calculating gear meshing stiffness based on potential energy method; Tian and Wu et al. improved the calculation method of comprehensive meshing stiffness considering shear stiffness; Zhou and Wan et al. further proposed to consider the flexible deformation of gear base.Analytical method for meshing stiffness calculation.

Above all, the analytical method based on potential energy is used to calculate the comprehensive TVMS of gears, including the predictive calculation of TVMS with fault defects and geometric modification, the algorithm improvement for meshing stiffness and the influence of excitation stiffness under dynamic model by Li Yapeng and Wan Zhiguo, etc. It can be seen that the analytical method has been widely used by scholars at home and abroad.

Finite element method and analysis-finite element hybrid method are used to discrete gear body and gear teeth into small elastic units and calculate TVMS of gear pair by elastic deformation response under external load according to relevant mechanical theory. The numerical model can simulate the actual geometry and boundary conditions of gear physical model and has been adopted by scholars and used to verify the proposed analytical method for calculating TVMS.Validity and accuracy of models and results.

The test method can accurately calculate TVMS of gear engagement pair by quasi-static analysis. In practical application, it does not have wide application conditions due to time consuming, consumables and application limitations of test equipment, but it is suitable for basic data research and mathematical model verification of new product gears.

In order to calculate TVMS effectively and accurately, a precise modeling method of TVMS based on geometry and potential energy method is proposed in this paper. By using actual meshing motion equation and geometric position relationship of spur gear pair, the meshing limit boundary of spur gear pair teeth is analyzed in real time, and as potential energy method, the calculation boundary condition of TVMS is analyzed and analyzed, so that TVMS can be constructed from physical model to mathematical model.The calculation and analysis model of type transition can accurately calculate TVMS of spur gear pair.

Compared with the tedious calculation process of finite element method and LTCA method, as well as the simple calculation by selecting stiffness coefficient in ISO standard, this modeling method can quickly realize the parametric modeling of meshing stiffness calculation of spur gear pair and detailed tooth profile parameter modeling (such as root fillet transition curve), fast modular modeling and optimal design of dynamic analysis, etc.In this paper, the modeling method is validated by establishing the corresponding finite element model, and the example analysis of spur gear pair and the influence of geometric parameters on TVMS are carried out. 