# Simulation and discussion of time-varying meshing stiffness of finite element gear pair

The quasi-static calculation method is adopted in the finite element simulation calculation, that is, the stiffness under the load at any meshing point is assumed to be independent, which is consistent with the time-varying meshing stiffness calculated by the above analytical method. Fig. 1 draws the time-varying meshing stiffness diagram calculated by finite element method. It can be seen from the diagram that the corresponding stiffness value also increases slightly with the increase of applied torque, which is consistent with the change trend of stiffness value in Fig. 2 (b). Due to the independence of each meshing point stiffness calculation, the stiffness curve under the same load fluctuates slightly, but the fluctuation value is quite weak, which shows the effectiveness of the finite element simulation model.

In Fig. 3, the time-varying meshing stiffness curves under different calculation methods are plotted. For the convenience of curve comparison, curve 1 represents the finite element calculation results, curve 2 represents the time-varying meshing stiffness considering the load action of semi empirical formula, and curve 3 represents the time-varying meshing stiffness considering Yang sun model independent of load action. The stiffness curve drawn in the figure shows that in the double meshing area, the mesh stiffness value is curve 1 ＜ curve 2 ＜ curve 3, while in the single meshing area, the mesh stiffness value is curve 2 ＜ curve 1 ＜ curve 3.

The stiffness values of the three curves are in the same series (100000000), and the trend of the stiffness curves is basically the same. In order to further compare the stiffness numerical results of the three curves, the average stiffness values of the three curves in the double single meshing region are taken for comparison. The average stiffness comparison in the table shows that the stiffness difference percentages of curve 1 and curve 2 in the double single meshing area are 8.1% and 8.8% respectively, while the stiffness difference percentages of curve 1 and curve 3 in the double single meshing area are 20.4% and 6.3% respectively. From the difference percentage value, except the percentage of curve 1 and curve 3 in the double meshing zone is larger, the other percentages are less than 10%. The results show that the stiffness calculation of the finite element method and the potential energy analytical method are consistent and verified with each other, and the method proposed in this paper has certain feasibility and reliability of the calculation model.

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