## 3 Comparative Analysis of Dynamic Meshing Force
Figure 4: Meshing error calculation and its curve drawing process based on TCA: (**a**) Step 1; (**b**) Step 2; (**c**) Step 3; (**d**) Step 4. Figure 5 shows the single pair of teeth meshing error curve and comprehensive meshing error curve obtained by the TCA method under different contact ratio \(\varepsilon\) with \(f_{fa}=+0.015\) mm, \(f_{r}=2\). As can be seen from Figure 5, the two error curves are coincident in the single-tooth meshing region; however, there is difference between them in the double-tooth meshing region, because the comprehensive meshing error curve only reflects the meshing state of the tooth pair with the larger meshing error. Moreover, with the increase of contact ratio, the double-tooth meshing region gradually expands and the difference between the two curves will become more and more obvious. For example, Figure 5c,d show the single pair of teeth meshing error and comprehensive meshing error obtained under different tooth profile deviations when contact ratio \(\varepsilon=1.92\), respectively. It should be noted that Figure 5d shows another single pair of tooth meshing error curve obtained by adjusting the tooth profile deviation during the double-tooth meshing region. By comparing Figure 5c,d, we can observe that although the single pair of teeth meshing errors in the two cases are obviously different, the corresponding comprehensive meshing errors curve are the same. It can be predicted that the dynamic performance and dynamic meshing force obtained by the traditional model (based on comprehensive meshing error) and the new model (based on a single pair of teeth meshing error) are likely to be different. To illustrate the difference of dynamic response and dynamic meshing force obtained by the two dynamic models, we take the meshing state reflected in Figure 5c,d as an example for analysis. Let \(\overline{\xi}=0.04,T=500\) N\(\cdot\)m, \(\Omega_{h}=0.6\). The single pair of teeth meshing error and comprehensive meshing error in Figure 5c,d are successively brought into the corresponding dynamic models and the time-history data of gear dynamic characteristics can be obtained by solving the dynamic equations. Figure 6 shows the time history diagram (only the variation of displacement from 80 to 100 meshing cycles is shown for saving space) and FFT spectrogram obtained by the traditional model and the new model. Here, in order to clearly show the change of dynamic transmission error in a meshing cycle, select \(\tau/\tau_{z}\) as the abscissa, where \(\tau_{z}\) is dimensionless period. It can be seen from Figure 6 that the dynamic responses obtained by the new model based on Figure 5c,d are different due to the difference of meshing state, but corresponding dynamic responses obtained by the  
Figure 5: The single pair of teeth meshing error and comprehensive meshing error with different contact ratio: (**a**) \(\varepsilon=1.28\); (**b**) \(\varepsilon=1.75\); (**c**) \(\varepsilon=1.92\); (**d**) \(\varepsilon=1.92\) (adjusted profile deviation). traditional model are the same. In addition, we can see that there is an obvious difference between the dynamic responses obtained by the new model and by the traditional model. Therefore, compared with the traditional dynamic model, the new model can effectively reflect the influence of different meshing errors on the dynamic performance in the whole meshing stage. By further bringing the solution results of the dynamic equation into Formulas (12) and (13), the total dynamic meshing force, which is the sum of the dynamic meshing forces of each pair of meshing teeth participating in meshing during different meshing regions, varying with the meshing cycle can be obtained, as shown in Figure 7. We can see that, the overall change trends of dynamic meshing force based on the two models are the same, but the amplitudes fluctuation of dynamic meshing force between them are quite different, especially in the double-tooth meshing region.