## 6 Simulation results and analysis  
Equivalent forces and contact tools must be inserted into the post-processing of simulation results. Then, the pressure in the contact tool is selected, and the simulation is solved according to Equation (2). The results of force convergence analysis show that the calculations are accurate and reliable in the nonlinear case. In Figure 3b, it can be seen that the stress distribution at the root of the gear tooth during meshing is in red color, which indicates that the stress received here is the highest, and the root of the gear tooth is usually the narrowest part of the tooth groove, which results in a small cross-sectional area of the material at the root of the tooth. As a result, the stresses experienced at the root of the tooth are usually higher. With the calculated equivalent stress and equivalent strain data, the stress-strain curves of the traction arc-toothed cylindrical gear pair in the working process are plotted as shown in Figure 4.  
Figure 5 shows the contact pressure distribution at different moments of a cycle under continuous operating conditions. Figure 5c shows the contact pressure distribution at the moment of engagement when the contact pressure was the largest, and the contact pressure was 2559.7 MPa. The force points were evenly distributed on the tooth surface, and the contact area was elliptical, with the greatest force at the center. A more accurate model is illustrated based on the characteristics of the curved cylindrical gears. The maximum contact pressure data for the meshing position of the gear pair under continuous operating conditions are listed in Table 4. \begin{table}
\begin{tabular}{c c} \hline
**Working Condition** & **Size** \\ \hline Torsion (N/m) & 841.5 \\ Rotation speed (rad/s) & 434 \\ \hline \end{tabular}
\end{table} 
Table 3: Continuous operation loads. Figure 4: Stress–strain curve. Figure 3: Mises stress cloud for gear contact under startup condition. In order to ensure the reliability of the simulation values, the maximum stress on the gear contact surface can be calculated according to the Hertz contact theory formula and the calculated results can be compared with the simulation results [15]. \[\sigma_{H}=Z_{\mathrm{E}}\sqrt{\frac{KF_{n}}{b\rho_{\Sigma}}} \tag{3}\]  
In the equation, \(Z_{\mathrm{E}}\) is the modulus of elasticity, \(F_{n}\) is the normal force, \(b\) is the tooth thickness, \(\rho_{\Sigma}\) is the integrated radius of curvature, \(K\) is the load distribution factor. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Name** & **Figure 5(a)** & **Figure 5(b)** & **Figure 5(c)** & **Figure 5(d)** & **Figure 5(e)** & **Figure 5(f)** \\ \hline \hline \begin{tabular}{c} meshing state \\ contact stress \\ \end{tabular} & \begin{tabular}{c} meshing \\ 632.56 \\ \end{tabular} & \begin{tabular}{c} meshing \\ 1920 \\ \end{tabular} & \begin{tabular}{c} engagement \\ 2559.7 \\ \end{tabular} & \begin{tabular}{c} disengage \\ 2225.2 \\ \end{tabular} & \
\begin{tabular}{c} meshing \\ 1461.8 \\ \end{tabular} &
\begin{tabular}{c} meshing \\ 1028.6 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum contact stress at different moments under continuous gear conditions (MPa). Figure 5: Maximum contact pressure distribution of the gear pair at different moments under continuous operating conditions. (**a–f**) show the meshing at different moments in a cycle. According to the simulation, the maximum tooth contact stress is 2559.7 MPa, and according to the Hertz theoretical formula, the maximum tooth contact stress is 2336 MPa. The theoretical and simulation error is 8.7%. The theoretical value is basically the same as the simulated value, which can be used to verify the validity of the results of the finite element model. By analyzing it through finite elements, the following conclusions can be obtained:  
1. The contact path of an arc-toothed cylindrical gear during meshing is along an elliptical region in the middle of the tooth flank, accounting for approximately 50% of the length of the flank.