## 4 Simulation Results
### Tooth Surface Contact Stress  
When load T = 1500 Nm, T = 4500 Nm, and T = 7500 Nm, the variation of contact stress on the convex face of the gear is shown in Figure 11.  
Figure 10: Comparison of loaded contact zone of the gear convex face: (**a**) Load T = 500 Nm, (**b**) Load T = 1500 Nm, and (**c**) Load T = 4500 Nm.  
Figure 9: Mesh generation of spiral level gear.  
As shown in Figure 11, the above results indicates that, with the increase of load T, the maximum contact stress \(\sigma_{H}\) of the gear convex face becomes larger, and the worst contact zone appears in the small end teeth top of the gear. When load increases from 1500 Nm to 7500 Nm, the comparisons between the simulation results and the calculated results of empirical formula are listed in Table 3 below. The empirical formula is according to ISO 10300 and ISO 6336, which is shown in Equations (41) and (42) [24,25]. The calculation is based on the contact (Hertzian) stress, in which the load is distributed along the lines of contact. Calculations are to be carried out for pinion and wheel together, and some parameters for contact stress calculation were determined by engineering experience, processing technology, and actual operation conditions, such as gear geometry, manufacturing accuracy, rigidity of flange, bearing and bearing seat, and torque transmitted. We can find that, with the increase of loading torque, the change trend of FEM and calculated results are similar. In addition, the results of empirical formulas are about 10-15% larger than analyzed ones. The authors believe that the FEM accuracy, constraint condition setting, and the parameters selection of theoretical formula cause the difference value:  
\[\sigma_{H}=\sigma_{H0}\sqrt{\mathbf{K}_{A}\mathbf{K}_{V}\mathbf{K}_{H\beta}\mathbf{K}_{H\alpha}} \tag{41}\]  
\[\sigma_{H0}=\sqrt{\frac{\mathbf{F}_{n}}{\mathbf{I}_{bm}\mathbf{\rho}_{rel}}}\mathbf{Z}_{M-B} \mathbf{Z}_{LS}\mathbf{Z}_{E}\mathbf{Z}_{K} \tag{42}\]  
\(\mathbf{K}_{A}\) is the application factor, \(\mathbf{K}_{V}\) is the dynamic factor, \(\mathbf{K}_{H\beta}\) is the face load factor for contact stress, \(\mathbf{K}_{H\alpha}\) is the transverse load factor for contact stress, \(\sigma_{H0}\) is the nominal value of the contact stress, \(\mathbf{F}_{n}\) is the nominal normal force of the virtual cylindrical gear at mean point, \(\mathbf{I}_{bm}\) is the length of contact line in the middle of the zone of action, \(\mathbf{\rho}_{rel}\) is the radius of relative curvature vertical to the contact line, \(\mathbf{Z}_{M-B}\) is the mid-zone factor, \(\mathbf{Z}_{LS}\) is the load sharing factor, \(\mathbf{Z}_{E}\) is the elasticity factor, and \(\mathbf{Z}_{K}\) is the bevel gear factor.  
Figure 11: Comparison of loaded contact stress of the gear convex face: (**a**) Load T = 1500 Nm, (**b**) Load T = 4500 Nm, and (**c**) Load T = 7500 Nm.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Load T/Nm** & **Calculated Results \(\sigma_{H}\)/MPa** & **Simulation Results \(\sigma_{H}\)/MPa** & **Values of Differences \(\Delta\%\)** \\ \hline
1500 & 632.79 & 573.83 & 9.32 \\
3000 & 815.48 & 758.25 & 7.02 \\
4500 & 1045.32 & 914.49 & 12.51 \\
6000 & 1324.21 & 1138.1 & 14.05 \\
7500 & 1569.74 & 1331 & 15.21 \\ \hline \end{tabular}
\end{table}
Table 3: Results comparison of FEM and empirical formula of maximum contact stresses.