## 4 Simulation Results
### Tooth Root Bending Stress  
When load T = 500 Nm, T = 1500 Nm and T = 4500 Nm, the maximum root bending stress variation of the gear and the pinion is shown in Figure 12.  
As shown in Figure 12, the maximum root bending stress appears near the transition line between tooth surface and tooth root. When the load T increases, the root bending stresses also increase, and the maximum root bending stress \(\sigma_{F2}\) of the gear is less than the maximum root bending stress \(\sigma_{F1}\) of the pinion. When torque increases from 1500 Nm to 7500 Nm, the maximum root bending stress \(\sigma_{F}\) comparisons between the FEM results and the calculated results of empirical formula are listed in Table 4 below. The empirical formula is according to ISO 10300 and ISO 6336, which is shown in Equations (43) and (44). The calculation is based on the maximum bending stress at the tooth root. It is determined separately for pinion and wheel. Some parameters for root bending stress calculation were determined by engineering experience and actual operation conditions, such as geometric similarity, operation, and manufacturing of actual gears:  
\[\mathbf{\sigma}_{F}=\mathbf{\sigma}_{F0}\mathbf{K}_{\alpha}\mathbf{K}_{\
u}\mathbf{K} _{F\beta}\mathbf{K}_{F\alpha} \tag{43}\]  
\[\mathbf{\sigma}_{F0}=\sqrt{\frac{\mathbf{F}_{\mathrm{vmT}}}{\mathbf{b}_{v}\mathbf{m}_{mn }}}\mathbf{Y}_{Fa}\mathbf{Y}_{Sa}\mathbf{Y}_{c}\mathbf{Y}_{BS}\mathbf{Y}_{LS} \tag{44}\]  
\(\mathbf{K}_{F\beta}\) is the face load factor for bending stress, \(\mathbf{K}_{F\alpha}\) is the transverse load factor for bending stress, \(\mathbf{\sigma}_{F0}\) is the nominal tooth root stress, \(\mathbf{F}_{\mathrm{vmT}}\) is the nominal tangential force of the virtual cylindrical gear, \(\mathbf{b}_{v}\) is the face width of virtual cylindrical gear, \(\mathbf{m}_{mn}\) is mean normal module, \(\mathbf{Y}_{Fa}\) is the tooth form factor, \(\mathbf{Y}_{Sa}\) is the stress correction factor, \(\mathbf{Y}_{Ec}\) is the contact ration factor, \(\mathbf{Y}_{BS}\) is the level spiral angle factor, and \(\mathbf{Y}_{LS}\) is the load sharing factor. As shown in Table 4, the results show that the results of finite element analysis and empirical formula are similar, and the empirical value is about 10-14% larger than that of finite element analysis. The authors believe that one reason is that the load T increases the deformation of the bevel gears, which increases the actual contact ratio of the bevel gears, but the empirical value does not take this into account. Another reason is that the finite element model constraint condition setting and the parameters selection of theoretical formula will affect the difference value. In order to observe the distribution of the maximum tooth root bending stress in a meshing period, the FEM analysis of the root bending stress is carried out by rotating the gear step by step. When load increases from 1500 Nm to 7500 Nm, the variation of the maximum value of root bending stresses of the pinion and the gear is shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{3}{*}{**Load T/Nm**} & \multicolumn{3}{c}{**Gear**} & \multicolumn{3}{c}{**Pinion**} \\ \cline{2-7}  & **FEM** & **Calculated** & **Values of** & **FEM** & **Calculated** & **Values of** \\  & **Results** & **Results** & **Differences** & **Results** & **Results** & **Differences** \\  & \(\sigma_{F2}\)/MPa & \(\sigma_{F2}\)/MPa & \(\Delta\)\% & \(\sigma_{F1}\)/MPa & \(\sigma_{F1}\)/MPa & \(\Delta\)\% \\ \hline
1500 & 296 & 343 & 13.71 & 338 & 389 & 13.11 \\
3000 & 456 & 529 & 13.79 & 567 & 643 & 11.82 \\
4500 & 621 & 710 & 12.54 & 791 & 883 & 10.42 \\
6000 & 805 & 890 & 9.55 & 925 & 1039 & 13.86 \\
7500 & 912 & 1067 & 12.52 & 1183 & 1352 & 12.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results comparison of FEM and empirical formula of bending stresses.