# Influence of Open Differential Design on the Mass Reduction Function
## 4 Numerical Analysis of Differential Transmission
### Analysis of Stress and Displacement of Bevel Gear and Pinion Gear  
Since the pinion and bevel gear are in contact, it was necessary to include a contact problem with the presence of friction as a nonlinear problem in the finite element analysis. Considering the complexity of the geometry, the meshing process requires a larger number of finite elements. A large number of elements has a direct impact on increasing the duration of the numerical analysis and later on the computational time for the optimization process itself. In order to avoid the mentioned problem, a coarser mesh of the finite elements of the gears in engagement was created as a global model, which is used to determine the displacement fields. Later, the submodeling technique was applied, where a finer mesh could be used in the area of the submodel, mainly at the point of contact, i.e., where stress concentrations occur. The displacement field of the global model is used as a boundary condition for the submodel. The contacts are defined at the contact surfaces between the gear teeth of the bevel gear and the pinion gear (Figure 5).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{Planetary gear} & \multirow{2}{*}{Stress on tooth root} & \(\sigma_{\text{FP3}}=512.15\) MPa & \multirow{2}{*}{\(\sigma_{\text{F3}}<\sigma_{\text{FP3}}\)} \\  & & \(F_{\text{m4D}}=15.391\) N & \\  & & \(\sigma_{\text{F3}}=274.93\) MPa & \\ \cline{2-4}  & Stress on tooth face & \(\sigma_{\text{HP3}}=1441.1\) MPa & \multirow{2}{*}{\(\sigma_{\text{H3}}<\sigma_{\text{HP3}}\)} \\  & flank & \(\sigma_{\text{HB3}}=1400.04\) MPa & \\ \hline \multirow{4}{*}{Sun gear} & \multirow{2}{*}{Stress on tooth root} & \(\sigma_{\text{FP4}}=512.5\) MPa & \multirow{2}{*}{\(\sigma_{\text{F4}}<\sigma_{\text{FP4}}\)} \\  & & \(F_{\text{Fm4D}}=15.391\) N & \\ \cline{1-1}  & & \(\sigma_{\text{F4}}=224.03\) MPa & \\ \cline{1-1} \cline{2-4}  & Stress on tooth face & \(\sigma_{\text{HP4}}=1441.1\) MPa & \\ \cline{1-1}  & flank & \(\sigma_{\text{FH4}}=954.44\) MPa & \\ \hline \hline \end{tabular}
\end{table}
Table 7: Amounts of stress in the tooth root and tooth face flank of planetary and sun bevel gears.  
Figure 5: Contact surfaces: (**a**) on the ring gear; (**b**) on the pinion gear.  
The global model was meshed with second-order SOLID187 tetrahedral finite elements. The size of the finite elements was set to 2 mm and scaled to 0.25 mm at the contact points. The size was determined from the convergence test for the equivalent von Mises stress. The mesh of finite elements of the global model consists of 157,398 finite elements and 230,134 nodes (Figure 6a). There are 7332 finite elements in contact on the contact surfaces of the gear teeth in engagement. After the gear teeth are meshed, the boundary conditions for the global model are defined. The locations where the bearings are installed is restricted in the radial and axial direction. The rotation is allowed so the tangential component of the displacement is free (Figure 6b). The given moment load is \(T_{1}\) (Figure 6b), the value of which is given in Table 1.  
The fields for the distribution of total displacements (Figure 7) and the fields for the equivalent stresses according to [PERSON] (Figure 8) were obtained by numerical calculations in the ANSYS Workbench program package. The obtained maximum value of equivalent stresses (Figure 8) was high due to the very coarse mesh of finite elements, while accurate displacements can be obtained even with this coarse mesh. In order to obtain a more accurate value of equivalent stress and compare it with the stress obtained by the analytical calculation for the pinion and bevel gear (Table 5), submodeling with a finer finite element mesh was performed.  
Figure 6: Global model: (**a**) tetrahedral finite element mesh; (**b**) boundary conditions.  
Figure 7: Distribution of total displacements in mm, pinion and bevel gear.