# Influence of Open Differential Design on the Mass Reduction Function
## 5 Optimization of Differential Transmission
### RSM Model  
The continuation of the article shows an analysis of the influence of four general design parameters (Table 8) on the response variables presented in Table 9. The objective of the RSM is to reduce the mass of the design elements of the ODT. Therefore, the objective function was set to reduce the mass of the design elements of the transmission (Table 9). The constraint functions in the form of maximum stresses were already included in the optimization as constraints of the topological optimization. The parameters were analyzed using the centrally composed experimental design, the values of which are listed in Table 10.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Name** & **Lower Value,** & **Upper Value,** \\  & & **mm** & **mm** \\ \hline \(a\) & Vertical chamfer of the ring gear & 1 & 7 \\ \(b\) & Horizontal chamfer of the ring gear & 1 & 13 \\ \(h\) & Overhang height of the ring gear & 1 & 3 \\ \(r\) & Inner radius of the drive shaft & 2.5 & 13.5 \\ \hline \hline \end{tabular}
\end{table}
Table 8: General design parameters and their values.  
Figure 14: General design parameters of the ring gear obtained by the TOM.  
Figure 15: Internal radius of the drive shaft.  
In order to obtain a mathematical model (description) of the influence of design parameters on response variables, functional dependencies were created that can contribute to faster and more efficient computer analysis in the process of designing a differential transmission. With the help of the regression analysis of the general design parameters and output mass parameters obtained by the centrally composed experimental design, which were carried out by numerical calculations, the mathematical functions of the response variables were obtained using TuringBot 2.16.1 [45] software by symbolic regression algorithm to derive mathematical formulas with high efficiency:  
\[m_{\text{all}}=16-\frac{r^{2}-r+b+h(26+a)-96}{99}, \tag{39}\]  
\[m_{1}=6\bigg{(}1-\frac{16h-78}{ab+276}\bigg{)}, \tag{40}\]  
\[m_{2}=9-\frac{r_{1}^{2}-42}{111}. \tag{41}\]  
The prediction plots (Figure 16) show for the response variables the dependence between the data obtained by numerical calculation and the data obtained by the mathematical model. The line drawn at an angle of 45\({}^{\circ}\) represents an ideal overlap of the mathematical model and the numerical results. The accuracy of the spread of the numerical model's response values relative to the predicted values should be evenly distributed on both sides of the diagonal. The diagram shows how the numerically determined values follow the line of ideal prediction (Figure 16).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \(a_{r}\) & \(b_{r}\) & \(h_{r}\) & \(r_{r}\) & \multirow{2}{*}{**No.**} & \(a_{r}\) & \(b_{r}\) & \(h_{r}\) & \(r_{r}\) \\  & **mm** & **mm** & **mm**  
The accuracy of the proposed mathematical model of the response variables for the R\({}^{2}\) and root mean square error (RMSE) values is shown in Table 11. From the graph in Figure 16 and the data in Table 11, some scatter in the results can be seen. This is due to the complexity of the geometry of the teeth in the contact between the pinion and ring gear (radii) and the density of the finite element mesh. These data should be taken into account when interpreting the above results. It can be concluded that the accuracy of the numerical and mathematical model is acceptable considering the above data. The accuracy was additionally confirmed by the numerical control calculation of the optimized design elements, which is presented in Section 6.