# Influence of Open Differential Design on the Mass Reduction Function
## 3 Design of the ODT
### Hypoid Bevel Gears Calculation  
#### 3.3.1 Dimensioning of the Driving and Driven Hypoid Bevel Gears  
The hypoid gear pair consists of two bevel gears (z1, z2) that are in mutual grip (Figure 2). The pinion (z1) is a part of the drive shaft of the differential transmission. By iterative calculation, variables needed to design the pinion and driven hypoid bevel gear (ring gear) were obtained. Iteration is carried out until the following condition is satisfied reached:  
\[|R_{\rm mint}-R_{\rm m1}|<0.0001\cdot R_{\rm m1}, \tag{16}\]  
where \(R_{\rm m1}\) is the mean cone distance of the pinion. \(R_{\rm m1}\) is calculated according to the expression:  
\[R_{\rm m1}=\frac{d_{\rm m1}}{2\cdot\sin\delta_{1}}, \tag{17}\]where \(d_{\rm m1}\) is pinion mean pitch diameter. Diameter \(d_{\rm m1}\) is determined by the expression:  
\[d_{\rm m1}=\frac{d_{\rm m2}}{i_{\rm d}}\cdot F. \tag{18}\]  
Ring gear mean pitch diameter is determined by the expression:  
\[d_{\rm m2}=d_{\rm e2}-b_{2}\cdot\sin\delta_{2}. \tag{19}\]  
\(R_{\rm{mint}}\) is determined using the following expression:  
\[R_{\rm{mint}}=\frac{A_{3}\cdot A_{4}}{A_{5}\cdot A_{6}+A_{3}\cdot A_{7}}, \tag{20}\]  
where \(A_{3}\), \(A_{4}\), \(A_{5}\), \(A_{6}\) and \(A_{7}\) are intermediate variables. Before the first iteration, the value of the hypoid dimensional factor is \(F=1\). Also, before the first iteration, the amount of ring gear pitch angle is  
\[\delta_{2}=\tan^{-1}\left(\frac{\sin\Sigma}{\frac{F}{i_{\rm d}}+\cos\Sigma} \right)=73.07^{\circ}, \tag{21}\]  
where \(\Sigma\) is the shaft angle. The amount of this angle is \(90^{\circ}\). The pinion gear pitch angle is  
\[\delta_{1}=\Sigma-\delta_{2}=16.93^{\circ}. \tag{22}\]  
Values of \(\delta_{1}\) and \(\delta_{2}\) obtained by Equations (21) and (22) are added in the first iteration, and their new values will be determined in the calculation of the first iteration (Table 3). Other variables and their values, which were determined by the iteration calculation, are presented in Table 3. Values determined in the last iteration represent the final solutions of the iterative calculation. The control calculation is determined by the final values of the following three geometric quantities, which are shown in Table 4. Pinion offset angle in the axial plane:  
\[\varsigma_{\rm m}=\sin^{-1}\left(\frac{2\cdot a}{d_{\rm m2}+d_{\rm m1}\cdot \frac{\cos\delta_{2}}{\cos\delta_{1}}}\right). \tag{23}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **First Iteration** & **Last Iteration** \\ \hline Ring gear mean pitch diameter (\(d_{\rm m2}\)), mm & 304.08 & 305.59 \\ Pinion offset angle in the axial plane (\(\zeta_{\rm m}\)), \({}^{\circ}\) & 11.28 & 10.51 \\ Pinion offset angle in the pitch plane (\(\xi_{\rm m}\)), \({}^{\circ}\) & 11.78 & 11.34 \\ Pinion gear pitch angle (\(\xi_{1}\)), \({}^{\circ}\) & 16.59 & 22.31 \\ Ring gear pitch angle (\(\xi_{2}\)), \({}^{\circ}\) & 67.57 & 67.69 \\ Mean normal module (\(m_{\rm n}\)), mm & 5.41 & 5.44 \\ Spiral angle of pinion (\(\beta_{\rm m1}\)), \({}^{\circ}\) & 46.78 & 46.34 \\ Hypoid dimensional factor (\(F\)) & 1.196 & 1.187 \\ Pinion mean pitch diameter (\(d_{\rm m1}\)), mm & 110.7 & 110.36 \\ Mean cone distance of the pinion (\(R_{\rm m1}\)), mm & 190.07 & 145.35 \\ Mean cone distance of the ring gear (\(R_{\rm m2}\)), mm & 158.93 & 165.16 \\ Auxiliary angle (\(\
u_{\rm m}\)), \({}^{\circ}\) & 10.8 & 9.74 \\ Iteration condition: (\(|R_{\rm{mint}}-R_{\rm m1}|<0.0001\cdot R_{\rm m1}\)) & \(|-47.537|<0.019\) & \(|-0.0101<0.0145\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Variable values determined by iterative calculation.