# Influence of Open Differential Design on the Mass Reduction Function
## 3 Design of the ODT
### Sun and Planetary Bevel Gears Calculation
#### 3.4.1 Dimensioning of the Planetary and Sun Bevel Gears  
Dimensioning includes the calculation of diameters, teeth width, teeth height and clearance. Values obtained by calculation are shown in Table 6. Expressions according to which the geometric quantities of the sun and planetary bevel gears are designed and presented in [35]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Geometric Quantities**} & \multicolumn{3}{c}{**Mark**} & \multicolumn{3}{c}{**Amount**} \\ \cline{2-6}  & \multicolumn{1}{c}{**Planetary**} & \multicolumn{1}{c}{**Sun**} & \multicolumn{1}{c}{**Planetary**} & \multicolumn{1}{c}{**Sun**} \\ \hline Pitch circle diameter, mm & \(d_{3}\) & & \(d_{4}\) & 102 & 136 \\ Length of the derivative of the dividing cone, mm & \(R_{\mathrm{a}}\) & & & 85 & \\ Tooth width, mm & \(b_{3}\) & & \(b_{4}\) & 26 & 26 \\ Middle pitch diameter of bevel gear, mm & \(d_{\mathrm{m3}}\) & & \(d_{\mathrm{m4}}\) & 85 & 115.2 \\ Clearance, mm & \(c\) & & & 2.125 & \\ Whole depth, mm & \(h_{3}\) & & \(h_{4}\) & 19.125 & 19.125 \\ Addendum depth for zero pair of bevel gears, mm & \(h_{\mathrm{a3}}\) & & \(h_{\mathrm{a4}}\) & 8.5 & 8.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Geometric quantities of planetary and sun bevel gears. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{Pinion gear} & \multirow{2}{*}{Stress on tooth root} & \(\sigma_{\mathrm{FP1}}=357.14\) MPa \\  & & \(F_{\mathrm{mH1}}=20,082.3\) N \\  & & \(\sigma_{\mathrm{F1}}=226.79\) MPa \\ \cline{2-3}  & Stress on tooth face & \(\sigma_{\mathrm{HP1}}=1310.14\) MPa \\  & flank & \(\sigma_{\mathrm{H1}}=1286\) MPa \\ \hline \multirow{4}{*}{Ring gear} & \multirow{2}{*}{Stress on tooth root} & \(\sigma_{\mathrm{FP2}}=357.14\) MPa \\  & & \(F_{\mathrm{mH2}}=23,987\) N \\  & & \(\sigma_{\mathrm{F2}}=149\) MPa \\ \cline{1-1} \cline{2-3}  & Stress on tooth face & \(\sigma_{\mathrm{HP2}}=1324\) MPa \\  & flank & \(\sigma_{\mathrm{H2}}=852.23\) MPa \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Amounts of stress in the tooth root and tooth face flank of pinion and ring bevel gear. 3.4.2 Stress Calculation of Planetary and Sun Bevel Gears in the Tooth Root and the Tooth Face Flank  
Allowed stress on planetary tooth root is determined by the expression:  
\[\sigma_{\rm FP3}=\frac{\sigma_{\rm Film3}}{S_{\rm Fmin}}\cdot Y_{\rm S}\cdot K_{ \rm FX}\cdot Y_{\rm N}\cdot Y_{\rm R}, \tag{35}\]  
where \(Y_{\rm N}\) is service life factor, \(Y_{\rm R}\) is roughness factor, \(Y_{\rm S}\) and \(K_{\rm FX}\) are factors according to [35]. Stress on planetary tooth root is  
\[\sigma_{\rm F3}=\frac{F_{\rm mt3}}{b\cdot m_{\rm m}}\cdot Y_{\rm F}\cdot Y_{\rm \varepsilon V}\cdot K_{\rm FA}\cdot K_{\rm FB}<\sigma_{\rm FP3}, \tag{36}\]  
where \(F_{\rm mt3}\) is tangential force on the middle pitch circle, \(Y_{\rm F}\), \(K_{\rm FA}\), \(K_{\rm FB}\) and \(Y_{\rm\varepsilon V}\) are factors according to [35]. Allowed Hertzian stress on planetary tooth face flank, is determined by the expression:  
\[\sigma_{\rm HP3}=\frac{\sigma_{\rm Film3}}{S_{\rm Hfilm}}\cdot K_{\rm L}\cdot K _{\rm HX}\cdot Z_{\rm R}\cdot Z_{\rm V}\cdot Z_{\rm N}\cdot Z_{\rm W}, \tag{37}\]  
where \(S_{\rm Hfilm}\) is minimum anti-pitting factor, \(Z_{\rm R}\) is roughness influence factor, \(Z_{\rm W}\) is influence factor of flank hardening, \(K_{\rm L}\), \(K_{\rm HX}\), \(Z_{\rm V}\) and \(Z_{\rm N}\) are factors according to [35]. Hertzian stress is determined by the expression:  
\[\sigma_{\rm H3}=Z_{\rm M}\cdot Z_{\rm HV}\cdot Z_{\rm\varepsilon V}\cdot\sqrt{ \frac{u_{\rm v}+1}{u_{\rm v}}\cdot\frac{F_{\rm mt3}}{b\cdot d_{\rm vm3}}\cdot K _{\rm H\alpha}\cdot K_{\rm H\beta}}<\sigma_{\rm HP3}, \tag{38}\]  
where \(Z_{\rm HV}\), \(Z_{\rm M}\), \(Z_{\rm\varepsilon V}\), \(K_{\rm FA}\) and \(K_{\rm FB}\) are factors according to [35]. Solutions for the planetary bevel gear, obtained by expressions (35)-(38), are shown in Table 7.  
Allowed stress on sun tooth root and stress on sun tooth root are determined according to expressions (35) and (36).