# Influence of Open Differential Design on the Mass Reduction Function
## 3 Design of the ODT
### Hypoid Bevel Gears Calculation
#### 3.3.2 Stress Calculation of Hypoid Bevel Gears in the Tooth Root and Tooth Face Flank  
Stress calculation in the tooth root was carried out for the most unfavorable case of tooth loading, i.e., when the force acts on the tooth crown. Loadability of the flanks represents the limit up to which the flanks of the teeth can be loaded without causing damage to the flanks of the teeth known as pitting [39; 40]. The material of the pinion gear is 17 CrNi6 and the selected material for ring gear is 15 CrNi6. For these materials, the amount of permanent dynamic strength for tooth root is \(\sigma_{\text{Film1}}=\sigma_{\text{Film2}}=500\) MPa and amount of permanent dynamic strength for tooth face flank is \(\sigma_{\text{Film1}}=\sigma_{\text{Film2}}=1630\) MPa.  
Allowed stress on pinion tooth root is determined by the expression:  
\[\sigma_{\text{FP1}}=\frac{\sigma_{\text{Film1}}}{S_{\text{Fmin}}}.K_{\text{FK}} \cdot Y_{\text{S}}, \tag{29}\]  
where \(S_{\text{Fmin}}\) is the minimum safety factor against root breakage, \(K_{\text{FK}}\) is size influence factor and \(Y_{\text{S}}\) is notch action factor.  
Bending stress on pinion tooth root is  
\[\sigma_{\text{FI}}=\frac{F_{\text{mt1}}}{b_{1}\cdot m_{\text{mm}}}.Y_{\text{F }}\cdot Y_{\text{ev}}\cdot Y_{\text{B}}\cdot K_{\text{FA}}\cdot K_{\text{FB} }<\sigma_{\text{FP1}}, \tag{30}\]  
where \(F_{\text{mt1}}\) is a tangential force on the dividing circle, \(Y_{\text{F}}\) is tooth shape factor, \(Y_{\text{ev}}\) is load share factor of auxiliary gears, \(Y_{\text{B}}\) is impact factor of tooth locking on the stress distribution in the root, \(K_{\text{FA}}\) is load distribution factor and \(K_{\text{FB}}\) is load distribution factor along the length of the tooth flank. Solutions obtained by expressions (29) and (30) are shown in Table 5.  
Allowed Hertzian stress on pinion tooth face flank, is determined by the expression:  
\[\sigma_{\text{HP1}}=\frac{\sigma_{\text{Hlim1}}}{S_{\text{Hmin}}}.K_{\text{L}} \cdot K_{\text{HK}}\cdot Z_{\text{V}}\cdot Z_{\text{N}}, \tag{31}\]  
where \(S_{\text{Hmin}}\) is minimum safety factor against tooth flank fracture, \(K_{\text{L}}\) is influence factor of lubricating oil, \(K_{\text{HK}}\) is factor of dimensions that influence the load capacity of the tooth flank, \(Z_{\text{V}}\) is speed influence factor and \(Z_{\text{N}}\) is service life factor.  
Hertzian stress in the kinematic pole of pinion is determined by the expression:  
\[\sigma_{\text{HI}}=Z_{\text{HV}}\cdot Z_{\text{M}}\cdot Z_{\text{eV}}\cdot \sqrt{\frac{F_{\text{mt1}}}{b_{1}\cdot d_{\text{ml}}}.\frac{\sqrt{i_{\text{d}} ^{2}+1}}{i_{\text{d}}}}\cdot K_{\text{HA}}\cdot K_{\text{HB}}<\sigma_{\text{ HP1}}, \tag{32}\]  
where \(Z_{\text{HV}}\) is tooth shape factor, \(Z_{\text{M}}\) is material factor, \(Z_{\text{eV}}\) is coverage factor, \(K_{\text{HA}}\) is load distribution factor and \(K_{\text{HB}}\) is load distribution factor along the length of the tooth flank. Solutions obtained by expressions (31) and (32) are shown in Table 5.  
Allowed stress on ring gear tooth root and bending stress on ring gear tooth root are determined according to expressions (29) and (30). Solutions for the ring gear, obtained using these expressions, are shown in Table 5.  
The allowed Hertzian stress on ring gear tooth face flank and Hertzian stress in the kinematic pole are determined according to expressions (31) and (32). Solutions for the ring gear, obtained using these expressions, are shown in Table 5. Table 5 shows how the pinion and ring hypoid bevel gear meet conditions from Equations (30) and (32). It can be concluded that both hypoid bevel gears can withstand maximum stresses in the tooth root and on the tooth face flank.