# Optimization Design of RV Reducer Crankshaft Bearing
## 2 Performance of crankshaft Bearings of Rotate Vector (RV) Reducers
### Fatigue Life of Crankshaft Bearings
#### 2.3.2 Roller-Raceway Contact Stress  
Hertzian contact is often used to solve the contact stress between two cylinders. However, it cannot reveal the stress concentration and the effect of the roller profile. Non-Hertzian contact method can reveal the effect of the roller profile. The distribution of the contact stress over the length of the rollers can be obtained by using non-Hertzian theory. Therefore, it is adopted in this study.  
The contact forces between the raceway and roller were calculated using the quasi-static model. When the roller is subjected to a normal load \(Q\), it is assumed that the contact area between the raceway and the roller is \(A\). The contact pressure at coordinate \((x,y)\) is \(P(x,y)\). The integral of pressure \(P(x,\ y)\) is equal to the contact force \(Q\).  
\[\int\limits_{A}P(x,y)dxdy=Q \tag{15}\]  
Contact stress occurs when two surfaces come into contact with each other. Assuming that the contact deformation between the two surfaces at coordinate \((x,\ y)\) is \(\delta(x,\ y)\), then  
\[\begin{cases}P(x,y)=0,&\text{if }\delta(x,y)<0\\ P(x,y)>0,&\text{if }\delta(x,y)\geq 0\end{cases} \tag{16}\]  
According to the half-space theory, when two bodies come into contact, the elastic deformation \(\delta(x,y)\) can be expressed by the following equation:  
\[\delta(x,y)=\frac{2}{E^{\prime}}\int\limits_{A}\frac{P(x^{\prime},y^{\prime}) dx^{\prime}dy^{\prime}}{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime} \right)^{2}}} \tag{17}\]  
where \(E^{\prime}\) is the equivalent Young's modulus.  
\[E^{\prime}=\frac{E}{1-\
u^{2}} \tag{18}\]  
where \(\
u\) is the [PERSON]'s ratio and \(E\) is the elastic modulus.  
When the roller is subjected to the normal load \(Q\), the contact surfaces are moved toward each other by a displacement \(\delta\). For a roller with a crowned profile, \(z(x,y)\) is the gap between the roller and raceway at the coordinate \((x,y)\) generated by the crown drop. \(f(x,y)\) is the initial distance generated by the geometry of the roller and raceway. Therefore, Equation (17) can be written as follows:  
\[\frac{2}{E^{\prime}}\int\limits_{A}\frac{P(x^{\prime},y^{\prime})dx^{\prime} dy^{\prime}}{\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}}}= \delta-f(x,y)-z(x,y) \tag{19}\]In these equations, the relative displacement \(\delta\), pressure distribution \(P(x,y)\), and contact area \(A\) are unknown. To simplify this problem, the contact area is subdivided into multiple discrete cells. It is assumed that the contact pressure distribution on each cell is constant. Therefore, the integral is replaced by a summation, represented as  
\[\sum_{j=1}^{r}P_{j}f_{ij}=D_{i},\ i=1,2\cdots r \tag{20}\]  
where \(f_{ij}\) is the influence coefficient representing the deflection of cell \(i\) because of a uniform pressure over the cell \(j\), which can be obtained according to reference [20]. \(D_{i}\) is the contact deformation of cell \(i\).  
Finally, the equilibrium condition of the applied contact force and the integral of the pressure over the contact yields:  
\[\frac{Q}{A}-\sum_{j=1}^{r}P_{j}=0 \tag{21}\]  
Equations (20) and (21) are solved simultaneously for the contact area and contact stress.