# Design of Flexible Bearing with Larger Inner Diameter
## 2 Materials and Methods
### Calculation of RREL of Bearing Collar  
From the geometry of the bearing [37]:  
\[\left\{\begin{array}{c}D=d_{\rm m}\mp D_{\rm w}\cos\alpha\\ \left[1+F(\rho)\right]\frac{D_{\rm w}}{2}\sum\rho=\frac{2}{1+\gamma}\\ \left[1-F(\rho)\right]\frac{D_{\rm w}}{2}\sum\rho=D_{\rm w}(\frac{1}{R}-\frac {1}{r})\end{array}\right. \tag{21}\]  
where, \(r\) is curvature radius of raceway (CRR), \(R\) is radius of curvature of the rolling elements generatrix, and \(\gamma\) is a dimensionless geometric parameter defined as:  
\[\gamma=\frac{D_{\rm w}\cos\alpha}{d_{\rm m}} \tag{22}\]  
where \(\alpha\) is the contact angle of the bearing and \(d_{\rm m}\) is the diameter of the bearing pitch circle.  
During the operation of flexible bearing, the number of stress cycles is not equal to the number of contact cycles, the FB is subjected to force on both sides in the major axis direction after assembling the cam WG, the number of stress cycles of FB is twice of the CRB. During HD operation, assuming that the outer ring of the FB is stationary, the inner ring rotation, the REL distribution range is \(2\phi_{1}\), then the number of stress cycles in the inner ring raceway:  
\[u_{\rm i}=Z(1+\gamma)\frac{\phi_{1}}{180} \tag{23}\]  
where \(Z\) is the number of rolling elements and \(\phi\) is the REL distribution range angle, as shown in Figure 1:  
The number of stress cycles in the outer raceway is:  
\[u_{\rm e}=Z(1-\gamma) \tag{24}\]  
Figure 1: FB load range schematic.  
Substituting Equations (21)-(24) into (20) yields:  
\[\Phi_{\rm i}=0.068555[1+F(\rho)]^{2.1}\bigg{(}\frac{T_{1}}{T}\bigg{)}^{3.1}\bigg{(} \frac{\xi}{\xi_{1}}\bigg{)}^{0.4}\frac{(1-\gamma)^{1.8}}{(1-\gamma)^{\frac{1}{ 3}}}\bigg{(}\frac{D_{\rm w}}{d_{\rm m}}\bigg{)}^{0.3}\bigg{(}\frac{Z\phi_{\rm i} }{90}\bigg{)}^{-\frac{1}{3}}n_{\rm a}^{\ 2.8}n_{\rm b}^{\ 3.5} \tag{25}\]  
\[\Phi_{\rm e}=0.054409[1+F(\rho)]^{2.1}\bigg{(}\frac{T_{1}}{T}\bigg{)}^{3.1} \bigg{(}\frac{\xi}{\xi_{1}}\bigg{)}^{0.4}\frac{(1+\gamma)^{1.8}}{(1-\gamma)^{ \frac{1}{3}}}\bigg{(}\frac{D_{\rm w}}{d_{\rm m}}\bigg{)}^{0.3}(Z)^{-\frac{1}{ 3}}n_{\rm a}^{\ 2.8}n_{\rm b}^{\ 3.5} \tag{26}\]  
In Equations (25) and (26), subscript \({\rm i}\) indicates the inner ring and subscript \({\rm e}\) the outer ring. According to bearing geometry, the following can be obtained:  
\[[1+F(\rho)]^{2.1}\bigg{(}\frac{T_{1}}{T}\bigg{)}^{3.1}\bigg{(}\frac{\xi}{\xi_ {1}}\bigg{)}^{0.4}n_{\rm a}^{\ 2.8}n_{\rm b}^{\ 3.5}\approx 1.3\bigg{[}\bigg{(}\frac{D_{\rm w}}{2R} \bigg{)}\frac{r-R}{r}(1\mp\gamma)\bigg{]}^{-0.41} \tag{27}\]  
The upper symbols in the above formula refer to the inner ring and the lower symbols to the outer ring. For ball bearing, the coefficient of curvature radius of raceway (CCRR) \(f\) is usually used to describe the degree of close contact between the rolling elements and the raceway:  
\[f=\frac{r}{D_{\rm w}} \tag{28}\]  
Substituting Equations (25)-(28) into (19) yields the formula for calculating the REEL of the inner ring:  
\[Q_{\rm ci}=A\bigg{(}\frac{2f_{\rm i}}{2f_{\rm i}-1}\bigg{)}^{0.41}\bigg{(} \frac{D_{\rm w}}{d_{\rm m}}\bigg{)}^{0.3}\bigg{(}\frac{\phi_{\rm i}}{90} \bigg{)}^{-\frac{1}{3}}\frac{\bigg{(}1-\frac{D_{\rm w}\cos\alpha}{d_{\rm m}} \bigg{)}^{1.39}}{\Big{(}1+\frac{D_{\rm w}\cos\alpha}{d_{\rm m}}\bigg{)}^{ \frac{1}{3}}}D_{\rm w}^{\ 1.8}(Z)^{-\frac{1}{3}} \tag{29}\]  
The formula for calculating the REEL of the outer ring is:  
\[Q_{\rm ce}=A\bigg{(}\frac{2f_{\rm e}}{2f_{\rm e}-1}\bigg{)}^{0.41}\bigg{(} \frac{D_{\rm w}}{d_{\rm m}}\bigg{)}^{0.3}\frac{\bigg{(}1+\frac{D_{\rm w}\cos \alpha}{d_{\rm m}}\bigg{)}^{1.39}}{\Big{(}1-\frac{D_{\rm w}\cos\alpha}{d_{\rm m }}\bigg{)}^{\frac{1}{3}}}D_{\rm w}^{\ 1.8}(2Z)^{-\frac{1}{3}} \tag{30}\]  
Equations (29) and (30), \(A\) for the material coefficient, for CRB steel, a large number of experiments to derive \(A=98.1\), \(f_{\rm i}\) for the CCRR of inner ring, \(f_{\rm e}\) for the CCRR of outer ring.