# Design of Flexible Bearing with Larger Inner Diameter
## Modeling
### Parameter Design of IFB
#### 3.3.2 Determination of CCRR  
Under the condition of meeting the installation conditions, the number of balls in the following design is taken as 28. According to Equations (29), (30) and (41)-(43) the relationship between the CCRR of the FB and the fatigue life can be calculated, as shown in Figure 6:  
Figure 6: Relationship between CCRR and bearing life: **(a)** Relationship between CCRR and bearing life for CFB. **(b)** Relationship between CCRR and bearing life of IFB.  
As shown in Figure 6, the smaller the CCRR of the FB, the longer the fatigue life obtained. However, the curvature radius of the raceway is closely related to the axial load carrying capacity of the FB. The CCRR of the inner ring of CFB is between 0.515 and 0.525, and the CCRR of the outer ring is between 0.53 and 0.55 [5], in which the fatigue life of the inner ring of the IFB can reach more than 10,000 h so the CCRR of inner ring can be kept unchanged. The following is intended to improve the CCRR of the outer ring of the IFB from the perspective of the axial load capacity of the bearing. The contact point between the rolling elements and the collar is at the most edge state of the raceway. The axial load in this state is called the ultimate axial load, and the safe contact angle under the ultimate axial load is calculated as follows [37]:  
\[\sin(\theta-\alpha^{\prime})=\frac{2C_{\mathrm{a}}\left[\frac{\theta}{K}\frac{ \left(\frac{\cos\alpha_{0}}{\cos\alpha^{\prime}}-1\right)}{\left(D_{\mathrm{w} }\sum\rho\right)^{\frac{1}{3}}}\right.}{\left(D_{\mathrm{w}}\sum\rho\right)^{ \frac{1}{3}}} \tag{47}\]  
where \(\theta\) is the included angle between the connection line and bearing radial plane, the connection line linked the bearing outer ring raceway shoulder edge and rolling elements center of the line and between the angle, \(C_{\mathrm{a}}\) with \(F(\rho)\) related constants, can be obtained by looking up the table, \(\alpha_{0}\) for the initial contact angle of the bearing, \(\alpha^{\prime}\) for the safety of the bearing contact angle, and \(K\) for the elastic deformation constants:  
\[K=2C_{\mathrm{a}}\left(4-\frac{1}{f}\right)^{\frac{1}{3}} \tag{48}\]  
\(B\) is the total curvature radius coefficient:  
\[B=f_{\mathrm{i}}+f_{\mathrm{e}}-1 \tag{49}\]  
The safe contact angle \(\alpha^{\prime}\) of the FB can be calculated by iterative method according to Equation (46), and the ultimate axial load is calculated as follows:  
\[F_{\mathrm{a}}=\left[\frac{B}{K}\left(\frac{\cos\alpha_{0}}{\cos\alpha^{\prime }}-1\right)\right]^{1.5}ZD_{\mathrm{w}}{}^{2}g\sin\alpha^{\prime} \tag{50}\]  
In Equation (50), \(g\) is the acceleration of gravity. Equation (50) shows that the ultimate axial load of the FB is directly proportional to the number of rolling elements. Substituting the parameters of CFB into Equation (50) yields the ultimate axial load of 1035 N approximately. This is according to the geometric relationship, taking the number of rolling elements of the FB as 28 and CCRR of the IFB as 0.52, and by substituting above parameter into Equation (50) to carry out the calculation of the ultimate axial load of the FB with different CCRR, as shown in Figure 7:  
As shown in Figure 7, the larger the CCRR of the outer ring in FB, the stronger the axial load carrying capacity of the bearing. In summary, following the aforementioned method, the basic parameters such as the diameter and the number of rolling elements, the coefficient of curvature radius of the raceways of the outer inner ring, and the outer ring in FB can be determined. For example, the basic parameters of the CFB and the IFB of type FB812 are listed in Table 4. Under the condition of limiting speed of 3000 rpm, the fatigue life of the IFB can reach over 8000 h.