## 2 Analytical Model of DACBB
### Analysis of Internal Deformation of DACBB
Therefore, at different azimuth positions, the additional radial displacements of \(O_{L}\) and \(O_{R}\) caused by angular misalignment \(\gamma\) is  
\[u_{mr\gamma i}=c_{1}\gamma L_{em}\cos\varphi_{i} \tag{5}\]  
where \(\overline{O_{h}O_{h}^{T}}=\overline{O_{m}O_{m}^{T}}=c_{1}\gamma L_{em}\), \(c_{1}\) is used to adjust the sign of the radial displacements of the left and the right row. \[c_{1}=\left\{ \begin{array}{cc}1&m=L\\ -1&m=R\end{array} \right. \tag{6}\]  
where \(L_{em}\) is the axial displacement between \(O_{L}\) and \(O_{R}\) and bearing geometrical center \(O\), which is  
\[L_{em}=L+c_{2}(r_{i}-0.5D)\sin\alpha_{0} \tag{7}\]  
where \(L\) is the axial distance between the geometric center of DACBB and bearing one row, \(\alpha_{0}\) is the unloaded contact angle, \(r_{i}\) is the radius of inner ring groove. \(c_{2}\) depends on the configuration of DACBB, which is  
\[c_{2}=\left\{ \begin{array}{cc}-1&\text{back-to-back arrangement}\\ 1&\text{face-to-face arrangement}\\ c_{1}&\tan\text{dem arrangement}\end{array} \right. \tag{8}\]  
Similarly, as seen in Figure 4, the rotation of \(\overline{O_{1}O_{L}}\) and \(\overline{O_{2}O_{R}}\) also leads to additional axial displacements of \(O_{L}\) and \(O_{R}\)  
\[u_{mzyi}=\overline{O_{h}O_{m}}\gamma=R_{m}\gamma\cos\varphi_{i} \tag{9}\]  
where \(R_{m}\) is the distance between the inner ring groove curvature center and \(z\)-axis, which is  
\[R_{m}=(0.5d_{m}+(r_{i}-0.5D))\cos\alpha_{0} \tag{10}\]  
where \(d_{m}\) is the pitch diameter of DACBB. Moreover, the radial and axial displacements of \(O_{L}\) and \(O_{R}\) caused by the external radial and axial load should also be considered. When the inner ring center of DACBB displaces the displacement \(\delta_{r}\), the reference points \(O_{L}\) and \(O_{R}\) will accordingly generate the local radial displacements. The local radial displacements of \(O_{L}\) and \(O_{R}\) at different azimuth position can be determined by  
\[\delta_{mri}=\delta_{r}\cos\varphi_{i} \tag{11}\]  
Similarly, for each row of DACBB, when the inner ring of DACBB is loaded by pure axial load, the reference points of DACBB at different azimuth positions have the same displacement \(\delta_{a}\). Therefore, the axial displacements of \(O_{L}\) and \(O_{R}\) caused by external axial load can be determined by  
\[\delta_{mzi}=\delta_{a} \tag{12}\]  
According to the aforementioned analysis, the total radial and axial displacements of \(O_{L}\) and \(O_{R}\) can be formulated as below:  
\[u_{mri}=u_{mr\gamma i}+\delta_{mri}=\delta_{r}\cos\varphi_{i}+c_{1}\gamma L_{ em}\cos\varphi_{i} \tag{13}\]  
\[u_{mzi}=u_{mzyi}+\delta_{mzi}=\delta_{a}+R_{m}\gamma\cos\varphi_{i} \tag{14}\]  
As seen in Figure 5, the internal positions and displacements of ball center and reference points are illustrated under external load. According to the assumption in Section 2.1, the inner and outer ring groove curvature centers and ball centers are always collinear under angular misalignment condition. Therefore, the actual contact deformation between ball and raceway can be calculated. For the sake of generality, the clearance between ball and raceway is considered in current study. The initial distances between the center of ball and the center of inner/outer ring groove are  
\[l_{0i}=r_{i}-0.5D-\Delta_{i} \tag{15a}\] \[l_{0e}=r_{e}-0.5D-\Delta_{e} \tag{15b}\]  
where \(\Delta_{i}\) and \(\Delta_{e}\) respectively denote the clearances between ball and inner/outer raceway. According to the geometrical relations shown in Figure 5, the actual distance between inner and outer ring groove centers is  
\[\left(l_{i}+l_{e}\right)_{mi}=\left[\left(\left(l_{0i}+l_{0e}\right)\cos \alpha_{0}+u_{mri}\right]^{2}+\left\{\left(l_{0i}+l_{0e}\right)\sin\alpha_{0}+ c_{3}u_{mzl}\right]^{2}\right]^{1/2} \tag{16}\]  
where \(c_{3}\) is a parameter related to the configuration of DACBB, which is expressed by:  
\[\text{For back-to-back}:\ c_{3}=\left\{ \begin{array}{cc}1&m=L\\ -1&m=R\end{array} \right. \tag{17a}\] \[\text{For face-to-face}:\ c_{3}=\left\{