## 2 Analytical Model of DACBB
### Analysis of Internal Deformation of DACBB  
A cylindrical system \(r\), \(\varphi\), \(z\) with its origin in the geometrical center of DACBB is introduced in Figure 3. The \(z\)-axis is the bearing rotational axis, \(r\) points to the radial direction of bearing, \(\varphi\) represents the azimuth angle of ball. The inner ring of DACBB is loaded externally by the load vector \(\left\{F\right\}^{T}=(F_{r},F_{a})\). \(F_{r}\) and \(F_{a}\) represent the external radial and axial loads, respectively. Angular misalignment between inner and outer rings is represented by the angle \(\gamma\). As a result, two corresponding translational displacements \(\left\{\delta\right\}^{T}=(\delta_{r},\delta_{a})\) of the bearing inner ring will be generated at the same origin. \(\delta_{r}\) and \(\delta_{a}\) denote the radial and axial displacements of inner ring, respectively. Meanwhile, the bearing will also produce an induced moment load \(M\). As shown in Figure 3b, \(O_{L}\) and \(O_{R}\) are the inner ring groove curvature centers of the left and the right rows of DACBB, respectively. According to the assumption in Section 2.1, only the inner ring groove curvature centers \(O_{L}\) and \(O_{R}\) move accordingly when the inner ring displaces. As a result, local displacement and contact load are generated, as shown in Figure 3b. Therefore, \(O_{L}\) and \(O_{R}\) are taking as the reference points in the following analysis. When the DACBB is subjected to combined external loads and angular misalignment, the local radial and axial displacements at reference points is  
\[\left\{u_{mi}\right\}^{T}=(u_{mri},u_{mzi}),(m=L,R) \tag{1}\]  
where \(u_{mri}\) and \(u_{mzi}\) are the local radial and axial displacements, respectively. The subscript \(m\) denotes the bearing rows, \(m=L\) and \(m=R\) represent the left and right row of DACBB, respectively. Meanwhile, at the reference points, the corresponding local contact load is generated  
\[\left\{Q_{mi}\right\}^{T}=(Q_{mri},Q_{mzi}) \tag{2}\]  
where \(Q_{mri}\) and \(Q_{mzi}\) are the radial and axial contact loads between ball and raceway, respectively. Moreover, the centers of rolling ball of each row will also move to the new positions. \[\left\{v_{mi}\right\}^{T}=(v_{mri},v_{mzi}) \tag{3}\]  
where \(v_{mri}\) and \(v_{mzi}\) are the radial and axial displacements of ball center, respectively. To calculate the internal deformation of each row of DACBB, deformations caused by the angular misalignment are firstly analyzed. As shown in Figure 4, \(O_{1}\) and \(O_{2}\) are the projection points of \(O_{L}\) and \(O_{R}\) on the bearing rotating axis, respectively. \(\varphi_{i}\) is the azimuth angle of ball. When the inner ring tilts \(\gamma\) relative to the outer ring. Accordingly, \(r\)-axis and \(z\)-axis will rotate to \(r^{T}\)-axis and \(z^{T}\)-axis, respectively. \(O_{h}(h=1,2)\) and \(O_{m}(m=L,R)\) will also rotate to \(O_{h}^{T}\) and \(O_{m}^{T}\). As described in Figure 4, it is observed that the rotation of \(\overline{O_{L}O_{R}}\) leads to the additional radial displacements of \(O_{L}\) and \(O_{R}\). Because the rotating angle \(\gamma\) is usually small, the radial displacements of \(O_{L}\) and \(O_{R}\) can be approximately represented by \(\overline{O_{L}O_{L}^{T}}\) and \(\overline{O_{R}O_{R}^{T}}\). Therefore, the additional radial displacements of \(O_{L}\) and \(O_{R}\) at the azimuth angle \(\varphi_{i}=0\) can be calculated by  
\[\left(u_{m\gamma}\right)_{\varphi=0}=\overline{O_{h}O_{h}^{T}}=\overline{O_{m }O_{m}^{T}} \tag{4}\]  
Figure 4: Deformations of each row of DACBB. Figure 3: Loads and displacements of DACBB. (**a**) external loads and displacements of inner ring; (**b**) local loads and displacements.