## 2 Analytical Model of DACBB
### Equilibrium Equation of DACBB  
According to Hertzian contact theory, the contact load between ball and raceway is determined by:  
\[\left\{ \begin{array}{c}Q_{mi}=K_{n}\delta_{mi}^{3/2},\delta_{mi}>0\\ Q_{mi}=0,\delta_{mi}\leq 0\end{array} \right.. \tag{23}\]  
Based on the Newton's third low, the reaction forces that the ball applied to the inner ring can be decomposed in the directions of radial and axial:  
\[\left\{ \begin{array}{c}Q_{mzi}=-Q_{mi}\sin\alpha_{mi}\\ Q_{mri}=-Q_{mi}\cos\alpha_{mi}\end{array} \right. \tag{24}\]  
where \(Q_{mri}\) and \(Q_{mzi}\) represent the radial and axial components of contact load between ball and raceway, respectively. Then summing all the radial and axial components of contact load at different azimuth position of ball, the equilibrium equation of DACBB can be established as  
\[\left\{ \begin{array}{c}F_{r}-\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}Q_{ mi}\cos\alpha_{mi}\cos\varphi_{i}=0\\ F_{a}-\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}c_{1}c_{4}Q_{mi}\sin\alpha_{mi} =0\end{array} \right. \tag{25}\]  
Moreover, the contact load between ball and raceway will produce an induced moment load to balance the effect of angular misalignment. The induced moment load is expressed as follow  
\[M-\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}Q_{mi}\cos\varphi_{i}(c_{1}L_{em} \cos\alpha_{mi}+c_{1}c_{4}R_{m}\sin\alpha_{mi})=0 \tag{26}\]  
where \(c_{4}\) is  
\[c_{4}=\left\{ \begin{array}{c}1\quad\text{back-to-back arrangement}\\ -1\quad\text{face-to-face arrangement}\\ c_{1}\quad\quad\text{tan dem arrangement}\end{array} \right.. \tag{27}\]It should be noted that the Equations (25) and (26) must satisfy the assumption that all the rolling balls of DACBB participate in the load transfer. However, in many situations, not all rolling balls inside the bearing are in contact with the raceway. In fact, when the external radial load and angular misalignment are very large or the axial preload is small, some rolling balls will be no longer constrained by the bearing inner ring [53]. This phenomenon will destroy the assumption of Equations (25) and (26). So, the rolling balls those do not participate in the load transfer should be excluded in the actual simulation. Evidently, as shown in Equation (21), if the contact deformation satisfies the relationship \(\delta_{mi}\leq 0\), ball and inner raceway will be in the critical or separate contact state, and the contact load will equal to zero. Therefore, the criteria for judging the contact state between ball and raceway is  
\[\left(l_{i}+l_{e}\right)_{mi}\leq\left(l_{0i}+l_{0e}\right)+\Delta_{i}+\Delta _{e} \tag{28}\]  
Therefore, the Equations (25) and (26) should be modified as  
\[\left\{ \begin{array}{c}F_{r}=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}Q_{mi }\cos\alpha_{mi}\cos\varphi_{i}-F_{rd}\\ F_{a}=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}c_{1}c_{4}Q_{mi}\sin\alpha_{mi }-F_{zd}\\ M=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{Z}Q_{mi}\cos\varphi_{i}(c_{1}L_{em} \cos\alpha_{mi}+c_{1}c_{4}R_{m}\sin\alpha_{mi})-M_{d}\end{array} \right. \tag{29}\]  
where  
\[\left\{ \begin{array}{c}F_{rd}=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{s}Q_{mi }\cos\alpha_{mi}\cos\varphi_{i}\\ F_{zd}=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{s}c_{1}c_{4}Q_{mi}\sin\alpha_{mi }\\ M_{d}=\sum\limits_{m=L}^{R}\sum\limits_{i=1}^{s}Q_{mi}\cos\varphi_{i}(c_{1}L_{ em}\cos\alpha_{mi}+c_{1}c_{4}R_{m}\sin\alpha_{mi})\end{array} \right. \tag{30}\]  
where \(s\) represents the number of inactive rolling balls, which is defined as the rolling ball that do not contact with the raceway. The above equilibrium equations are nonlinear. Therefore, Newton-Raphson iteration algorithm is adopted to solve the equations. In current simulation, the external radial and axial loads \(F_{r}\) and \(F_{a}\), and angular misalignment \(\gamma\) are known. Therefore, there are only two unknowns (\(\delta_{r}\) and \(\delta_{a}\)) should be determined.