## 2 Dynamic Model of the Feed System
### Calculation of Restoring Force of Bearings
#### 2.2.1 Restoring Force of Angular Contact Ball Bearing  
The coordinates of the inner and outer raceway centers for the left angular contact ball bearing in \(\text{CS}_{\text{Ab}}\) are:  
\[\left[ \begin{array}{c}x_{Lij}^{\text{A}}\\ y_{Lij}^{\text{A}}\\ z_{Lij}^{\text{A}}\\ 1\end{array} \right]=\text{Trans}\big{(}R_{\text{Am}}\cos\theta_{\text{bj}}+ \delta_{Sx},R_{\text{Am}}\sin\theta_{\text{bj}}+\delta_{Sy},-L_{\text{LA}}+ \delta_{Sz}\big{)}\cdot\text{Rot}_{z}\Big{(}\theta_{\text{bj}}+\frac{\pi}{2} \Big{)}\cdot\left[ \begin{array}{c}0\\ -(R_{\text{i}}-R_{\text{bb}})\cos\alpha_{\text{bb}0}\\ (R_{\text{i}}-R_{\text{bb}})\sin\alpha_{\text{bb}0}\\ 1\end{array} \right] \tag{31}\]  
\[\left[ \begin{array}{c}x_{Loj}^{\text{A}}\\ y_{Loj}^{\text{A}}\\ z_{Loj}^{\text{A}}\\ 1\end{array} \right]=\text{Trans}\big{(}R_{\text{Am}}\cos\theta_{\text{bj}},R_{ \text{Am}}\sin\theta_{\text{bj}},-L_{\text{LA}}\big{)}\cdot\text{Rot}_{z} \Big{(}\theta_{\text{bj}}+\frac{\pi}{2}\Big{)}\cdot\left[ \begin{array}{c}0\\ (R_{\text{o}}-R_{\text{bb}})\cos\alpha_{\text{bb}0}\\ -(R_{\text{o}}-R_{\text{bb}})\sin\alpha_{\text{bb}0}\\ 1\end{array} \right] \tag{32}\]  
where \(\delta_{Sx}\), \(\delta_{Sy}\) and \(\delta_{Sz}\) are the displacement of the screw shaft along the \(x\), \(y\) and \(z\) directions; \(R_{\text{Am}}\) is the pitch radius; \(R_{\text{i}}\) and \(R_{\text{o}}\) are the inner and outer raceway radius; \(R_{\text{bb}}\) is the ball radius; \(\alpha_{\text{bb}}\) and \(\theta_{\text{bj}}\) are the initial contact angle and \(j\)th ball azimuth angle; \(L_{\text{LA}}\) is the axial  
Figure 4: Assembly model between bearings and screw shaft: (**a**) Angular contact ball bearing; (**b**) Deep groove ball bearing. distance between the ball center and the CS\({}_{\text{Ab}}\)'s origin for the left angular contact ball bearing, respectively. The coordinates of the raceway centers for the right angular contact ball bearing in CS\({}_{\text{Ab}}\) are:  
\[\left[ \begin{array}{c}x_{\text{Rij}}^{\text{A}}\\ y_{\text{Rij}}^{\text{A}}\\ z_{\text{Rij}}^{\text{A}}\\ 1\end{array} \right]=\text{Trans}\left(R_{\text{Am}}\cos\theta_{\text{bj}}+\delta _{Sx},R_{\text{Am}}\sin\theta_{\text{bj}}+\delta_{Sy},L_{\text{RA}}+\delta_{Sx} \right)\cdot\text{Rot}_{z}\left(\theta_{\text{bj}}+\frac{\pi}{2}\right)\cdot \left[ \begin{array}{c}0\\ -(R_{\text{i}}-R_{\text{bb}})\cos\alpha_{\text{b0}}\\ -(R_{\text{i}}-R_{\text{bb}})\sin\alpha_{\text{b0}}\\ 1\end{array} \right] \tag{33}\]  
\[\left[ \begin{array}{c}x_{\text{Roj}}^{\text{A}}\\ y_{\text{Roj}}^{\text{A}}\\ z_{\text{Roj}}^{\text{A}}\\ 1\end{array} \right]=\text{Trans}\left(R_{\text{Am}}\cos\theta_{\text{bj}},R_{ \text{Am}}\sin\theta_{\text{bj}},L_{\text{RA}}\right)\cdot\text{Rot}_{z} \left(\theta_{\text{bj}}+\frac{\pi}{2}\right)\cdot\left[ \begin{array}{c}0\\ (R_{\text{o}}-R_{\text{bb}})\cos\alpha_{\text{b0}}\\ (R_{\text{o}}-R_{\text{bb}})\sin\alpha_{\text{b0}}\\ 1\end{array} \right] \tag{34}\]  
where \(L_{\text{RA}}\) is the axial distance between the ball center and the CS\({}_{\text{Ab}}\)'s origin for the right angular contact ball bearing, respectively. According to Equations (31)-(34), the radial distances (\(\sigma_{L\text{Aj}}\),\(\sigma_{\text{RAj}}\)) of the raceway centers for the left and right angular contact ball bearings are expressed by:  
\[\sigma_{L\text{Aj}}=\left(x_{L\text{ij}}^{\text{A}}-x_{L\text{oj}}^{\text{A} }\right)\cos\theta_{\text{bj}}+\left(y_{L\text{ij}}^{\text{A}}-y_{L\text{oj} }^{\text{A}}\right)\sin\theta_{\text{bj}} \tag{35}\]  
\[\sigma_{RAj}=\left(x_{R\text{ij}}^{\text{A}}-x_{R\text{oj}}^{\text{A}}\right) \cos\theta_{\text{bj}}+\left(y_{R\text{ij}}^{\text{A}}-y_{R\text{oj}}^{\text{ A}}\right)\sin\theta_{\text{bj}} \tag{36}\]  
According to the position relationship of raceway centers in CS\({}_{\text{Ab}}\), the contact force \(Q_{L\text{Aj}}\) and contact angle \(\alpha_{L\text{Aj}}\) for the left angular contact ball bearing can be obtained by using the Hertz contact theory, given as:  
\[Q_{L\text{Aj}}=\left\{