## 2 Dynamic Model of the Feed System
### Calculation of Restoring Force for Ball Screw
#### 2.1.2 Restoring Force for the Screw-Nut Right Section  
Similar to the restoring force calculation for the screw-nut left section method, the position vectors of the raceway centers corresponding to the \(i\)th ball for the screw-nut right section in CS\({}_{2}\) can be given as:  
\[\mathbf{R}_{\text{Rs}}=\left[ \begin{array}{cc}0&-(R_{\text{s}}-R_{\text{b}}) \cos\alpha_{0}&-(R_{\text{s}}-R_{\text{b}})\sin\alpha_{\text{sn0}}&1\end{array}  \right]^{\text{T}} \tag{18}\]  
\[\mathbf{R}_{\text{Rn}}=\left[ \begin{array}{cc}0&(R_{\text{n}}-R_{\text{b}}) \cos\alpha_{0}&(R_{\text{n}}-R_{\text{b}})\sin\alpha_{\text{sn0}}&1\end{array}  \right]^{\text{T}} \tag{19}\]  
The coordinates of the raceway centers for the screw-nut right section in CS\({}_{\text{S}}\) and CS\({}_{\text{W}}\) can be expressed as:  
\[\left[ \begin{array}{cc}X_{\text{Rsi}}^{\text{S}}&Y_{\text{Rsi}}^{\text{S}}& Z_{\text{Rsi}}^{\text{S}}&1\end{array} \right]^{\text{T}}=\text{Trans}(0,0,L_{ \text{Rbi}})\cdot\mathbf{T}_{1-2}\cdot\mathbf{R}_{\text{Rs}} \tag{20}\]  
\[\left[ \begin{array}{cc}X_{\text{Rn}i}^{W}&Y_{\text{Rni}}^{W}&Z_{\text{Rn}i }^{W}&1\end{array} \right]^{\text{T}}=\text{Trans}(-H_{\text{W}},0,L_{\text{ Rbi}})\cdot\mathbf{T}_{1-2}\cdot\mathbf{R}_{\text{Rn}} \tag{21}\]  
where \(L_{\text{Rbi}}\) is the displacement of the \(i\)th ball center for the screw-nut right section along the \(z\)-axis direction in CS\({}_{\text{S}}\)(CS\({}_{\text{W}}\)), which can be expressed as:  
\[L_{\text{Rbi}}=\frac{L_{D}}{2}-(N_{\text{b}}-i)\Delta L \tag{22}\]  
By the homogeneous coordinate transformation, the coordinates of the raceway centers for the screw-nut right section in CS\({}_{0}\) can be expressed as:  
\[\left[ \begin{array}{c}x_{\text{Rsi}}^{\text{S}}\\ y_{\text{Rsi}}^{\text{S}}\\ z_{\text{Rsi}}^{\text{S}}\\ 1\end{array} \right]=\text{Trans}(0,0,L_{\text{OW}})\cdot\text{Trans}\big{(} \delta_{SNx},\delta_{SNy},\delta_{SNz})\cdot\left[ \begin{array}{c}X_{\text{ Rsi}}^{\text{S}}\\ Y_{\text{Rsi}}^{\text{S}}\\ Z_{\text{Rsi}}^{\text{S}}\\ 1\end{array} \right] \tag{23}\]  
\[\left[ \begin{array}{c}x_{\text{Rni}}^{W}\\ y_{\text{Rni}}^{\text{S}}\\ z_{\text{Rn}i}^{W}\\ 1\end{array} \right]=\text{Trans}(H_{\text{W}},0,L_{\text{OW}})\cdot\text{Trans} \big{(}\delta_{Wx},\delta_{Wy},\delta_{Wz})\cdot\left[ \begin{array}{c}X_{ \text{Rni}}^{W}\\ Y_{\text{Rni}}^{W}\\ Z_{\text{Rn}i}^{W}\\ 1\end{array} \right] \tag{24}\]  
According to the position relationship of raceway centers, the radial distance \(\sigma_{\text{Rsn}i}\) of the raceway centers for the screw-nut right section can be expressed as:  
\[\sigma_{\text{Rsn}i}=\left(x_{\text{Rsi}}^{\text{S}}-x_{\text{Rni}}^{W} \right)\cos\theta_{\text{sn}i}+\left(y_{\text{Rsi}}^{\text{S}}-y_{\text{Rni }}^{W}\right)\sin\theta_{\text{sn}i} \tag{25}\]  
Considering that the ball may lose contact with the raceway, the contact force \(Q_{\text{Rsn}i}\) can be obtained by using the Hertz contact theory, given by:  
\[Q_{\text{Rsn}i}= \begin{cases}\left(\frac{\sqrt{\sigma_{\text{Rsn}i}^{2}+\left( x_{\text{Rni}}^{W}-\text{S}_{\text{Rni}}^{\text{S}}\right)^{2}}-A_{0}+\delta_{ \text{sn0}}}{\zeta_{\text{sn}i}+\zeta_{\text{cn}i}}\right)^{2/3}&\delta_{\text {Rsn}i}+\delta_{\text{sn0}}>0\\ 0&\delta_{\text{Rsn}i}+\delta_{\text{sn0}}\leq 0\end{cases}  \tag{26}\]  
where \(\delta_{\text{Rsn}i}\) is \(i\)th ball contact deformation for the screw-nut right section.