## 2 Dynamic Model of the Feed System
### Calculation of Restoring Force for Ball Screw
#### 2.1.1 Restoring Force for the Screw-Nut Left Section
=\text{Trans}(0,0,L_{OW})\cdot\text{Trans}\big{(}\delta_{SNx}, \delta_{SNy},\delta_{SNz}\big{)}\cdot \begin{bmatrix}X_{Issi}^{S}\\ Y_{Issi}^{S}\\ Z_{Issi}^{S}\\ 1\end{bmatrix}  \tag{9}\]  
Combining Equations (8) and (9), the radial distance \(\sigma_{L\text{sni}}\) of raceway centers for the screw-nut left section can be expressed as:  
\[\sigma_{L\text{sni}}=\left(x_{Issi}^{S}-x_{L\text{n}i}^{W}\right)\cos\theta_{ \text{sni}}+\left(y_{Lsi}^{S}-y_{L\text{n}i}^{W}\right)\sin\theta_{\text{sni}} \tag{10}\]  
With the increase of excitation amplitudes, the ball may lose contact with the raceway, which affects the nonlinear contact characteristics of the feed system. To judge whether each ball is loaded or not, if \(\delta_{l\text{sni}}\leq 0\), then the \(i\)th ball is not loaded, and the contact force is \(Q_{L\text{sni}}=0\). If \(\delta_{l\text{sni}}>0\), then the \(i\)th ball is loaded, and the contact force \(Q_{L\text{sni}}\) can be obtained by using Hertz contact theory, given by:  
\[Q_{l\text{sni}}= \begin{cases}\left(\frac{\delta_{l\text{sni}}+\delta_{\text{sni }}}{\text{C}_{\text{si}}+\text{C}_{\text{ni}}}\right)^{2/3}&\delta_{L\text{sni }}+\delta_{\text{sni}0}>0\\ 0&\delta_{L\text{sni}}+\delta_{\text{sni}0}\leq 0\end{cases}  \tag{11}\]  
where \(C_{\text{si}}\) and \(C_{\text{ni}}\) are the Hertz contact constant of the screw and nut; \(\delta_{\text{sni}0}\) is contact deformation under the preload; \(\delta_{L\text{sni}}\) is contact deformation of \(i\)th ball for the screw-nut left section, which is composed of the ball-screw contact deformation \(\delta_{l\text{s}i}\) and the ball-nut contact deformation \(\delta_{l\text{n}i}\), given by:  
\[\delta_{l\text{sni}}=\delta_{l\text{s}i}+\delta_{L\text{n}i}=\sqrt{\sigma_{ L\text{sni}}^{2}+\left(z_{L\text{n}i}^{W}-z_{L\text{s}i}^{S}\right)^{2}}-A_{0} \tag{12}\]  
where \(A_{0}\) is the initial distance of raceway centers, given by:  
\[A_{0}=R_{\text{s}}+R_{\text{n}}-2R_{\text{b}} \tag{13}\]  
According to the position relationship of raceway centers in CS\({}_{0}\), the contact angle \(\alpha_{L\text{sni}}\) for the screw-nut left section can be expressed as:  
\[\alpha_{L\text{sni}}=\cos^{-1}\biggl{(}\frac{\sigma_{L\text{sni}}}{A_{0}+ \delta_{L\text{sni}}+\delta_{\text{sni}0}}\biggr{)} \tag{14}\]  
In order to calculate the restoring force for the screw-nut left section, the ball contact force should be transformed into equivalent force. According to the homogeneous coordinates transformation, the restoring forces of ball screws along the \(x\), \(y\) and \(z\) directions can be expressed by:  
\[F_{L\text{snx}}=\sum_{i=1}^{N_{\text{b}}}Q_{l\text{sni}}(\cos\theta_{\text{sni }}\cos\alpha_{L\text{sni}}+\sin\alpha_{L\text{sni}}\sin\theta_{\text{sni}}\sin\lambda) \tag{15}\]  
\[F_{L\text{sny}}=\sum_{i=1}^{N_{\text{b}}}Q_{l\text{sni}i}(\cos\alpha_{L\text{ sni}}\sin\theta_{\text{sni}}-\sin\alpha_{L\text{sni}}\cos\theta_{\text{sni}} \sin\lambda) \tag{16}\]  
\[F_{L\text{snz}}=\sum_{i=1}^{N_{\text{b}}}Q_{L\text{sni}i}\sin\theta_{\text{sni }}\cos\lambda \tag{17}\]  
where \(N_{\text{b}}\) is the number of loaded balls.