## 2 Dynamic Contact Model of SNBSP
### Dynamic Contact Mechanics Model of SNBSP  
In the quasi-static model in which the screw speed is not considered, it is considered that the normal forces and contact angles of the \(i\)th ball and the screw and nut raceway are equal, that is,  
\[\alpha_{si}=\alpha_{ni}\;\text{and}\;Q_{si}=Q_{ni} \tag{18}\]  
Then the relationship between the axial load and the normal force of contact pair of balls and raceway can be represented as [12],  
\[Fa=\sum_{i=1}^{z}Q_{ci}\sin\alpha_{zi}\cos\lambda \tag{19}\]  
where \(Fa\) denotes the axial force; \(Q_{ci}\) is the normal contact force of the \(i\)th contact pair; \(\alpha_{zi}\) denotes the contact angle of the \(i\)th contact pair. \(\varsigma=S,N\). \(\lambda\) is the lead angle of screw. But the rotating screw will drive the balls to generate the angular velocity of the revolution around screw axis and the angular velocity of the ball rotation, respectively, which will make the balls produce outward centrifugal force \(F_{\mathrm{C}}\), and produce the transverse frictional force \(F_{x^{\prime\prime}}\) and longitudinal frictional force \(F_{y^{\prime\prime}}\) along the contact surface, which leads to the contact angles of the nut and screw being no longer equal, that is \(\alpha_{si}\
eq\alpha_{ni}\). The directions of \(F_{x^{\prime\prime}}\) and \(F_{y^{\prime\prime}}\) are along the \(-y^{\prime\prime}_{z}\) and \(-z^{\prime\prime}_{z}\) axes, respectively. Figure 3 exhibits the force state of the \(i\)th ball in \(\text{CSO}^{\prime\prime}_{z}x^{\prime\prime}_{z}y^{\prime\prime}_{z}z^{\prime\prime}_ {z}\), and the dynamic balance equation of the \(i\)th ball can be obtained from the figure:  
\[Q_{Si}\cos\alpha_{Si}-Q_{Ni}\cos\alpha_{Ni}+F_{x^{\prime\prime}Si}\sin\alpha_{Si }-F_{x^{\prime\prime}Ni}\sin\alpha_{Ni}+F_{Ci}=0 \tag{20a}\] \[Q_{Si}\sin\alpha_{Si}-Q_{Ni}\sin\alpha_{Ni}-F_{x^{\prime\prime}Si}\cos\alpha_ {Si}+F_{x^{\prime\prime}Ni}\cos\alpha_{Ni}=0 \tag{20b}\]  
where \(F_{x^{\prime\prime}ci}\) denotes the transverse friction of the \(i\)th contact pair, and \(F_{Ci}\) denotes the centrifugal force generated by the rotation of the \(i\)th ball. This can be represented as:  
\[F_{Ci}=\frac{1}{2}m_{B}d_{m}\omega_{mi}^{2} \tag{21}\]  
where \(m_{B}\) is the ball mass; \(d_{m}\) is the screw helix diameter; and \(\omega_{mi}\) is the angular velocity of ball revolution along the screw helix of the \(i\)th ball. Then, the relationship between the axial force of the screw nut pair and the loads of the ball and raceway contact pairs can be represented as follows:  
\[Fa=\sum\limits_{i=1}^{z}Q_{\odot i}\sin\alpha_{\odot i}\cos\lambda+\sum\limits _{i=1}^{z}F_{x^{\prime\prime}\odot i}\cos\alpha_{\odot i}\cos\lambda+\sum \limits_{i=1}^{z}F_{y^{\prime\prime}\odot i}\sin\lambda \tag{22}\]  
Figure 4 exhibits the load state of the screw \(F_{Si}\) and nut \(F_{Ni}\) with the \(i\)th ball in the screw nut pair. \(F_{Si}\) and \(F_{Ni}\) can be represented as follows:  
\[ \begin{array}{ll}F_{c\dot{c}}&=Fa-\sum\limits_{j=1}^{i-1}Q_{\odot j}\sin \alpha_{\odot j}\cos\lambda-\sum\limits_{j=1}^{i-1}F_{x^{\prime\prime}\odot j }\cos\alpha_{\odot j}\cos\lambda-\sum\limits_{j=1}^{i-1}F_{y^{\prime\prime} \odot j}sin\lambda\\ &=\sum\limits_{j=i}^{z}Q_{\odot j}\sin\alpha_{\odot j}\cos\lambda+\sum\limits _{j=i}^{z}F_{x^{\prime\prime}\odot j}\cos\alpha_{\odot j}\cos\lambda+\sum \limits_{j=i}^{z}F_{y^{\prime\prime}\odot j}sin\lambda\end{array}  \tag{23}\]  
It is known from the deformation relationship between the \(i\)th and (\(i\)-1)th contact pairs that the deformation increment of the contact pairs of the adjacent balls can be expressed respectively as:  
\[\Delta_{Si}=\frac{\Delta LF_{Si}}{E_{S}A_{S}} \tag{24a}\] \[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]  
Figure 3: The force balance of ball in contact coordinate system. where \(\Delta L\) denotes the axial distance between adjacent balls, as shown in Figure 5. \(E_{S}\) and \(E_{N}\) denote the Young modulus of the screw and nut materials, respectively.