## 2 Dynamic Contact Model of SNBSP
### Dynamic Contact Mechanics Model of SNBSP
\(A_{S}\) and \(A_{N}\) denote effective cross-sectional area of the screw and nut, respectively. \[\Delta L=\frac{d_{\text{m}}\Delta\theta}{2}\tan\lambda \tag{25}\]  
Substituting Equation (23) into Equation (24a), it can be obtained as follows:  
\[\Delta_{\text{Si}}=\frac{d_{\text{m}}\Delta\theta\tan\lambda}{2E_{\text{S}}A_ {\text{S}}}(\sum_{j=i}^{z}Q_{\text{S}j}\sin\alpha_{\text{S}j}\cos\lambda+\sum _{j=i}^{z}F_{x^{\prime\prime}\text{S}j}\cos\alpha_{\text{S}j}\cos\lambda+\sum _{j=i}^{z}F_{y^{\prime\prime}\text{S}j}\sin\lambda)\] (26a) Substituting Equation ( 23 ) into Equation ( 24b ), it can be obtained as follows: \[\Delta_{\text{Ni}}=\frac{d_{\text{m}}\Delta\theta\tan\lambda}{2E_{\text{N}}A_ {\text{N}}}(\sum_{j=i}^{z}Q_{\text{N}j}\sin\alpha_{\text{N}j}\cos\lambda+\sum _{j=i}^{z}F_{x^{\prime\prime}\text{N}j}\cos\alpha_{\text{N}j}\cos\lambda+\sum _{j=i}^{z}F_{y^{\prime\prime}\text{N}j}\sin\lambda) \tag{26b}\]  
Figure 4: Single nut ball screw pair (SNBSP) load condition. Figure 5: Contact deformations of two adjacent balls in the raceway. According to the deformation coordination relationship, the deformation increment of the \(i\)th and the (\(i\)-1)th ball screw contact pair and ball nut contact pair can be represented as:  
\[\Delta_{\text{Si}}=(\varepsilon_{\text{S}(i-1)}\sin\alpha_{\text{S}(i-1)}- \varepsilon_{\text{Si}}\sin\alpha_{\text{Si}})\cos\lambda \tag{27a}\] \[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]0  
Then, according to the Hertz contact theory, there is a relationship between the deformation and the contact force [14; 15]:  
\[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]1  
\[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]2  
where \(v_{B}\), \(v_{S}\) and \(v_{N}\) denote the Poisson ratio of the ball, screw and nut, respectively. \(E_{B},E_{S},E_{N}\) denote the modulus of elasticity of the ball, screw and nut, respectively. \(\sum\rho_{\text{<}B},\sum\rho_{\text{>}B}\) denote the summation of principal curvatures of the ball and screw and the ball and nut, respectively. \(K(e)\) denotes complete elliptic integral of the first kind, and \(m_{a}\), \(m_{b}\) and \(\frac{2K(e)}{\pi m_{a}}\) can be obtained through the table in reference [16]. Simultaneous Equations (26a) and (27a), and substituting Equation (28) into the equation, it can be obtained as follows:  
\[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]3  
In the same way, simultaneous Equations (26b) and (27b), and substituting Equation (28) into the equation, it can be obtained as follow:  
\[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]4  
where:  
\[\Delta_{Ni}=\frac{\Delta LF_{Ni}}{E_{N}A_{N}} \tag{24b}\]5  
As a high-speed ball screw nut pair, on the one hand, it mainly bears axial force during operation. On the other hand, the influence of centrifugal force caused by the high-speed rotation of the ball should also be considered. The effect of contact angle, contact force and centrifugal force is considered comprehensively, and the deformation is shown in Figure 6a [17; 18]. For analyzing the motion state and deformation of the ball screw nut pair in the motion process, the center of the nut raceway \(O_{N}\) is fixed. As shown in Figure 6b, when the preload is not considered, there is no force between the ball and the screw and nut raceway before the SNBSP under load. So, there is no contact deformation, and the ball center is aligned with the screw and nut raceway curvature center. The internal and external contact angles on both sides of the ball and the raceway are the initial design angle \(\alpha_{0}\).