## 2 Dynamic Contact Model of SNBSP  
### Setting and Transformation of the Coordinate System  
For researching the movement and contact characteristics of a screw and nut conveniently, three coordinate systems are introduced in this paper. Rotating coordinate system _Oxyz_: The direction of \(z\) axis is consistent with that of screw, and its rotating speed will be the same as the screw speed (Figure 1). Figure 1: Rotating and local coordinate systems. Local coordinate system \(O^{\prime}\tau nb\): In order to describe the contact pair during movement, the Frenet frame is selected and the Cartesian coordinate system is carried out to establish \(CSO^{\prime}\tau nb\). The center of the coordinate is the ball center. The direction of the \(n\) axis is consistent with the screw radial direction at the \(O^{\prime}\) point. The tangent direction of the helix at point \(O^{\prime}\) is the \(\tau\) axis. The direction of the \(b\) axis is determined by the \(\tau\) axis and \(n\) axis through the right-hand rule (Figure 1). Contact coordinate system \(O^{\prime}_{\varsigma}x^{\prime}_{\varsigma}y^{\prime}_{\varsigma}z^{\prime \prime}_{\varsigma}\) is established (Figure 2). When \(\varsigma=S\), it denotes the contact area between the ball and the screw raceway. When \(\varsigma=N\), it denotes the contact area between the ball and the nut raceway. Take \(CSO^{\prime}_{\varsigma}x^{\prime\prime}_{\varsigma}y^{\prime\prime}_{ \varsigma}z^{\prime\prime}_{\varsigma}\) as an example--\(O^{\prime}_{\varsigma}\) denotes the contact point between the ball and the screw. The direction of the \(x^{\prime\prime}_{\varsigma}\) axis coincides with the tangent direction, which crosses the \(O^{\prime\prime}_{\varsigma}\) point, of the contact curve between the rolling element and the lead screw. The direction of the \(z^{\prime\prime}_{\varsigma}\) axis coincides with the normal direction of the contact curve crossing the \(O^{\prime\prime}_{\varsigma}\) point between the rolling element and the lead screw. The \(y^{\prime\prime}_{\varsigma}\) axis direction is determined by the right-hand rule [12]. The transformation relationship between the rotating coordinate system \(CSOxyz\) and the local coordinate system \(CSO^{\prime}\tau nb\) can be obtained by the following formula:  
\[\left\{ \begin{array}{c}x\\ y\\ z\\ 1\end{array} \right\} = Trans(r_{m}\cos\theta,r_{m}\sin\theta,r_{m}\theta\tan\lambda)Rot \Big{(}z,\theta+\frac{\pi}{2}\Big{)}Rot(y,-\lambda)\left\{ \begin{array}{c} \tau\\ n\\ b\\ 1\end{array} \right\} \tag{1}\] \[= \left[ \begin{array}{cccc}-\cos\lambda\sin\theta&-\cos\theta& \sin\theta\sin\lambda&r_{m}\cos\theta\\ \cos\lambda\cos\theta&-\sin\theta&-\sin\lambda\cos\theta&r_{m}\sin\theta\\ \sin\lambda&0&\cos\lambda&r_{m}\theta\tan\lambda\\ 0&0&0&1\end{array} \right]\left\{ \begin{array}{c}\tau\\ n\\ b\\ 1\end{array} \right\}\]  
where \(\lambda\) denotes the lead angle of the ball screw, \(r_{m}\) denotes the screw pitch radius, and \(\theta\) denotes the position angle of the ball. Let the transformation matrix between \(CSOxyz\) and \(CSO^{\prime}\tau nb\) be \(T_{0-1}\). Equation (1) can be written as follows,  
\[\{x,y,z,1\}^{T}=T_{0-1}\{\tau,n,b,1\}^{T} \tag{2}\]  
The relationship between \(CSO^{\prime}\tau nb\) and \(CSO^{\prime\prime}_{\varsigma}x^{\prime\prime}_{\varsigma}y^{\prime\prime}_{ \varsigma}z^{\prime\prime}_{\varsigma}\) can be represented as follows,  
\[\{\tau,n,b,1\}^{T}=T_{1-2\varsigma}\{x^{\prime\prime}_{\varsigma},y^{\prime \prime}_{\varsigma},z^{\prime\prime}_{\varsigma},1\}^{T} \tag{3}\]  
Figure 2: Coordinate systems. where \(\mathbf{T}_{1-2\zeta}\) denotes the transformation matrix. \[\mathbf{T}_{1-2\zeta}=\left[ \begin{array}{cccc}0&1&0&0\\ -\sin\alpha_{z}&0&\cos\alpha_{z}&\eta r_{B}\cos\alpha_{z}\\ \cos\alpha_{z}&0&\sin\alpha_{z}&\eta r_{B}\sin\alpha_{z}\\ 0&0&0&1\end{array} \right] \tag{4}\]  
where, when \(\varsigma=S\), \(\eta=1\). When \(\varsigma=N\), \(\eta=-1\). \(\alpha_{z}\) denotes the contact angle between the ball and the screw (or nut) raceway.