## Nomenclature  
\(A_{S}A_{N}\): Net sectional area of screw or nut  
CSOxyz: Rotating coordinate system  
CSO'\(m\): Local coordinate system  
CSO'\(x_{i}^{*}x_{j}^{*}z_{i}^{*}z_{i}^{*}\): Contact coordinate system  
\(d_{\rm B}\): Diameter of the ball  
\(d_{\rm m}\): Diameter of the screw helix  
\(E_{S}E_{N}\),\(E_{B}\): Young modulus of the screw, nut and ball materials  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Screw Speed (rpm)**} & \multirow{2}{*}{**Direction**} & \multicolumn{4}{c}{**Theoretical Value**} & \multicolumn{2}{c}{**Measured Value**} \\ \cline{3-6}  & & \(F_{a}({\bf N})\) & \(K_{p}\) (\({\bf N}\)/m) & \(K_{f}\) (\({\bf N}\)/m) & \(f_{\rm n}\) (\({\bf Hz}\)) & \(f\) (\({\bf Hz}\)) \\ \hline \multirow{6}{*}{1000} & \multirow{2}{*}{Z1—20} & 636.8 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.3 \\  & & 635.6 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.6 \\  & & 637 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.5 \\ \cline{2-6}  & \multirow{2}{*}{Z0—Z1} & 637 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.4 \\  & & 635.6 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.6 \\  & & 637.5 & \(3.439\times 10^{8}\) & (1.4790–1.4776) \(\times 10^{8}\) & 107.31–107.36 & 108.5 \\ \hline \multirow{6}{*}{50} & \multirow{2}{*}{Z1—20} & 684.8 & \(3.89\times 10^{8}\) & (1.5551–1.5566) \(\times 10^{8}\) & 110.09–110.15 & 112.1 \\  & & 683.6 & \(3.89\times 10^{8}\) & (1.5551–1.5566) \(\times 10^{8}\) & 110.09–110.15 & 111.8 \\  & & 684.8 & \(3.89\times 10^{8}\) & (1.5551–1.5566) \(\times 10^{8}\) & 110.09–110.15 & 112.1 \\ \cl  
\begin{tabular}{l l} \(f_{n}\) & Natural frequency of the feed system \\ \(f_{c}\) & Frictional coefficient of the ball and screw or ball and nut \\ \(Fa\) & Axial force of single nut ball screw pair \\ \(F_{Ci}\) & Centrifugal force generated by the rotation of the \(i\)th ball \\ \(F_{Sii}\), \(F_{Ni}\) & The load on screw or nut at the \(i\)th contact pair \\ \(F_{x^{\prime\prime}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
Ref. [6]. Here, these core equations are listed. (Here, we unify the symbols in the equations in Refs. [6,21] with those in this paper).  
\[\eta=\frac{W_{Ra}}{T_{ai}\omega}=\frac{F_{a}L_{p}}{2\pi T_{ca}}\] (A1)  
\[W_{Ra}=\frac{F_{a}\omega L_{p}}{2\pi}\] (A2)  
\[T_{aii}=\sum_{i=1}^{z}T_{cai}\] (A3)  
\[T_{cai}=r_{m}Q_{Si}\bigg{\{}\sin(\lambda)\cos(\rho_{i})\bigg{[}\sin(\alpha_{Si })+f_{S}\bigg{(}\frac{r_{B}}{r_{m}}-\cos(\alpha_{Si})\bigg{)}\bigg{]}+\cos( \lambda)\sin(\rho_{i})\bigg{\}}\] (A4)  
\[\rho_{i}=\tan^{-1}\bigg{(}\frac{f_{S}}{r_{B}\sin(\alpha_{Si})}\bigg{)}\] (A5)  
Classical equation of transmission efficiency is also used to set initial value:  
\[\eta=\frac{\tan(\lambda)}{\tan(\lambda+\rho)}\] (A6)  
Therefore, combined with the model and method proposed in this paper, the above equations are used to calculate the transmission efficiency \(\eta\) of ball screw pair. The algorithm flow chart is shown in the Figure A1.  
Firstly, the initial value \(\eta^{0}\) is given according to the classical equation of transmission efficiency. According to Equation (47) in this paper, \(Fa^{0}\) is calculated. The parameters such as \(Fa^{0}\) and screw speed are substituted into the dynamic contact model, and results such as \(Q_{Si}^{0}\) and \(a_{Si}^{0}\) are obtained. Substituting the results into Equations (A1)-(A5), \(\eta^{temp}\) is obtained. According to Equation (47), the axial force \(F_{a}{}^{temp}\) is calculated. The error of the temporary value and the initial value of \(\eta^{temp}\) and \(F_{a}{}^{temp}\) are compared, respectively, and the iterative solution is carried out until the accuracy requirements are met. Finally, \(\eta\) and \(Fa\) are obtained.