# Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws
## 2 Materials and Methods
### Non-Hertzian Normal Contact Stress Calculation  
Due to the asymmetry of the contact region of the helical raceway of the high-speed BSs, there is a big error in the calculation of the contact stress distribution and contact \begin{table}
\begin{tabular}{c c c c} \hline \hline Screw radius, \(r\) (mm) & 16 & Ball radius, \(r_{b}\) (mm) & 2.9765 \\ \hline Radius of the normal section & & Radius of the normal section & \\ on the screw raceway, \(r_{5}\) (mm) & 3.215 & on the nut raceway, \(r_{N}\) (mm) & 3.215 \\ \hline Helix angle, \(\alpha\) (\({}^{\circ}\)) & 5.68 & Pitch, \(L_{0}\) (mm) & 10 \\ \hline Initial contact angle \(\beta_{0}\) (\({}^{\circ}\)) & 40.26 & Preload, \(F_{pre}\) (N) & 1500 \\ \hline Screw speed, \(\Omega^{\gamma}\) (mm/s) & 83 & Number of working balls, \(n_{bb}\) & 168 \\ \hline [PERSON]’s modulus, \(E_{1}=E_{2}\) (GPa) & 200 & Density \(\rho_{0}\) (kg/m\({}^{3}\)) & 970 \\ \hline Poisson’s ratio, \(v_{1}=v_{2}\) & 0.3 & Viscosity \(\eta_{0}\) (Pa-s) & 0.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of double-nut ball screws. Figure 1: Force analysis of ball screws. (**a**) Force balance analysis. (**b**) Contact mechanics analysis. region in the screw raceway contact point \(A\) and the nut raceway contact point \(B\) by using the traditional Hertzian contact theory. The non-Hertzian contact region between the ball and raceway is inclined compared with the Hertzian contact region, which is shown in Figure 1b. Based on the minimum excess principle, a method was proposed to analyze and calculate the normal contact stress between the ball and the raceway of the high-speed DNBSs. In terms of the numerical solution process of the contact stress distribution between the ball and raceway, the contact region and deformation compatibility equation should be discretized. The contact gap and contact stress of the two contact surfaces should satisfy the following equation [21]. \[c_{i,j}=d_{0}+c_{gi,j}+c_{ri,j}+V_{i,j},\;0\leq i\leq I,0\leq j\leq J \tag{3}\]  
where \(c\) is the contact surface gap matrix. \(c_{i,j}\) is the contact surface gap distribution in the mesh region, which is the element of the contact surface gap matrix \(c\). \(d_{0}\) denote the approaching distance of the contact surface. \(c_{g}\) is the contact surface geometry matrix. \(c_{gi,j}\) indicates the contact surface geometry in the mesh region, which is the element of the contact surface geometry matrix \(c_{g}\). \(c_{r}\) is the contact surface roughness matrix. \(c_{ri,j}\) is the contact surface roughness distribution, which is the element of the contact surface roughness matrix \(c_{r}\). \(V\) is the surface elastic deformation matrix. \(V_{i,j}\) is the surface elastic deformation distribution in the mesh region, which is the element of the surface elastic deformation matrix \(V\). \(I\) and \(J\) indicate the mesh grid number in the \(x\) and \(y\) directions, respectively. The discretized contact deformation equation is expressed as  
\[V_{i,j}=\sum_{k=0}^{M}\sum_{l=0}^{N}K_{i-k,j-l}P_{i,j} \tag{4}\]  
where \(p_{i,j}\) is the element of the contact stress matrix \(P\). \(K_{i-k,j-l}\) is the element of deformation influence coefficient matrix \(K\). The boundary condition of the contact equation is expressed as  
\[c_{i,j}\geq 0,\;p_{i,j}\geq 0 \tag{5}\]  
In the contact region, the boundary condition of the contact equation is \(c_{i,j}=0,p_{i,j}\geq 0\); Out of the contact region, the boundary condition of the contact equation is \(c_{i,j}>0,p_{i,j}=0\). According to the above two conditions, \(c_{i,j}\), and \(p_{i,j}\) cannot be both zero, thus  
\[c_{i,j}p_{i,j}=0 \tag{6}\]  
The solution of the contact stress and the contact geometry is the key to the contact equation.