# Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws
## 2 Materials and Methods
### Non-Hertzian Elastohydrodynamic Contact Stress Calculation  
The viscosity flow behavior of the lubricant film between the ball and raceway can be described by the Reynolds equation shown in Figure 4b. The initial film contact stress distribution in the Reynolds equation is the non-Hertzian contact stress distribution, which can further observe the variation of the elastohydrodynamic lubrication characteristics. The assumptions of the Reynolds equation in the analysis are as follows:  
1. The lubricant and the solid are isothermal, and the contact stress and the film thickness do not vary with time. 2. The curvature radius of the solid surface is much larger than the lubricant film thickness. 3. No relative sliding between the lubricant film and solid surface at the common interface. 4. Compared with the film shear stress, the inertial force and other bulk forces of the lubricant film are negligible. 5. Due to the lubricant film thickness being very thin, it can be assumed that the film contact stress remains constant along the film thickness direction, namely no squeeze film effect. 6. The inlet of the contact region is flooded fully by the lubricant film. Figure 4: Diagram of the elastohydrodynamic contact mechanics. (**a**) The fractal rough surface. (**b**) Elastohydrodynamic contact mechanics. Figure 3: A comparison of the non-Hertzian and Hertzian solutions for the contact stress distribution between the ball and the screw raceway. The lubricant is the Newtonian fluid and obeys [PERSON]'s law of viscosity. The coupling relationship of the elastohydrodynamic contact stress distribution \(p\), film thickness distribution \(h\), film viscosity \(\eta\), film density \(\rho\), and film entrainment velocity \(u_{s}\) could be accurately determined by the Reynolds equation [24], which is written as  
\[\frac{\partial}{\partial x}\left(\frac{\rho h^{3}}{\eta}\frac{\partial p}{ \partial x}\right)+\frac{\partial}{\partial y}\left(\frac{\rho h^{3}}{\eta} \frac{\partial p}{\partial y}\right)=6(u_{1}+u_{2})\frac{\partial(\rho h)}{ \partial x} \tag{14}\]  
where \(p\) denotes the non-Hertzian elastohydrodynamic contact stress in the contact region. \(u_{1}\) and \(u_{2}\) indicate the linear velocity of the ball and the raceway in the tangential direction (x-direction) of the helix raceway track, respectively. The entrainment velocity of the lubricant film is \(u_{s}=(u_{1}+u_{2})/2\). The output parameters of point \(A\) and point \(B\) calculated by Equation (2), such as \(Q_{A}\), \(\beta_{A}\), \(Q_{B}\), and \(\beta_{B}\), are substituted into the Reynolds Equation (14), the elastohydrodynamic characteristics of point A and point B can be determined, respectively. In terms of the elastohydrodynamic lubrication theory [25], the film thickness distribution is expressed as  
\[h(x,y)=h_{0}+\frac{x^{2}}{2R_{x}}+\frac{y^{2}}{2R_{y}}+V(x,y)+ROU(x,y) \tag{15}\]  
The lubricant film viscosity proposed by [PERSON] [26] is written as  
\[\eta=\eta_{0}\exp\left\{(\ln\eta_{0}+9.67)\left[\left(1+5.1\times 10^{-9}p\right) ^{0.68}-1\right]\right\} \tag{16}\]  
The lubricant film density from [PERSON] and [PERSON] [27] is shown in Equation (17). \[\rho=\rho_{0}\left(1+\frac{0.6\times 10^{-9}p}{1+1.7\times 10^{-9}p}\right) \tag{17}\]  
where \(h_{0}\) indicates the central film thickness in the contact region. \(\eta_{0}\) and \(\rho_{0}\) denote the initial value of the film viscosity and density, respectively. \(ROU(x,y)\) indicates the roughness of the contact surface calculated by the fractal theory [28] shown in Figure 4a. The major parameters of the rough fractal rough surface are the fractal dimension \(D\) and scale parameter \(G\), \(D=2.47\), \(G=0.41\). The rough fractal surface is the contact region of the raceway. The hardness range of the ball is 62-64 HRC higher than that of the screw raceway 58-62 HRC; The processing technology of the working ball is mature, and the surface of the working ball is smooth. According to the above two reasons, the ball contact surface is regarded as a smooth surface.