## 3 The EHL Model of the Textured Raceway of Ball Screws
### The EHL Model of the Textured Raceway of Ball Screws
The oil film thickness distribution at the contact point of the textured raceway of the ball screws is characterized as follows:  
\[h(x,y)= \begin{cases}ho(x,y)&(x,y)\in\Omega_{o}\\ ho(x,y)+hp(x,y)&(x,y)\in\Omega_{h}\end{cases}  \tag{19}\]  
\[ho(x,y)=h_{0}+\frac{x^{2}}{2R_{x}}+\frac{y^{2}}{2R_{y}}+V(x,y)+R(x,y),(x,y)\in \Omega_{o} \tag{20}\]  
where \(h_{o}\) and \(h_{p}\) are the oil film thickness distribution in the non-textured area and textured area on the raceway surface of the ball screws, respectively. \(\Omega_{o}\) and \(\Omega_{h}\) are the oil film bearing area of the non-textured and textured in the elliptical contact area, respectively. \(h_{0}\) is the oil film thickness in the center of the elliptical contact area. \(R_{x}\) and \(R_{y}\) are the equivalent curvature radius in the \(x\) and \(y\) directions of the contact area, \((Ry/Rx)2/3=b/a\). \(a\) and \(b\) are the semi-major axis and semi-minor axis of the elliptic contact area, respectively. \(R(x,y)\) is the rough surface of the grinding raceway according to the grinding surface topography model of the screw raceway mentioned above. \(V(x,y)\) is the elastic contact deformation in the elliptical contact area between the ball and the screw raceway. \(V(x,y)\) is described as follows:  
\[V(x,y)=\frac{2}{\pi E}\iint\limits_{\Omega}\frac{p_{f}(s,t)+p_{a}(s,t)}{ \sqrt{\left(x-s\right)^{2}+\left(y-t\right)^{2}}}dsdt \tag{21}\]  
where \(\Omega\) is the elliptical contact area between the ball and the screw raceway. \(p_{a}\) and \(p_{f}\) are the bearing stress distribution of the asperity and the oil film in the elliptical contact area, respectively. \(s\) and \(t\) are additional coordinates of \(x\) and \(y\), respectively. The texture is located at the grid center of the contact area, and the mathematical expression of the thickness distribution of the oil film in the texture area is established as follows:  
\[hp(x,y)=h_{t}\sqrt{1-\left(x(i,j)/r_{t}\right)^{2}-\left(y(i,j)/r_{t}\right)^{ 2}} \tag{22}\]According to the [PERSON] viscosity formula [36], the viscosity equation of the lubricating oil is expressed as follows:  
\[\eta=\eta_{0}\exp\biggl{\{}\left(\ln\eta_{0}+9.67\right)\biggl{[}\left(1+5.1 \times 10^{-9}p\right)^{0.68}-1\biggr{]}\biggr{\}} \tag{23}\]  
The density equation of the lubricating oil [37] is indicated as follows:  
\[\rho=\rho_{0}\biggl{(}1+\frac{0.6\times 10^{-9}p}{1+1.7\times 10^{-9}p}\biggr{)} \tag{24}\]  
where \(\rho_{0}\) and \(\eta_{0}\) indicate the environmental density and environmental viscosity of lubricating oil, respectively. The boundary conditions for the Reynolds equation are denoted as follows:  
\[\left\{ \begin{array}{l}p(x_{in},y)=p(x_{out},y)=0\\ p(x,y_{in})=p(x,y_{out})=0\\ p(x,y)\geq 0,(x_{in}<x<x_{out},\ y_{in}<y<y_{out})\end{array} \right. \tag{25}\]  
where \(x_{in}\) and \(y_{in}\) are the coordinates at the entrance of the contact area between the ball and the raceway. \(x_{out}\) and \(y_{out}\) represent the coordinates at the exit of the contact area between the ball and the raceway. The bearing capacity of the oil film in the contact area is equal to the normal contact force of the ball based on the load balancing conditions of the Reynolds equation, and the load balance equation of the Reynolds equation is derived as follows:  
\[\iint\limits_{\Omega}p(x,y)dxdy=Q \tag{26}\]  
The cavitation phenomenon in the oil film area is considered in the process of solving the EHL characteristics of the textured raceway of the ball screws. The boundary conditions of the cavitation phenomenon are introduced when establishing the EHL model of the textured raceway of the ball screws. The Sommerfeld, Half-Sommerfeld, Reynolds, and JFO conditions are the commonly used cavitation conditions at home and abroad, and the Reynolds condition has the advantages of higher prediction accuracy and faster prediction speed.