## 3 The EHL Model of the Textured Raceway of Ball Screws
### Calculation of Friction Coefficient and Friction Torque  
[PERSON] conducted a large number of friction experiments on rolling bearings to obtain the Stribeck friction characteristic curve, and lubrication is divided into three stages: boundary lubrication, mixed lubrication, and EHL, as shown in Figure 10. The Stribeck friction characteristic curve describes the relationship between the friction coefficient and the friction characteristic parameters such as the motion speed, the viscosity of the lubricating oil, and the normal contact force. The raceway contact surface of the contact system is rough and uneven, as shown in Figures 6b and 7b. The contact area is divided into the asperity contact area and the EHL contact area, and the load is shared by two areas, as shown in Figure 11.  
Figure 10: Stribeck friction characteristic curve. Figure 9: Calculation flowchart of EHL characteristics of the textured raceway. The lubrication state of the contact system of the ball screws is distinguished by the film thickness ratio \(\Lambda=h_{min}/R_{q}\) under different working conditions of the ball screws, where \(h_{min}\) is the minimum oil film thickness of the contact area between the ball and the screw raceway. \(R_{q}\) is the root mean square of the roughness of the screw raceway, \(R_{q}=0.05\)\(\mu\)m. The friction coefficient of the contact system of the ball screws is the accumulation of the contact friction coefficient of the asperity and the shear friction coefficient of the oil film. The friction coefficient of the asperity contact areas between the ball and screw raceway is expressed as follows:  
\[f_{a}=\mu_{a}Q_{a}/Q \tag{27}\]  
where \(Q\) is the total normal contact force of the ball, and \(Qa\) represents the normal force of the asperity contact. \(\mu_{a}\) is the static friction coefficient of the raceway surface, \(\mu_{a}=0.12\). The Ree-Eyring constitutive equation describes the relationship between the shear force and the shear strain rate of the oil film in the contact area between the ball and screw raceway. The shear strain rate of the lubricating oil film is expressed as follows:  
\[\dot{\gamma}=\frac{\tau_{0}}{\eta_{0}}\text{sinh}\bigg{(}\frac{\tau}{\tau_{0} }\bigg{)} \tag{28}\]  
The entrainment speed of the oil film in the normal direction of the raceway spiral is much smaller than that in the tangential direction. The shear strain rate of the oil film in the normal direction of the raceway spiral is neglected, and the shear strain rate is obtained according to the force balance of the asperity in the contact area as follows:  
\[\dot{\gamma}=\frac{h}{2\eta}\frac{\partial p}{\partial x}+\frac{\Delta u}{h} \tag{29}\]  
Combining Equations (37) and (38), the shear force distribution of the oil film in the grid of the contact area is written as follows:  
\[\tau_{i,j}=\tau_{0}\text{arcsinh}\bigg{(}\frac{\eta_{0}h_{i,j}}{2\eta_{i,j} \tau_{0}}\frac{\partial p_{i,j}}{\partial x_{i,j}}+\frac{\eta_{0}\Delta u_{i, j}}{\tau_{0}h_{i,j}}\bigg{)} \tag{30}\]  
The shear friction coefficient of the oil film in the contact area between the ball and the screw raceway is calculated as follows:  
\[f_{o}=\iint_{A}\tau_{i,j}dA_{i,j}/Q \tag{31}\]  
Figure 11: Contact area of the asperity and oil film. where \(\tau\) and \(\tau_{0}\) are the shear stress and the characteristic shear stress of the lubricating oil film. \(A\) is the contact area of the lubricating oil film. The friction coefficient and friction force between the ball and the raceway are determined as \(\mu=f_{a}+f_{o}\) and \(Fs=\mu\ Q\), respectively. Equations (27)-(31) are used to solve the friction coefficient of the contact area between the ball and the textured raceway. The friction torque of the ball screws is mainly composed of the friction torque between the ball and the screw raceway and the friction torque between the ball and the nut raceway, and the friction torque of the ball screws is expressed as follows:  
Figure 11.0  
where \(n_{b}\) is the number of working balls in a nut.