## 2 Grinding Surface Topography of the Screw Raceway
### Modeling of the Geometric Model of the Screw Raceway  
The raceway surface of the ball screws is a helical surface with a specific angle, and the angle is numerically equal to the angle \(\alpha\) between the grinding wheel axis and the screw axis, as shown in Figure 3. The grinding wheel and screw rotate around their respective spindles, and the screw is fed in the axial direction of the grinding wheel to grind the spiral raceway surface with a circular cross-section. The speed direction of the screw and the grinding wheel are different at the center of the contact surface between the grinding wheel and the screw. When applying the theory of surface grinding to the grinding of the screw raceway, it is necessary to convert the grinding parameters of the screw and grinding wheel into surface grinding parameters in an equivalent manner. The \(A\)-\(A\) plane is perpendicular to the grinding wheel axis and passes through the center \(D\) of the contact surface. The equivalent diameters of the grinding wheel and the screw in the \(A\)-\(A\) plane are expressed as follows:  
\[d_{s,e}=d_{s},d_{w,e}=d_{w}/\cos^{2}\alpha \tag{12}\]  
where the diameters of the grinding wheel and the screw are \(d_{s}\) and \(d_{w}\). \(\alpha\) is the spiral angle of the ball screws. The equivalent velocity of the grinding wheel and the screw at the contact point \(D\) is denoted as follows:  
\[v_{s,e}=v_{s},v_{w,e}=v_{w}\cos\alpha \tag{13}\]  
where the speeds of the grinding wheel and the screw are \(v_{s}\) and \(v_{w}\), and the equivalent speeds of the grinding wheel and the screw are \(v_{s,e}\) and \(v_{w,e}\), respectively. The equivalent diameter of the grinding wheel in the grinding process of the screw is expressed as follows:  
\[d_{e}=\frac{d_{s,e}d_{w,e}}{d_{s,e}+d_{w,e}}=\frac{d_{s}d_{w}}{d_{w}+d_{s}\cos ^{2}\alpha} \tag{14}\]  
Compared with the raceway of the bearing, the raceway of the ball screws is asymmetric, and the asymmetry of the contact surface of the ball screws increases with the increase in the spiral angle. To accurately construct the grinding surface topography of the screw raceway, it is necessary to establish the geometric model of the screw raceway to describe the geometric features of the screw raceway. The motion relationship between the screw, the raceway, and the nut is taken into account in the construction of the geometric model of the screw raceway. It is, therefore, essential to define the coordinate system of the ball screws, as shown in Figure 4.  
The coordinate system \(O_{W}X_{W}Y_{W}Z_{W}\) is used as the inertial reference system, and the \(Z_{W}\) axis is consistent with the axis direction of the ball screws. The coordinate system  
Figure 4: The coordinate system of the ball screws. Figure 3: Schematic diagram of the grinding of the screw raceway. _OXYZ_ is fixedly connected to the screw and rotates with the central axis of the screw. The coordinate system \(O_{H}tnb\) is the Frenet-Serret coordinate system. The motion trajectory of the ball center \(O_{H}\) is a spiral line, and the position of the origin \(O_{H}\) is related to the bottom angle \(\theta\) of the spiral line. The parameters \(t\), \(n\), and \(b\) are the tangent, principal normal, and secondary normal directions of the Frenet-Serret coordinate system. The geometry of the contact area of the ball screws is determined by the parameters such as the arc radius of the raceway section, the bottom radius of the spiral line, the spiral angle, the ball diameter, and the contact angle. The coordinate system of the contact point \(A\) between the ball and the screw raceway is determined to accurately describe the geometric shape of the screw raceway, as shown in Figure 5. The origin \(A\) of the coordinate system is the initial contact point between the ball and the screw raceway.