# A Novel Optimization Design Method of Form Ginding Wheel for Screw Rotor
## 2 Theoretical Background  
### The Coordinate Relationship between the Screw Rotor and the Grinding Wheel  
In order to obtain precise installation parameters, the coordinate transformation and meshing relationship between the screw rotor and the grinding wheel needs to be established according to grinding space geometry and engagement theory. Given that the cross-section of the screw rotor is composed of multiple discrete points \((x_{t},y_{t})\), as shown in Figure 1, the helical surface equation can be expressed as follows [3]:  
\[\left\{\begin{array}{l}X=x_{t}\cos\theta-y_{t}\sin\theta\\ Y=x_{t}\sin\theta+y_{t}\cos\theta\\ Z=p\theta\end{array}\right. \tag{1}\]  
where \(p\) is the screw parameter of the rotor, and \(p=S/2\pi\) (\(S\) is the lead of the screw rotor).  
Noting that the grinding wheel shaft is mathematically described by multiple discrete points \((Z_{c},R_{t})\), as shown in Figure 2, the revolution surface equation of the grinding wheel can be expressed as follows [3]:  
\[\left\{\begin{array}{l}X_{c}=R_{t}\cos\phi\\ Y_{c}=R_{t}\sin\phi\\ Z_{c}=f(R_{t})\end{array}\right. \tag{2}\]  
Figure 1: Three-dimensional (3D) structure of the screw rotor.  
where \(X_{c}\), \(Y_{c}\), and \(Z_{c}\) are the revolution surface equation of the grinding wheel; \(R_{t}\) is the radius when the width of the grinding wheel is \(Z_{c}\); \(\phi\) is the angle between \(R_{t}\) and the \(X_{c}O_{c}Z_{c}\) plane; and the \(X_{c}\) to \(Y_{c}\) direction is defined as the positive direction.  
The screw surface is formed by relative meshing movement enabled by the grinding wheel and the screw rotor. Figure 3 shows the geometric relationship between a grinding wheel and a screw rotor. The mounting angle \(\omega\) is formed due to the crossing axes of the grinding wheel and the screw rotor. The distance between the axes and a point on the contact line is denoted by \(T\) and \(M\), respectively. \(O-XYZ\) represents the screw rotor coordinate system, while \(O-X_{c}Y_{c}Z_{c}\) represents the grinding wheel coordinate system.  
The geometric relationship between the grinding wheel coordinate system \(O-X_{c}Y_{c}Z_{c}\) and the screw rotor coordinate system \(O-XYZ\) is as follows:  
\[\left\{\begin{array}{l}X=X_{c}\cos\omega+Z_{c}\sin\omega\\ Y=Y_{c}+T\\ Z=Z_{c}\cos\omega-X_{c}\sin\omega\end{array}\right. \tag{3}\]  
Figure 3: Coordinate systems of the grinding wheel and the screw rotor.  
Figure 2: Schematic diagram of the grinding wheel structure.  
\[\left\{\begin{array}{c}\overrightarrow{i}=\cos\omega\dot{i_{c}}+\sin\omega \overrightarrow{k_{c}}\\ \overrightarrow{j}=\dot{j_{c}}\\ \overrightarrow{k}=\cos\omega\overrightarrow{k_{c}}-\sin\omega\dot{i_{c}} \end{array}\right. \tag{4}\]  
\[\left\{\begin{array}{c}X_{c}=X\cos\omega-Z\sin\omega\\ Y_{c}=Y-T\\ Z_{c}=X\sin\omega+Z\cos\omega\end{array}\right. \tag{5}\]  
\[\left\{\begin{array}{c}\overrightarrow{i_{c}}=\cos\omega\overrightarrow{i }-\sin\omega\overrightarrow{k}\\ \overrightarrow{j_{c}}=\overrightarrow{j}\\ \overrightarrow{k_{c}}=\cos\omega\overrightarrow{k}+\sin\omega\overrightarrow{ i}\end{array}\right. \tag{6}\]  
where \(\overrightarrow{i_{c}}\), \(\overrightarrow{j_{c}}\), \(\overrightarrow{k_{c}}\) is the unit vector in the grinding wheel coordinate system \(O-X_{c}Y_{c}Z_{c}\), and \(\overrightarrow{i}\), \(\overrightarrow{j}\), and \(\overrightarrow{k}\) are the unit vectors in the screw rotor coordinate system \(O-XYZ\).