# A Novel Optimization Design Method of Form Ginding Wheel for Screw Rotor
## 2 Theoretical Background
### Rotor Profile Generation Model  
The profile of the screw rotor can be derived once the form wheel profile is known. The contact line equation can be expressed as [3]:  
\[\left(\overrightarrow{k}\times\overrightarrow{r}+p\overrightarrow{k} \right)\overrightarrow{n}=0 \tag{13}\]  
where \(\overrightarrow{r}=\overrightarrow{OM}\) is the radial vector of \(M\) in \(O-XYZ\). In the screw rotor coordinate system \(O-X_{c}Y_{c}Z_{c}\), the vector \(\overrightarrow{r}\) can be expressed as follows:  
\[\overrightarrow{r}=\overrightarrow{R}+T\overrightarrow{j_{c}}=R_{t}\cos\phi \overrightarrow{i_{c}}+R_{t}\sin\phi\overrightarrow{j_{c}}+f(R_{t}) \overrightarrow{k_{c}}+T\overrightarrow{j_{c}} \tag{14}\]  
Substituting Equation (6) into Equation (14), in the screw rotor coordinate system \(O-XYZ\), the vector \(\overrightarrow{r}\) can be expressed as:  
\[\overrightarrow{r}=\left[R_{t}\cos\phi\cos\omega+f(R_{t})\sin\omega\right] \overrightarrow{i}+\left(R_{t}\sin\phi+T\right)\overrightarrow{j}+\left[f(R_ {t})\cos\omega-R_{t}\cos\phi\sin\omega\right]\overrightarrow{k} \tag{15}\]  
The term \(\overrightarrow{k}\times\overrightarrow{r}+p\overrightarrow{k}\) can be calculated as follows:  
\[\overrightarrow{k}\times\overrightarrow{r}+p\overrightarrow{k}=\left[R_{t} \cos\phi\cos\omega+f(R_{t})\sin\omega\right]\overrightarrow{j}-\left(R_{t }\sin\phi+T\right)\overrightarrow{i}+p\overrightarrow{k} \tag{16}\]  
The normal vector \(\overrightarrow{n}\) at each coordinate axis can be figured out according to:  
\[\overrightarrow{n}=\frac{\partial\overrightarrow{r}}{\partial R_{t}}\times \frac{\partial\overrightarrow{r}}{\partial\phi}=\left|\begin{array}{ccc} \overrightarrow{i}&\overrightarrow{j}&\overrightarrow{k}\\ \frac{\partial X}{\partial R_{t}}&\frac{\partial Y}{\partial R_{t}}&\frac{ \partial Z}{\partial R_{t}}\\ \frac{\partial X}{\partial\phi}&\frac{\partial Y}{\partial\phi}&\frac{ \partial Z}{\partial\phi}\end{array}\right| \tag{17}\]  
Taking the partial derivative of \(R_{t}\) and \(Z_{c}\), the following relationship can be obtained:  
\[\left\{\begin{array}{c}\frac{\partial X}{\partial R_{t}}=\cos\phi\cos \omega+f^{\prime}(R_{t})\sin\omega\\ \frac{\partial Y}{\partial R_{t}}=\sin\phi\\ \frac{\partial Z}{\partial R_{t}}=f^{\prime}(R_{t})\cos\omega-\cos\phi\sin \omega\end{array}\right. \tag{18}\]  
\[\left\{\begin{array}{c}\frac{\partial X}{\partial\phi}=-R_{t}\sin\phi\cos \omega\\ \frac{\partial Y}{\partial\phi}=R_{t}\cos\phi\\ \frac{\partial Z}{\partial\phi}=R_{t}\sin\phi\sin\omega\end{array}\right. \tag{19}\]  
Substituting Equations (18) and (19) into Equation (17) yields the following:  
\[\left\{\begin{array}{c}n_{x}=R_{t}\sin\omega-f^{\prime}(R_{t})\cos\omega R _{t}\cos\phi\\ n_{y}=-f^{\prime}(R_{t})R_{t}\sin\phi\\ n_{z}=R_{t}\cos\omega+f^{\prime}(R_{t})\sin\omega R_{t}\cos\phi\end{array}\right. \tag{20}\]  
Then, the substitution of Equations (16) and (20) into Equation (13) yields:  
\[\left(T+p\tan\omega\right)\cos\phi-\left[f(R_{t})+\frac{1}{f^{\prime}(R_{t})} R_{t}\right]\tan\omega\sin\phi+\frac{1}{f^{\prime}(R_{t})}\left(p-T\tan\omega \right)=0 \tag{21}\]where \(f^{\prime}(R_{t})\) is the first-order derivative of \(Z_{c}\) versus \(R_{t}\). Since each \(R_{t}\) can be used to deduce two different values of \(Z_{c}\), the \(R_{t}\) value can be regarded as a function of \(Z_{c}\), as shown in Figure 2. The term \(1/f^{\prime}(R_{t})\) is provided by the inverse function theorem:  
\[\frac{1}{f^{\prime}(R_{t})}=\left[f^{-1}(Z_{c})\right]^{\prime} \tag{22}\]  
On the one hand, it can be observed from Equation (21) that angle \(\phi\) is the only variable to be determined; on the other hand, it is worth noticing that \(T\) can be used to determine angle \(\phi\). As such, the space contact line can be figured out, based on which the screw rotor profile can also be determined.