# Reverse Engineering Used to Profile a Gerotor Pump Rotor
## 3 Results
### Calculation of the Enwrapping Profile
\begin{array}{c}X_{1}\cdot(\cos\varphi_{1}\cdot\cos\varphi_ {2}+\sin\varphi_{1}\cdot\sin\varphi_{2})-\\ X_{1}(\sin\varphi_{1}\cdot\cos\varphi_{2}-\cos\varphi_{1}\cdot\sin\varphi_{2}) +\\ -Y_{1}(\sin\varphi_{1}\cdot\cos\varphi_{2}-\cos\varphi_{1}\cdot\sin\varphi_{2} )+A_{12}\cdot\cos\varphi_{2}\\ +Y_{1}(\cos\varphi_{1}\cdot\cos\varphi_{2}+\sin\varphi_{1}\cdot\sin\varphi_{2} )-A_{12}\cdot\sin\varphi_{2}\end{array} \right)=\] \[=\left( \begin{array}{c}X_{1}\cdot\cos(\varphi_{1}-\varphi_{2})-Y _{1}\cdot\sin(\varphi_{1}-\varphi_{2})+A_{12}\cdot\cos\varphi_{2}\\ X_{1}\cdot\sin(\varphi_{1}-\varphi_{2})+Y_{1}\cdot\cos(\varphi_{1}-\varphi_{2} )-A_{12}\cdot\sin\varphi_{2}\end{array} \right).\]  
Using the \"virtual pole\" method, the profile of the \(Z_{4}\) rotor was recalculated so that it was reciprocally enwrapping around the circle arc profile of the \(Z_{5}\) rotor. The coordinates of the points on the profile of this rotor are presented in Table 1, and the trajectories passed by these points in the relative movement described by the \(Z_{4}\) rotor compared to the reference system of the \(Z_{5}\) rotor are highlighted in Figure 13.  
Figures 14 and 15 show the surfaces between the two rotors, which will lead to the existence of different volumes of pumped fluid, depending on the angular positioning of \(\varphi_{2}\) at \(0^{\circ},30^{\circ},60^{\circ},90^{\circ},120^{\circ},150^{\circ}\), and \(180^{\circ}\). The values that define the sections of the chambers (S), in [mm\({}^{2}\)], and the distances between the outer and inner rotor profiles, right and left, in [mm], both for the initial version and for the redesigned one, are specified in Table 2, respectively, in Figures 16-18. \begin{table}
\begin{tabular}{c c} \hline \hline
**X [mm]** & **Y [mm]** \\ \hline \(-16.26\) & 0 \\ \(-16.2964\) & 1.122976 \\ \(-16.4052\) & 2.241001 \\ \(-16.5857\) & 3.34923 \\ \(-16.8364\) & 4.443046 \\ \(-17.1556\) & 5.518244 \\ \(-17.5408\) & 6.571349 \\ \(-17.9893\) & 7.600286 \\ \(-18.4988\) & 8.606171 \\ \(-19.0703\) & 9.599994 \\ \(-19.7401\) & 10.66095 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coordinates of the redesigned \(Z_{4}\) rotor profile points. Figure 13: Trajectories of the points on the rotor profile \(Z_{4}\) in relative movement to \(Z_{5}\). Figure 14: Defining the surfaces occupied by the fluid in rolling movement (red color)—initial version. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\mathbf{\varphi_{2}}\) & **Initial \(S\)** & **Initial \(d\)—** & **Initial \(d\)—** & **Redesigned \(S\)** & **Redesigned \(d\)—** & **Redesigned \(d\)—** \\ \([^{\circ}]\) & **[mm\({}^{2}\)]** & **Left [mm]** & **Right [mm]** & **[mm\({}^{2}\)]** & **Left [mm]** & **Right [mm]** \\ \hline \(0^{\circ}\) & 199.983 & 0.366 & 0.363 & 224.089 & 0.106 & 0.106 \\ \(30^{\circ}\) & 183.651 & 0.289 & 0.437 & 204.739 & 0.012 & 0.238 \\ \(60^{\circ}\) & 140.856 & 0.328 & 0.449 & 153.756 & 0.047 & 0.233 \\ \(90^{\circ}\) & 87.181 & 0.461 & 0.392 & 89.807 & 0.182 & 0.108 \\ \(120^{\circ}\) & 41.141 & 0.484 & 0.259 & 36.575 & 0.259 & 0.061 \\ \(150^{\circ}\) & 16.8 & 0.412 & 0.143 & 10.382 & 0.181 & 0.06 \\ \(180^{\circ}\) & 10.533 & 0.383 & 0.39 & 4.282 & 0.073 & 0.07 \\ \hline \end{tabular}
\end{table} 
Table 2: The values of the sections occupied by the fluid and the distances between the profiles—initial and redesigned. Figure 15: Defining the surfaces occupied by the fluid in rolling movement (grey color)—redesigned version. In order to achieve the sealing between the two rotors, it is necessary that the radius of connection between the lobes of the \(Z_{4}\) rotor be modified and, simultaneously, the  
Figure 16: The variation of the values of the \(S\) occupied by the fluid depending on the angular positioning \(\varphi_{2}\). Figure 17: The variation of the values of the distances between the profiles depending on the angular positioning \(\varphi_{2}\). Figure 18: Distances \(d\) between the outer and inner rotor profiles: right and left (**a**); right (**b**); left (**c**).