# Reverse Engineering Used to Profile a Gerotor Pump Rotor
## 3 Results
### Verification of the Minimum Rolling Radius  
In order to be able to ensure enwrapping along the entire length of the two profiles, the rolling radius of each of the two centrodes should exceed the value of the minimum rolling radius. The minimum rolling radius for a certain profile is the radius of the circle that admits a tangency point with the normal to the considered profile, the normal taken through the limit point of the profile, Figure 11[48].  
Figure 10: Reference systems for the \(Z_{4}\)–\(Z_{5}\) rotors.  
In order to determine the minimum rolling radius, it starts from the equations of the analyzed profile, \(\Sigma_{1}\) profile:  
\[\Sigma_{1}\left|\begin{array}{l}X_{1}=-R_{0}+r\cdot\cos u;\\ Y_{1}=r\cdot\sin u.\end{array}\right. \tag{2}\]  
In Equation (2), \(u\) is the angular parameter describing the \(\Sigma_{1}\) profile.  
The tangent to the \(\Sigma_{1}\) profile has the equation:  
\[t-t\left|\frac{X-X(u)}{\dot{X}_{u}}=\frac{Y-Y(u)}{\dot{Y}_{u}}\right.; \tag{3}\]  
\[\dot{X}_{u}=-r\cdot\sin u;\dot{Y}_{u}=r\cdot\cos u. \tag{4}\]  
Taking into account the profile Equation (2), Equation (3) becomes:  
\[\frac{X+R_{0}-r\cdot\cos u}{-r\cdot\sin u}=\frac{Y-r\cdot\sin u}{-r\cdot\cos u} \tag{5}\]  
or, developed:  
\[Y=\frac{r-R_{0}\cdot\cos u-X\cdot\cos u}{\sin u}. \tag{6}\]  
The normal to the \(\Sigma_{1}\) profile, which passes through the \(O_{1}\) origin, has the equation:  
\[n-n:X\cdot\dot{X}_{u}+Y\cdot\dot{Y}_{u}=0; \tag{7}\]  
\[-X\cdot r\cdot\sin u+Y\cdot r\cdot\cos u=0; \tag{8}\]  
\[-X\cdot\sin u+\frac{r\cdot\cos u}{\sin u}-\frac{R_{0}\cdot\cos^{2}u}{\sin u} -\frac{X\cdot\cos^{2}u}{\sin u}=0; \tag{9}\]  
\[\frac{-X}{\sin u}=\frac{R_{0}\cdot\cos^{2}u-r\cdot\cos u}{\sin u}; \tag{10}\]  
Figure 11: Defining the minimum rolling radius.  
\[X =r\cdot\cos u-R_{0}\cdot\cos^{2}u=(r-R_{0}\cdot\cos u)\cdot\cos u; \tag{11}\] \[Y =\frac{r-R_{0}\cdot\cos u-(r-R_{0}\cdot\cos u)\cdot\cos u}{\sin u}=( r-R_{0}\cdot\cos u)\cdot\sin u.\]  
The minimum rolling radius is mathematically defined as the maximum distance between the current point on the \(\Sigma_{1}\) profile and the intersection point between the tangent to \(\Sigma_{1}\) at the current point and the normal to \(\Sigma_{1}\), which passes through the origin of the reference system (\(O_{1}\)). If this distance is noted by \(\Delta\), this is obtained:  
\[\Delta =\left[\left(-R_{0}+r\cdot\cos u-r\cdot\cos u+R_{0}\cdot\cos^{2}u \right)^{2}+(r\cdot\sin u-r\cdot\sin u+R_{0}\cdot\sin u\cdot\cos u)^{2}\right] ^{1/2}, \tag{12}\] \[\Delta =\left(R_{0}^{2}\cdot\sin^{4}u+R_{0}^{2}\cdot\sin^{2}u\cdot\cos^ {2}u\right)^{1/2},\] \[\Delta =R_{0}\cdot\sin u.\]  
The value of the minimum rolling radius will be:  
\[R_{\text{min}}=R_{0}\cdot\sin u_{\text{max}}. \tag{13}\]  
For the previously calculated rolling radius value \(R_{r1}\) (\(R_{r1}\) = 19 mm), it means that the last point on the \(\Sigma_{1}\) profile in which it is in a reciprocally enwrapping relation with the \(\Sigma_{2}\) profile corresponds to the angle \(u_{\text{max}}\) = 36.42\({}^{\circ}\).  
The profile area (\(S\)) between the points \(A\)--corresponding to \(u\) = 36.42\({}^{\circ}\) and \(B\)--corresponding to \(u\) = 42.1\({}^{\circ}\) will not be in contact during rolling with the \(\Sigma_{2}\) profile; therefore, there will be a volume of uncirculated fluid.  
For the dimensions measured, this fluid volume is approximately 200 mm\({}^{3}\) (Figure 12).