## 4. Machining of a Non-Circular Gear Using the Fellows Method and a Shaper Cutter  
The Warsaw University of Technology developed an innovative method of forming non-circular gears using the Fellows method [14; 15]. This method is used in situations in which other methods do not allow the machining of a concave profile [16; 17].  
A kinematic model of teeth shaping using this method is shown in Figure 9.  
Figure 8. Wear width of flank surface for two kinds of hobbing [12].  
Figure 7. (a) Virtual hobbing of a non-circular helical gear based on the diagonal hobbing process; (b) 3-order elliptic helical gear hobbed using diagonal hobbing [12].  
Figure 9. Examples of technological transmission for a shaper and a non-circular gear [14; 15].  
Let us consider two curves rolling on each other without sliding (Figure 10). The first curve is rigidly connected with the coordinate system \(x_{1},\ O_{1},\ y_{1}\) rotating with constant angular velocity by the angle \(\varphi_{1}\). The second curve is rigidly connected with the coordinate system \(x_{2},\ O_{2},\ y_{2}\) rotating with variable angular velocity by the angle \(\varphi_{2}\). Both curves may be described in their own systems using variable parameters \(\vartheta_{1},\ \vartheta_{2}\) as:  
\[\left.\begin{array}{l}x_{1}=x_{1}(\vartheta_{1})\\ y_{1}=y_{1}(\vartheta_{1})\end{array}\right\}\text{or}\ \widetilde{r}_{1}=\left[ \begin{array}{l}x_{1}(\vartheta_{1})\\ y_{1}(\vartheta_{1})\end{array}\right] \tag{1}\]  
\[\left.\begin{array}{l}x_{2}=x_{2}(\vartheta_{2})\\ y_{2}=y_{2}(\vartheta_{2})\end{array}\right\}\text{or}\ \widetilde{r}_{2}=\left[ \begin{array}{l}x_{2}(\vartheta_{2})\\ y_{2}(\vartheta_{2})\end{array}\right] \tag{2}\]  
and vector \(\left[\begin{array}{l}a\\ 0\end{array}\right]\) represents the center distance in the transmission.  
In the general case, the center distance \(a\) is also variable (Figure 9), although in a specific case, it can be constant. The task is to compute the kinematic functions \(\varphi_{2}=f_{1}(\varphi_{1})\) and \(a_{2}=f_{2}(\varphi_{1})\) describing the relative motions (rotational and rectilinear) of both gears.  
To solve this task, let us consider any two points M\({}_{1}\), M\({}_{2}\) being in contact. Let us take as an example the part of a hydraulic motor transmission consisting of a toothed triangle rotor (with rounded corners) and a circular satellite. The roll curves are shown in Figure 10.  
They can be described as follows:  
* Circular rolling curve of the satellite: \[\left.\begin{array}{l}x_{2}=r\ \cos\ \vartheta_{2}\\ y_{2}=r\ \sin\ \vartheta_{2}\end{array}\right\}\] (3)  
\[\widetilde{n}_{2}=\left[\begin{array}{l}n_{x2}\\ n_{y2}\end{array}\right]=\left[\begin{array}{l}\cos\ \vartheta_{2}\\ \sin\ \vartheta_{2}\end{array}\right] \tag{4}\]
* Rectilinear part of the triangle: \[\left.\begin{array}{l}x_{1}=b=const\\ y_{1}=y_{1}(\text{equivalent of variable parameter}\ \ \vartheta_{1})\end{array}\right\}\] (5)  
Figure 10: The roll curves of a hydraulic motor: (**a**) At the beginning (\(\varphi_{1}=0\)); (**b**) after rotation of the triangle rotor by any angle \(\varphi_{1}\)[14; 15].  
* Rounded corners of the triangle (point \(M_{1}\) in Figure 4): \[\begin{array}{c}x_{1}=b-\rho(1-\cos\vartheta_{1})\\ y_{1}=\left[(b-\rho)\text{ctg}30^{0}+\rho\ \sin\vartheta_{1}\right]\\ \end{array}\] (6) \[\begin{array}{c}\vec{n}_{1}=\left[\begin{array}{c}n_{x1}\\ n_{y1}\end{array}\right]=\left[\begin{array}{c}\cos\vartheta_{1}\\ -\sin\vartheta_{1}\end{array}\right]\end{array}\] (7)  
Equations of rolling curves and normal vectors are the starting point for determining the coordinates of tooth profiles [14].