## 3 Results and Discussion  
### Design of the Pitch Curve Shape  
These non-circular gear sets were developed considering the law of motion of the driven gear, for variable gear ratio and the elliptical shape of pitch curve of the driving gear wheel [32]. The first step in a gearing design is the design of a pitch curve. Regarding \"standard\" gears, the pitch curve is created by a pitch circle with the center situated on the axis of rotation. Based on the results of the solution of the given problem using gears with circular shape and eccentrically placed centers of rotation, the elliptical shape of the gears was chosen. The geometrical gear wheel center is not same as the rotation center of this gear wheel. The center of rotation of the gear was chosen in the focus of the ellipse. The dimensions of the chosen pitch ellipse of the gear are visible in Figure 4.  
The length of the semimajor axis of the pitch ellipse a\({}_{\rm e}\) = 45 mm was computed from the given axial distance a = 90 mm. The centers of rotation of the gears O\({}_{1}\) and O\({}_{2}\) were determined according to the demand to create a transmission with the time-variable gear ratio in the range from 0.5 through 1.0 up to 2.0. When r\({}_{1}\) + r\({}_{2}\) = 90 mm, where r\({}_{1}\) = r\({}_{2}\) and r\({}_{2}\) / r\({}_{1}\) = 1.0, the gear ratio equals 1.0 and the length of the semiminor axis of the ellipse b\({}_{\rm e}\) can be determined: b\({}_{\rm e}\) = 42.426 mm (Figure 4). The points O\({}_{1}\) and O\({}_{2}\) are the foci of the pitch ellipses as well. According to the property of an ellipse, which is true for each point X of the ellipse, the sum of the distances O\({}_{1}\)X and O\({}_{2}\)X equals the twice the length of the semimajor axis, in this case the sum equals the axial distance. In Figure 5, the pitch ellipses of the eccentrically placed intermeshing elliptical gears and also the pitch ellipse division into 24 pitches (according to the number of teeth z = 24), which are equal in length, are shown. Both gears are identical, therefore the same marking of teeth was chosen on both intermeshing gears. The rotational speed of the driving elliptical gear wheel has the symbol n\({}_{1}\) in the Figure 5, n\({}_{2}\) is the rotational speed of the driven wheel. There are values of pitch radii and values of variable gear ratio for designed elliptical gear set in Table 1.  
In Table 1 are the values of pitch radii, in particular mesh points, denoted by the symbols r\({}_{1\text{-}i}\) or r\({}_{2\text{-}j}\) where the index 1 stands for the driving gear, index 2 for the driven gear, index i, or j is the order number of the mating tooth (Figure 5) at one revolution of the driving and driven gear. This table shows the variable gear ratio of the intermeshing gears as well, in the range from 0.5 (the first pair of teeth mates, the tooth 24 of the driving gear is in mesh with the tooth 12 of the driven gear) through the gear ratio = 1.0 (if the tooth 6 of the driving gear is in mesh with the tooth 6 of the driven gear), up to the gear ratio = 2.0 (if the tooth 12 of the driving gear is in mesh with the tooth 24 of the driven gear) and back through the gear ratio = 1.0 to the gear ratio = 0.5. Figure 4: Dimensions of the designed pitch ellipse. For each tooth, the pitch ellipse has to meet the condition that the sum of radii equals the axial distance:  
\[\mathrm{r_{1-i}+r_{2-j}=a=90 mm} \tag{9}\]  
The pitch ellipse is given by the parametric equations (\(\varphi\) is the eccentric anomaly of an ellipse \(\varphi\in 0,2\pi\)) [33]:  
\[ \begin{array}{l}\mathrm{x=a_{e}\ *\cos\varphi}\\ \mathrm{y=b_{e}\ *\sin\varphi}\end{array}  \tag{10}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Tooth of Driving** & **Pitch Radius** & **Tooth of Driven** & **Pitch Radius** & **Gear Ratio** & **Axial Distance** \\
**Gear—Order Number** & **r-1 (mm)** & **Gear—Order Number** & **r2-j (mm)** & **\(u_{i}=\mathbf{r_{2-j}}\)/r1-i** & **r1-i + r2-j (mm)** \\ \hline  
\hline \hline \end{tabular}
\end{table}