## 3 Results and Discussion
### Creation of the Geometrical Model of the Elliptical Gear Set  
The tooth side profile curve consists of two portions, the involute and the non-involute portion. Only the involute portion of the tooth profile is allowed to be active during meshing of a gear [35]. The function of the non-involute portion of the tooth profile is to create a smooth rounded transition between the involute portion of the tooth and the root cylinder. In general, an involute is created as the trajectory of a point on a piece of taut string as the string is unwrapped from a curve. The geometrical locus of the curvature centers of an involute is called evolute (Figure 6). Regarding standard involute gears, in order to create the teeth profiles the involute is used, of which the evolute is a circle. Therefore, the created straight line is the normal of the involute and also the tangent to the base circle (evolute) with the point of contact in the curvature center of the involute. In this case, the involute is unambiguously determined by one parameter, namely the base circle radius. In the first solution, the involute was created by rolling the created straight line on the base circle despite fact that in this case we dealt with elliptical gears. The center of the base circle was always coincident with the center of rotation in the eccentrically placed gear. It means that an involute was created by each central point of rolling. The involute was created by rolling the created straight line on the base circle of variable diameter according to the relation:  
\[\mathrm{d_{bi}}=\mathrm{d_{i}}*\cos\alpha_{n} \tag{13}\]  
where \(\mathrm{d_{i}}\) is the pitch circle size for individual teeth and the pressure angle in the normal plane \(\alpha_{n}=20^{\circ}\) (Figure 7). For the drawing of the involute portion of the tooth side profile, the trochoidal method of involute construction was used. Figure 6: Involute, evolute. The diameters of the base circles for the left and the right side of a tooth are different. The resulting shape of the gear created this way is shown in Figure 8a and it is identical with the shape of the model provided by the customer. These gears are not functional (Figure 8b), because the teeth interfere and the conditions of correct meshing are not met. Based on this finding, a more accurate tooth design was needed, so the involute portion of the tooth side profile was created by rolling the tangent on the base ellipse as the trajectory of a point. For this solution, the first step of constructing the involute portion of the teeth side profiles was the construction of tangents in the central points of contact on the pitch ellipse. In order to create the tangents, a separate construction was used (Figure 9). In Figure 9, the construction of the normal (denoted by the letter n) and the tangent (denoted by the letter t) for the active side of the tooth denoted by the number 4 in the central point of mesh of this tooth, so the construction of the normal and tangent to the pitch ellipse is shown. Figure 8: (**a**) Resulting shape of the gear; (**b**) the teeth interfere—the conditions of correct meshing are not met. Figure 7: Example of involute construction from base circle for the tooth with order number 3. The points 1p, 2p, 3p, etc., marked in red are the points for the passive sides of individual teeth. The center of rotation of the gear is denoted by the letter O. This center is also one of the foci--F\({}_{1}\) of the pitch ellipse. The second focus of the pitch ellipse is denoted by the symbol F\({}_{2}\). The second step was the determination of the evolute of the involute, so the geometrical locus of the points N\({}_{\rm i}\) as the last points of mesh (Figure 10). Therefore, it was necessary to construct straight lines tilted at the pressure angle \(\alpha_{\rm n}=20^{\circ}\) in the central point of mesh for each tooth individually and to determine the position of the involute curvature centers S\({}_{\rm i}\) and S\({}_{\rm pi}\) (for the left and also right side of each tooth).