## 3 Results and Discussion
### Creation of the Geometrical Model of the Elliptical Gear Set
The curvature centers of the ellipse lie on the normal and they are determined by the curvature centers \(\tau_{\rm ie}\). \[\tau_{\rm ie}=\frac{1}{\alpha_{\rm e}^{4}b_{\rm e}^{4}}\sqrt{\left(a_{\rm e}^ {4}y_{\rm i}^{2}+b_{\rm e}^{4}x_{\rm i}^{2}\right)^{3}} \tag{14}\]  
Figure 10: Construction of the last points of mesh \(N_{i}\). Figure 9: Construction of the tangent “\(t\)” and normal “\(n\)” to the pitch ellipse. where \(x_{i}\), \(y_{i}\) are the coordinates of a point on the pitch ellipse. The left and the right side of each tooth were solved individually. The points \(N_{i}\) for the right sides of the teeth determine the evolute. The points \(N_{pi}\) determine the second evolute for the left sides of the teeth. Figure 10 illustrates the construction of the evolutes for the left and right side of the tooth number 4. In Figure 11 is the trochoidal method of construction of the involute curve of the tooth side profile if the evolute of the involute is the ellipsis. Using the trochoidal method of involute construction (Figure 11), the points of the involute for each side of a tooth were laboriously created. The more involute points are created using the above described trochoidal method, the more accurate the shape of the involute portion of the tooth side profile we acquire. The difference in the construction of these tooth profiles from the reference [36] is in the formation of the active curve of the tooth profile. The methods of involute construction used in the references used the base circle as the evolute. The procedure described in this article is different. The evolute has the shape of a (basic) ellipse (Figure 11). The difference in the procedure is also visible in the position--the location of the wheel teeth with respect to the center of rotation. The principal difference between the shape of teeth due to the differential creation of the involute is in Figure 12.  
A root transition is the surface between the involute surface of a tooth and the root cylinder. The function of the non-involute portion of the profile of a tooth is to create a  
Figure 11: Example of involute construction from base ellipse for the tooth with number 4. Figure 12: Principal difference between the shape of teeth due to the differential creation of the involute. smooth transition between the involute portion of the toothing and the root cylinder. It is a very important region, because it determines considerably the flexural strength and the interference phenomena during meshing [37]. The shape of the non-involute portion of the profile of a tooth depends on the manufacturing method of the toothing. The manufacture of a toothing by a rack cutter is one of the most used manufacturing methods, where the non-involute portion of the tooth side profile is the envelope of the positions of the rack cutter which rolls on the pitch circle of a gear. Such creation of the non-involute portion of the tooth side profile is not suitable for the given elliptical eccentrically placed gear. For the geometrical design, the following formula for computing the fillet radius of the non-involute portion of the tooth side profile was used [38]:  
Figure 6:0  
where \(\mathrm{m_{n}}\) is the module in the normal plane. For spur gears (with straight teeth), the value of \(\mathrm{m_{n}}\) equals the value of the module in the transversal plane, \(\mathrm{m_{n}}=\mathrm{m_{t}}=\mathrm{m}\). Non-involute portions of the tooth profile (its left and right side) were constructed using the same fillet radius \(\mathrm{r_{f}}=1.425\) mm. By connecting the involute and the non-involute portion of a tooth side profile we create one tooth side profile. In order to create one complete tooth profile, we need further inputs such as the dimensions of the tip and root cylinders (for circular gears) which create the height boundaries of the tooth.