## 3 Results and Discussion
### Velocity Relations in the Designed Elliptical Gear Set  
When researching the kinematics of the designed gear set with eccentrically chosen centers of rotation, we followed the condition of a correct mesh, which defined that circumferential velocities in the central point of mesh are equal and their projections to the profile normal are equal. Studied kinematical parameters are processed for the driving elliptical gear--index 1 (with the center of rotation in the point O\({}_{1}\)) and for the driven elliptical gear--index 2 (with the center of rotation in the point O\({}_{2}\)). The graphical  
Figure 16: Asymmetry of tooth No. 6. Figure 17: Designed model of elliptical gear set. representation of kinematical dependences of both gears is chosen in a single graph (on the horizontal axis are the driving gear teeth--\(z_{1}\)). For solving the given issue, the constant angular velocity of the driving elliptical gear, \(\omega_{1}\) = 100 s\({}^{-1}\) was chosen. In Figure 18, the courses of angular velocity of the driving (\(\omega_{1}\) = 100 s\({}^{-1}\)) and the driven (\(\omega_{21}\)) elliptical gear are shown. Unlike standard circular gears, where the angular velocity of the driving and also of the driven gear is constant, in this case, the angular velocity of the driven eccentrically placed gear is not constant, but it varies depending on the continuously changing gear ratio (Figure 19). For example, if the teeth are meshing where a gear ratio of 0.5 is achieved (tooth number 24 on the driving wheel and tooth number 12 on the driven wheel), at that moment the angular velocity of driving wheel has the value \(\omega_{1}\) = 100 s\({}^{-1}\), but on the driven wheel there is an angular velocity \(\omega_{2}\) = 200 s\({}^{-1}\). These values apply if the point of teeth contact is at the pitch point. When tooth number 6 on the driving wheel is meshing with tooth number 6 on the driven wheel, the angular velocity is the same for both the driving wheel and the driven wheel (\(\omega_{1}\) = \(\omega_{2}\) = 100 s\({}^{-1}\)). The angular velocity of the driving wheel is constant, \(\omega_{1}\) = const. The angular velocity of the driven wheel is not constant, it depends on the changing gear ratio, \(\omega_{2}\) = f(u). Simultaneously, it follows from Figure 20 that the gear ratio of the designed, eccentrically placed elliptical gear transmission depends on the angle of rotation \(\varphi\) of the driving gear (Figure 21) and thus the gear ratio of the designed gear transmission is the function of this angle of rotation:  
\[\mathrm{u}=\mathrm{f}(\varphi) \tag{16}\]  
Figure 22 shows the length of part of contact of tooth number 24 of the driving wheel with tooth number 12 of the driven wheel. The point A is the first point of meshing of tooth number 24 of driving wheel. The point E is the last point of meshing of tooth number 24 of the driving wheel. The change of the gear ratio was investigated for this tooth--see Figure 23.  
Figure 19: The course of changing gear ratio in the designed elliptical gear set. Figure 18: The course of angular velocities in the designed elliptical gear set. Figure 21: The course of the gear ratio dependent on the angle of rotation of the driving gear. Figure 22: The length of part of contact. The standard spur gears have the constant (same) contact ratio for all teeth of meshing. The value of the contact ratio (CR) of meshing for spur gears is in the range 1 \(<\) CR \(<\) 2. This is not the case with the designed elliptical gearing.