## 2 Materials and Methods
### Conditions of Correct Meshing of Gears  
The quality of gear transmissions is determined mainly by their geometrical design [21]. In the case of an incorrect geometrical design, even the use of highest quality materials does not ensure the reliability of a gear transmission. On the contrary, the excellent geometrical design of a gearing can sometimes save the costs of an expensive material [22; 23].  
For the design of correctly intermeshing gears without a backlash, the essential conditions of correct meshing have to be fulfilled, according to [24; 25]:  
* The condition of common profile normal at each mesh point of intermeshing gears, which has to pass through the pitch point.
* The condition of mesh continuity, which means the condition of the existence of the mesh of two consecutive profiles. Providing that the pitches measured on both working circles of two intermeshing gears (for standard circular gears) are equal, this condition is fulfilled.
* The condition of the contact of teeth along the whole face width of a gearing. Providing that the helix angles on working circles are equal, this condition is fulfilled.
* The condition of circumferential velocities, which means the projections of circumferential velocities to the common profile normal have to be equal in each mesh point.
* The condition of working circle contact (the sum of the working radiuses of intermeshing gears in each mesh point equals the center distance).  
The industrial standard for the involute tooth shape has been chosen for non-circular gears. Therefore, involute gearing standards and methods, which already exist, can be used and adopted [26]. Basic conditions that are required of the gearing determine the correct design for the meshing. The profiles of the teeth forming a shape bond are designed to retain a requirement for continuous meshing [27]. Transmitting of a uniform rotary motion between shafts using gear teeth, requires the normals to these teeth profiles. Every point of contact has to intersect with a virtual fixed point. For transmission of rotary motion between two shafts by gear teeth means, the instantaneous normals of the teeth profiles have to pass through a virtual fixed point [28; 29].  
Figure 2 represents non-circular gears like cylinders rolling together without slipping and given addendum undergone no modification. This means that nominal axle distance is applied. The gear set are depicted by the pitch curves \(k_{1}\) and \(k_{2}\), which have centers in the points \(\mathrm{O}_{1}\) and \(\mathrm{O}_{2}\). The pitch curve radii \(\mathrm{r}_{1}(\varphi)\) and \(\mathrm{r}_{2}(\varphi)\), which are variable, are determined for the non-circular gear set with gear ratio:  
\[\mathrm{i}(\varphi)=\frac{\mathrm{d}\varphi}{\mathrm{d}\psi}=\frac{1}{\psi^{ \prime}(\varphi)}=-\frac{\mathrm{r}_{2}(\varphi)}{\mathrm{r}_{1}(\varphi)}= \frac{\omega_{1}(\varphi)}{\omega_{2}(\varphi)}, \tag{1}\]  
where \(\omega_{1}(\varphi)\) and \(\omega_{2}(\varphi)\) represent functions for the angular velocity of the gear 1 and 2.  
Figure 1: Model of the gear set.  
Equation for the center distance is the following:  
\[\text{a}=\text{constant}=\text{r}_{1}(\varphi)+\text{r}_{2}(\varphi), \tag{2}\]  
where \(\psi\)' (\(\varphi\)) is the transmission function, which describes pitch curves of the non-circular gear set relation.