## 2 Materials and Methods
### Velocity Relations in Ideal Intermeshing Spur Gears  
Standard (circular) spur gears perform a rotary motion of angular velocities \(\omega_{1,2}\) about their axes of rotation. Individual points of gears move in circular paths at the circumferential velocity, which is determined in the central point of mesh \(C\) by the distance of pitch radii \(\text{r}_{1,2}\) according to the equation [30]:  
\[\text{v}=\text{v}_{1}=\text{v}_{2}=\text{r}_{1}^{X}*\omega_{1}=-\text{r}_{2}^ {X}*\omega_{2}, \tag{3}\]  
where \(\text{r}_{1}^{X}\) and \(\text{r}_{2}^{X}\) represents connecting lines of the point \(X\) to the centers of rotation \(\text{O}_{1}\) and \(\text{O}_{2}\) (Figure 3). The circumferential velocity of intermeshing gears is an important indicator in the velocity relations. Depending on the circumferential speed (v), g  
Figure 3: Velocity relations in intermeshing spur gears. Figure 2: Example of non-circular pitch curves. three groups: low-speed gears (v \(\leq\) 5 m.s\({}^{-1}\)); medium speed gears (5 m.s\({}^{-1}\) \(<\) v \(<\) 20 m.s\({}^{-1}\)) and high-speed gears (v \(\geq\) 20 m.s\({}^{-1}\)) [31]. In any mesh point \(X\) on the line of action \(\tau\), the circumferential velocity of the point of contact of the driver (with the center of rotation in the point O\({}_{1}\)) is v\({}_{1}\) and the circumferential velocity of the point of contact of the driven gear (with the center of rotation in the point O\({}_{2}\)) is v\({}_{2}\) (Figure 3). The directions of these velocities are perpendicular to the connecting lines r\({}_{1}\)\({}^{X}\) r1X and r\({}_{2}\)\({}^{X}\) of the point X to the centers of rotation O\({}_{1}\) and O\({}_{2}\). In the transverse plane, the velocities v\({}_{1}\) and v\({}_{2}\) can be decomposed into two perpendicular components, according to the equations:  
\[\vec{\mathrm{v}}_{1}=\vec{\mathrm{v}}_{\mathrm{b}}+\vec{\mathrm{v}}_{\mathrm{ p1}}\text{ and }\vec{\mathrm{v}}_{2}=\vec{\mathrm{v}}_{\mathrm{b}}+\vec{\mathrm{v}}_{ \mathrm{p2}} \tag{4}\]  
The velocities v\({}_{\mathrm{p1}}\) and v\({}_{\mathrm{p2}}\) are sliding velocities. They are situated on the common tangent to the profiles of both gears and are perpendicular to the line of action. The velocity v\({}_{\mathrm{b}}\) is a driving velocity. It is situated on the line of action \(\tau\) and it is the velocity of the point X, which is moving in the path of action (in the straight line \(\tau\)). From the condition of the continuous contact of the profiles of both teeth in any point X on the line of action, it follows that the driving velocity v\({}_{\mathrm{b}}\) of both intermeshing gears has to be equal. The sliding velocities v\({}_{\mathrm{p1}}\) and v\({}_{\mathrm{p2}}\) are not equal and their vector subtraction equals the relative velocity of the instantaneous mesh point v\({}_{\mathrm{s}}\):  
\[\mathrm{v}_{\mathrm{s21}}=\mathrm{v}_{2}-\mathrm{v}_{1}=\mathrm{v}_{\mathrm{ p2}}-\mathrm{v}_{\mathrm{p1}}=-\mathrm{v}_{\mathrm{s12}} \tag{5}\]  
The velocities v\({}_{1}\) and v\({}_{2}\) are circumferential velocities on the radii r\({}_{1,2}\)\({}^{X}\) and their magnitude is given by Equation (3). The velocity v\({}_{\mathrm{b}}\) follows from velocity triangles and using Equation (3) we acquire:  
\[\mathrm{v}_{\mathrm{b}}=\mathrm{v}_{1}*\cos\alpha_{1}^{X}=\mathrm{v}_{2}*\cos \alpha_{2}^{X}=\mathrm{r}_{\mathrm{b1}}*\omega_{1}=-\mathrm{r}_{\mathrm{b2}}* \omega_{2} \tag{6}\]  
The velocities v\({}_{\mathrm{p1}}\) and v\({}_{\mathrm{p2}}\) follow from velocity triangles:  
\[\mathrm{v}_{\mathrm{p1}}=\mathrm{v}_{1}*\sin\alpha_{1}^{X}=\mathrm{r}_{1}^{X}* \sin\alpha_{1}^{X}=\mathrm{\rho}_{1}*\omega_{1} \tag{7}\]  
\[\mathrm{v}_{\mathrm{p2}}=\mathrm{v}_{2}*\sin\alpha_{2}^{X}=-\mathrm{r}_{2}^{X}* \sin\alpha_{2}^{X}=-\mathrm{\rho}_{2}*\omega_{2}, \tag{8}\]  
where \(\mathrm{\rho}_{1,2}\) are the radii of curvature in the mesh point X. In Equation (7) are the minus signs in order to the corresponding velocities were positive with regard to the sense of r\({}_{2}\)\({}^{X}\) and \(\omega_{2}\).