## 2 Problem Formulation
### Uniform Wear of the Teeth  
The profile of the gear teeth wears down during operation, changing the size and shape of the teeth. As a result, several undesired phenomena, including vibrations and noise, happen during the transmission. Achieving uniform wear of the tooth profile of the coupled gear pair requires achieving the equality of the specific sliding in the locations that are furthest from the present pole of the relative speeds. Hence, it is imperative to examine the kinematic aspect of the coupling of gear pairs with internal and external gearing to derive the expressions for specific sliding. Given the sliding speed, the specific sliding of the profile of the teeth of coupled gears can be defined as the ratio of the intensity of the sliding speed of the observed pair of teeth \(v_{sliding,1-2}\) to the intensity of the relative speed \(v_{rel,M}\) as follows  
\[v_{M}=\frac{v_{sliding,1-2}}{v_{rel,M}}=\frac{v_{rel,2}}{v_{rel,1}}-1, \tag{23}\]  
where \(v_{rel,i}\), \(i=1,2\) denotes the relative speed of the driving and driven gear, respectively, at any point along the LOA. The relative speed of points \(M_{1}\) and \(M_{2}\), which belong to the respective profiles of the pinion and driven gear, can be determined as  
\[ \begin{array}{l}v_{rel,M_{1}}=v_{M_{1}}\tan\!\alpha_{M_{1}},\\ v_{rel,M_{2}}=v_{M_{2}}\tan\!\alpha_{M_{2}},\end{array}  \tag{24}\]  
where the respective angles \(\alpha_{M_{1}}\) and \(\alpha_{M_{2}}\) can be determined from  
\[ \begin{array}{l}\tan\alpha_{M_{1}}=\frac{\rho_{M_{1}}}{\rho_{M_{1}}},\\ \tan\alpha_{M_{2}}=\frac{\rho_{M_{2}}}{\rho_{M_{2}}},\end{array}  \tag{25}\]  
where the \(\rho_{M_{1}}\) and \(\rho_{M_{2}}\) are corresponding curve radii on the tooth profile of pinion and driven gear. Substituting into (23), we obtain the expression for determining the specific sliding on the pinion and driven gear  
\[ \begin{array}{l}v_{M_{1}}=-1+\frac{\rho_{M_{2}}}{u\rho_{M_{1}}},\\ v_{M_{2}}=u\frac{\rho_{M_{1}}}{\rho_{M_{2}}}-1,\end{array}  \tag{26}\]  
where \(u\) is the gear ratio of the considered gear pair. Since the specific sliding on the gear profile is highest on the points furthest from pole C, determining the specific sliding in points \(A_{1}\) and \(E_{2}\) with equations  
\[ \begin{array}{l}v_{A_{1}}=\frac{\rho_{E_{2}}}{u\rho_{A_{1}}}-1,\\ v_{A_{2}}=u\frac{\rho_{E_{1}}}{\rho_{M_{2}}}-1,\end{array}  \tag{27}\]is of interest, where the \(\rho_{A_{1}}\), \(\rho_{E_{1}}\) and \(\rho_{A_{2}}\), \(\rho_{E_{2}}\) are corresponding curve radii on the tooth profile of the pinion and driven gear. In order to analyze the specific sliding in the characteristic points on the involute profile of the gears in coupling, Figure 2 shows the dependence of the specific sliding of simultaneously coupled teeth, for a gear pair with the parameters: \(z_{1}=20\), \(z_{2}=100\), \(m_{n}=3\) mm, \(x_{1}=x_{2}=0\), \(\alpha=20^{\circ}\). Based on the geometric interpretation of the obtained results, in Figure 2, it follows that the value of specific sliding is the largest at the beginning of the meshing at the base of the profile of the pinion \(E_{1}\) and the top of the profile of the tooth of the driven gear \(A_{2}\). In the kinematic pole (point C) the specific sliding is equal to zero. When the coupled profiles pass through pole \(C\), the sliding speeds change direction to the corresponding points on the top of the pinion and the base \(E_{2}\) of the driven gear. Therefore, it is of interest to analyze how to modify the value of specific sliding. Therefore, in Figure 3, we presented the sensitivity analysis of the maximum value of specific sliding on the pinion and driven gear as a function of profile shift coefficients. Figure 3: Maximum value of the specific sliding of pinion and driven gear as a function of profile shift coefficients \(x_{1}\) and \(x_{2}\). Figure 2: Specific sliding at different points on the pinion and driven gear along the line of action.