# Comparison of Selected Parameters of a Planetary
## 5 Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Gear Pair**} & \multirow{2}{*}{
\begin{tabular}{c} **Point of the Path** \\ **of Contact** \\ \end{tabular} } & \multicolumn{2}{c}{**Contact Pressure \(p\) [MPa]**} & \multicolumn{2}{c}{**Flink Type Ratio**} \\ \cline{3-5}  & & **Involute Teeth** & **Convex–Concave** & \(p_{\text{C-C}}/p_{\text{INV}}\) **(-)** \\ \cline{3-5}  & & **Flanks** & **Teeth Flanks** & \\ \hline \multirow{4}{*}{1-2} & A & 12.3086 & 3.6178 & 0.2939 \\  & B\({}_{2}\)/B\({}_{1}\) & 5.9996/8.4645 & 3.8418/5.4372 & 0.6407/0.6423 \\  & C & 7.9379 & 7.9379 & 1.0000 \\  & D\({}_{1}\)/D\({}_{2}\) & 7.9999/5.658 & 5.2443/3.6961 & 0.6555/0.6533 \\  & E & 7.6604 & 3.4491 & 0.4502 \\ \hline \multirow{4}{*}{2-3} & A & 42.9729 & 8.7852 & 0.2044 \\  & B\({}_{2}\)/B\({}_{1}\) & 14.2245/20.0888 & 9.1338/12.9432 & 0.6421/0.6443 \\  & C & 16.9851 & 16.9851 & 1.0000 \\  & D\({}_{1}\)/D\({}_{2}\) & 16.2260/11.4516 & 9.5872/6.7698 & 0.5909/0.5912 \\  & E & 8.1294 & 6.3054 & 0.7756 \\ \hline \multirow{4}{*}{4-5} & A & 41.7424 & 7.7110 & 0.1847 \\  & B\({}_{2}\)/B\({}_{1}\) & 12.8809/18.0976 & 7.9674/11.2820 & 0.6185/0.6234 \\ \cline{1-1}  & C & 15.3391 & 15.3391 & 1.000 \\ \cline{1-1}  & D\({}_{1}\)/D\({}_{2}\) & 14.7914/10.4142 & 8.4263/5.9422 & 0.5697/0.5706 \\ \cline{1-1}  & E & 7.2949 & 5.6199 & 0.7704 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Contact pressures of the partial gear pairs of the calculated gearbox. Figure 9: Specific sliding ratio \(v_{\text{ss}}\) that depends on the angle \(\phi_{pinion}\) between the gear pairs with involute and convex–concave teeth flanks: (**a**) gear pair 1-2, (**b**) gear pair 2-3, and (**c**) gear pair 4-5. The gear pair specific sliding ratios of the convex-concave teeth flanks dropped significantly, as shown in Figure 9 and Table 4, where the values of the slide ratios are described in detail. The points A and E represent the start and end of the contact path (indexed with 1, 2, and 4 for the pinion and 2, 3, and 5 for the gear), and point C represents the pitch contact point lying on the gear pair centers' link. Table 4 also presents the flank type slide ratio \(v_{\text{sSC-C}}\)/\(v_{\text{ssl}NV}\) between the convex-concave (\(v_{\text{sSC-C}}\)) and the involute (\(v_{\text{ssl}NV}\)) teeth flank values. The analysis of the involute gear teeth flank geometry of the observed gearbox led to the reduction of the tip diameters of the ring gears (Figure 6). The involute gear profile with the standard addendum and dedendum coefficients was used for the external gear pair 1-2, as well as for the internal gear pairs 2-3 and 4-5. The contact pressures in the convex-concave gears were significantly lower than for the involute gears, especially for the gear pairs with internal gearing (Figure 8). The drop of the contact pressure was most significant for the gear pair 4-5 at the starting point of the path of contact A, where the pressure value in the convex-concave teeth flanks dropped to 18.47% of the value calculated for the involute teeth flanks (Table 3). The drop of the contact pressure was accompanied by the drop of the contact ratio, which had a value close to 1 for the convex-concave teeth flanks. The comparison of the contact ratio values \(\varepsilon_{\alpha}\) using the involute and convex-concave teeth flanks in the observed gearbox are presented in Table 5.  
Past research has shown the preferable sliding behavior of the convex-concave-shaped tooth flanks relative to the involute ones. The calculations were made just for the external gear pairs [51]. The most important features of polymeric S-gears, which were expected, were the lower sliding due to the relatively larger contact lengths of dedendum-addendum of a gear pair (and consequently lower friction and contact temperature); the convex-concave contact zones in the vicinity of meshing start and end; and the higher reduced radii of curvature, particularly in the meshing start and end zones [31].