# Comparison of Selected Parameters of a Planetary
## 3 Reduced Planetary Gearbox Parameters  
For the purpose of the comparison of selected parameters of planetary gears with involute and convex-concave gear profiles, a specific type of the gearbox was selected, namely a rear mirror positioner gearbox designed as a reduced Wolfrom stage (Figure 3). This type of the reduced planetary stage offers a high-speed reduction in a small built-in space and a satisfactory value for the torque ratio and gear contact efficiency.  
The kinematic scheme of the gearbox is presented in Figure 4.  
The gearbox had no carrier and had the same teeth number for gears 2 and 4. The axis distances of the planet gear 2 and ring gear 3, and for the planet gear 4 and the ring gear 5, had to be the same. Thus, the moduli ratio of the gear pairs 2-3 and 4-5 was defined by the following equation:  
\[\frac{m_{45}}{m_{23}}=\frac{z_{3}-z_{2}}{z_{5}-z_{4}}=\frac{51-18}{48-18}=1.1. \tag{4}\]  
Due to the defined value of the smaller module \(m_{23}=0.45\) mm, the module of the output part of the stage had to be \(m_{45}=0.5\) mm. The teeth number and the moduli of the gearbox gears are defined in Table 1.  
Figure 4: Kinematic scheme of the Wolfrom stage gearbox.  
Figure 3: Rear-view mirror drive gearbox, reproduced from [46].  
The overall gearbox ratio was defined as [47]:  
\[i=\frac{i_{035}-i_{015}}{i_{035}-1}=\frac{0.9412-(-3.2)}{0.9412-1}=-70.4. \tag{5}\]  
The partial stood gear stage II ratio \(i_{035}\) was defined as:  
\[i_{035}=+\frac{z_{5}z_{2}}{z_{4}z_{3}}=\frac{48\times 18}{18\times 51}=0.9412. \tag{6}\]  
The partial stood gear stage III ratio \(i_{015}\) was defined as:  
\[i_{015}=-\frac{z_{5}z_{2}}{z_{4}z_{1}}=-\frac{48\times 18}{18\times 15}=-3.2. \tag{7}\]  
Due to the proper calculation of the torques, the reduced planetary gearbox had to be divided into two planetary stages, which had an equivalent efficiency to the solved gearbox. For the researched case, the scheme for the torque calculation considering the gear contact efficiency between the internal and external gear pair is shown in Figure 5.  
The stood gearbox efficiency were defined by considering the efficiency of the internal gear contact \(\eta_{i}\) = 0.99 and the efficiency of the external gear contact \(\eta_{e}\) = 0.98 according to the equations:  
\[\eta_{035}=\eta_{i}\,\eta_{i}=0.99\times 0.99=0.9801, \tag{8}\]  
\[\eta_{015}=\eta_{e}\,\eta_{i}=0.98\times 0.99=0.9702. \tag{9}\]  
The torque ratio was defined as [47]:  
\[\mu=\frac{T_{B}}{T_{A}}=-\frac{i_{035}\,\eta_{05}-i_{05}\,\eta_{015}}{i_{03}\, \eta_{035}-1}=-\frac{0.9412\times 0.9801-(-3.2)\times 0.9702}{9.9412\times 0.9801 -1}=\ . \tag{10}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Gear** & **Teeth Number z (-)** & **Module \(m\) (mm)** \\ \hline Sun gear 1 & 15 & 0.45 \\ Planet gear 2 & 18 & 0.45 \\ Ring gear 3 & 51 & 0.45 \\ Planet gear 4 & 18 & 0.50 \\ Ring gear 5 & 48 & 0.50 \\ \hline \end{tabular}
\end{table}
Table 1: Teeth numbers and moduli of the Wolfrom stage gears.  
Figure 5: Equivalent efficiency scheme of the Wolfrom stage gearbox.  
The overall gearbox efficiency was calculated as:  
\[\eta=-\frac{\mu}{i}=-\frac{51.9269}{-70.4}=0.7376. \tag{11}\]  
The relative speeds and the torques of the gearbox central elements were calculated using the relative input speed \(\omega_{A}=1\) and the relative input torque \(T_{A}=1\). The results of the calculation presented in Table 2 were used for the contact pressures calculation.  
The results show the power division using the equivalent two-stage gearbox scheme: about 94% of the power flowed from the sun gear 1 across the carriers S\({}_{2}\) and S\({}_{1}\) to the ring gear 5', and just about 6% of the power flowed directly to the output ring gear 5'' and then together to the output B.