## 2 Automatic Transmission Modeling
### Bearing Installations and Other Modelings  
An automatic transmission generally includes three types of bearings for supporting rotating parts, which are thrust bearings for axial loads, radial bearings (journal and needle roller bearings) for radial loads and inclined contact bearings (ball and tapered roller bearing) for both thrust and radial direction loads. For all of these type bearings, the respective bearing models are used in building the transmission model in this study. As shown in Figure 4, a total of 20 bearings are installed while the model is in construction, and it consists of 11 thrust bearings (highlighted in yellow), 8 radial bearings (tinted in green), and 1 of the inclined contact bearing (colored in blue). However, it should be noted that concept bearings are applied rather than practical bearings from bearing catalog, since the purpose of this modeling is to analyze thrust loads acting to bearings. Therefore, the concept bearings corresponding to thrust bearings and radial bearings are modeled as including the bearing stiffness in an axial and radial direction, respectively. For replacing the inclined contact bearing, the bearing model is set up to contain both axial and radial bearing models which are independent to each other. In this transmission model, 10\({}^{9}\) N/m is defined for a value of the bearing stiffness in the axial and radial directions, and 10\({}^{6}\) Nm/rad is set for the moment stiffness of the bearings. For installing the bearings in the model, pitch diameters of the bearings are defined as shown in Table 4. For setting planet bearings which are not labelled in Figure 4, the concept bearings supporting radial loads are applied to pin shafts of planetary gear sets as well. However, since the scope of this study is the influences of thrust loads on bearing power losses, the planet bearings are not discussed precisely. As a central member of one planetary gear set is not closely connected to a member of another gear set, shafts, hubs, drums and other miscellaneous parts are used to link them mechanically for simultaneous rotations. In completing the transmission model, these parts are modeled with referring a cross section presented in [PERSON] et al. (2021).  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Bearing Number** & **1–8** & **9–11** & **12** & **13** & **14** & **15** & **16** & **17** & **18** & **19** & **20** \\ \hline \(D_{p}\) (mm) & 60.0 & 38.5 & 40.0 & 32.0 & 34.0 & 44.5 & 34.0 & 56.5 & 25.5 & 18.5 & 58.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Pitch diameters of bearing 1 to 20.  
Figure 4: Bearing installations in the transmission model.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Item** & **S1 \& S2** & **P1 \& P2** & **R1 \& R2** & **S3** & **P3** & **R3** & **S4** & **P4** & **R4** \\ \hline \(Z\) & 48 & 24 & 96 & 60 & 18 & 96 & 28 & 38 & 104 \\ \hline \(m\) (mm) & & 1.22 & & & 1.26 & & & 1.26 & \\ \(PA\) (deg.) & & 18 & & & 20 & & 18 & & \\ \(HA\) (deg.) & & 22 & & & 18 & & & 22 & \\ \(CD\) (mm) & & 48.2 & & & 52.3 & & & 45.4 & \\ \(FW\) (mm) & 21.5 & 21.5 & 21.5 & 17 & 17 & 17 & 17 & 17 & 17 & 17 \\ \(D_{e}\) (mm) & 66.7 & 34.6 & 126.1 & 83.0 & 26.7 & 126.2 & 41.0 & 54.8 & 140.6 \\ \(D_{r}\) (mm) & 61.3 & 29.2 & 131.5 & 77.4 & 21.1 & 131.8 & 35.4 & 49.3 & 146.1 \\ \(TT\) (mm) & 2.07 & 2.07 & 2.07 & 1.87 & 2.28 & 1.87 & 2.13 & 2.13 & 2.13 \\ \(TTD\) (mm) & 64.3 & 32.1 & 128.5 & 80.5 & 24.1 & 128.7 & 38.5 & 52.3 & 143.1 \\ \hline \(TE\) (\(\mu\)m) & & 0.38 & - & & 0.74 & - & & 1.67 & - \\ \cline{2-9}  & - & & 0.30 & - & & 0.66 & - & & 0.77 & \\ \hline \multirow{3}{*}{\(\sigma_{c}\) (MPa)} & \multicolumn{3}{c}{3936} & - & \multicolumn{3}{c}{2455} & - & \multicolumn{3}{c}{5425} & - \\ \cline{2-9}  & - & & 2737 & - & & 3255 & - & & 2718 & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Designs of planetary gear sets consisting of the example transmission model.