## 2 Automatic Transmission Modeling
### Planetary Gear Set Designs  
In designing gear geometry profiles of a planetary gear set, spatial requirements of minimum (or maximum) diameters of gears, face widths, and so on, and performance analysis of contact stress, transmission error, and others must be considered simultaneously, since a restricted space is allowed in the transmission and qualified performance must be guaranteed for gear durability and vibration. The planetary gear set consists of the sun, ring and multiple planet gears, and these sun-planet and ring-planet gears rotate simultaneously with forming the external and internal gear meshes. Therefore, identifying individual gear profiles must examine geometrical violations which would occur between gears in mesh and adjacent planet gears. This design complexity makes it more difficult designing a planetary gear set.  
For the design of planetary gear sets in the automatic transmission, a methodology of a design search of a planetary gear set proposed by [PERSON] et al. (2018) is utilized in this study. The methodology is comprised of three separate stages of (1) generating a group of gear designs, (2) analyzing performances of designs, (3) selecting the best designs by considering confined ranges of performance indices. The first stage is to generate a large group of design candidates which have acceptable gear geometries within defined design parameters of ring-to-sun gear ratio, pressure angle, backlash and others and do not violate any geometrical interferences such as undercut, root/tip interference and so on. The next is to analyze loaded tooth transmission errors, contact and root stresses of each design candidate of a generated design group by a loaded tooth contact analysis model developed by [PERSON] and [PERSON] (2018). Lastly, the best designs are searched by eliminating the designs in the complete design group, which do not meet any of multiple bounded ranges of selected performances.  
Designs of 4 decks of planetary gear sets are made by using the design search methodology, and the gear profiles and analysis results are shown in Table 3. In the table, as the gear geometry parameters, \(Z\), \(m\), \(PA\), \(HA\), \(CD\) and \(FW\) denote the number of teeth of a gear, module, pressure and helix angles, center distance and face width, respectively. \(D_{o}\) and \(D_{r}\) mean outer and root diameters of a gear. \(TT\) represents the transverse tooth thickness at the diameter called as tooth thickness diameter \(TTD\). As the performance parameters, \(TE\) and \(\sigma_{c}\) imply the motion transmission error and contact stress. Here, the motion transmission error accepting as a gear vibration excitation source is a difference of the maximum and minimum translation amplitudes occurring in loaded gear rotations. For the loaded tooth contact analysis, the torques applied to gears are calculated by using the results of the power flow analysis in Table 2b, and center distances which size the gear sets are defined by referring to previous research ([PERSON] and [PERSON], 2018). Here, it should be noted that the helix angle directions of the gear set designs are not yet determined in left or right-hand direction.