## 3 Contact Analysis of Topologically Modified Internal Helical Gear Pair
### Temperature Field and Thermal Deformation Simulation Analysis
This behavior is primarily attributed to the enhanced convective heat transfer at the end faces, which provide superior heat dissipation conditions, while the convective heat transfer coefficient in the middle region of the tooth width is lower, resulting in relatively poorer heat dissipation conditions. Figure 11 illustrates that, along the meshing line, the temperature is higher at the upper portion of the tooth. The temperature at the middle of the tooth width exceeds that at the ends, and the overall temperature variation diminishes progressively. From the addendum to the dedendum, the temperature decreases gradually, with the sequence of the distribution following: addendum, pitch circle, and then dedendum. At the pitch circle, the high temperature region shows a significant shrink  
Figure 10: The temperature distribution along the tooth width of the driven wheel after three-dimensional topology modification and the gear without modification. the relative sliding velocity at the pitch circle is zero theoretically and the heat dissipation condition is relatively poor, so that a higher temperature level can be maintained. When the gear with a three-dimensional topology modification is engaged in meshing transmission, because the meshing surface clearance makes the lubricating oil contact the tooth surface and take away the heat, the temperature of the meshing area is reduced, indicating that the modification improves the heat dissipation effect, optimizes the load distribution of the tooth surface, and makes it more uniform. The non-uniformity of the tooth profile temperature may cause impact and vibration during the transmission of the internal helical gear pair, which in turn affects its transmission characteristics. Therefore, the temperature factor needs to be considered in the transmission analysis. Through thermal-structural coupling simulation analysis, Figure 12 presents the thermal deformation results of the gears subjected to a steady-state temperature field. The analysis reveals that the deformation gradually decreases from the addendum to the dedendum, and the temperature of the addendum is higher due to the large friction heat flow. The maximum deformation of the gear without a three-dimensional topology modification and the gear with a three-dimensional topology correction is 0.43324 \(\upmu\)m and 0.27498 \(\upmu\)m, respectively. The topology optimization reduces the deformation by 36.5%. It can be seen that topology optimization can effectively reduce thermal deformation. Additionally, the fit of the gear bore to the shaft is subject to axial constraints, while the tooth top remains unconstrained. Along with the accumulation of radial thermal expansion effects, this results in the most significant deformation at the addendum. Figure 11: The temperature distribution of the modified gear and the unmodified gear along the direction of the meshing line. Figure 12: Thermal deformation analysis of driven gear after three-dimensional topology modification and without modification. The deformation data in the tooth width direction can be extracted by analyzing the results from each node on the meshing surface, as shown in Figure 13. On the meshing surface of the gear, the deformation at the addendum and pitch circle appears an upward bulge shape, while the dedendum shows more dramatic deformation in the middle region of the tooth width. Overall, the deformation at the addendum is generally more outstanding than that at the pitch circle and dedendum; especially, the central region appears as a particularly prominent deformation, while the sides appear as a relatively slight deformation. This phenomenon is primarily attributed to the higher temperature field in the central region. To the standard gear, the deformation at the middle of the tooth top is 0.433 \(\upmu\)m, while the maximum deformation at the middle of the pitch circle and tooth root are 0.292 \(\upmu\)m and 0.216 \(\upmu\)m, respectively.