## 3 Contact Analysis of Topologically Modified Internal Helical Gear Pair
### Temperature Field and Thermal Deformation Simulation Analysis  
From Section 3, it can be seen that Case 3 is the best group of modification effects. Case 3 is compared with the unmodified gears. The ambient temperature is set to 20 \({}^{\circ}\)C, the rotational speed of the driving wheel is 200 rpm, and the load- torque is 2 N\(\cdot\)m. Under these conditions, the calculated friction heat flux and convective heat transfer coefficient are applied to the corresponding surface of the two meshing teeth surface to obtain the temperature field change diagram. Based on the steady temperature field, the thermal deformation calculation of a single tooth is carried out. The steady-state temperature field data are exerted to the model of the static analysis structure, and the deformation of the shaft hole area and the end face are restricted. The steady-state temperature field of the gear is shown in Figure 9.  
From Figure 9a, it can be seen that the peak temperature of the tooth profile of the standard gear reaches 50.892 \({}^{\circ}\)C, while the minimum temperature is 33.408 \({}^{\circ}\)C; in contrast, the maximum temperature of the tooth profile of the topologically optimized gear in Figure 9b is 47.647 \({}^{\circ}\)C, and the minimum temperature is 32.462 \({}^{\circ}\)C. It can be seen that the maximum temperature decreased by 6.376% and the minimum temperature decreased by 2.832% The highest frictional heat flow density appears at the addendum region. The low overall convective heat transfer coefficient leads to less efficient heat dissipation and results in relatively high temperatures near the tooth top within the engagement region. Figure 8: Single-tooth discretized MultiZone mesh map. Figure 9: The steady-state temperature distribution cloud diagram of the modified gear and the unmodified gear. The highest temperature is concentrated at the center of the tooth width, with a significant decrease in temperature towards the shaft bore and tooth ends. The temperature distribution at the center r varies more significantly than that of the addendum. The temperature of the tooth surface with a three-dimensional topology modification is 3 \({}^{\circ}\)C lower than that of the tooth surface without a modification. Because the contact area of the tooth surface of the topology optimization is mainly concentrated in the middle of the tooth surface, the heat dissipation condition in the middle of the tooth width is poor, so the high temperature area of the meshing surface moves down obviously. According to the calculation formula of the friction heat flow, the friction heat flow generated by the gear pair during transmission is affected by the relative sliding speed, average contact stress, energy conversion coefficient, and friction coefficient. The formulas of average contact stress and relative sliding velocity demonstrate that, under the same working conditions, the two parameters of the gear with three-dimensional topology modification are lower than those of the gear without three-dimensional topology modification, while the friction coefficient and energy conversion coefficient are basically constant. Therefore, the frictional heat flow of the gear with three-dimensional topology modification is lower than that of the gear without modification, resulting in a lower temperature of the meshing surface of the former. The convective heat transfer coefficient of the internal helical gear pair is affected by the characteristics of the lubricating oil and the working conditions, so there is little difference in the convective heat transfer coefficient between the two kinds of gear. Therefore, the temperature distribution trends of the two tooth profiles are similar, and are gradually reduced from the meshing surface to both ends. As shown in Figure 10, the temperature distribution along the tooth width exhibits an upward convex pattern, initially increasing and subsequently decreasing.