## 1 Introduction  
The internal helical planetary gear train is usually used in high-speed rail, wind turbine, ship, high-speed machine tools, etc., due to its advantages of stable operation, high bearing capacity and wear resistance. However, in the transmission process, the relative sliding between the tooth surfaces will inevitably produce friction heat. Because the lubricating oil cannot completely take away the friction heat, the temperature of the gear will gradually increase and a corresponding thermal deformation will occur. Especially under poor lubrication or extreme conditions, the friction heat effect is more prominent, which may cause failure phenomena such as pitting and scuffing on the tooth surface, thus affecting the meshing performance, transmission efficiency, and service life of the gear. Finally, the transmission accuracy of the reducer will be affected [1]. Tooth surface modification is one of the important means to improve the meshing performance of gears. Through micro-geometric correction, the edge contact stress and friction heat concentration can be effectively reduced, so as to optimize the temperature field distribution and suppress the thermal deformation. However, at present, research on the steady-state heat effect of the modification of the internal helical gear pair under thermo-elastic coupling is still limited. Therefore, it is of great academic value and engineering significance to improve the performance and reliability of the reducer by studying the temperature rise effect of the modification on the internal helical planetary transmission system during the working process and its influence on the transmission accuracy. A large amount of research on the temperature field and thermal deformation of gears has been carried out by scholars at home and abroad. The effects of thermoelastic deformation, contact deformation and oil film thickness of the meshing point on backlash under thermoelastic coupling conditions were considered by [PERSON] et al. [2]. The temperature distribution within the gear is examined in conjunction with the calculated thermal input, the penetration depth of the lubricant, and the thermal conductivity coefficient. The contact stress of gear teeth in the process of gear transmission is analyzed by [PERSON] et al. [3] using the finite element method, and the influence of gear modification on the gear transmission system is studied. The thermoelastic coupling finite element model of the gear pair before and after reshaping is established by [PERSON] et al. [4]. Through the comparison of stress analysis, it is observed that the maximum equivalent stress and contact stress of the gear pair after reshaping are significantly reduced, which improves the bearing condition of the gear meshing process. The thermal-elastic coupling analysis was conducted by [PERSON] et al. [5], by considering the elastic deformation of the gears, the forces between the contacting tooth pairs, and the heat generation during contact. The finite element method was used by [PERSON] et al. [6] to study the steady-state thermal characteristics of the double involute gear transmission system. By constructing a multi-parameter stepped simulation model, the influence of different tooth profile characteristic parameters on the peak temperature distribution in the meshing area was systematically analyzed. The simplified boundary conditions handling method of a straight-toothed cylindrical gear model was proposed by [PERSON] et al. [7], based on the boundary element method, to analyze the bulk temperature of the gear. The steady-state temperature distribution of the gear was obtained by [PERSON] and others [8], who divide the engaging tooth surface into several strip areas along the width direction to apply frictional heat flux at different engaging positions, and they analyze the impact of the number of strip areas on the temperature field. Thermal coupling analysis was conducted by [PERSON] et al. [9] to solve the thermal deformation of gears resulting from transmission temperatures.