## 2 Steady-State Thermal Analysis of Internal Helical Gears Pair
### Calculation of Frictional Heat Flow in the Working Face  
The magnitude of the frictional heat flow \(q_{k}\) on the meshing surface is determined by the contact pressure \(F_{k}\), relative sliding speed \(u_{c}\), friction coefficient \(f\) and the ratio \(\gamma\) of frictional work converted into heat, on the meshing surface of the internal helical gear pair. Consequently, the frictional heat flux is calculated by the following formula:  
\[q_{k}=F_{k}u_{c}f\gamma \tag{23}\]  
The tooth contact pressure in an internal helical gear pair varies with the location of the contact point. To mitigate the influence of end face effects, the average contact stress is selected as a crucial parameter for calculating the frictional heat flow density. Based on Hertzian contact theory, the transmission process can be approximated as the contact between two cylinders; the radius of cylinders is regarded as the equivalent radius of curvature at the meshing point. \[p_{max}=\sqrt{\frac{F_{max}E}{2\pi\rho_{\sum k}(1-v^{2})}} \tag{24}\]  
According to the literature, the average contact stress of the internal helical gear pair is equal to \(\pi/4\) times the maximum contact stress [24]. Hence, the average contact stress of the internal helical gear pair is:  
\[F_{k}=\frac{\pi}{4}\sqrt{\frac{w_{k}E}{2\pi\rho_{\sum k}(1-v^{2})}} \tag{25}\]  
where \(E\) is the modulus of elasticity of the two cylindrical materials, \(v\) is the Poisson's ratio of the materials of the two cylindrical, and \(\rho_{\sum k}\) is the radius of curvature of any meshing point. As shown in Figure 2, Point \(O\) is the center of the internal helical gear and \(\theta\) is the angle of the velocity direction. The internal helical gear rotates counterclockwise to drive the external helical gear to rotate in the same direction. \(OQ\) is the pitch circle radius R\({}_{0}\) of the inner helical gear, \(V_{1}\) is the absolute velocity of the \(Q\) point on the pitch circle, \(V_{2}\) is the velocity of the outer helical gear at this point, and \(V_{3}\) is the absolute velocity of the \(P\) point on the inner helical gear in the vertical direction of the pitch circle; the angular velocity of the outer helical gear is \(W_{1}\) and the angular velocity of the inner helical gear is \(W_{2}\). When the inner helical gear acts, there is the following kinematic relationship:  
The absolute speed of the external helical gear at point \(Q\) is:  
\[V_{1}=L_{\text{OQ}}\overrightarrow{w_{1}}=R_{0}\overrightarrow{w_{1}} \tag{26}\]  
The absolute velocity of the internal helical gear at point \(P\) is:  
\[V_{3}=L_{op}\overrightarrow{w_{1}}=(R_{0}+l_{PQ})\overrightarrow{w_{1}} \tag{27}\]  
The speed of the internal helical gear at point \(Q\) is:  
\[V_{2}=V_{3}+l_{PQ}\overrightarrow{\omega_{2}}=(R_{0}+l_{PQ})\overrightarrow{ \omega_{1}}+l_{PQ}\overrightarrow{\omega_{2}} \tag{28}\]Combining Formulas (26) and (28), the relative sliding velocity of the internal helical gear relative to the external helical gear at point \(Q\) is obtained:  
\[\Delta V=V_{2}-V_{1}=l_{PQ}(\overset{\rightarrow}{\omega_{1}}+\overset{ \rightarrow}{\omega_{2}}) \tag{29}\]  
Then, the relative sliding velocity at any contact point \(M\) of the two gears can be approximately solved by Equation (29). At this time, the rotation radius at \(M\) on the external helical gear is:  
\[L_{OM}=L_{QQ}\cos\theta \tag{30}\]  
The coefficient of friction is determined experimentally, influenced by the material characteristics and lubrication conditions. The friction coefficient of the internal helical gears pair is expressed as follows:  
\[f=0.12\Bigg{(}10^{-3}\frac{wR_{a}}{\eta_{f}\Delta V\rho_{\Sigma}}\Bigg{)}^{0.25} \tag{31}\]  
In the equation above, \(w\) denotes the average line load per unit length along the tooth contact, \(R_{a}\) represents the roughness of the two gear tooth surfaces, \(\eta_{f}\) is the lubricant viscosity at the body temperature, and \(\rho_{\Sigma}\) is the synthesis radius of curvature at the meshing points.