## 2 Semi-Analytical Calculation Model for Gear Surface Flash Temperature
### Calculation Model for Tooth Surface Load and Flash Temperature
The tangential velocity components \(v_{g1}\) and \(v_{g2}\) of the tooth surfaces of involute cylindrical gears are related to the local geometric parameters and rotational speed. Therefore, the aforementioned equation can be replaced equivalently with the following equation:  
\[\Theta_{fl}=2.52\cdot\mu\cdot\frac{X_{M}}{50}\cdot X_{J}\cdot\sqrt{\left(X_{ \Gamma}\cdot w_{n}\right)^{3}}\cdot\sqrt{n_{1}/60}\cdot\frac{abs\left(\sqrt{ \rho_{y1}}-\sqrt{\rho_{y2}/u}\right)}{\sqrt[4]{p_{r}}_{rly}} \tag{7}\]  
where \(X_{J}\) represents the engage-in coefficient, \(X_{M}\) denotes the thermoelastic coefficient, and \(X_{\Gamma}\) denotes the load-sharing factor. The specific calculation methods for these three coefficients are detailed in Ref. [7]. The local comprehensive curvature radius is as follows:  
\[\rho_{y}=\frac{\rho_{y1}\cdot\rho_{y2}}{\rho_{y1}+\rho_{y2}} \tag{8}\]  
In this paper, the friction coefficient used for calculating the TSFT does not adopt the calculation equation recommended by Ref. [7], but instead uses a more accurate empirical equation derived from the EHL theory through extensive experimental regression analysis [13] as follows:  
\[\left\{ \begin{array}{l}\mu=e^{f(\xi,P_{h},v_{0},R_{q})}Ph_{c_{2}}|\xi|^{c_{ 3}}V_{e}^{c_{6}}v_{0}^{c_{8}}p_{y}^{c_{8}}\\ f(\xi,P_{h},v_{0},R_{q})=c_{1}+c_{4}|\xi|P_{h}\log_{10}(v_{0})+c_{5}e^{-|\xi|P _{h}\log_{10}(v_{0})}+c_{9}e^{R_{q}}\end{array} \right. \tag{9}\]  
where \(\xi\) denotes the slip-roll ratio; \(P_{h}\) represents the maximum Hertzian stress in GPa; \(v_{0}\) represents the dynamic viscosity of the lubricating oil in cP; \(R_{q}\) denotes the comprehensive roughness on the tooth surface, in \(\mu\)m; \(V_{e}\) denotes the entrainment velocity, in m/s. Coefficients \(b_{i}\) are the regression coefficients, the values of which are shown in the following Table 1.  
\[\xi=V_{s}/V_{e} \tag{10}\]  
\[R_{q}=\sqrt{\frac{R_{q1}^{2}+R_{q2}^{2}}{2}} \tag{11}\]  
where \(R_{q1}\) and \(R_{q2}\) represent the RMS roughness of the tooth surfaces for pinion and gear, respectively, and \(w_{n}\) represents the normal unit load. The normal unit load, as defined in ISO 6336-20:2022, is the ratio of the normal force at the current meshing position to the tooth width. This value represents a mean value and is therefore unable to reflect the distribution of load along the instantaneous contact line. As a consequence of meshing misalignment and tooth surface modification, the load distribution along the instantaneous contact line is necessarily uneven. In order to calculate the actual tooth surface load distribution, this paper employs Loaded Tooth Contact Analysis (LTCA) based on the FEM to study the distribution of \(w_{n}\), as illustrated in Figure 4.  
Figure 4: The LTCA model-based FEM for gear pair meshing. If a meshing cycle is divided into \(N\) instantaneous positions and the tooth width direction is divided into \(M\) grid nodes, then after achieving load balance at the \(i\)-\(th\) instantaneous meshing position, the normal load vector on the instantaneous contact line can be obtained according to Equation (12). \[w_{i}^{n}=(w_{11}^{n},w_{12}^{n},w_{13}^{n},\cdots\cdots,w_{IM}^{n}) \tag{12}\]  
After solving for the \(N\) meshing positions, they are assembled into the form of a load matrix as shown in Equation (13). \[w^{n}= \begin{bmatrix}w_{11}^{n}&w_{12}^{n}&w_{13}^{n}&\cdots&w_{1M}^{n}\\ w_{21}^{n}&w_{22}^{n}&w_{23}^{n}&\cdots&w_{24}^{n}\\ w_{31}^{n}&w_{32}^{n}&w_{33}^{n}&\cdots&w_{3M}^{n}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ w_{N1}^{n}&w_{N2}^{n}&w_{N3}^{n}&\cdots&w_{NM}^{n}\end{bmatrix}  \tag{13}\]  
Then, the transverse unit load \(w_{t}\) can be calculated according to Equation (14). \[w_{t}=w_{n}\cdot\cos\alpha_{\rm{wn}}\cdot\cos\beta_{\rm{w}} \tag{14}\]  
In summary, the specific calculation process for the flash temperature distribution on the tooth surface involves the following steps: Firstly, the gear LTCA, which is based on the FEM model, is employed for the calculation of the load and stress distribution on the micro-modified tooth surface.