## 3 Analysis and Optimization of Friction Flash Temperature of Gears Tooth Surface  
### Semi-Analytical Method for Solving Tooth Surface Friction Flash Temperature  
Due to the existence of friction between tooth surfaces, flash temperature will be generated during gear meshing. When the flash temperature is higher than the allowable temperature, tooth surface scuffing will occur. Scaffing is different from fatigue in that it occurs instantly. There are usually two methods for calculating tooth surface flash temperature: the Blok flash temperature method and the integrated temperature method. The integrated temperature is the average of the flash temperatures and adds the influence factor to the load sharing factor, the resulting value is close to the maximum contact temperature, so the scuffing risk assessment is approximately the same as the flash temperature method. Situations in which there are local temperature peaks, such as gear pairs; low contact ratios; tooth tip and root edge contact; or contact at other geometrically sensitive locations, the integrated temperature method has low sensitivity and cannot reflect the tooth surface flash temperature distribution. In order to study the flash temperature distribution of the tooth surface when the gear pair undergoes unbalance load after elastic deformation, it is more appropriate to use the improved flash temperature method for analysis. The flash temperature at the instant meshing position of the tooth surface is described as follows [25]:  
\[\Theta_{fl}=1.11\cdot\frac{\mu_{m}\cdot X_{T}\cdot X_{I}\cdot w_{n}}{\sqrt{2 \cdot b_{H}}}\cdot\frac{abs\big{(}v_{g1}-v_{g2}\big{)}}{B_{M1}\cdot\sqrt{v_{g 1}+B_{M2}\cdot\sqrt{v_{g2}}}} \tag{8}\]  
For involute cylindrical gears, the tooth surface tangential velocity components \(v_{g1}\) and \(v_{g2}\) are related to the local radius of curvature of the tooth surface and the rotation speed, so the Equation (8) can also be equivalently replaced by Equation (9). \[\Theta_{fl}=2.52\cdot\mu_{m}\cdot\frac{X_{M}}{50}\cdot X_{I}\cdot\sqrt[4]{(X_{ I}\cdot w_{n})^{3}}\cdot\sqrt{n_{1}/60}\cdot\frac{abs\big{(}\sqrt{p_{y1}}-\sqrt{p_{g2} /u}\big{)}}{\sqrt[4]{p_{rely}}} \tag{9}\]  
The friction coefficient for calculating the tooth surface flash temperature, which is obtained through a large number of experimental fittings based on the EHL theory, is defined as follows in Equation (10). \[\mu_{m}=0.06\cdot\bigg{(}\frac{w_{t}}{v_{s}\cdot p_{y}}\bigg{)}^{0.2}\cdot X_{ L}\cdot X_{R} \tag{10}\]  
In the above equations, \(w_{n}\) and \(w_{t}\) represent the normal and transverse unit loads, respectively. Since the transverse unit load calculated using ISO 6336-20:2022 is the ratio  
Figure 5: The misalignments of all gear pairs. of the tangential force to the tooth width, it is an average value and cannot show the load distribution on instantaneous contact line. Due to the existence of meshing misalignment and tooth surface modification, the load on the contact line must be unevenly distributed. In order to investigate the real load distribution on the tooth surface, this paper uses gear loaded tooth contact analysis (LTCA) based on the finite element method to calculate the \(w_{n}\) distribution. If a meshing cycle (an angular pitch, \(2\pi/\mathrm{z}\)) is equally divided into \(N\) instantaneous positions, and \(M\) grid nodes along the tooth width direction, then after finite element calculation converges, the normal load on the current contact line at the \(i\)th instant meshing position can be obtained as in Equation (11), where the superscript \"\(n\)\" represents the normal direction. \[w_{i}^{n}=(w_{i1}^{n},w_{i2}^{n},w_{i3}^{n},\dots\dots,w_{iM}^{n}) \tag{11}\]  
After solving all \(N\) meshing positions, the normal load matrix can be assembled in the form of Equation (12). \[w^{n}=