## 2 Semi-Analytical Calculation Model for Gear Surface Flash Temperature  
### Local Geometry and Velocity Parameters of Gear Tooth Surface  
During the gearset operation, there is both rolling and sliding between the meshing tooth surfaces, but pure rolling only occurs at pitch point \(C\), as shown in Figure 1. When the pinion rotates clockwise with an angular velocity \(\omega_{1}\) to drive the gear, the line segment \(\text{T}_{1}\text{T}_{2}\) is their line of action. When the current contact point is at an arbitrary point \(Y\), a pair of conjugate tooth surfaces of the pinion and the gear are tangent at point \(Y\). At this time, the circumferential velocity of the pinion at this point is \(v_{11}\), and the circumferential velocity of the gear is \(v_{12}\). They have the same normal velocity \(v_{n}\) at point \(Y\), but the tangential velocities \(v_{g1}\) and \(v_{g2}\) of the pinion and the gear at point \(Y\) are not equal. Their difference is the relative sliding velocity \(v_{s}\) at point \(Y\).  
Based on the properties of involute and gear meshing relationships, the sliding velocity at any meshing point on the tooth surface is calculated using the following equation, with the unit \(\text{m}/\text{s}\).  
\[v_{s}(i)=\text{abs}(v_{g1}(i)-v_{g2}(i))=\text{abs}\big{(}\omega_{1}\cdot \rho_{y1}-\omega_{2}\cdot\rho_{y2}\big{)}/1000 \tag{1}\]  
Figure 1: Relative sliding speed of meshing point \(Y\) on the involute gearset tooth surface.  
The local radii of curvature at the contact point for the pinion and the gear are given by the following equations, respectively:  
\[\rho_{y1}=\frac{\sqrt{d_{y1}^{2}-d_{b1}^{2}}}{2} \tag{2}\]  
\[\rho_{y2}=\frac{\sqrt{d_{y2}^{2}-d_{b2}^{2}}}{2} \tag{3}\]  
According to the definition of the tooth surface-specific sliding [24], the specific sliding for the pinion and the gear at any contact point are shown below:  
\[\zeta_{1}(i)=\frac{v_{s}(i)}{v_{g1}(i)}=\text{abs}(1-\frac{\rho_{y2}}{u\rho_{ y1}}) \tag{4}\]  
\[\zeta_{2}(i)=\frac{v_{s}(i)}{v_{g2}(i)}=\text{abs}(1-\frac{u\rho_{y1}}{\rho_{ y2}}) \tag{5}\]  
where \(u=\left\lfloor z_{2}/z_{1}\right\rfloor\) represents the gear ratio.  
It can be seen that gears with different geometric parameters have different local radii of curvature on their tooth surfaces, which results in different tangential velocities and relative sliding velocities during rotation, thus leading to different tooth surface-specific sliding.  
As shown in Figure 2, during the operation of a pair of meshing gears, the tooth of the pinion enters meshing from the root to the tip and then exits meshing, with the tooth surface sliding velocity \(v_{s}\) changing from large to small and then back to large. At the pitch point, the tangential velocity components of the tooth surfaces are equal, indicating pure rolling, while at other positions there is relative sliding velocity, which increases further away from the pitch point, therefore the maximum value of \(v_{s}\) occurring near the tooth tip and root. The same pattern applies to a single instantaneous contact line on the tooth surface, where the sliding velocity is zero along the pitch line and then gradually increases towards the tooth tip and root as it moves along the height of the tooth.