# Dynamic Induction Heating Temperature Field Analysis of Spiral Bevel Gears
## 2 Modeling of Induction Heating Coils for Spiral Bevel Gears  
### Induction Coil Modeling  
Our induction heating mathematical model establishes an induction coil geometry model for the spiral bevel gear spatial shape characteristics and studies the \"electromagnetic-thermal-motion\" multi-physical field coupling mechanism of the induction coil and gear [17; 18]. In this paper, the pinion is selected as the research object, because the pinion is more prone to tooth surface wear and fatigue failure in the transmission process. Therefore, we consider the temperature field distribution uniformity of the pinion after induction heating as the research objective to prepare for subsequent shot peening of the tooth surface. The structure of the spiral bevel gear is shown in Figure 1, and relevant parameter information is listed in Table 1.  
The structural design of the heating coil of the spiral bevel gear induction is based on the spiral bevel gear geometry and the constraints of the shot peening equipment. The distance between the induction heating coil and the outer conical surface of the gear, that is, the air gap h, should be considered.  
To avoid damage to the induction coil during shot peening, the induction coil must be reserved for an opening in the design process. The coil opening angle is represented by \(\theta_{0}\). Part of the coil near the shot-peening area is bent along the direction of the conical surface generatrix, and its bending height is represented by \(H_{0}\). \(d_{1}\) represents the maximum coverage diameter of the projectile on the tooth surface and the \(L_{1}\) represents the tooth width. The assembly of the spiral bevel gear and induction heating coil is shown in Figure 2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Number of teeth & 8 \\ Modulus/m & 6.8610 \\ Helix angle \(\beta/^{\circ}\) & 40\({}^{\circ}\) \\ Pressure angle \(\gamma/^{\circ}\) & 22\({}^{\circ}\) \\ Tooth width \(L_{1}\)/mm & 44 \\ Tooth top height factor & 0.8500 \\ Top gap coefficient & 0.1880 \\ Hand of helix & Right \\ \hline \hline \end{tabular}
\end{table}
Table 1: Spiral bevel gear parameters.  
Figure 1: Spiral bevel gear structure.  
Figure 2: Assembly relationship between the spiral bevel gear and induction heating coil.  
Based on the relationship between the nozzle and spiral bevel gear assembly in the shot peening process, if the induction coil opening angle is too small, the projectile interferes with the induction coil and damages it. In contrast, if the coil's opening angle is too large, the coverage of the coil on the gear is reduced, and the heating efficiency of the coil decreases. The angle \(\theta_{0}\) of the induction heating coil opening is calculated as follows:  
\[\theta_{0}\;\text{min}=\arcsin\biggl{(}\frac{d_{1}/2}{R_{a}}\biggr{)} \tag{1}\]  
where \(d_{1}\) is the diameter of the shot-peening coverage area and \(R_{a}\) is the tooth top circle radius of the small end face of the spiral bevel gear.  
Combining the calculated basic parameters of the coils, a transverse coil (TC) and longitudinal coil (LC) structures were designed for induction heating shot peening of spiral bevel gears, as shown in Figure 3. The induction heating coil material is metallic copper with a square cross section and a hollow coil interior. A water stream is connected inside the coil to cool the heat generated by the coil conductor during induction heating. The basic parameters of the two coils are shown in Table 2.