# Dynamic Induction Heating Temperature Field Analysis of Spiral Bevel Gears
## 2 Modeling of Induction Heating Coils for Spiral Bevel Gears
### Mathematical Model  
The eddy currents induced by the alternating magnetic field generated by the induction coil provide the main heat for the dynamic induction heating of the spiral bevel gear. Therefore, to obtain the temperature model of the spiral bevel gear surface, it is necessary to first calculate the eddy currents induced on the gear surface based on the electromagnetic field generated by the induction coil, and take the heat generated by the eddy work as the internal heat source to obtain the temperature field model of the gear surface. The electromagnetic field characteristics of induction heating can be represented by [PERSON]'s equations, [PERSON]'s law, which describes the relationship between the current and the direction of magnetic induction lines, [PERSON]'s law of induction which describes the quantitative relationship between the amount of electricity passed on the electric level and the precipitated material, [PERSON]'s law of flux, which describes the magnetic field as a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Transverse Coil** & **Longitudinal Coil** \\ \hline Materials & Copper & Copper \\ Section size/mm & \(8\times 8\) & \(8\times 8\) \\ Opening angle/\({}^{\circ}\) & \(90^{\circ}\) & \(90^{\circ}\) \\ Wall thickness/mm & \(1.5\) & \(1.5\) \\ Bend height H\({}_{0}\)/mm & \(30\) & \(60\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Induction heating coil parameters. Figure 3: Two induction coil structures: (**a**) transverse coil, (**b**) longitudinal coil. passive field, and [PERSON]'s law of flux, which describes the relationship between the charge and electric flux. The differential expression is:  
\[\left\{ \begin{array}{l}\
abla_{H}\times\overset{\rightarrow}{H}=\overset{ \rightarrow}{J}+\frac{\partial\overset{\rightarrow}{D}}{\partial t}\\ \
abla_{H}\times\overset{\rightarrow}{E}=-\frac{\partial\overset{\rightarrow} {B}}{\partial t}\\ \
abla_{H}\overset{\rightarrow}{B}=0\\ \
abla_{H}\overset{\rightarrow}{D}=\rho\end{array} \right. \tag{2}\]  
where \(\
abla_{H}\) is the hamiltonian, \(\overset{\rightarrow}{H}\) is the magnetic field strength vector, \(\overset{\rightarrow}{J}\) is the current density vector, \(\overset{\rightarrow}{D}\) is the electric induction intensity vector, \(t\) is the induction heating tie, \(\overset{\rightarrow}{E}\) is the electric field strength vector, \(\overset{\rightarrow}{B}\) is the magnetic induction strength and \(\rho\) is the charge density. When applying [PERSON]'s system of equations, the electromagnetic field quantities and the gear steel must satisfy the following intrinsic relationship equation:  
\[\left\{ \begin{array}{l}\overset{\rightarrow}{D}=\varepsilon\overset{ \rightarrow}{E}\\ \overset{\rightarrow}{B}=\mu\overset{\rightarrow}{H}\\ \overset{\rightarrow}{J}=\sigma\overset{\rightarrow}{E}\end{array} \right. \tag{3}\]  
where \(\varepsilon\) is the dielectric constant of gear steel, \(\mu\) is the relative magnetic permeability of gear steel, and \(\sigma\) is the electrical conductivity of gear steel. In the dynamic induction heating process of a spiral level gear, the heat generated by the induction eddy current as the internal heat source heats the surface layer of the gears rapidly, while a part of the heat is transferred to the core of the spiral level bears and the gear shaft by heat conduction. The remaining part of the heat is dissipated into the surrounding environment by heat radiation and heat convection.