## 2 Steady-State Thermal Analysis of Internal Helical Gears Pair
### Finite Element Analysis of Gear Body Temperature Field  
The finite element method is applied to express the continuous temperature field with a finite number of discrete nodal temperatures, and is equivalent to solving the extremum problem of the corresponding generalized function. This can be approximated using the variational principle of generalized functions to obtain an approximate solution, as outlined in the literature [23]:  
The function space mapping for the steady-state temperature field under the second type of boundary conditions is expressed as follows:  
\[J[T(x,y)]=\frac{\lambda}{2}\iint\limits_{\Omega}\bigg{[}\bigg{(}\frac{\partial T }{\partial x}\bigg{)}+\bigg{(}\frac{\partial T}{\partial y}\bigg{)}\bigg{]}dxdy+ \int\limits_{\Gamma_{1}}qTds \tag{12}\]  
where the thermal conductivity \(\lambda\) and the heat flow density \(q\) at the boundary are known quantities. The function space mapping for the steady-state temperature field under the third type of boundary conditions is expressed as follows:  
\[J[T(x,y)]=\frac{\lambda}{2}\iint\limits_{\Omega}\bigg{[}\bigg{(}\frac{\partial T }{\partial x}\bigg{)}+\bigg{(}\frac{\partial T}{\partial y}\bigg{)}\bigg{]}dxdy+ \int\limits_{\Gamma_{2}}\bigg{(}\frac{1}{2}hT^{2}-hT_{0}T\bigg{)}ds \tag{13}\]  
The temperature of the medium \(T_{f}\), the heat transfer coefficient \(h\), and the thermal conductivity of the solid \(\lambda\) are known quantities. The function space mapping for the temperature field of the internal helical gear pair is expressed as follows  
\[J[T(x,y)]=\frac{\lambda}{2}\iint\limits_{\Omega}\bigg{[}\bigg{(}\frac{ \partial T}{\partial x}\bigg{)}+\bigg{(}\frac{\partial T}{\partial y}\bigg{)} \bigg{]}dxdy+\int\limits_{\Gamma_{2}}\bigg{(}\frac{1}{2}hT^{2}-(hT_{0}-q)T \bigg{)}ds \tag{14}\]Select any cell and the temperature at a point on the boundary line can be expressed as a linear interpolation of the temperatures at the two end nodes:  
\[T=\left(1-\frac{s}{s_{k}}\right)T_{i}+\frac{s}{s_{k}}T_{j} \tag{15}\]  
The temperature interpolation function is expressed as follows:  
\[T(x,y)=N_{i}T_{i}+N_{j}T_{j}+N_{k}T_{k} \tag{16}\]  
In the formula above, \(N_{i}=\frac{1}{\Delta t}(a+b_{i}x+c_{i}y),i,j,k\). The interpolating function of the temperature field indicates that solving the temperature field involves determining the temperature values at each node, a process involving the application of multivariate function operators. Thus, the solution to the temperature field can be viewed as an extremum problem for a multivariate function. Assuming the computational region consists of \(n\) nodes, the expression for \(J[T(x,y)]\) is converted into the form of \(J[T_{1},T_{1}\cdots T_{n}]\). The extremum condition is:  
\[\frac{\partial J}{\partial T_{m}}=\sum\frac{\partial J^{e}}{\partial T_{m}}=0 \quad(m=1,2,\ldots n) \tag{17}\]  
By substituting the temperature interpolation function into the spatial mapping of the temperature field function for derivation, the resulting extremum conditions are:  
\[\frac{\partial J^{e}}{\partial T_{i}}=\iint\limits_{e}\lambda\left[\frac{ \partial T}{\partial x}\frac{\partial}{\partial T_{i}}\left(\frac{\partial T }{\partial x}\right)+\frac{\partial T}{\partial y}\frac{\partial}{\partial T _{i}}\left(\frac{\partial T}{\partial y}\right)\right]dxdy+\int\limits_{jk}h(T -T)\frac{\partial T}{\partial T_{i}}=0 \tag{18}\]  
In Equation (18), the first term of the equation represents the temperature of the internal cell, which is characterized by a rigid array of temperatures. \[\frac{\partial J^{e}}{\partial T_{i}}=\frac{\lambda}{4A}\left[\left({b_{i}}^{ 2}+{c_{i}}^{2}\right)T_{i}+\left(b_{i}b_{j}+c_{i}c_{j}\right)T_{j}+\left(b_{i }b_{k}+c_{i}c_{k}\right)T_{k}\right] \tag{19}\]  
Its matrix form is notated as:  
\[ \begin{bmatrix}\frac{\partial J^{e}}{\partial T_{i}}\\ \frac{\partial J^{e}}{\partial T_{i}}\\ \frac{\partial J^{e}}{\partial T_{k}}\end{bmatrix} =\frac{\lambda}{4A}