# Analytical Model of Gear Thermal Stiffness Based on the Potential Energy Method
## 4 Case Study and Discussion  
To better verify the accuracy of the model, the same gear parameters as those in previous studies [21; 24] are adopted in the calculation example of this paper, as shown in Table 2.  
Meanwhile, to better represent the distribution of each parameter along the line of action, the position coordinates of the arbitrary contact point K are denoted by the normalized coordinate \(\Gamma_{K}\). As shown in Figure 4, the line of action is the coordinate axis, and the pitch point C is the coordinate origin when establishing the coordinate axis with the direction from point C to point N1 as the negative direction. The normalized coordinates of point K can be expressed as \(\Gamma_{\text{K}}=\frac{\overline{\text{N}_{1}\text{K}}-\overline{\text{N}_{ 1}\text{C}}}{\overline{\text{N}_{1}\text{C}}}=\frac{\tan a_{\text{K}}}{\tan a _{\text{C}}}-1\). In the formula, \(\alpha_{\text{C}}\) is the pressure angle of the pitch point, and \(\alpha_{\text{K}}\) is the pressure angle of the contact point K on the pinion.  
The temperature of contact point K to K\({}^{\prime}\) is assumed to be linearly distributed. According to the model proposed in this paper, the curve distribution of the gear thermal stiffness along the line of action without considering the influence of thermal stress is calculated by using MATLAB software programming, and the result is shown in Figure 5.  
According to Figure 5, the gear mesh stiffness proposed in this paper is basically consistent with \"K-Dai\" that obtained by using the method of Reference [25]. Without considering the influence of thermal stress, the thermal stiffness decreases as a whole compared with the elastic stiffness. The decrease is small at the meshing-in and -out points, while the decrease is largest near the pitch point. This is consistent with the conclusions in Reference [21]. This effectively proves the correctness of the proposed model. Compared with the method for calculating the gear thermal stiffness by introducing the \"thermal stress correction coefficient\" in Reference [21], the method proposed in this paper is more general and is a fully analytical method. The more details can be found in the Appendix A.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of** & **Module \(m\) (mm)** & **Pressure Angle** & **The Face Width** & **Power \(P\) (kW)** & **Speed \(n_{1}\)** \\
**Teeth \(z_{1}\)/\(z_{2}\)** & **Module \(m\) (mm)** & **\(\alpha\) (\({}^{\circ}\))** & **\(b\) (mm)** & **Power \(P\) (kW)** & **(r-min\({}^{-1}\))** \\ \hline
27/35 & 3 & 20 & 25 & 80 & 2000 \\ \hline Addendum & Clearance coefficient \(h_{a}^{*}\) & Linear expansion coefficient \(c^{*}\) & \begin{tabular}{c} General initial \\ \(\alpha_{T}\) (\({}^{\circ}\)C\({}^{-1}\)) \\ \end{tabular} & \begin{tabular}{c} Steady-state \\ temperature rise of \\ temperature (\({}^{\circ}\)C) \\ \end{tabular} & \
\begin{tabular}{c} Steady-state \\ temperature rise of \\ mesh tooth \\ surface (\({}^{\circ}\)C) \\ \end{tabular} &
\begin{tabular}{c} Steady-state \\ temperature rise of \\ nonmesh tooth \\ surface (\({}^{\circ}\)C) \\ \end{tabular} \\ \hline
1 & 0.2 & \(1.13\times 10^{-5}\) & 20 & 60, 75, 90 & 50, 60, 70 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the spur gear pair.  
Figure 4: Normalized coordinates of the line of action. A—the meshing-in point; C—the pitch point; E—the meshing-out point; K—arbitrary contact point on the line of action; N1—ideal meshing-in point; N2—ideal meshing-out point; r1—pitch radius of the pinion; r2—pitch radius of the gear; r1—radius of addendum of the pinion; ra2—radius of addendum of the gear; rb1—base radius of the pinion; rb2—base radius of the gear.