# Analytical Model of Gear Thermal Stiffness Based on the Potential Energy Method
## 3 Gear Thermal Stiffness Based on Potential Energy Method
### Effect of Temperature on Contact Stiffness
#### 3.2.3 Contact Thermal Stiffness Considering Thermal Stress  
Thermal deformation and thermal stress inevitably occur due to the increase in gear temperature. This affects the distribution of contact stress along the \(y\)-axis. On the premise that the coupling effect is not considered, and the basic assumption of elasticity is satisfied, the distribution of thermal stress \(\sigma_{\rm T}\) along the \(y\)-axis is shown in Equation (17) [31]. The detailed derivation of Equation (17) can be found in the Appendix B.  
\[\sigma_{\rm T}=-\frac{\alpha_{\rm T}E}{1-v}\frac{1}{(\rho_{k}-y)^{2}}\int_{0} ^{l_{k}}\Delta T(\rho_{k}-y)(\rho_{k}-y)dy+\frac{E}{1+v}\left(\frac{C_{1}}{1-2 v}-\frac{C_{2}}{(\rho_{k}-y)^{2}}\right) \tag{17}\]  
where \(l_{k}\) is the distance between the line of action and the intersection of the involute tooth profile curves on both sides (as shown in Figure 2) and is determined by Equation (18); \(C_{1}\) and \(C_{2}\) are undetermined coefficients and are determined by boundary conditions. \[l_{k}=r_{\rm b}[\tan(\alpha_{k})-\tan(\alpha_{k}-\varphi_{k}-\varphi_{k^{ \prime}})] \tag{18}\]  
In the formula, \(\alpha_{k}\) is the pressure angle of the contact point K, and \(\varphi_{k}\) and \(\varphi_{k^{\prime}}\) are determined by the Formulas (19) and (20). \[\varphi_{k}=\frac{\pi}{2z}-(\mathrm{inv}(\alpha_{k})-\mathrm{inv}(\alpha_{\rm C })) \tag{19}\]  
\[\varphi_{k^{\prime}}=\arcsin(\frac{abr_{k}+\sqrt{\left(abr_{k}\right)^{2}-(a^{ 2}+2r_{k}^{2})(b^{2}-r_{k}^{2})}}{a^{2}+2r_{k}^{2}}) \tag{20}\]where \(z\) is the number of gear teeth, \(r_{k}\) is the radius of the contact point K, and \(a\) and \(b\) are transitional variables that are determined by Equations (21) and (22). \[a=\frac{r_{k}\cos(\varphi_{k})-r_{\rm b}\cos(\alpha_{k}-\varphi_{k})}{r_{k}\sin( \varphi_{k})+r_{\rm b}\sin(\alpha_{k}-\varphi_{k})} \tag{21}\]  
\[b=r_{k}\cos(\varphi_{k})-\frac{r_{k}\cos(\varphi_{k})-r_{\rm b}\cos(\alpha_{k} -\varphi_{k})}{r_{k}\sin(\varphi_{k})+r_{\rm b}\sin(\alpha_{k}-\varphi_{k})}r_ {k}\sin(\varphi_{k}) \tag{22}\]  
The thermal expansion at the contact point belongs to the restricted expansion as the temperature of the gear increases. Reference [30] gives the calculation formula of the contact pressure generated under the restricted expansion of the object, as shown in Equation (23). \[P=\frac{1}{2}\alpha_{\rm T}E\Delta T \tag{23}\]  
where \(\Delta T\) is the temperature rise of the object. The contact pressure \(\sigma_{\rm Tk}=\frac{1}{2}\alpha_{\rm T}E\Delta T(0)\) caused by the temperature rise at contact point K (\(y=0\)) in the coordinate system is shown in Figure 3. As shown in Figure 2b, the thermal deformation at the intersection point K\({}^{\prime}\) (the intersection point of the line of action and the back-side tooth profile curve) belongs to free expansion in the case of sufficient gear backlash. The contact pressure at K\({}^{\prime}\)\(\sigma_{\rm TK^{\prime}}=0\). The coordinate of point K\({}^{\prime}\) in the coordinate system shown in Figure 3 is (\(y=l_{k}\)). The undetermined coefficients \(C_{1}\) and \(C_{2}\) can be obtained by substituting the boundary conditions into Equation (17) as follows:  
\[C_{1}=\frac{\alpha_{\rm T}(1+v)(1-2v)\Big{[}\frac{1}{2}\Delta T(0)\rho_{k}^{2 }+\frac{A}{(1-v)}\Big{]}}{\rho_{k}^{2}-\left(\rho_{k}-l_{k}\right)^{2}} \tag{24}\]  
\[C_{2}=\frac{\alpha_{\rm T}(1+v)(\rho_{k}-l_{k})^{2}\Big{[}\frac{1}{2}\Delta T (0)\rho_{k}^{2}+\frac{A}{(1-v)}\Big{]}}{\rho_{k}^{2}-\left(\rho_{k}-l_{k} \right)^{2}} \tag{25}\]  
where \(A=\int_{0}^{k_{k}}\Delta T(\rho_{k}-y)(\rho_{k}-y)dy\). After obtaining the thermal stress distributed along the \(y\)-axis under restricted expansion, the distribution of the contact stress along the \(y\)-axis considering the influence of thermal stress \(\sigma_{\rm yT}=\sigma_{y}+\sigma_{\rm T}\) can be obtained from the principle of stress superposition.