## 3 Construction of LPTN-Based VML
### Calculation and Classification of TRs
#### 3.3.2 Variable TRs  
Applied radial loads and rotational speeds induce variations in specific TRs, which mainly include the thermal shrinkage TRs between the inner/outer race raceway and the rolling elements, and the convective TRs between the oil-grease-air mixture inside the bearing cavity and rolling elements (including cage), as well as inner/outer ring surface. * Shrinkage TRs  
The contact heat transfer between the inner/outer ring raceways and rolling elements is related to the moving heat source. Considering factors such as contact area and pressure, the mathematical expression of the shrinkage TRs model proposed by [PERSON] et al. [50] based on Hertz contact theory is as follows:  
\[R_{c}=\frac{1}{\pi ka}\Big{(}\frac{a}{b}\Big{)}\frac{1}{\sqrt{Pe}}=\frac{\sqrt {\alpha}}{\pi k}\sqrt{\frac{1}{ab^{2}u_{c}}} \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Name and Application Location** & **Description** \\ \hline Radial conduction TRs of inner and outer rings & \(R_{1}=\frac{\ln(d_{out}/d_{in})}{2\pi k}\) \\ Solid ball conduction TRs & \(R_{2}=\frac{1}{k_{b}\pi r_{b}}\) \\ Contact TRs between inner ring/sleeve and & \(R_{3}=\frac{L_{s}(k_{1}+k_{2})}{2 AA^{2}k_{1}k_{2}}\) \\ Axial contact TRs between outer ring/end cup and & \(R_{4}=\frac{L_{s}(k_{1}+k_{2})}{2 AAc_{1}k_{2}+A_{c}(k_{1}+k_{2})}\) \\ chamber, inner ring, and sleeve—clearance fit & \(R_{4}=\frac{L_{s}(k_{1}+k_{2})}{2 AAc_{1}k_{2}+A_{c}(k_{1}+k_{2})}\) \\ Axial conduction TRs of bearing house & \(R_{5}=\frac{L}{5}\) \\ Natural convection TRs between the outer surface of the & \(R_{6}=\frac{1}{h_{c}S}\) \\ chamber/bench/endcap and air & \(R_{4}=\frac{1}{h_{c}S}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation of constant TRs. where \(k\) is the thermal conductivity of the medium and \(Pe\) is the Peclet number. \(\alpha\) and \(u_{c}\) are the diffusion coefficient and the linear velocity of the contact area, respectively. \(a\) and \(b\) are the semi-major and semi-minor axes of the elliptical, respectively, which can be expressed by the following formula. \[\left\{ \begin{array}{l}a=a^{*}\cdot\sqrt{\frac{3Q}{2\Sigma\rho }\left(\frac{1-{v_{1}}^{2}}{{E_{1}}}+\frac{1-{v_{2}}^{2}}{{E_{2}}}\right)}\\ b=b^{*}\cdot\sqrt{\frac{3Q}{2\Sigma\rho}\left(\frac{1-{v_{1}}^{2}}{{E_{1}}}+ \frac{1-{v_{2}}^{2}}{{E_{2}}}\right)}\end{array} \right. \tag{6}\]  
where \(a^{*}\) and \(b^{*}\) are the dimensionless quantities obtained from the lookup table. \(E_{1}\), \(E_{2}\), \(v_{1}\) and \(v_{2}\) are, respectively, the elastic modulus and Poisson's ratio of the two contact materials. \(\rho\) and \(Q\) are the curvature and normal contact load at the contact point. In the deep groove ball bearings that only bear radial loads, the normal contact load is proportional to the radial load. Assuming that the size and physical parameters of the bearing material remain basically unchanged during the temperature rise process, substituting Equation (6) into Equation (5), the contact TRs is related to the radial load and rotational speed of the bearing as follows:  
\[R\propto a^{-\frac{1}{2}}b^{-1}u_{c}^{-\frac{1}{2}}\propto Q^{-\frac{1}{2}}u_{ c}^{-\frac{1}{2}}\propto{F_{r}}^{-\frac{1}{2}}n_{s}^{-\frac{1}{2}} \tag{7}\]  
Let the proportional coefficient of the contact TRs in the bearing contact area be \(K_{Bc1}\) and \(K_{Bc2}\), respectively. Then, the simple expression is  
\[R_{c}(F_{r},n_{s})=K_{c1}+K_{c2}F_{r}^{-\frac{1}{2}}n_{s}^{-1} \tag{8}\]  
In the formula, \(K_{c1}\) and \(K_{c2}\) are related to the size and physical parameters of each bearing component. When the influence of temperature rise on these parameters is ignored, the proportional coefficient is approximately constant and set as the parameter to be identified. * Convective TRs The bearing cavity is filled with a mixture of grease, precipitated lubricating oil, and air (the vast majority). As the bearing rotates, the oil-grease-air mixture undergoes convective heat transfer with the inner/outer rings, cage, and rolling elements at all times.