## 5 Results and Analysis
### Thermal Prediction Under Variable Speed Conditions  
#### 5.2.1 Comparison of Predicted and Real Temperature  
The case study employs a 6311 deep groove ball bearing, with fundamental dimensions detailed in Table 3 and material parameters detailed in Table 4. Under a fixed radial load (2650 N), rotational speed is utilized as the input variable for thermal parameter identification. During the temperature rise test, six operational speed conditions are defined: 1500 rpm, 2400 rpm, 4200 rpm, 4900 rpm, 5600 rpm, and 6200 rpm. Each speed condition is maintained for designated durations to acquire comprehensive temperature datasets across the entire operational range. Reference nodes are defined as the inner surface temperature of the inner ring \(T_{ir}\) and the outer surface temperature of the outer ring \(T_{or}\) of the deep groove ball bearing. As illustrated in Figure 10, temperature comparisons at reference nodes obtained through the proposed digital twin-driven virtual-real hybrid framework are presented and the identified values of unknown parameters \(p_{1}\) are showed in Equation (27)  
\[\left\{ \begin{array}{l}f_{2}=0.7,K_{c1}=9998,K_{c2}=0.099\\ K_{11}=5,K_{12}=4.4\times 10^{-4},K_{13}=0.99\\ K_{21}=10.66,K_{22}=2.46,K_{23}=0.998\\ K_{31}=5,K_{32}=0.057,K_{33}=0.97\\ K_{41}=5,K_{42}=0.074,K_{43}=0.99\end{array} \right. \tag{27}\]  
The predicted temperature generated by the VML exhibit strong congruence with the real temperature acquired from the PEL, as shown in Figure 10. The temperature discrepancies at the inner surface of the inner ring and outer surface of the outer ring are plotted in the upper and lower subplots, respectively, with maximum errors of approximately 1.2 \({}^{\circ}\)C and 0.5 \({}^{\circ}\)C. This empirically confirms the accuracy of the proposed methodology in identifying bearing thermal parameters. Due to sensor number limitations, experimental validation \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & \multirow{2}{*}{**Ring**} & \multirow{2}{*}{**Ball**} & \multicolumn{2}{c}{**House/Bench/Shaft/Sleeve/End**} & \multirow{2}{*}{**Cage**} \\ \cline{2-2} \cline{4-5}  & & & **Cap** & \\ \cline{2-2} \cline{4-5}  & **GCr15** & **GCr13** & **Structural Steel** & **Copper** \\ \hline Density/kg\(\cdot\)m\({}^{-3}\) & 7810 & 7750 & 7810 & 8300 \\ Coefficient of thermal & & & & \\ expansion/\({}^{\circ}\)C\({}^{-1}\) & \(12.5\times 10^{-5}\) & \(10.5\times 10^{-5}\) & \(12.0\times 10^{-5}\) & \(18.0\times 10^{-5}\) \\ Young’s modulus/GPa & 207 & 200 & 200 & 110 \\ Poisson’s ratio & 0.3 & 0.29 & 0.3 & 0.34 \\ Thermal conductivity/W-m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) & 60.5 & 28.5 & 60.5 & 401 \\ Specific heat capacity/J\(\cdot\)kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) & 460 & 502 & 434 & 385 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Parameters Name**} & \multicolumn{1}{c}{**Symbol**} & \multicolumn{1}{c}{**6311**} & \multicolumn{1}{c}{**6016**} \\ \hline External diameter/mm & \(d_{e}\) & 120 & 125 \\ Internal diameter/mm & \(d_{i}\) & 55 & 80 \\ Inner raceway diameter/mm & \(d_{ir}\) & 71.7 & 89.7 \\ Pitch diameter/mm & \(d_{m}\) & 87.5 & 102.5 \\ Width/mm & \(w\) & 29 & 22 \\ Ball number & N & 8 & 18 \\ Ball radius/mm & \(d_{br}\) & 20 & 11.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Bearing fundamental dimensions. data for non-reference nodes are unavailable. Consequently, the parameter \(p_{1}\) identified in Figure 10 was integrated into the VML twin structure to generate twin temperature datasets \(T_{\text{TD1}}\) for all bearing nodes, as illustrated in Figure 11. The thermal distribution reveals significantly elevated temperatures at the inner ring and shaft sleeve compared to other components. The bench's large-volume components exhibit enhanced heat dissipation capacity due to their substantial surface area, resulting in minimal temperature rise.