## 5 Results and Analysis
### Thermal Prediction Under Variable Radial Load Conditions  
The proposed digital twin-driven virtual-real hybrid framework is applied to predict high-risk thermal behaviors under heavy loads for a 6016 deep groove ball bearing (dimensional specifications in Table 3). Under constant rotational speed (5800 rpm), reference node temperatures acquired from the PEL under light-load (2650 N) conditions are utilized to identify unknown parameters in the LPTN. These identified parameters are subsequently imported into the VML to generate twin temperature predictions under heavy-load conditions (4000 N and 5500 N), which are rigorously validated against \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Scheme Name**} & \multirow{2}{*}{**Number of Reference Nodes**} & \multicolumn{2}{c}{**Average Euclidean Distance of All Nodes/K**} \\  & & **Distance of All Nodes/K** \\ \hline N\_Ref\_4 & T1/T2/T3/T4 & 13.7 \\ N\_Ref\_6 & T1/T2/T3/T4/T5/T6 & 12.7 \\ N\_Ref\_8 & T1/T2/T3/T4/T5/T6/T7/T8 & 5.0 \\ N\_Ref\_10 & T1/T2/T3/T4/T5/T6/T7/T8/T9/T10 & 3.4 \\ N\_Ref\_13 & T1/T2/T3/T4/T5/T6/T7/T8/T9/T10/ & 0.7 \\  & T11/T12/T13 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Detailed scheme specifications and average Euclidean distances. Figure 15: Similarity of temperature nodes with different numbers of reference nodes. experimental measurements. The unknown parameter \(p_{2}\) values extracted during the identification process are  
\[\left\{ \begin{array}{l}f_{2}=0.7,K_{c1}=1889,K_{c2}=2.7\times 10^{-4}\\ K_{11}=5,K_{12}=6.4\times 10^{-4},K_{13}=0.16\\ K_{21}=5,K_{22}=0.106,K_{23}=0.99\\ K_{31}=24.9,K_{32}=1.58,K_{33}=0.98\\ K_{41}=5.67,K_{42}=1.6\times 10^{-3},K_{43}=0.672\end{array} \right. \tag{28}\]  
Figure 16 and Table 8 present the temperature transient variations under variable radial load conditions and the maximum prediction errors relative to experimental temperatures, respectively. As depicted in Figure 16, in the parameter identification process with the 2650 N light-load sample data, the maximum temperature differences at reference nodes during the rise phase are 1.6 \({}^{\circ}\)C and 1.2 \({}^{\circ}\)C, whereas these differences reduce to 0.6 \({}^{\circ}\)C and 0.4 \({}^{\circ}\)C at the stabilization stage. The steady-state temperature errors are 1.1% and 1.0%, respectively. Figure 16: Temperature comparison under variable radial load conditions. For the heavy-load conditions of 4000 N and 5500 N, the temperature curves generated by the digital twin-driven virtual-real hybrid framework exhibit strong congruence with experimental measurements. Under 4000 N radial load of the 6016 deep groove ball bearing, maximum temperature differences at reference nodes during the rise stage are 5.4 \({}^{\circ}\)C (inner ring) and 1.1 \({}^{\circ}\)C (outer ring). These differences reduce to 1.4 \({}^{\circ}\)C and 0.3 \({}^{\circ}\)C at the stabilization stage. Similarly, under 5500 N radial load, maximum transient differences are 4.3 \({}^{\circ}\)C and 1.8 \({}^{\circ}\)C, stabilizing to 2.6 \({}^{\circ}\)C and 0.6 \({}^{\circ}\)C at the stabilization stage. In this case, it can be seen that the temperature difference on the inner surface of the inner ring is consistently greater than that on the outer surface of the outer ring during thermal prediction. Thermal equilibrium attenuates these discrepancies, yet elevated loads exacerbate these differences. Critically, the proposed framework achieves robust predictive accuracy under extreme loading, with stabilization stage errors \(\leq\) 3.5% even at 5500 N, validating its applicability for thermal prediction under high-risk behaviors with variable radial load conditions.