## 5 Results and Analysis  
### Comparison of Parameter Identification Results  
To evaluate the efficacy and applicability of the NOA-based thermal parameter identification method, a simplified LPTN model is designed as shown in Figure 7. The model comprises five temperature nodes and seven TRs, where Node 1 represents ambient temperature. A linear heat source is applied in a 1:1 ratio to Node 2 and Node 3. TRs \(R_{1}\),\(R_{4}\), \(R_{5}\) and \(R_{7}\) are predefined constants, while \(R_{2}\) and \(R_{3}\) are unknown constant TRs. To emulate the nonlinear exponential characteristics of thermal parameters in full-scale bearing LPTN models, \(R_{6}\) is formulated as a time-varying TR expressed through a power law function with unknown coefficients. Mathematical expressions for all unknown parameters are provided below:  
\[ \begin{split}& f=k_{1}n_{s}+k_{2}F_{r}+k_{3}\\ & R_{2}=k_{4},\ R_{3}=k_{5},\ R_{6}=k_{6}n_{s}-k_{7}\end{split}  \tag{26}\]  
From this, it can be inferred that the set of thermal parameters that need to be identified is \(p=[k_{1},k_{2},k_{3},k_{4},k_{5},k_{6},k_{7}]\). The model inputs are defined as the bearing's rotational speed \(n_{s}\) and radial load \(F_{r}\), with outputs corresponding to real-time temperatures at reference Nodes 1 and Node 2. The experimental procedure is initiated by giving predetermined target thermal parameters \(p_{t}=[1,0.29,50,0.2,0.35,0.5,0.18]\), followed by forward-solving the LPTN model to generate a benchmark dataset under these target parameters. This dataset, combined with a predefined range for thermal parameter values, is then imported into the NOA-based identification framework for unknown parameter identification. Throughout the identification process, the convergence process of each parameter relative to its target value is illustrated in Figure 8.  
Figure 8: Comparison of convergence process and results of identification value \(p\) relative to target value \(p_{t}\). Figure 7: Simple LPTN model. As illustrated in Figure 8, the per-unit values of all parameters progressively approach unity as the identification process iterates, indicating that the identified values converge toward their target values. Ultimately, the identified parameters exhibit near-complete agreement with the target values, conclusively demonstrating the feasibility of the NOAA-based identification method in resolving time-varying and nonlinear unknown parameters within the LPTN model. To evaluate the efficiency of the NOA-based identification strategy, a comparative analysis was conducted under identical computational resource allocations, search space dimensions, and maximum iteration constraints. The PSO algorithm was applied to identify unknown parameters in the simplified LPTN model, with comparative results detailed in Figure 9. Parameter \(k_{4}\) served as the benchmark metric: the NOA-based method achieved the required accuracy threshold within 103 s, whereas the PSO-based approach required 1045 s to meet convergence criteria, demonstrating a tenfold disparity in identification efficiency. These findings conclusively demonstrate the superior identification efficacy and convergence performance of the NOA-based method. The integration of this approach with the proposed digital twin-driven hybrid virtual-real framework enables the systematic exploration of parameter identification capabilities and thermal prediction accuracy for deep groove ball bearings in traction motor applications. It is critical to note that the experimental datasets in the bearing test rig play a pivotal role in accurately identifying unknown parameters within the LPTN model. The selection of an appropriate sample dataset must account for comprehensive bearing operation, steady-state thermal equilibrium, and environmental variations. However, the hydraulic loading system in the test platform exhibits inherent pressure hysteresis, limiting rapid load control during transient operations.