## 2 Digital Twin-Assisted Gearbox Monitoring
### Bearing Faults
#### 2.1.1 Model-Based Development
Inner race motion in the \(y\)-direction:  
\[m_{i}\ddot{y}_{i}+c_{i}\dot{y}_{i}+k_{i}y_{i}+F_{cy}+F_{ky}=m_{i}g \tag{2}\]  
This equation models the vertical displacement of the inner race. The right-hand side includes the gravitational force \(m_{i}\dot{g}\), while the left-hand side includes inertia, damping, stiffness, and vertical contact forces. Outer race motion in the \(x\)-direction:  
\[m_{o}\ddot{x}_{o}+c_{o}\dot{x}_{o}+k_{o}x_{o}-F_{cx}-F_{kx}=0 \tag{3}\]  
This equation captures the horizontal motion of the outer race. The contact forces \(F_{cx}\) and \(F_{kx}\) act in the opposite direction compared to the inner race due to [PERSON]'s third law. Figure 4: Bearing dynamic model with: (**a**) 5 DOFs [50], (**b**) 4 DOFs [40]. Unit resonator motion in the \(y\)-direction:  
\[m_{r}\hat{y}_{r}+c_{r}(y_{r}-y_{o})+k_{r}(y_{r}-y_{o})=m_{r}g \tag{4}\]  
This equation defines the vertical motion of the resonator relative to the outer race. The resonator experiences restoring and damping forces relative to the outer race displacement and is also subject to gravity. Outer race motion in the \(y\)-direction:  
Figure 1:0  
This equation governs the vertical dynamics of the outer race, accounting for gravitational force, its own damping and stiffness, vertical contact forces, and the coupling interaction with the resonator via both damping and stiffness terms. At this stage, the fundamental principle of dynamics is applied to the system (inner race, outer race, and rolling elements), resulting in a system of differential equations equal to the number of DOFs. Solving this system of differential equations using numerical solvers allows for the determination of the outputs, which are the vibrations corresponding to the five DOFs. Subsequently, it becomes possible to simulate faults that may affect a bearing and obtain the vibrational signature of each fault. Regarding the use of Finite Element Modeling (FEM), it has been the subject of several studies on bearing condition monitoring based on the DT approach. [PERSON] et al. [52] used FEM to simulate bearing fault signals in order to compensate for the lack of real fault data. Time-domain features were extracted and used to train an SVM classifier. The method achieved high accuracy across six fault types, demonstrating that FEM-generated signals can enhance fault diagnosis when experimental data are scarce. [PERSON] et al. [15] developed an FEM-based model to simulate bearing degradation under multiple conditions. A domain adaptation technique was applied to align simulated and real data, improving the prediction of RUL. This hybrid approach enhanced prediction accuracy, showcasing the benefit of combining FEM with transfer learning for robust diagnostics. [PERSON] et al. [53] addressed the high computational cost of FEM in real-time applications by proposing the KKR-DT system, which integrates k-nearest neighbors, Kriging, and Radial Basis Functions. A reduced-order model was constructed from FEM data and extended to various working conditions. The system predicts the RUL of roller bearings and enables rapid, real-time computations. Testing on industrial equipment confirmed its effectiveness for dynamic fault prediction. [PERSON] et al. [49] introduced multi-fidelity information fusion, a method for continuous bearing fault diagnosis using digital twins and class-incremental learning. A low-fidelity DT generates synthetic data, which are fused with a limited number of high-fidelity measurements. This system maintains strong diagnostic performance while significantly reducing training time compared to traditional joint training. It effectively mitigates catastrophic forgetting and supports adaptive fault diagnosis over time.