## 2 Digital Twin-Assisted Gearbox Monitoring
### Bearing Faults  
#### 2.1.1 Model-Based Development  
The role of the bearing in this transmission system is to minimize contact and energy loss. Bearings generally operate at high speeds, causing a rise in temperature due to excessive friction. Several fault modes can affect these components, such as electric melting, plastic deformation, corrosion, and wear, as shown in Figure 2. Corrosion occurs due to lubricant degradation from chemical factors such as oxygen-induced oxidation or water infiltration, particularly in marine environments (offshore turbines). An additional electrochemical factor involves stray currents within generators. Bearing corrosion initially appears as gray-to-black surface pitting that progressively expands over time. Wear can be initiated by corrosion (which increases surface susceptibility to abrasion) or caused by physical lubricant contamination from wind-borne particles or internal component wear debris, typically accompanied by abnormal vibration and noise. Electric melting results from high current passing between races through rolling elements, generating localized heating that leads to contact zone melting and surface micro-welding, identifiable by characteristic dark fluting patterns on rollers and raceways. Plastic deformation primarily stems from excessive cyclic loading or improper installation, causing permanent geometric changes in contact zones between rolling elements and bearing races [46,47]. Figure 1: Wind turbine components [43]. These faults can affect the three parts of a bearing: inner race fault, outer race fault, and rolling elements fault, as shown in Figure 3. Each of these faults has a characteristic frequency determined by its geometric parameters and rotation speed [48]. The model adopted for the dynamic modeling of bearing faults is the nonlinear dynamic model of a rolling bearing with five degrees of freedom (DOFs), as shown in Figure 4a, knowing that some reduce to four DOFs while maintaining almost the same axes, as shown in Figure 4b. This model has been extensively studied in previous frameworks, particularly in the context of DT applications, due to its simplicity and fidelity. The model includes two DOFs for the inner race (\(x_{i}\)\(y_{i}\)) and the outer race (\(x_{o}\)\(y_{o}\)), and one DOF for the unit resonator (\(y_{r}\)), where \(x\) and \(y\) represent the horizontal and vertical axes, respectively [49]. The five-DOF model includes one additional degree of freedom for the  
Figure 3: Bearing fault types: (**a**) rolling bearing fault, (**b**) inner race fault, (**c**) outer race fault [48]. Figure 2: Bearing fault modes: (**a**) corrosion, (**b**) fatigue, (**c**) plastic deformation, (**d**) electric melting [48]. resonator mass \(m_{r}\), compared to the four-DOF model. This added degree of freedom enhances the model's fidelity, allowing for more realistic simulations and capturing the interactions between the rotor and stator. On the other hand, the five-DOF representation has some drawbacks, and it requires more parameters to be identified before the simulation can be run. This additional complexity increases computation time and demands higher storage capacity. The dynamic behavior of the system is governed by a set of second-order differential equations that model the motion of the inner race, outer race, and the unit resonator. The parameters include the shaft speed (\(\omega_{s}\)), the stiffness of the inner race (\(k_{i}\)), the outer race (\(k_{o}\)), and the resonator (\(k_{r}\)), along with their respective damping coefficients (\(c_{i}\), \(c_{o}\), and \(c_{r}\)) [51]. Inner race motion in the \(x\)-direction:  
\[m_{i}\ddot{x}_{i}+c_{i}\dot{x}_{i}+k_{i}x_{i}+F_{cx}+F_{kx}=0 \tag{1}\]  
This equation describes the translational motion of the inner race along the horizontal axis. The terms represent, respectively, the inertial force, damping force, and restoring force due to the stiffness of the inner race. The contact forces \(F_{cx}\) and \(F_{kx}\) arise from interaction with the other elements such as rolling elements [22].