## 2 Digital Twin-Assisted Gearbox Monitoring
### Gear Faults
#### 2.2.1 Model-Based Development
This level of detail makes it particularly well suited for  
Figure 5: Gear fault modes: (**a**) healthy, (**b**) root crack, (**c**) chipped tooth, (**d**) missing tooth, (**e**) surface wear [58]. predictive maintenance and advanced fault detection. However, such comprehensiveness comes at a significant computational cost, requiring longer simulation times and a detailed understanding of the system's physical parameters for proper calibration. As a result, this model is best suited for specialized applications where high precision is essential. Figure 6: Gearbox dynamic model with 6 DOFs [63]. Figure 7: Gearbox dynamic model with 21 DOFs [64]. Despite its simplicity and the limited number of physical parameters involved, the six-DOF model has proven effective in reproducing the vibratory behavior of gear systems. The modeling process begins with the calculation of various forms of energy involved in power transmission--namely, Hertzian contact, bending, axial compression, and shear energy--based on the physical and geometric parameters depicted in Figure 6. These energy components require a range of physical and geometric inputs to ensure accurate estimation [23]. List of model parameters:  
\(I_{m}\): Moment of inertia associated with the motor. \(I_{b}\): Moment of inertia of the applied load. \(I_{m}\): Moment of inertia of the pinion. \(I_{m}\): Moment of inertia of the gear. \(M_{1}\): Torque generated by the motor. \(M_{2}\): Torque exerted by the load. \(m_{1}\): Mass of the pinion. \(m_{2}\): Mass of the gear. \(R_{b1}\): Radius of the pinion's base circle. \(R_{b2}\): Radius of the gear's base circle. Coupling Characteristics:  
\(k_{p}\): Torsional rigidity of the input coupling. \(k_{g}\): Torsional rigidity of the output coupling. \(c_{p}\): Damping factor of the input coupling. \(c_{g}\): Damping factor of the output coupling. Bearing Properties:  
\(k_{x1}\): Radial stiffness in the horizontal direction for the input bearing. \(k_{x2}\): Radial stiffness in the horizontal direction for the output bearing. \(k_{y1}\): Radial stiffness in the vertical direction for the input bearing. \(k_{y2}\): Radial stiffness in the vertical direction for the output bearing. Figure 8: Gearbox dynamic model with 34 DOFs [65]. \(c_{x1}\): Horizontal viscous damping coefficient of the input bearing. \(c_{x2}\): Horizontal damping coefficient of the output bearing. \(c_{y1}\): Vertical damping coefficient of the input bearing. \(c_{y2}\): Vertical damping coefficient of the output bearing. Gear Mesh Dynamics:  
\(k_{t}\): Stiffness of the gear meshing. \(c_{t}\): Damping coefficient at the mesh interface. Translational Displacements:  
\(x_{1}\): Axial displacement of the pinion along the meshing direction (\(x\)-axis). \(x_{2}\): Axial displacement of the gear along the \(x\)-axis. \(y_{1}\): Radial displacement of the pinion in the perpendicular direction (\(y\)-axis). \(y_{2}\): Radial displacement of the gear on the \(y\)-axis. Angular Displacements:  
\(\theta_{m}\): Motor shaft rotation angle. \(\theta_{1}\): Angular rotation of the pinion. \(\theta_{2}\): Angular rotation of the gear. \(\theta_{b}\): Angular rotation of the load. Equations of motion derived from [PERSON]'s second law:  
Vertical translation of the pinion:  
\[m_{1}\ddot{y}_{1}+F_{u}+F_{uc}=-F_{k}-F_{c} \tag{6}\]  
Vertical translation of the gear:  
\[m_{2}\ddot{y}_{2}+F_{l}+F_{lc}=F_{k}+F_{c} \tag{7}\]  
Rotation of the pinion:  
\[I_{1}\ddot{\theta}_{1}=M_{pk}+M_{pc}-R_{b1}(F_{k}+F_{c}) \tag{8}\]  
Rotation of the gear:  
Figure 6.0  
Rotation of the motor:  
Figure 6.1  
Rotation of the load:  
Figure 6.2  
Definitions of forces and moments:  
Vertical support forces:  
Figure 6.3  
Figure 6.4  
Figure 6.5  
Figure 6.6  
Gear mesh contact force:  
Figure 6.7Figure 6.8  
Moments from bearing stiffness and damping:  
Figure 6.9  
Figure 5:0  
Figure 5:1  
Figure 5:2  
An overview of FEM-based approaches for gear monitoring supported by DT technology follows. [PERSON] et al.