## 2 Digital Twin-Assisted Gearbox Monitoring
### Gear Faults  
#### 2.2.1 Model-Based Development  
Gears generate vibrations and noise even in a healthy state, particularly in the case of straight-toothed gears. These vibrations occur at the meshing frequency, which corresponds to each contact between gear teeth. This fundamental frequency is \(z1^{*}f1\) and equal to \(z2^{*}f_{2}\), where \(z\) represents the number of teeth and \(f\) the rotational frequency [57]. Several fault modes can affect gears, caused by the load supported by the gear teeth or, in some cases, manufacturing defects that lead to these faults. Among these fault modes are cracks, pitting, wear, chipped teeth, or missing teeth, as shown in Figure 5.  
Accurate analysis of these vibratory phenomena relies on precise modeling of gear tooth contact. Our recent work has led to significant advances in this domain. On one hand, we developed a simplified formulation of the fundamental equations of gear contact, reducing the classical system of five nonlinear equations to only two [59]. This approach provides better numerical stability and enables parametric analysis of misalignment effects--both in-plane and out-of-plane--on the contact path, transmission error, and pressure distribution. On the other hand, a three-dimensional generalization of the contact problem, based on a novel parameterization of the contacting surfaces, was introduced [60]. This allows for efficient handling of the relative motion between two toothed surfaces rotating about fixed axes in space and improved accuracy. In a faulty state, two scenarios may arise. If the fault is distributed across all the teeth, the impact between teeth becomes more pronounced, which amplifies vibrations at the meshing frequency. Conversely, if the fault is localized, it generates vibrations at the rotational frequency of the gear with the defective tooth [61]. In this context, the development of a digital twin dedicated to gear monitoring relies on computing the energies resulting from deformation and contact between one or two pairs of teeth, including Hertzian, bending, shear, and axial compressive energies. The relationship between these energies and tooth stiffness is then used to estimate the gear mesh stiffness [62]. A widely used model for simulating single-stage gear transmissions is the six-DOF model, illustrated in Figure 6. Although more complex models, such as the 21-DOF and 34-DOF models shown in Figures 7 and 8, respectively, are available, the 6-DOF model remains popular due to its simplicity and computational efficiency. It is particularly effective for analyzing systems where torsional effects dominate. Its main strengths lie in its ease of implementation and low computational cost, which make it especially suitable for basic applications and educational purposes. However, this simplicity comes at the expense of accuracy. The model cannot capture complex interactions such as vibrations or nonlinear behaviors, and it assumes perfectly rigid bearings--an assumption that is often unrealistic in industrial settings. Moreover, it is not suitable for analyzing high-frequency phenomena. In contrast, the 21-DOF model offers a good balance between complexity and accuracy. It captures the coupling between torsional and translational dynamics, as well as gear mesh forces, and it includes the modeling of gear transmission errors, making it a valuable tool for studying gear noise and vibration. The computational effort remains reasonable for simulations requiring moderate precision. Nonetheless, it has its limitations: it assumes linear damping, neglects nonlinearities in the bearings, and does not account for contact losses. As a result, it is less effective for fine fault detection or for analyzing vibrations in the gearbox housing. The 34-DOF model represents the most advanced approach, capable of capturing a wide range of complex phenomena. It considers nonlinear bearing characteristics, such as variable stiffness, models localized defects, and enables detailed analysis of housing resonances.