## 2 Modeling Methods
### Modeling of Damages--Surface Fatigue (Pitting)  
One of the most important damages that occurs in gears and that has a progressive evolution that can be monitored by means of SHM is surface fatigue; namely, pitting. This phenomenon is generated in the contact regions where the Hertzian pressure distribution generates very high shear stresses just under the surface. These local shear-stress peaks can, especially in presence of microdefects or subsuperficial inclusions, promote the nucleation of microcracks. These, with repeated loading cycles, propagate up to the surface, causing the detachment of small portions of material, called pits (Figure 6). While the initial stage leads to vibrations and noise only, with the progression of the damage, more and more pits are generated. The coalescence of multiple pits leads to the formation of big cavities on the surface. This phenomenon, once initiated, is accelerated by the presence of the lubricant that, during the contact, is pumped inside the cracks, promoting their propagation. The presence of cavities on the surfaces is in turn responsible for a nonhomogeneous distribution of the contact pressures on the surfaces (because, in correspondence with the pits, the surfaces are not in contact). Consequently, highly loaded areas generate promotion of an acceleration of the damage evolution. Finally, when the damage reaches a certain value, a complete failure of the system is also possible. In the present work, the presence of damage on the surface was modeled by selecting the position (radial and tangential coordinates S and T) on the teeth flank (Figure 7), the main diameters of the pits (intended as an ellipse), and their depths. Figure 5: Comparison between the numerical and experimental data for the back-to-back test rig (housing). Figure 6: Example of pitting. Table 1 reports the different conditions that were simulated. In addition to the healthy condition, 7 damages (PitX) were chosen. Their dimensions varied, as shown in Table 1. Moreover, these 7 damage severities were modeled in different positions on the flank (position indexes shown in Table 1). A total of 40 transient simulations were performed. Figure 8 clearly shows the effect of the presence of a pit on the pressure distribution in the contact. While the stresses were zero due to the missing contact in correspondence with the damage, in the surrounding region, stress peaks were generated. While the vibrational spectrum already had a characteristic frequency related to the GMF (which is also the main source of excitation), the presence of a pit on a single tooth promoted the excitation of an additional frequency (1/n), where n is the rotational speed (Figure 9). The vibrational spectra were obtained with a double derivative of the displacement data obtained from the simulations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Code**} & **Radial** & **Axial** & **Depth** & **Position** & **Radial** & **Axial** \\  & **Dimension** & **Dimension** & **[mm]** & **Index** & **Position** & **Position** \\  & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** \\ \hline Healthy & 0 & 0 & 0 & 0 & 24 & 0 \\ PitA & 1 & 2 & 0.5 & 1 & 24 & 1 \\ PitB & 2 & 5 & 1 & 2 & 24 & \(-1\) \\ PitC & 2.5 & 7 & 1 & 3 & 36 & 0.5 \\ PitD & 3 & 8 & 2 & 4 & 36 & \(-0.5\) \\ PitE & 3.5 & 9 & 2 & 5 & 12 & 0.5 \\ PitF & 5 & 10 & 1 & 6 & 12 & \(-0.5\) \\ PitG & 20 & 14 & 3 & & & \\ \hline \hline \multicolumn{2}{c}{**Position Index (PI)**} & **PitA** & **PitB** & **PitC** & **PitD** & **PitE** & **PitF** & **PitG** \\ \hline
0 & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\
1 & x & x & x & x & ✓ & ✓ \\
2 & x & x & x & x & ✓ & ✓ \\
3 & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\
4 & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\
5 & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\
6 & ✓ & ✓ & ✓ & ✓ & ✓ & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulated pit parameters, positions indexes, and simulations performed. Figure 7: (**a**) Representation of a pit on the face of a tooth; (**b**) position indexes. S is the radial reference axis and T is the axial one.