## 3 Modeling, Implementation, and Numerical Simulation
### Numerical Simulation  
The model setup (Figure 3), for healthy and damaged cases, considers a speed of 3000 rpm, a torque of 1 N, and 6000 steps. To address a feasible number of steps for each simulation, the following observation was considered: the greater the number of steps, the more the time to simulate the desirable system. Then, some assumptions and investigations regarding aspects of the spectrum and the number of steps were also performed for the system presented in this paper. First, the computational capacity for simulating each system was considered so that the work would be performed in an appropriate time. Second, at least four periods of rotation were desired in order to achieve a proper visualization of spectra and their characteristics in the time  
Figure 3: Numerical model of the test rig system considering the contact between the meshing surfaces of the gear teeth of the helical gear pair (**right** side) and the spur gear pair (**left** side), where the main analysis is performed for the healthy system and cases of damage. domain. Third, delimitation of the range of the data acquired from each FEM simulation was accomplished in order to obtain a suitable sampling (discretization) time. Then, each system simulation generated data with reference frame deflection in the z-direction. Fourth, the data were used to post-process a clean signal in the time and frequency domains, considering the visualization of the harmonics, sidebands, and gear mesh frequencies (GMFs). The latter defines the resolution of the analysis, which is very important for accurate results and the determination of sampling time. The calculation of the GMFs of the two gear pairs (test and service gearboxes) can be expressed as:  
\[GMF=\frac{Z_{min}\cdot n_{min}}{2\pi}, \tag{2}\]  
where \(z\) is the number of teeth and \(n\) is the rotational speed. Finally, the last investigation was related to the delimitation of the range of the frequencies to be used in the ANN application to avoid overfitting. A healthy system and 15 cases of damage (tooth root cracks) were considered in this research (Table 2). Moreover, three cases were exploited. First case: one healthy system and one damaged system (classification case); second case: one healthy and two damaged systems (first multi-class classification case); third case: one healthy and four damaged systems (second multi-class classification case). The tooth root of the spur gear (pinion) with 17 teeth in the test gearbox was used as a reference location for the damage (crack). By disconnecting the elements located in the mesh, with the FE model, the damage can be simulated (Figure 4) [55]. An outlook for the damaged tooth in each condition is visible in Figure 5 for the first classification case and in Figure 6 (damage cases 1 and 2) and Figure 7 (damage cases 3 and 4) for the second multi-class classification case. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Set of Simulations**} \\ \hline \multirow{2}{*}{**Damage Index**} & **Axial Dimension (AD) Longitudinal** & **Radial Dimension (RD)** \\ \cline{2-3}  & \multicolumn{2}{c}{**N\({}^{*}\) Elements**} & **Clearance (0 mm)** \\ \hline Crack 1 & 1 & ✓ \\ \hline Crack 2 & 2 & ✓ \\ \hline Crack 3 & 3 & ✓ \\ \hline Crack 4 & 4 & ✓ \\ \hline Crack 5 & 5 & ✓ \\ \hline \multicolumn{3}{c}{**Simulated Systems**} & **Damages** \\ \hline
1 & No crack & - \\ \hline
2 & Pos 1-AD1 & Crack 1 \\ \hline
3 & Pos 2-AD1 & Crack 2 \\ \hline
4 & Pos 3-AD1 & Crack 3 \\ \hline
5 & Pos 4-AD1 & Crack 4 \\ \hline
6 & Pos 5-AD1 & Crack 5 \\ \hline
7 & Pos 6-AD2 & Crack 1 \\ \hline
8 & Pos 7-AD2 & Crack 2 \\ \hline
9 & Pos 8-AD2 & Crack 3 \\ \hline
10 & Pos 9-AD2 & Crack 4 \\ \hline
11 & Pos 10-AD2 & Crack 5 \\ \hline
12 & Pos 11-AD3 & Crack 1 \\ \hline
13 & Pos 12-AD3 & Crack 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Damage index. Figure 4: Elements and positions for the dynamic FEM simulations of the cases of damage.