## 5 Results  
The numerical analysis aimed at the extraction of displacements and, after a double integration, the extraction of vibrations. The time discretization (sampling) must be selected accurately for the performance of the extraction. To avoid aliasing phenomena, the time discretization should be enough dense; however, at the same time, it should be conserved as far as possible to limit the computational effort. Moreover, the time-step, which defines the resolution of the analysis, can be affected by the rotational speed of the system, the number of teeth, the number of meshing pairs, the number of harmonics, and the number of time-steps for each meshing period [43]. In this regard, GMFs can be used to determine the sampling time of the analysis. After post-processing the data, the GMFs of the system (Spur Gear Pair: 850 Hz; Helical Gear Pair: 1700 Hz) are visible. Peaks of 50 Hz away from each other were visibly found in the spectra for all simulated systems. They have a sharp definition in the damaged systems, whose amplitudes at each peak became higher each time the damage level was increased. This was expected. Examples of some of the signals in the time and frequency domains are shown in Figure 16a,b for the classification problem and in Figure 17a,b for the multi-class classification problem. The system simulation considers five harmonics, which gives the possibility of observing frequencies in the range of 8500 Hz. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Neural Pattern Recognition (nprtool)** & \\ \hline
**Results** & **Samples** & **CE** & **\%E** \\ \hline Training & 81 & 0.794494 & 1.60494 \\ \hline Validation & 27 & 1.56895 & 1.85185 \\ \hline Testing & 27 & 1.58625 & 3.33333 \\ \hline \hline \end{tabular} Note 1: Important parameters from the “nprtool” interface. Minimizing Cross-Entropy (CE) results in good classification. Lower values are better. Zero means no error. **Note 2.** Percent Error (%E) indicates the fraction of samples that are misclassified. A value of 0 means no misclassifications; 100 indicates the maximum of misclassifications.  
\end{table} 
Table 3: Important parameters from the “nprtool” interface. Figure 16: Example of signals for the classification problem in the (**a**) time and (**b**) frequency domains. Both domains show comparisons of vibrational spectra on a bearing between a healthy transmission and a damaged one. Figure 17: Example of signals for the multi-class classification problems in the (**a**) time and (**b**) frequency domains. Both domains show comparisons of vibrational spectra on a bearing between a healthy transmission and cases of damage. The sideband is another important feature in this analysis. Since the wave generated by the meshing gears is generally not a pure sinusoidal function, because the relative amplitude of the different harmonics determines the quality of the sinusoidal wave, the spectra usually show additional harmonics of the GMFs with their own sideband group. Then, all real sinusoidal waves can have some number of harmonics present in each one. Thus, when determining two or more complex sinusoidal waves, it must be seen whether any pair of harmonics lies within a critical bandwidth. In fact, producing a pure wave is rather difficult because it is very hard to remove all of these clustered harmonics, which may cause noise signals. Therefore, in a healthy system, the amplitude of the side bands can be low, while in damaged cases the sidebands increase due to the vibrational spectra, which increasingly differ from the ideal sine function. Since the first ANN approach applied in this research was the classification to determine systems with no damage and with damage conditions (Figure 7), the first case was used. Initially, systems exhibiting healthy or damage behavior, with only one element, were simulated.