## 2 Materials and Methodology
### Fault Diagnosis Methodology
#### 2.2.2 Time Frequency Analysis  
Vibration signals obtained from failed bearings have characteristics that can be obtained by spectrum analysis. Therefore, we begin by representing the vibration signals from the bearings in the Matlab environment together with their respective scalograms. This is done to visualize the relationship between the signal in the time domain and its representation in the time-frequency domain by means of the scalogram. It is important to  
Figure 4: Representation of the original vibration signals: (**a**) Inner track failure; (**b**) No failure; (**c**) Outer track failure.  
Figure 5: Representation of the standardized vibration signals: (**a**) Inner track failure; (**b**) No failure; (**c**) Outer track failure.  
remember that a scalogram is a two-dimensional representation of a signal that shows how its energy varies over time and in different frequency bands. Therefore, each point in the scalogram corresponds to a specific moment in time and to a particular frequency. This allows to observe how the characteristics of the signal, the vibration peaks or oscillations, manifest themselves at different frequencies as it evolves over time.  
This representation is particularly useful since bearing failures generate specific vibration patterns over time and at different frequencies. For example, a failure on the outer race might manifest as a series of high-energy spikes at certain frequencies, while a failure on the inner race might have a different pattern.  
Figure 6 shows the vibration signal from a normal bearing without failure. Despite not having any defects, the signal in Figure 6 has a certain amplitude of vibration that is completely normal and corresponds to its base spectrum. In addition, the scalogram shows a more uniform energy, contrary to what was observed in previous cases.  
Figure 7 shows the vibration signal from a bearing with a failure on the outer race. During the 0.1 s shown in Figure 7, the vibration signal contains 16 pulses because the BPFI of the bearing is 159.63 Hz. If we divide this number of pulses by the 0.1 s of representation the result is 160 Hz. Consequently, the scalogram shows the alignment of the 16 peaks with the pulses of the vibration signal.  
Figure 8 shows the vibration signal of a fault in the outer track with its scalogram. In this case, although barely noticeable to the naked eye, the outer race failure shows 7 distinct peaks during the first 0.1 s, which is consistent with the ball passing frequencies of 75.06 Hz. Because the pulses in the time domain signal are not as prominent as in the inner race failure case, the distinct peaks in the scalogram show less contrast with the background.  
Figure 6: Sign and scalogram of a healthy bearing.  
The number of distinct peaks is a good characteristic to differentiate between inner race failures, outer race failures and normal conditions. Therefore, a scalogram can be a good candidate for classifying bearing failures.  
Subsequently, to automate this process of transforming signals into scalograms, a helper function has been created in Matlab. This function allows to input a vibration signal and obtain its scalogram quickly and efficiently. This facilitates the analysis of a large volume of vibration signal data and helps to identify fault patterns in a more systematic way.