## 2 Materials and Methodology  
### Materials  
In this work, fault diagnosis techniques have been applied to the Case Western Reserve University (CWRU) bearing fault database [23]. The test bench used consists of a 2 HP motor, a torque transducer or encoder, a dynamometer, an electronic control system and test bearings that support the motor shaft (Figure 1). It is worth noting that the bearings used at the transmission end were SKF 6205-2 RS JEM (AB SKF, Gothenburg, Sweden), in which bores of 0.007, 0.014 and 0.021 inches were made. In addition, at the fan end, SKF 6203-2 RS JEM were used, in which bores of 0.028 and 0.040 inches were made. Thus, the characteristics are listed in Table 1. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Bearing Type**} & **Inside** & **External** & \multirow{2}{*}{**Thickness**} & **Ball Diameter** & **Pitch** & **Number** \\  & **Diameter** & **Diameter** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **of Balls** \\ \hline SKF 6205-2 RS JEM (transmission shaft) & 25.00122 & 51.99888 & 15.00124 & 7.94004 & 39.03980 & 9 \\ \hline SKF 6203-2 RS JEM (fan) & 17.00022 & 39.99992 & 11.99896 & 6.74624 & 28.49880 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bearing construction features. Figure 1: Test bench. Vibration data were collected using accelerometers, which were fixed to the housing with magnetic bases located above the shaft at both the drive end and the fan end of the motor housing. Vibration signals were collected using a 16-channel DAT recorder and further processed. In terms of sampling rate, digital data were obtained at both 12,000 and 48,000 samples per second. Speed and power data were collected using the torque transducer and recorded by hand. Rolling bearings consist of several different components: inner ring, balls or rollers, cage and outer ring (Figure 2). Damage to any of these elements causes one or more characteristic frequencies to appear in the frequency spectrum, allowing to identify them. The four possible frequencies of wear characteristic of a bearing and their mathematical expressions are as follows:  
* Ball pass frequency of the outer race or BPFO: The frequency with which defects in the outer ring pass through the rolling elements. Physically, this is the number of balls or rollers that pass a point on the outer race each time the shaft rotates one complete revolution. If there is a crack or a pitting defect in the outer race, every time a rolling element moves over it, a small impact occurs, generating vibrations at this frequency. When performing vibration analysis, an increase in amplitude at this frequency suggests an outer race fault. \[\text{BPFO}=\text{rpm}\frac{\text{N}_{\text{b}}}{2}\bigg{(}1-\frac{\text{b}} {\text{P}}\text{cos}\ \beta\bigg{)}\] (1)
* Ball pass frequency of the inner race or BPFI: The frequency with which rolling elements cross defects in the inner ring. Physically, this is the number of balls or rollers that pass a point on the inner race each time the shaft completes a full revolution. A defect on the inner race produces periodic impacts as each rolling element moves over it, generating vibrations at this frequency. Unlike outer race defects, inner race defects usually change frequency with load or speed variations because the inner ring rotates with the shaft. \[\text{BPFI}=\text{rpm}\frac{\text{N}_{\text{b}}}{2}\bigg{(}1+\frac{\text{b}} {\text{P}}\text{cos}\ \beta\bigg{)}\] (2)  
Figure 2: Construction parameters of the bearing: D is the bearing outer diameter, d is the bearing inner diameter, b is the diameter of each ball, P is the pitch diameter. Ball spin frequency or BSF is the failure frequency of the rollers. Physically, this is the number of revolutions a bearing ball makes each time the shaft completes one full revolution. If a rolling element itself has a defect (such as a dent or crack), it will generate vibrations at this frequency as it rolls and contacts the races.