# Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection
## 3 Validation Experiment
### Case 2: Single Fault Signal  
In this subsection, this paper evaluates the ability of the proposed method to identify different faults of bearings by using single-fault bearing vibration signals obtained from the CRWU Bearing Data Center [34]. The experimental setup schematic is shown in Figure 7. This experimental setup comprises a bearing and accelerometer(located on the left side of the figure), a torque transducer/dencoder (connected in the middle of the figure), and a dynamometer (positioned on the right side of the figure). Additionally, the experimental setup is driven by a 1.5 kW (2 horsepower) electric motor (not depicted in the figure). The vibration signal used in this work is from a 6205-2 RS JEM SKF bearing, and the geometric parameters of the bearing are shown in Table 1. In this study, the vibration signal used corresponds to faults in the rolling elements, inner race, and outer race. In each case, the vertical acceleration on the drive end bearing housing was measured, and sampling rate of 12 kHz was used for these cases. Figure 5: Envelope spectrum obtained by LMWT (**a**) and EWT (**b**). Figure 6: Relative energy ratio obtained by LMWT and EWT. According to the physical characteristics shown in Table 1, the characteristic frequencies related to outer race fault can be calculated using the following formula:  
\[BPFO=\frac{n_{b}}{2}F_{r}(1-\frac{d}{D}\cos(\alpha)) \tag{11}\]  
To calculate the fault frequency associated with the inner race fault, the following formula can be used:  
\[BPFI=\frac{n_{b}}{2}F_{r}(1+\frac{d}{D}\cos(\alpha)) \tag{12}\]  
To calculate the fault frequency associated with the rolling element fault, the following formula can be used:  
\[BSF=\frac{D}{2d}(1-(\frac{d\cos(\alpha)}{D})^{2}) \tag{13}\]  
where \(n_{b}\) is the number of rolling elements, \(D\) is the pitch diameter of the bearing in mm, \(d\) is the diameter of the rolling element in mm, and \(F_{r}\) is the bearing's rotating frequency in Hz. Table 2 presents the fault frequencies of different faults for the bearing 6205-2 RS JEM SKF at 1730 rpm in the CRWU bearing dataset. Figure 8 shows the vibration signals corresponding to three types of faults, i.e., the rolling element fault Figure 8a, the inner race fault Figure 8b, and the outer race fault Figure 8c, at a speed of 1730 RPM. Figure 9 shows the corresponding spectra and direct envelope spectra of these signals. In the envelope spectrum of the rolling element fault, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Bore diameter & 25 mm & Outside diameter & 52 mm \\ Overall width & 15 mm & Pitch diameter & 39 mm \\ Ball diameter & 8 mm & Rolling element number & 9 pc. \\ Contact angle & 0 rad & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric characteristics of the 6205-2 RS JEM SKF bearing. Figure 7: The schematic of the CRWU bearing dataset. From left to right include accelerometer (1), bearing (2), torque transducer and encoder (3) and dynamometer (4). it is difficult to find frequency information corresponding to the fault frequency band. Compared with the rolling element fault, the fault frequency bands of the inner race and outer race are more easily found in the envelope spectra, and the corresponding peak frequencies are 156.1325 Hz and 103.3675 Hz, respectively. This indicates that for inner race and outer race faults, the characteristic frequencies are more obvious in the low-frequency region and are easier to extract and analyze. For rolling element faults, the characteristic frequencies are not easily extracted and analyzed accurately due to the influence of background noise. In order to comprehensively capture the fault feature information in the vibration signals of bearings, the LMWT method proposed in this paper is applied to analyze the vibration signals of bearings with different faults shown in Figure 8. First, the vibration  
Figure 8: The vibration signals of bearing with different fault in CWRU bearing dataset.