# Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection
## 3 Validation Experiment
### Case 3: Compound Fault Signal  
In this section, the proposed method is applied to identify compound faults. The experimental object is a vibration signal generated by a motor from the University of Paderborn in Germany [35]. The experimental setup is shown in Figure 14. This experimental setup consists of a bearing housing and an electric motor. The motor powers the shafts of 6203-type test bearings installed within the bearing housing, and these test bearings rotate under a radial load applied by a spring-screw mechanism. The vibration signal used in this paper is generated by a 6203 bearing, which is one of the components that experienced faults during the experiment, the physical and geometric parameters of the bearing are shown in Table 4. Figure 15 shows the corresponding spectra and direct envelope spectra of these signals. In the envelope spectrum of the compound fault, it is difficult to find frequency information corresponding to the frequency band of the inner and outer race. By applying the LMWT method, this paper aims to detect and identify different types and locations of bearing faults to verify the feasibility and effectiveness of the proposed method. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Diameter of inner race & 24 mm & Diameter of outer race & 33.1 mm \\ Pitch circle diameter & 28.55 mm & Number of rolling elements & 8 pc. \\ Rolling element diameter & 6.75 mm & Length of rolling element & 6.75 mm \\ Nominal pressure angle & 0 rad & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Geometric characteristics of 6203 bearing. Figure 14: The schematic of the Paderborn bearing dataset, which includes a drive motor (1), housing (2), adjusting nut (3), and spring packge (4). On the experimental test bench shown in Figure 14, a 6203 bearing numbered KB23 bearing with inner and outer race compound fault was tested; the specific fault information of this signal is given in Table 5. And the frequencies of different faults for the 6203 bearing have been shown in Table 6. The vibration signal was measured at a sampling rate of 64 kHz and a constant speed of 900 RPM. Based on its physical and geometric parameters, the rotation frequency \(F_{r}=15\) Hz can be calculated using the formula. The BPFO of the outer race fault is 46.585 Hz, and the BPFI of the inner race fault is 73.9415 Hz, as shown in Table 6. These calculated characteristic frequencies are based on the kinematic and dynamic knowledge of the bearing and can help to identify and locate different types and locations of bearing faults, providing a foundation and basis for subsequent signal processing and analysis. In order to fully capture the fault feature information in the bearing vibration signal, the proposed LMWT method is applied to analyze the vibration signal shown in Figures 16 and 17. Firstly, a one-level decomposition is performed on the vibration signal \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**RPM (r/min)**} & \multicolumn{3}{c}{**Characteristic Frequency of Bearing Fault**} \\ \cline{2-4}  & **BFPO (Hz)** & **BFPI (Hz)** & **\(F_{r}\) (Hz)** \\ \hline
900 & 46.0585 & 73.9415 & 15 \\ \hline \end{tabular}
\end{table} 
Table 6: The fault characteristic frequency of 6203 bearing. Figure 15: (**a**) The vibration signals of compound fault signal in Paderborn bearing dataset and its (**b**) spectrum, (**c**) envelope spectrum. In Table 7, the relative energy ratios extracted by EWT are almost close to 0 for inner race faults, and the maximum relative energy ratio for outer loop faults is only 0.016. And the average relative energy ratios of the inner race fault obtained by LMWT is 0.068, and the average relative energy ratio corresponding to the outer race fault is 0.278, which is significantly higher than the EWT method.