## 3 Diagnosis and Verification Analysis of Fault Signal Based on Acoustic Testing of MWA  
In order to verify the effectiveness of the acoustic measurement and analysis, an experiment was performed by the use of a normal MWA and an MWA with a wear retainer. Two sound microphones were positioned at a distance of 90 cm in front of the MWA with 0 degrees and 45 degrees of the rotor's axis separately, as illustrated in Figure 7. The experiment was executed three times for each MWA with a sampling frequency of 102.4 kHz and a sampling time of 60 s. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Order** & **Frequency (Hz)** & **Damping Ratio** & **Order** & **Frequency (Hz)** & **Damping Ratio** \\ \hline
1 & 607.2 & 0.26 & 11 & 1616.1 & 0.37 \\
2 & 730.7 & 0.45 & 12 & 1780.6 & 0.33 \\
3 & 817.2 & 0.20 & 13 & 1965.5 & 0.32 \\
4 & 863.7 & 0.45 & 14 & 2331.7 & 0.33 \\
5 & 935.2 & 0.12 & 15 & 2615.5 & 0.21 \\
6 & 1062.6 & 0.49 & 16 & 2776.2 & 0.26 \\
7 & 1164.1 & 0.34 & 17 & 2989.9 & 0.30 \\
8 & 1244.2 & 0.10 & 18 & 3083.9 & 0.37 \\
9 & 1293.5 & 0.38 & 19 & 3347.6 & 0.19 \\
10 & 1483.0 & 0.44 & 20 & 3740.2 & 0.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The first 20 th resonant frequencies and damping ratios. Figure 6: The vibration waterfall diagram. Cage failure is most common in spacecraft bearing failures [21]. However, the rolling elements, and internal and external raceway pitting faults cannot be ignored either [22]. When there is a local fault in the inner ring, the outer ring, or the rolling element of the rolling bearing, an impact of a certain characteristic frequency is generated, causing periodic pulsating vibration of the bearing element [23]. This periodic pulse has a short acting time and a steep shape, and its impact frequency is determined by factors such as the geometry of the bearing, the fault location, and the rotational speed of the shaft. For a particular bearing, the unique fault characteristic frequency produced by each component is determined under known steady speed. Therefore, these characteristic frequencies are usually used to diagnose and isolate the fault component of rolling bearing [24]. Here, the characteristic frequencies are used to check whether the noise signal contains the pulse vibration information raised by defects. The theoretical calculation formulas are listed as follows for the characteristic frequency of the cage \(f_{c}\), the characteristic frequency of the outer ring \(f_{\sigma}\), the characteristic frequency of the inner ring \(f_{\mu}\), and the characteristic frequency of the rolling elements \(f_{b}\)[25]:  
\[f_{c}=\left(\frac{1}{2}\right)\star\left\{f_{0}\left(1+\frac{D_{b}\cos[ \alpha]}{D_{m}}\right)+f_{i}\left(1-\frac{D_{b}\cos[\alpha]}{D_{m}}\right)\right\} \tag{1}\]  
\[f_{\sigma}=\left(\frac{Z}{2}\right)\star\left(1+\frac{D_{b}\cos(\alpha)}{D_{m }}\right)\star\left|f_{0}-f_{i}\right| \tag{2}\]  
\[f_{\mu}=\left(\frac{Z}{2}\right)\star\left(1-\frac{D_{b}\cos(\alpha)}{D_{m}} \right)\star\left|f_{0}-f_{i}\right| \tag{3}\]  
\[f_{b}=\left(\frac{D_{m}}{2D_{b}}\right)\star\left(1-\frac{\left(D_{b}\cos( \alpha)\right)^{2}}{D_{m}^{2}}\right)\star\left|f_{0}-f_{i}\right| \tag{4}\]  
where \(f_{i}\) and \(f_{0}\) represent the rotation frequency of the inner and outer rings of the bearing, respectively. \(Z\) is the number of rolling elements, which is equal to 9 in this paper. \(D_{\text{m}}\) is the diameter of the bearing pitch and \(D_{\text{m}}=31\) mm. \(D_{\text{b}}\) is the diameter of rolling elements and \(D_{\text{b}}=6\) mm. \(\alpha\) is the contact angle and \(\alpha=15^{\circ}\). If the characteristic frequency is found in the test signal, it indicates that the corresponding component of the bearing has a fault. Empirical modal decomposition (EMD) might be considered as the most common approach for the faulty feature extraction. EMD is an adaptive signal processing method which was first put forward by [PERSON] in 1998 [26].