## 2 Acoustic Properties Analysis of an MWA's  
As the acoustic signal of an MWA is transmitted through its shell, it is essential to capture the acoustic properties before carrying out the fault diagnosis. A shell of an MWA is shown in Figure 2a, which is the core component for acoustic radiation. Therefore, the modal test of the shell can be a basis for the acoustic analysis as well as the health monitoring and evaluation of MWAs. Due to the light weight and small damping of the shell itself, the hammering method with multiple input single output (MISO) was used to obtain the modals of the shell, as illustrated in Figure 2b, and 367 nodes were selected to construct the shell structure during the modal test, as shown in Figure 3.  
Figure 1: The block diagram of the proposed method. Figure 3: Measurement points. Figure 2: The modal test of a moment wheel assembly’s (MWA’s) shell: (**a**) an MWA shell with grid; (**b**) modal test of an MWA. The modal analysis was performed by a modal test system by the China Orient Institute of Noise and Vibration. The exciting force was input by an force hammer. The data was collected by an acquisition instrument of the type INV306G(H), and the modal analysis was performed by the DASP-AutoModal modal analysis system. Figure 4a shows the lumped display of the transfer functions with all the frequency response function curves for the reference of modal determination. Since a high number of frequency response function curves make the information unclear, this paper averaged all the frequency response functions to obtain an averaged curve to determine the modal orders, which is shown in Figure 4b. Finally, the reliability of the experimental results was verified by the vibration mode correlation matrix by the DASP AutoModal analysis system. The row number and column number of the mode correlation matrix element represent two orders of mode, respectively. The extent of orthogonality of the two orders of mode shape represented by the value of element which is calculated by the normalizing scalar product of two orders of mode. The smaller the value, the better the extent of orthogonality. Ideally, the matrix is symmetric about the main diagonal, and the elements of the main diagonal are all \"1\". In practice, in addition to the main diagonal elements, the modal analysis results of the mode correlation matrix with very small values of other elements are usually more reliable. In this paper, the first nine modes are selected to verify the mode correlation matrix which is shown in Figure 5, in which the Z-axis value represents the correlation coefficient of two modes. It can be seen from Figure 5 that the values of diagonal elements in the matrix are much larger than others, indicating that the modal orthogonality obtained by the test is correct, and the test result is effective. Table 1 provides the first 20 th order resonant frequencies and damping ratios based on the DASP AutoModal analysis system with a frequency domain global modal parameter identification estimator. It can be seen from Table 1 that numerous resonance frequencies of the shell are mainly around the range of 600 Hz\(\sim\)4000 Hz. The vibration sensor at the shaft end was used to reveal the excitation signal when the bearing was running, and the waterfall diagram shown in Figure 6 was obtained. Figure 4: The frequency response functions: (**a**) transfer functions; (**b**) average transfer function. Figure 5: The correlation matrix of modals. It can be seen that there are more excitation components at the range of 600 Hz\(\sim\)4000 Hz, including the resonance frequency components that do not change with the rotation speed and the harmonic components that change with the rotation speed. Moreover, these resonant frequencies are not only in the range of resonant frequencies of the bearings but also in the range of frequencies that the human hearing is sensitive to [20]. This is why some faults in MWAs and CMGs can be detected by engineers with experience.