**5. Study on Micro-Vibration Signal Denoising of Reaction Wheels**

### *5.1. Micro-Vibration Test*

As important attitude control components of a satellite, reaction wheels have the general characteristics of rotating machinery. The specific structure of a reaction wheel is depicted in Figure 9. It primarily consists of a rotor supported by ball bearings encased in a housing and driven by a brushless direct current (DC) motor. Influenced by some factors such as the internal rotor imbalance, bearing imperfections, and structural modes, a reaction wheel generates disturbance forces and moments during running. The negative impact of the disturbances is unacceptable for the normal operation of the payloads in satellites [39]. To ensure the successful implementation of the operations in space, it is necessary to analyze the micro-vibration characteristics of reaction wheels.

An on-ground micro-vibration test is frequently performed as an approach to study the micro-vibration characteristics of reaction wheels. It is conducted using the Kistler micro-vibration test device, as depicted in Figure 10. During the operation of a reaction wheel, the disturbance response is transmitted to the force measurement platform through the transfer tool. Then, the micro-vibration signals collected by piezoelectric sensors are transmitted to a data acquisition (DAQ) system via a charge amplifier, which are displayed and processed on a computer.

**Figure 9.** Structure of a reaction wheel. DC: direct current.

**Figure 10.** Micro-vibration test of a reaction wheel. DAQ: data acquisition.

The micro-vibration signals of the reaction wheel are collected at different rotational speeds (0–2000 rpm). Performing the fast Fourier transform on the time domain signals, three-dimensional waterfall diagrams of the radial and axial disturbances are obtained, as shown in Figure 11. The vibration of the reaction wheel mainly concentrates on the radial disturbance forces *Fx*, axial disturbance forces *Fz*, and radial disturbance torque *Mx*. Relatively, the magnitude of the axial disturbing moment *Mz* is small, which can be ignored. Therefore, the analysis of the micro-vibration signals is carried out in *Fx*, *Fz* and *Mx*.

**Figure 11.** Waterfall diagrams of a reaction wheel: (**a**) *Fx*; (**b**) *Fz*; (**c**) *Mx*; (**d**) *Mz*.

### *5.2. Analysis of Micro-Vibration Denoising*

Excluding the environmental factors, the noise of micro-vibration signals is also derived from the internal torque fluctuations and frictional interference. To separate the noise component from the micro-vibration signals, a general processing method called peak threshold denoising is used in reaction wheels at present. Based on the amplitude statistical characteristics of the noise, the threshold value to remove the noise from the original signal is determined. It is described as [40]

$$DT = \mu + N\_{\delta} \cdot \delta\_{\prime} \tag{22}$$

where μ and δ are the mean and standard deviation of the spike amplitude, respectively, and *N*<sup>δ</sup> is a user-defined tolerance level, which also depends on the SNR of the sampling signals. Generally, the value of *N*<sup>δ</sup> can be 2 or 3.

In the study of a reaction wheel under the ultimate working conditions, the micro-vibration signals at 1800 and 2000 rpm are selected for the denoising analysis. The frequency of interest is set within 500 Hz, which is the main frequency band that causes satellite jitter. The threshold values are calculated according to different tolerance levels, as shown in Figure 12. Similarly, WTD and EMD–SG are used to suppress noise in micro-vibration signals, as shown in Figure 13.

**Figure 12.** Peak threshold denoising of the micro-vibration signals: (**a**) 1800 rpm; (**b**) 2000 rpm.

**Figure 13.** Spectra of micro-vibration signals by WTD and EMD–SG: (**a**) 1800 rpm; (**b**) 2000 rpm.

As exhibited in Figure 12, the magnitude of the user tolerance level directly affects the final effect of the signal denoising. In Figure 12a, a reaction wheel generates a large disturbance force at 131.5 Hz owing to the coupling of the harmonic responses and structural modes, which increases the threshold value to filter out some critical frequency features. Some obvious feature frequencies are equally easy to be removed in Figure 12b, such as 20 Hz. As shown in Figure 13, WTD and EMD–SG mainly act on high-frequency of test signals, which appear under-denoising in the low-frequency range and lose super-harmonics. The filtered details frequently indicate that the system experiences a significant motion mechanism, which is not conducive to the subsequent characteristic analysis and fault diagnosis.

Owing to inappropriate parameter setting and resonance coupling, these denoising methods can easily cause phenomena of over-denoising and under-denoising. Therefore, AIC–SVD is introduced into the pre-processing of the micro-vibration signals of the above-mentioned reaction wheel. By constructing the Hankel matrix of micro-vibration signals, the singular values are solved by SVD. Owing to the presence of colored noise in the micro-vibration signals, the improved AIC is used to determine the order of the effective singular value by correcting the eigenvalues. According to the calculation results as shown in Figure 14, the indices of minimum AIC value is selected to reconstruct

the approximate matrixes. Once the time series signals are restored by the averaging method, denoised frequency spectra are obtained, as shown in Figure 15.

**Figure 14.** AIC diagrams at different rotational speeds: (**a**) 1800 rpm; (**b**) 2000 rpm.

**Figure 15.** AIC–SVD denoised frequency spectrum of *Fx*: (**a**) 1800 rpm; (**b**) 2000 rpm.

By comparing Figures 12, 13 and 15, it is observed that AIC–SVD can effectively eliminate the noise from the micro-vibration signals. The denoised signals are convenient in the extraction of harmonic features. As shown in Figures 16 and 17, the micro-vibration signals of *Fz* and *Mx* are processed by AIC–SVD. And the related parameters of the reaction wheel are listed in Table 3.

**Figure 16.** AIC–SVD denoised frequency spectra of *Fz*: (**a**) 1800 rpm; (**b**) 2000 rpm.

**Figure 17.** AIC–SVD denoised frequency spectra of *Mx*: (**a**) 1800 rpm; (**b**) 2000 rpm.

**Table 3.** Characteristic parameters of micro-vibration signals by AIC–SVD.


The data listed in Table 3 provide all the harmonic coefficients and related frequencies. The average running time of AIC–SVD is 443 s, which is mainly caused by SVD of matrices at the high sampling frequency. Combined with the analysis of the disturbance mechanism, it reveals that the denoised signals include a fundamental harmonic caused by the rotor imbalance, a sub-harmonic of 0.6 times frequency caused by the bearing cage defects, and super-harmonics. Super-harmonics contain 4.4, 5, 5.6, 9.4, and 14.4 times frequency in both *Fx* and *Mx*, 5, 7.1, 7.5, and 10 times frequency in *Fz*, which are caused by the coupling of bearing imperfections.
