## 4 Validation of the Proposed Disturbance Torque Model  
### Experimental Results  
The Kistler 6-component piezoelectric sensors have been widely used in testing aerospace micro-vibration [3; 31; 32]. In the research, the disturbance force/torque of the sun-tracking solar array is measured by a micro-vibration test platform based on the Kistler 6-component piezoelectric sensors (HR-FP3406, Kistler, Winterthur, Zurich, Switzerland), as shown in Figure 8.  
Figure 7: A structure model of the SAS. A micro-disturbance force/torque measurement platform was built, which was composed of a Kistler 6-component piezoelectric sensor, a marble countertop, and an air isolation system. During the tests, the SAS was suspended by a rope to simulate the zero-gravity environment on orbit. The SADA was attached to the Kistler 6-component piezoelectric sensor fixed on the marble countertop, and the air isolation system was installed below the marble countertop to avoid external vibration during the test. The sampling frequency of the system was 4096 Hz, and the sampling time was 16 s.  
The internal driving source of the SADA studied in this work was a two-phase stepping motor. The beat number of the motor b was 4, the tooth number z was 50, and a 16-subdivision drive was adopted. The speed was 6\({}^{\circ}\)/s. Therefore:  
\[\Delta\theta=\frac{360^{\circ}}{bz}=1.8^{\circ} \tag{25}\]  
\[f_{d}=\frac{\omega_{m}}{\Delta\theta}n=53.33\;\mathrm{Hz} \tag{26}\]  
where \(f_{d}\) is the motor driving frequency. To investigate the coupling relationship between the SADA and the different types of solar arrays, the disturbance force/torque generated by the SADA-driven SAS in different states were measured. Table 2 shows the five test states of the SAS. Among them, states 1-3 have the same moment of inertia and different modal frequencies. States 1, 4, and 5 have the same modal frequency and different moments of inertia. Figure 9 shows the time-domain curve obtained by the state 1 test. As shown in Figure 9a, the amplitudes of the disturbance forces of the X, Y, and Z directions are of small magnitude, no greater than 0.725 N. According to engineering experience, the disturbance \begin{table}
\begin{tabular}{c c c} \hline
**State No.** & **Modal Frequency (Hz)** & **Rotational Inertia (kg-m2)** \\ \hline
1 & 1.19 & 8.18 \\
2 & 1.39 & 8.18 \\
3 & 1.59 & 8.18 \\
4 & 1.19 & 15.69 \\
5 & 1.19 & 22.99 \\ \hline \end{tabular}
\end{table} 
Table 2: List of modal frequencies and rotational inertias from the test in different states. Figure 8: Disturbance force/torque test. forces of this magnitude have little impact on spacecraft such as satellites [33]. As shown in Figure 9b, the amplitude of the disturbance torque in the Z direction is much bigger than that in the other two directions, and their peaks can reach 1.643 N-m, while the amplitudes of the disturbance torque in the X and Y directions are one magnitude smaller than that in the Z direction, and the peak value of both is not greater than 0.166 N-m. Therefore, it can be considered that the main form of disturbance generated by the SADA-driven SAS is the disturbance torque along the rotation axis. The time-domain signal of the disturbance torque along the z-direction during the stable operation of the SADA was intercepted, which was processed by fast Fourier transform and characterized in the frequency domain. Since the spacecraft is mainly susceptible to the disturbance in the middle and low frequency bands [7], the frequency band 0-200 Hz was analyzed, as shown in Figure 10.  
As shown in Figure 10, within the frequency range of 0-200 Hz, there are four peaks of the disturbance torque: 1.1875 Hz, 53 Hz, 106 Hz, and 159.0625 Hz. By classifying the four peaks, the frequency components of the disturbance torque can then be divided into two categories: (1) The first-order torsional natural frequency of the solar array, which is 1.1875 Hz; (2) The motor-driving frequency and its high-order harmonics (53 Hz, 106 Hz, and 159 Hz, etc.).