## 3 Solar Array Simulator
### Structure of the SAS  
Based on the dynamic characteristics of the multi-fold solar array with a connecting rod analyzed in Section 3.1, the SAS shown in Figure 6 is proposed. The SAS is composed of a central plate, a series of springs, an inertia frame, and four counterweights. After output by the SADA, the driving torque is transferred to the inertia frame through the central plate and a series of springs. When the spring extends and compresses in reaction to the SADA output torque, the inertia frame will rotate at the required frequency. Therefore, the torsional oscillation characteristics of the solar array can be simulated.  
According to the dynamic characteristics analysis of the structure, the kinetic energy of the system is as follows:  
\[T=\frac{1}{2}J_{SAS}\dot{a}^{2} \tag{21}\]  
where \(T\) is the kinetic energy of the system, \(J_{SAS}\) is the moment of inertia of the SAS along the axis of the drive shaft, and \(\alpha\) is the angle displacement of inertia frame.  
The potential energy of the system is:  
\[V=4\times\left[\frac{1}{2}k(d+l\times\alpha)^{2}-\frac{1}{2}kd^{2}+\frac{1}{2}k (d-l\times\alpha)^{2}-\frac{1}{2}kd^{2}\right]=4 kl^{2}\alpha^{2} \tag{22}\]  
where \(d\) is the initial deformation of the springs, \(k\) is the stiffness of the springs, and \(l\) is the distance of the spring from the center axis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Frequency (Hz)** & **Modal Effective Mass (kg\(\cdot\)m\({}^{2}\))** & **Modal Effective Mass Fraction** \\ \hline
1 & 0.22 & \(3.82\times 10^{-16}\) & \(4.97\times 10^{-16}\) \\
2 & 0.55 & \(2.80\times 10^{-4}\) & \(3.64\times 10^{-5}\) \\
3 & 1.18 & \(7.44\) & \(0.96  
The Lagrange function was constructed from Equations (21) and (22), and substituted into the Lagrange equation to obtain the dynamics equation of the SAS:  
\[J_{SAS}\ddot{\alpha}+8 kl^{2}\alpha=0 \tag{23}\]  
Furthermore, the fundamental frequency of the system can be obtained, which is the first torsional mode frequency:  
\[f=2l\sqrt{\frac{2k}{J_{SAS}}} \tag{24}\]  
Figure 7 shows the SAS designed in detail according to the structure sketch in Figure 6. The system consists mainly of a central shaft, a central plate, an inertia frame, four counterweights, four counterweight seats, four spring adjusting sliders, four spring seats, and a SADA connector. The SADA connector is fixedly connected at one end of the central axis, and the central plate is fixed to the central axis by bolts. The inertia frame and the central axle are connected through the bearing in the center of the inertia frame so that the inertia frame can rotate freely along the central axle. The inertia frame is oscillated by the spring connection between the central plate and the inertia frame. Therefore, the driving torque output by the SADA is transmitted according to the path: the SADA interface-the central shaft-the central plate-the springs-the inertia frame. Finally, the inertia frame is driven to rotate.  
There are adjustment mechanisms in the SAS to simulate different types of solar arrays. The counterweight seats and the spring-fixed seats can slide along the inner inertia frame, and the spring-adjusting seats can slide along the sliding column on the central plate. According to Equation (24), the torsional mode frequency of the SAS can be adjusted by changing the distance \(l\) of the spring from the central axis, spring stiffness \(k\), and moment of inertia \(J_{SAS}\). The distance \(l\) can be adjusted by moving the spring-fixed seats and the spring-adjusting seats, so the torsional mode frequency of the structure changes accordingly. The spring stiffness \(k\) can be adjusted by replacing the springs with different levels of stiffness. The moment of inertia \(J_{SAS}\) of the system can be adjusted by replacing the counterweights with different weights or by changing the distance of the counterweights from the center axis by moving the counterweight seats.