## 3 Micro-Vibration Test Principle  
### The Quasi-Zero Stiffness Characteristics Analysis  
There are two gravity unloading methods: rigid support and flexible suspension. Both states will introduce additional perturbation, which will affect the micro-vibration test accuracy. Compared with rigid support, the disturbance peak caused by the flexible suspension system appears at a low frequency, close to 0 Hz [34]. Therefore, this paper adopted a set of adjustable elastic ropes and a high-stiffness sling to achieve flexible suspension. In addition, during the actual measurement, traction ropes were added on both sides of the load. As studies have shown in the past years, the suspension system can be simplified as a simple pendulum system or a double pendulum system to make the stiffness study [5; 9; 35]. Although there may be a risk of forward and backward torsion on the pendulum, due to the strict symmetry of the disturbance source and the introduction of the traction rope, this problem is negligible compared to the longitudinal vibration along the X direction and simple pendulum motion. After analysis, the AQZSSD suspending the HPSOP as shown in Figure 4a can be simplified to a simple pendulum system as in Figure 4b.  
Combining the kinetic energy equation \(T=\frac{1}{2}m_{OL}\left(\dot{x}^{2}+(l_{0}+x)^{2}\dot{\theta}^{2}\right)\) and the potential energy equation \(V=\frac{1}{2}k_{C}x^{2}+m_{OL}g(l_{0}+x)(1-\cos\theta)\), the dynamic equation of the simple pendulum system can be expressed as follows:  
Figure 3: Automatic adjustment mechanism based on AQZSSD.  
\[\left\{\begin{array}{c}\ddot{x}+\frac{k_{C}}{m_{OL}}x+\frac{g(1-\cos\theta)}{m_{ OL}}-(l_{0}+x)\dot{\theta}^{2}=0\\ \ddot{\theta}+\frac{g}{l_{0}+x}\sin\theta+\frac{2}{l_{0}+x}\dot{x}\dot{\theta}=0 \end{array}\right. \tag{2}\]  
where \(x\) stands for the changes in an elastic rope; \(\theta\) represents the swing angle of the simple pendulum system; \(m_{OL}\) expresses the mass of the HPSOP and AQZSSD equivalent to 300 kg (\(m_{OL}=m_{HP}+m_{AQ}\), where \(m_{HP}\) and \(m_{AQ}\), measured by the electronic hanging scale, indicate the mass value of the HPSOP and AQZSSD, respectively); \(k_{C}\) indicates the comprehensive stiffness of the parallel suspended system composed of 16 elastic ropes (since the stiffness value test of each elastic rope made of latex is equal to 130 N/m, it can be obtained that \(k_{C}=16\times 130\) N/m from the stiffness calculation theory of spring parallel systems); and \(l_{0}\) represents the initial length 3.5 m of the latex elastic rope without the optical payload.  
Since the swing angle of the suspension system changes caused by micro-vibrations is little, Equation (2) can be simplified as:  
\[\left\{\begin{array}{c}\ddot{x}+\frac{k_{C}}{m_{OL}}x+\frac{g}{2}\theta^{2}-( l_{0}+x)\dot{\theta}^{2}=0\\ \ddot{\theta}+\frac{2}{l_{0}+x}\dot{x}\dot{\theta}+\frac{g}{l_{0}+x}\theta=0 \end{array}\right. \tag{3}\]  
Obviously, the vertical eigenfrequency \(f_{v}=\frac{1}{2\pi}\sqrt{\frac{k_{C}}{m_{OL}}}\approx 0.42\) Hz can be obtained by solving \(\ddot{x}+\frac{k}{m}x=0\), an equation simplified by Equation (3) while only considering the vertical vibration of the simple pendulum system corresponding to \(\theta=0\). Similarly, the horizontal eigenfrequency \(f_{h}=\frac{1}{2\pi}\sqrt{\frac{g}{l_{0}+m_{OL}S/k_{C}}}\approx 0.22\) Hz can also be obtained by solving \(\ddot{\theta}+\frac{g}{l_{0}+x}\theta=0\), another equation simplified by Equation (3) while only considering the horizontal vibration of the simple pendulum system corresponding to \(x\approx\frac{m_{OL}g}{k_{C}}\).