## 3 Numerical Analysis
### Center of Mass Calibration  
In the previous two sections, the influence of the reference tilt on the stiffness of the torsional pendulum was modeled and analyzed using Simscape Multibody. Feasible methods for amplifying the reference tilt based on counterweights for the inverted pendulum were also provided. However, it is difficult to completely eliminate the effects of microgravity drag by leveling the reference under the condition of default center of mass offset. Therefore, by modulating the tilt angle of the reference surface and calibrating the relative position of the center of mass of the torsional pendulum, the microgravity coupling and thrust noise of the pendulum frame can be monitored and compensated. To calculate the center of mass offset of the pendulum frame during torsion, a model shown in Figure 9 (front view) and Figure 10 (top view) can be used to represent the mass distribution of the pendulum frame:  
1. Divide the mass of the pendulum frame into symmetric parts that are balanced on the left and right sides, and an unbalanced part (counterweight block). The sum of the mass of the balanced parts on both sides is \(m_{1}\), and the mass of the unbalanced counterweight block is \(m_{2}\). Figure 8: Tilt angles amplified under different counter weights. Define the center of mass position of \(m_{1}\) as the coordinate origin (i.e., the pivot point), and the horizontal distance between the center of mass of the counterweight \(m_{2}\) and the pivot point as \(l_{cm_{2}}\). At this point, the overall mass of the pendulum frame is \(M=m_{1}+m_{2}\), and the center of mass position caused by the unbalanced counterweight is: \[l_{cm}=\frac{m_{2}}{m_{1}+m_{2}}\cdot l_{cm_{2}}.\] (10) Here, the overall center of mass offset of the pendulum frame is equivalent to the tilting moment caused by the counterweight alone. 3. In particular, the horizontal distance between the center of mass of the counterweight block and the pivot point on the vertical plane in the direction perpendicular to the pendulum arm is \(l_{cm_{2},y}\). The center of mass offset caused by the counterweight block and the coupling due to the reference pitch tilt is: \[Mg\sin\phi_{y}\cdot l_{cm,y}=m_{2}g\sin\phi_{y}\cdot l_{cm_{2},y}.\] (11)
4. Similarly, the horizontal distance between the center of mass of the counterweight block and the pivot point on the horizontal plane in the direction of the pendulum arm is \(l_{cm_{2},x}\). The center of mass offset caused by the counterweight block and the coupling due to the reference roll tilt is: \[Mg\sin\phi_{x}\cdot l_{cm,x}=m_{2}g\sin\phi_{x}\cdot l_{cm_{2},x}.\] (12)  
Using the models described above, it is possible to flexibly describe the center of mass offset of the pendulum frame and calculate the positions and weights of different structures, such as thrusters and counterweight blocks, which will help analyze and compensate for the coupling of microgravity in the torsional pendulum. Additionally, based on existing research, it is known that the effects of reference roll and pitch can be decoupled. Therefore, in the following analysis, only the tilt angle \(\alpha\) in the pitch direction will be considered, and the resulting changes in influence will be understood. Similarly, using the Simscape Multibody modeling, the coupling variations of different center of mass positions in the pendulum frame will be compared based on pitch and thrust. For example, by periodically adjusting the pitch angle of the platform at 25 \(\upmu\)rad and 0.1 Hz, and simultaneously applying synchronized thrust on the pendulum frame at 1 \(\upmu\)N and 0.11 Hz, the displacement amplitude spectral density (ASD) changes of the torsional pendulum will be observed when the counterweight is positioned at distances of 1 mm, 2 mm, 3 mm, 4 mm, and 5 mm from the axis. As shown in Figure 11, as the  
Figure 9: Equivalent model of centroid shift of torsion pendulum (front view).