## 3 Results and Analysis of Multi-Physics Coupling
### Analysis of Cavity Length Change under the Random Vibration Experiment  
During aerospace missions, it is imperative to perform vibration tests on the FP cavity, which is one of the weakest components of a USL. Table 1 displays the parameters for random vibration testing using the Long March 5 rocket platform [36]. The test consists of three successive stages: rising, holding, and falling. Moreover, the parameters are more stringent than actual launch conditions to guarantee payload reliability. \begin{table}
\begin{tabular}{c c} \hline
**Frequency Range/Hz** & **Power Spectral Density** \\ \hline
10–50 & 3 db/oct (rising slope) \\
50–300 & 0.25 g\({}^{2}\)/Hz (holding value) \\
300–2000 & \(-12\) db/oct (falling slope) \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters of the random vibration tests. Figure 4: EEMD results of cavity length change along the 45\({}^{\circ}\) and \(-45^{\circ}\) directions. Figure 5: Kurtogram of cavity length change along the 45\({}^{\circ}\) and \(-45^{\circ}\) directions. The displacement power spectrum densities of the cavity length change responses at different points along the circle of radius 5 mm and 2.5 mm are shown in Figure 6. Various PSD responses along the logarithmic frequency curves at eight points every 45\({}^{\circ}\) around the circumference are presented, which first increase, then remain constant, and finally decrease as the test progresses. The PSD value at 180\({}^{\circ}\) is the highest, with radius 5 mm, while the lowest value occurs at 315\({}^{\circ}\). By contrast, the PSD values along radius 2.5 mm at 45\({}^{\circ}\), 135\({}^{\circ}\), 225\({}^{\circ}\), and 315\({}^{\circ}\) are far more than others (almost zero), which indicates a more complicated \(\Delta L\) distribution under the vibration test. Therefore, the parameter of 100 Hz which corresponds to the maximum PSD value of the FP cavity is chosen for the next analysis. Figure 7 shows the displacement and Von Mises stress distribution under the frequency of 100 Hz, where the color map is logarithmically scaled. Interestingly, the variation of the distribution is non-uniform, i.e., there are local maxima in addition to the four supporting areas, especially on the mirror surface. The Z component displacement of the plane mirror, the concave mirror, and the cavity length change are shown in Figure 8. It can be observed that the cavity length displacement has local maxima and minima around the circumference at 2.5 mm and 5 mm radii. Similarly, two stripe regions of data in Figure 8 with a width of 2 mm along the 45\({}^{\circ}\) and \(-\)45\({}^{\circ}\) were extracted separately and converted to a one-dimensional signal. Figure 6: Displacement power spectrum density of the cavity length change. Figure 7: Displacement and Von Mises stress distributions at frequency 100 Hz. The EEMD results of the cavity length change along the \(45^{\circ}\) and \(-45^{\circ}\) directions under the vibration test are shown in Figure 9. In the EEMD process, white noise with a standard deviation of 0.6 is added for the calculation and the number of realizations is set to 200. The IMFs are more complex compared to the fixed force condition's cavity length change, and additionally suffer from frequency aliasing (IMF3-6). The pattern of the EEMD results is almost identical along both directions, as the two sets of data appear to be highly similar. The kurtogram of the cavity length change \(\Delta l\) data along the \(45^{\circ}\) and \(-45^{\circ}\) directions during the vibration test are respectively computed and visualized in Figure 10. It can be seen that there is a higher maximum K value in the \(-45^{\circ}\) direction compared to the \(45^{\circ}\) direction, with both directions exhibiting low levels at \(K_{max}\), indicating that transient fluctuations of the signal have little influence, even if it appears highly nonstationary. Figure 8: Z component displacement of mirrors under random vibration test.