# Parallel Load-Bearing and Damping System Design and Test for Satellite Vibration Suppression
## 4 FE Modeling and Analysis for the Damping System
### FE Analysis for the Damping System  
In order to obtain the appropriate parameters for the damping system, FE analysis was necessary. By changing the number and the damping coefficient of the dampers, five combinations were considered, as shown in Table 2. By comparing these analysis results, the first and fourth results' damping effects were shown to be not good enough. The fifth result's damper number was too large. The best two results were the second and the third. Considering that a higher damping coefficient means higher weight of the structure, which is not expected, result two could fulfill the requirements. The final damping system was composed of eight dampers with the coefficient of 2.8 million N-s/m.  
In order to confirm that the introduction of the dampers on the PAF will not affect the structure's original characteristics, FE analysis results of the original state and the viscous damping state were compared, as shown in Figure 6. Results of the finite element analysis are shown in Table 3. The skins and stringers affected the structure's transverse bending frequency and longitudinal frequency, respectively. As seen from Table 3, after adding dampers in a viscous damping state, transverse and longitudinal frequencies reduced by less than 0.1 Hz compared with the original model. As a result, the addition of dampers has little effect on the stability of an attitude control system and a fairing's dynamic clearance.  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**No.**} & \multirow{2}{*}{
\begin{tabular}{c} **Damping Coefficients** \\ **(* Number of the** \\ **Dampers)/(N-s/m)** \\ \end{tabular} } & \multirow{2}{*}{**Axial/(m/s\({}^{2}\))**} & \multirow{2}{*}{**Lateral/(m/s\({}^{2}\))**} & \multirow{2}{*}{**Attenuation Percentage**} \\ \cline{5-5}  & & & & \\ \hline / & No dampers & 37.7 & 27.3 & / & / \\
1 & 700,000 (* 8) & 32.5 & 23.4 & 13.79\% & 14.29\% \\
2 & 2,800,000 (* 8) & 22.5 & 20.1 & 40.32\% & 26.37\% \\
3 & 3,200,000 (* 8) & 21.3 & 20.3 & 43.50\% & 25.64\% \\
4 & 2,800,000 (* 4) & 27.8 & 22.0 & 26.26\% & 19.41\% \\
5 & 1,200,000 (* 16) & 22.5 & 20.1 & 40.32\% & 26.37\% \\ \hline \end{tabular}
\end{table}
Table 2: Damping coefficient designs and results.  
Figure 5: Finite element modeling of the damping system on the Payload Adaptor Fitting (PAF).  
As shown in Section 2, the axial peak frequency and acceleration transmissibility of the original state in the experiment were 28.33 Hz and 14.05 m/s\({}^{2}\), respectively. Compared with the experiment results, the FEA results were 28.56 Hz and 14.1 m/s\({}^{2}\), respectively. The frequency difference was 0.8%, while the transmissibility difference was 0.4%. The discrepancies could be negligible, considering the error factors.  
The removal of the partial skins may affect the strength of the PAF. Therefore, two harsh working conditions were chosen to indicate whether the strengths were within the requirement: (1) the second stage engine shut down; (2) the satellite is in a transonic condition. The material of the model was aluminum alloy 2A14, and the parameters are shown in Table 4. The maximum stresses were 57.9 MPa and 131 MPa, respectively, which are less than the yield stress of material 2A14. The results prove that the PAF would not be damaged after introducing the damping system.