## 5 Conclusions  
This study focuses on thermally induced dynamical behaviors and control strategy of the satellite's attitude motion and hoop-truss antenna's vibration due to thermal shock from solar flux.  
The rigid-flexible coupled dynamic model of spacecraft is derived by using the law of conservation of momentum and angular momentum. An analytical approach based on a finite element method is developed for thermal-structural dynamic analysis of the antenna. The modal shapes and natural frequencies of antenna obtained from the finite element method and the results obtained from Multiphysics FEA verify the proposed method. The numerical results indicate that the thermally induced deformations and vibrations of the antenna are mainly caused by average temperature, rather than perturbation temperatures.  
The coupled responses demonstrate that the vibrations have significant influences on the attitude motion; the attitude motions also in turn affect the vibrations. The double-loop sliding mode variable structure control method is adopted to reduce the vibration while  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time/s** & **Rolling Bias/Deg** & **Yawing Bias/Deg** & **Pitching Bias/Deg** \\ \hline
0.5 & 3.840 \(\times\) 10\({}^{-3}\) & 3.959 \(\times\) 10\({}^{-3}\) & 5.570 \(\times\) 10\({}^{-3}\) \\
0.7 & 2.647 \(\times\) 10\({}^{-3}\) & 2.617 \(\times\) 10\({}^{-3}\) & 4.496 \(\times\) 10\({}^{-3}\) \\
0.85 & 2.348 \(\times\) 10\({}^{-3}\) & 2.348 \(\times\) 10\({}^{-3}\) & 4.228 \(\times\) 10\({}^{-3}\) \\
1 & 2.319 \(\times\) 10\({}^{-3}\) & 2.319 \(\times\) 10\({}^{-3}\) & 4.198 \(\times\) 10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Angular velocity biases.  
Figure 18: Diagram of rolling moment, yawing moment, and pitching moment.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time/s** & **Rolling Bias/Deg** & **Yawing Bias/Deg** & **Pitching Bias/Deg** \\ \hline
0.5 & 3.130 \(\times\) 10\({}^{-3}\) & 1.674 \(\times\) 10\({}^{-3}\) & 4.298 \(\times\) 10\({}^{-3}\) \\
0.65 & 2.189 \(\times\) 10\({}^{-3}\) & 9.872 \(\times\) 10\({}^{-4}\) & 7.625 \(\times\) 10\({}^{-4}\) \\
1 & 1.313 \(\times\) 10\({}^{-3}\) & 6.569 \(\times\) 10\({}^{-4}\) & 4.055 \(\times\) 10\({}^{-4}\) \\
2 & 7.461 \(\times\) 10\({}^{-4}\) & 3.649 \(\times\) 10\({}^{-4}\) & 3.892 \(\times\) 10\({}^{-4}\) \\
7 & 5.087 \(\times\) 10\({}^{-5}\) & 4.853 \(\times\) 10\({}^{-5}\) & 5.472 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Angular biases.  
maintaining the attitude. The control system is composed of the inner angular velocity loop and the outer angular loop. The simulation results show the controller can accurately track the attitude angles and restrain the thermal vibrations efficiently.  
Conceptualization and methodology, Y.-J.Q.; software, X.L.; validation, Y.L.; formal analysis, H.-L.Y. All authors have read and agreed to the published version of the manuscript.  
This work research was funded by the National Natural Science Foundation of China (Project No. 11772009), the Beijing Municipal Natural Science Foundation (Project No. 1192002), and the State Key Laboratory of Mechanical System and Vibration (Grant no. MSV202107).  
Not applicable.  
The data that support the plots within this paper and other findings of this study are available from the corresponding author on request.  
This work was supported in part by the National Natural Science Foundation of China (Project No. 11772009), the Beijing Municipal Natural Science Foundation (Project No. 1192002), and the State Key Laboratory of Mechanical System and Vibration (Grant no. MSV202107).  
The authors declare no conflict of interest.