## 3 Thermal-Induced Structural Dynamic Analysis of Antenna
### Thermal Analysis  
Since the heat transfer function is dependent on the local deformation of the structure, the classical FEM cannot treat this type of coupling well. The proposed method in this section combines the FEM and generalized method to take into account the local deformation-dependent heat transfer effect. The beams of the hoop-truss antenna structure are simplified as thin-walled beams as shown in Figure 6. The beams are subjected to solar flux \(S_{0}\) as the hoop-truss antenna exits the Earth's shadow. For the thin-walled beams that are subject to solar radiation in the space, their exterior surface would absorb solar radiation heat flux, at the same time, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{E}\) (Pa)} & **Poisson’s** & \multirow{2}{*}{\(\boldsymbol{\rho}\) (kg/m\({}^{3}\))} & \multirow{2}{*}{\(\boldsymbol{L}\) (m)} & \multirow{2}{*}{\(\boldsymbol{A}\) (m\({}^{2}\))} & \multirow{2}{*}{\(\boldsymbol{J}\) (m\({}^{4}\))} \\  & **Ratio** & & & & \\ \hline \(2.1\times 10^{11}\) & 0.3 & 7860 & 2 & \(1.2\times 10^{-5}\) & \(1.1\times 10^{-9}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the hoop-truss antenna. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Natural Frequency** & **1 st Order Q\({}_{1}\)** & **2 nd Order Q\({}_{2}\)** & **3 rd Order Q\({}_{3}\)** & **4 th Order Q\({}_{4}\)** & **5 th Order Q\({}_{5}\)** & **6 th Order Q\({}_{6}\)** \\ \hline Results by Equation (15) & 0.0956 & 0.1450 & 0.2656 & 0.5135 & 0.5621 & 0.9534 \\ Multiphysics FEA & 0.0945 & 0.1453 & 0.2660 & 0.5151 & 0.5632 & 0.9541 \\ Relative error & 1.15\% & 0.2\% & 0.15\% & 0.3\% & 0.2\% & 1.0\% \\ Vibration type & Roll & Bend & Respiration & Torsion & Respiration & Respiration \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The frequencies of hoop-truss antenna obtained by Equation (15) method and Multiphysics FEA. According to [PERSON]'s law for the heat conduction and neglecting the temperature variation across the wall thickness. The transient heat transfer in the thin-walled beams of the hoop-truss is given by:  
\[\frac{\partial\mathbf{T}}{\partial t}-\frac{k_{x}}{\rho c}\frac{\partial^{2} \mathbf{T}}{\partial x^{2}}-\frac{k_{\varphi}}{\rho cR^{2}}\frac{\partial^{2} \mathbf{T}}{\partial\varphi^{2}}+\frac{\sigma\varepsilon}{\rho ch}\mathbf{T}^{ 4}=\frac{\alpha_{s}S_{0}\cos\beta}{\rho ch}\varsigma\cos\varphi \tag{17}\]  
where \(\mathbf{T}\) is the temperate, \(c\) is the specific heat, \(k_{x}\) and \(k_{\varphi}\) are respectively the thermal conductivity along axis-direction \(x\) and circumferential direction \(\varphi\), \(R\) is the radius of the beam, \(\sigma\) is the Stefan-Boltzmann constant, \(\varepsilon\) is the surface emissivity, \(\alpha_{s}\) is the surface absortivity, and \(h\) is the thickness of the wall. The heat flux \(S\) is the projection of solar flux \(S_{0}\), which is always perpendicular to the normal direction \(n\) of the cross section, as shown in Figure 6. The \(\alpha\) is the angle between solar flux \(S_{0}\) and the \(z\) direction of the cross section, which is different for each element beam as shown in Figure 6. The parameter \(\varsigma\) is introduced to indicate whether the beam surface is exposed to solar flux. \[\varsigma=\left\{ \begin{array}{ll}1,&\frac{\pi}{2}-\alpha<\varphi<\frac{3\pi }{2}-\alpha\\ 0,&-\frac{\pi}{2}-\alpha<\varphi<\frac{\pi}{2}-\alpha\end{array} \right. \tag{18}\]  
Particularly, in our study, the solar radiation angle \(\beta\) for different beams can be divided into four types according to the angle between solar radiation \(S_{0}\) and the perpendicular direction to the normal direction \(n\).