## 3 Thermal-Induced Structural Dynamic Analysis of Antenna
### Thermal Analysis
When the solar radiation angle of the horizontal beam \(\beta\) is equal to \(\beta_{0}\), they are \(\beta=\beta_{0}\)\({}^{\circ}\), \(\beta=90\)\({}^{\circ}\) - \(\beta_{0}\), \(\beta=135\)\({}^{\circ}\) - \(\beta_{0}\), \(\beta=\beta_{0}-45\)\({}^{\circ}\), as shown in Figure 7.  
By following the steps in reference [20], the thermal load vectors on the thin-walled beam caused by thermal shock can be expressed as:  
\[\mathbf{F}_{T}=EA\alpha_{T}\left(\frac{-\overset{-}{\mathbf{T}_{i}}+\overset{ -}{\mathbf{T}_{j}}}{2}-\mathbf{T}_{0}\right)\left[ \begin{array}{cccccccccccc} -1&0&0&0&0&0&1&0&0&0&0&0\end{array} \right]^{\mathrm{T}} \tag{19}\]  
\[\mathbf{M}_{T}=\frac{EI\alpha_{T}}{RI}\Big{[} \begin{array}{cccccccccccc}0&0&0 &0&-I\overset{(1)}{\mathbf{T}_{i}}&I\overset{(2)}{\mathbf{T}_{i}}&1&0&0&0&I \overset{(1)}{\mathbf{T}_{j}}&-I\overset{(2)}{\mathbf{T}_{j}}\end{array} \Big{]} ^{\mathrm{T}} \tag{20}\]  
where node \(i\) and node \(j\) are in the same beam, \(\overset{-}{\mathbf{T}_{i}}\) and \(\overset{-}{\mathbf{T}_{i}}\) are defined as the average temperatures for node \(i\) and \(j\), and \(\overset{(1)}{\mathbf{T}_{i}}\) and \(\overset{(2)}{\mathbf{T}_{i}}\) are the perturbation temperatures on the cross section. \(\overset{-}{\mathbf{T}_{0}}\) is the initial temperature. The \(\alpha_{T}\) is the coefficient of thermal expansion. The correspondingly thermal-induced generalized force \(\overset{-}{\mathbf{P}_{local}}\) for node \(i\) and node \(j\) is derived as:  
\[\mathbf{P}_{\mathrm{local}}=\mathbf{F}_{T}+\mathbf{M}_{T} \tag{21}\]  
In Equation (21), the first six elements represent the generalized forces of node \(i\), and the rest of the elements represent the generalized forces of node \(j\). Then, based on Equations (15) and (16), the thermal-induced vibration response in the general coordinate system for the antenna can be obtained. To explain the causes of thermally induced dynamical behaviors of the antenna, thermal-structural dynamics of the single hoop-truss are analyzed. All the parameters used in the simulations are listed in Table 4.  
Considering the solar radiation angle of the horizontal beam is \(\beta=\beta_{0}=60^{\circ}\), the corresponding solar radiation angle for vertical beam, left inclined beam, and right inclined beam are \(\beta=30^{\circ}\), \(\beta=15^{\circ}\), and \(\beta=75^{\circ}\), respectively. Figures 8 and 9 respectively show the thermal response of the node's average temperatures and perturbation temperatures for beams with different solar radiation angles. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Units** & **Parameter** & **Value** & **Units** \\ \hline \(E\) & \(2.1\times 10^{11}\) & Pa & \(k_{x}\) & \(16.6\) & W/m-K \\ \(v\) & 0.3 & - & \(k_{\varphi}\) & \(16.6\) & W/m-K \\ \(\rho\) & 7860 & kg/m\({}^{3}\) & \(\varepsilon\) & \(0.13\) & - \\ \(c\) & 502 & J/kg-K & \(\alpha_{s}\) & \(0.6\) & - \\ \(l\) & 2 & m & \(\alpha_{T}\) & \(1\times 10^{-8}\) & \(1/\)K \\ \(R\) & 0.009733 & m & \(S_{0}\) & \(1350\) & W/m\({}^{2}\) \\ \(r\) & 0.00953 & m & \(\beta_{0}\) & \(60\) & degree \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Thermal structure data of spatial beam thin-walled beam. Figure 7: The spatial beam imposed on solar radiation. Taking the horizontal beam that \(\beta=60^{\circ}\) with node 9 and node 25 as an example, the thermal responses of average temperature \(\mathbf{T_{9}}\) for node 9 denoted as the blue line are increased until they reach their steady-state values, as shown in Figure 8. The perturbation temperatures \(\mathbf{T_{9}}^{(1)}\), \(\mathbf{T_{9}}^{(2)}\) are shown in Figure 9a,b, which are also increased with time until reaching their steady-state values. It can be found that the perturbation temperature responses are much faster than the average temperature responses which are stable in 100 s. The temperature responses for node 25 can be performed in a similar way.