## 3 Thermal-Induced Structural Dynamic Analysis of Antenna
### Modal and Frequency Analysis for Antenna by Finite Element Method  
Generally, structures similar to plate or beam in the infinite system can be discretized by the modal shapes of the corresponding static structure. However, the structure of the hoop-truss antenna is relatively complex, and it is difficult to find the appropriate assumed modal shapes. In this study, the hoop-truss antenna is assembled by 64 thin-walled beams and 32 nodes, as shown in Figure 3. Each thin-walled beam element is regarded as a two-node element and subjected to the axial force, shear force, bending moment, and torsional moment. A finite element model is established to obtain the coordinates of the nodes, and the corresponding modal shapes matrix for the hoop-truss antenna can be found. In our model, the length of the transverse or vertical element beam is \(L\) and the cross-section area of the beam element is \(A\). Considering constraints between the hoop-truss antenna and the connecting boom, node 1 and node 17 are set as fixed points. Taking the thin-walled beam element with node \(i\) and node \(j\) as an example, as shown in Figure 4, each node has six degrees of freedom including displacements \(u\), \(v\), and \(w\) and rotation angles of the plane cross section \(\theta_{x}\), \(\theta_{y}\), \(\theta_{z}\) in its local coordinate system \(O\)-\(xyz\). The element has two nodes and twelve generalized coordinates that can be expressed as:  
\[\mathbf{\delta}_{\text{local}}=\left[ \begin{array}{ccccccccccccc}u_{i}&v_{i}&w_{ i}&\theta_{xi}&\theta_{yi}&\theta_{zi}&u_{j}&v_{j}&w_{j}&\theta_{xj}&\theta_{yj}& \theta_{zj}\end{array} \right]^{\text{T}} \tag{13}\]  
and node generalized force can be expressed as:  
\[\mathbf{P}_{\text{local}}=\left[ \begin{array}{ccccccccccccc}U_{i}&V_{i}&W_{ i}&M_{xi}&M_{yi}&M_{zi}&U_{j}&V_{j}&W_{j}&M_{xj}&M_{yj}&M_{zj}\end{array} \right]^{ \text{T}} \tag{14}\]  
in which \(U\) denotes the axial force, \(V\), \(W\) respectively denote the sheer forces in \(x\)-\(z\) plane and \(x\)-\(y\) plane, \(M_{x}\) denotes the torque, and \(M_{y}\) and \(M_{z}\) denote the bending moments in \(x\)-\(z\) plane and \(x\)-\(y\) plane, respectively. Figure 4: Schemes follow the same formatting. Beam element and local coordinate system \(O\)-\(xzy\). Figure 3: Deployable antenna model assembled by 64 thin-walled beams and 32 nodes. Neglecting the internal damping and the microgravity of space, the equations of motion for the antenna in the global coordinate system \(O_{b}\)-\(X_{b}Y_{b}Z_{b}\) are given by:  
\[\mathbf{M}\ddot{\mathbf{\delta}}+\mathbf{K}\mathbf{\delta}=\mathbf{P} \tag{15}\]  
where \(\mathbf{\delta}\) is the nodal displacement vector, \(\mathbf{M}\) is the mass matrix, \(\mathbf{K}\) is the stiffness matrix in the global coordinate system, and:  
\[\mathbf{K}=\lambda^{\mathrm{T}}\mathbf{K}_{\mathrm{local}}\lambda,\mathbf{M}= \lambda^{\mathrm{T}}\mathbf{M}_{\mathrm{local}}\lambda,\mathbf{P}=\lambda^{ \mathrm{T}}\mathbf{P}_{\mathrm{local}},\mathbf{\delta}=\lambda\mathbf{\delta}_ {\mathrm{local}} \tag{16}\]  
in which \(\mathrm{M}_{\mathrm{local}}\) and \(\mathrm{K}_{\mathrm{local}}\) are the mass matrix and the stiffness matrix in the local coordinate system which are both 12 \(\times\) 12. \(\lambda\) is the transformation matrix between the local coordinate system and the global coordinate system. The detailed explanations for Equations (15) and (16) and the expressions for \(\mathbf{K}\), \(\mathbf{M}\), \(\mathbf{P}\), \(\boldsymbol{\lambda}\) can be found in reference [17]. The parameters of the hoop-truss antenna are listed in Table 1. Since the beam is where node 1 and node 17 are fixed, all the degrees of freedom of node 1 and node 17 are set as 0 during calculation in this study.