## 2 Finite Element Analysis of Antenna Structures  
As shown in Figure 1b, the whole structure of the radio telescope antenna can be briefly separated into two parts: the reflector and the alidade. Above the alidade is the antenna's reflector, which mainly includes the primary reflector surface, the sub-reflector, the BUS, the quadruped, the elevation gear, and the umbrella structure. These structures are primarily composed of a large number of bars. According to the antenna's motion characteristics, in the antenna structure finite element analysis, the antenna structural bar can be discretized into bar elements or beam elements [19] and solved by the finite element stiffness equation.  
We illustrate the basic idea of antenna finite element analysis with a beam element as an example. In finite element analysis, [PERSON]'s law can be used to calculate the elongation of the elastic beam under tensile load or the shortening of the rod due to compressive load. The law is  
\[\sigma=E\epsilon \tag{1}\]  
where \(\sigma\) represents the stress, \(E\) represents the elastic modulus, and \(\epsilon  
where \(J\) represents the torsional moment inertia on the beam's cross-section. \(G\) represents the shear modulus; according to elastic mechanics, its expression is  
\[G=\frac{E}{2(1+v)} \tag{5}\]  
where \(v\) represents the Poisson's ratio. On the oxy plane, the stiffness matrix corresponding to the nodal displacement \(v_{1}\), \(\theta_{z1}\), \(v_{2}\), and \(\theta_{z2}\) generated by the bending problem is  
\[K^{e}_{oxy}=\frac{EA}{l^{3}}\begin{bmatrix}12&6l&-12&6l\\ 6l&4l^{2}&-6l&2l^{2}\\ -12&-6l&12&-6l\\ 6l&2l^{2}&-6l&4l^{2}\\ \end{bmatrix} \tag{6}\]  
By replacing \(I_{z}\) in Formula (6) with \(I_{y}\), the stiffness matrix expressions corresponding to the nodal displacement \(w_{1}\), \(\theta_{y1}\), \(w_{2}\), and \(\theta_{y2}\) generated by the bending problem on the \(oxz\) plane can be obtained.  
The elements in the torque stiffness matrix, bending moment stiffness matrix, and axial force stiffness matrix are assembled based on the order of node displacement in Equation (1), and the beam element global stiffness matrix can be obtained:  
\[K^{e}=K^{e}_{u_{1}u_{2}}\oplus K^{e}_{\theta_{x1}\theta_{z2}}\oplus K^{e}_{ oxy}\oplus K^{e}_{oxz} \tag{7}\]  
where \(\oplus\) represents the appropriate combination of matrices to ensure that the corresponding influence of the degree of freedom of each node is correctly considered. From the Formulas (1), (2), and (7), the beam element stiffness equation can be obtained as follows:  
\[K^{e}q^{e}=P^{e} \tag{8}\]  
where \(q^{e}\) and \(P^{e}\) are the node force vector and displacement vector of the element respectively. The beam element stiffness equation in the global coordinate system can be obtained using the direction cosine between the local coordinate \(oxyz\)'s axis and the overall coordinate \(OXZ\)'s axis.  
When the element's stiffness equations are solved, these stiffness matrixes are assembled according to the determined connection relationship, and the antenna FE model's stiffness equation is obtained:  
\[Kq=P \tag{9}\]  
where \(q\) represents the displacements at each node in the overall coordinate system, \(K\) represents the global stiffness matrix, and \(P\) represents each node's nodal force vector.  
Considering the influence of uncertainties such as manufacturing errors and assembly defects on the antenna structure's deformation characteristics, the antenna structure global stiffness equation of the antenna is as follows:  
\[(K+\Delta K)\;\mathbf{u}=\mathbf{f} \tag{10}\]  
where \(\Delta K\) represents the stiffness matrix error between the FE model and the antenna. According to the Formula (10), when the node force vector \(f\) of the FE model in the global coordinate system remains constant, the overall stiffness matrix \(K\)'s change will affect the node displacement vector. Therefore, updating the global stiffness matrix can increase the closeness between the FE model deformation characteristics and the antenna.