# Design and Experiment of Magnetic Antenna Vibration and Noise Reduction System
## 3 Analysis and Design of Damping System  
### Principle Model of Vibration Damping System and Matching Analysis of Main Parameters  
According to the geometric parameters of the magnetic antenna and the mounting base, as well as the arrangement and interface constraints of the damping elastic elements, the schematic diagram of the antenna damping system is shown in Figure 2. It is necessary to design a vibration damping system in a horizontal cylindrical space with a diameter of \(D\) to isolate the large-scale vibration transmitted from the mounting base to the antenna. The vibration transmission rate along the radial direction of the antenna at a frequency of 60 Hz should be less than 0.01 (\(-\)40 dB), and ensure that the minimum distance between the antenna and the mounting base \(\delta\geq\) 0.02 m in the static balance state. The relevant parameters are listed in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline
**Physical Quantity** & **Symbol** & **Unit** & **Value** \\ \hline Antenna quality & \(m\) & kg & 6.50 \\ \hline Moment of inertia of antenna & \(I_{yy}=I\) & kgm\({}^{2}\) & 0.348 \\ \hline Diameter of antenna & \(D_{1}\) & m & 0.06 \\ \hline length of antenna & \(L_{1}\) & m & 0.80 \\ \hline The vertical distance between the damping fixed point and the center of the antenna & \(a\) & m & 0.018 \\ \hline The lateral distance between the damping fixed point and the center of the antenna & \(b\) & m & 0.036 \\ \hline The axial distance between the damping fixed point and the center of the antenna & \(L\) & m & \(0.3\leq L\leq 0.35\) \\ \hline Diameter of the distribution circle of the vibration damping fixed point on the & \(D\) & m & 0.15 \\ \hline The minimum distance between the antenna and the inner wall of the base in the & \(\delta\) & m & \(\delta\geq 0.02\) \\ \hline The height difference between the antenna and the center of the base in the static & \(h\) & m & \((D-D_{1})/2-\delta\) \\ \hline The angle between the elastic element and the vertical profile in the static & \(\theta\) & deg & to be determined \\ \hline The length of the elastic element in static balance state & \(l\) & m & \(\leq(D/2+c)/cos\theta\) \\ \hline The stiffness of the elastic element & \(k\) & N/m & to be determined \\ \hline \end{tabular}
\end{table} 
Table 2: The structure and mechanical parameters of the antenna and its damping system. Figure 2: Structure and dimension diagram of the antenna and its damping system. The antenna damping system shown in Figure 2 consists of four elastic elements with tensile stiffness. In the static balance state, the coordinates of the suspension point on the antenna relative to the centroid of the antenna are given as:  
\[\left\{ \begin{array}{l}(\overline{x}_{1},\overline{y}_{1},\overline{z}_{1})=(L,-b,-a)\\ (\overline{x}_{2},\overline{y}_{2},\overline{z}_{2})=(L,b,-a)\\ (\overline{x}_{3},\overline{y}_{3},\overline{z}_{3})=(-L,-b,-a)\\ (\overline{x}_{4},\overline{y}_{4},\overline{z}_{4})=(-L,b,-a)\end{array} \right. \tag{5}\]  
In the state of static equilibrium, the angle relationship between the axis of each elastic element and the vertical plane passing through the rotation center of the antenna is as the following formula:  
\[\varphi_{1}=\varphi_{3}=-\varphi_{2}=-\varphi_{4}=\theta \tag{6}\]  
The translational stiffness matrix of the elastic element in its own local coordinate system is denoted as:  
\[\mathbf{K}_{\mathrm{T}}=\mathrm{diag}(0,0,k) \tag{7}\]  
Then, the six-degree-of-freedom stiffness matrix of the elastic element in its own local coordinate system is:  
\[\mathbf{K}_{0}=\mathrm{diag}(\mathbf{K}_{\mathrm{T}},0)=\mathrm{diag}(0,0,k,0,0,0) \tag{8}\]  
The directional cosine matrix [19] from the global coordinate system to the local coordinate system of each elastic element is:  
\[A_{i}=\mathbf{A}(\varphi_{i},0,0)=\left[ \begin{array}{ccc}1&0&0\\ 0&\cos\varphi_{i}&\sin\varphi_{i}\\ 0&-\sin\varphi_{i}&\cos\varphi_{i}\end{array} \right] \tag{9}\]  
According to literature [20], there is a rotation matrix:  
\[\mathbf{R}_{i}=\left[