# 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
\begin{array}{cc}\mathbf{A}_{i}&0\\ 0&\mathbf{A}_{i}\end{array} \right] \tag{10}\]  
In addition, there is a transformation matrix [20]:  
\[\mathbf{T}_{i}=\left[ \begin{array}{cc}\mathbf{I}&\mathbf{U}_{i}\\ 0&\mathbf{I}\end{array} \right] \tag{11}\]  
In the Formula (11), \(I\) is the third-order unit matrix, and \(U_{i}\) is determined by the relative coordinates shown in Equation (5), according to the following rules:  
\[\mathbf{U}_{i}=\mathbf{U}(\overline{x}_{i},\overline{y}_{i},\overline{z}_{i})=\left[ \begin{array}{ccc}0&\overline{z}_{i}&-\overline{y}_{i}\\ -\overline{z}_{i}&0&\overline{x}_{i}\\ \overline{y}_{i}&-\overline{x}_{i}&0\end{array} \right] \tag{12}\]  
According to Equation (13), the four elastic elements can be equivalent to the stiffness matrix relative to the antenna centroid coordinate system [20]:  
\[\mathbf{K}_{s} =\sum\limits_{i=1}^{4}\mathbf{T}_{i}^{\mathrm{T}}\mathbf{R}_{i}^{\mathrm{ T}}\mathbf{K}_{0}\mathbf{R}_{i}\mathbf{T}_{i}=\sum\limits_{i=1}^{4}\left[ \begin{array}{ cccc}\mathbf{A}_{i}^{\mathrm{T}}\mathbf{K}_{\mathrm{T}}\mathbf{A}_{i}&\mathbf{A}_{i}^{ \mathrm{T}}\mathbf{K}_{\mathrm{T}}\mathbf{A}_{i}\mathbf{U}_{i}\\ \mathbf{U}_{i}^{\mathrm{T}}\mathbf{A}_{i}^{\mathrm{T}}\mathbf{K}\mathbf{A}_{i}&\mathbf{U}_{i}^{ \mathrm{T}}\mathbf{A}_{i}^{\mathrm{T}}\mathbf{K}_{\mathrm{T}}\mathbf{A}_{i}\mathbf{U}_{i}\end{array}  \right]\] (13) \[=4k\left[\begin{array}{cccccc}0&0&0&0&0&0\\ 0&S_{\theta}{}^{2}&0&aS_{\theta}{}^{2}+bS_{\theta}{}C_{\theta}&0&0\\ 0&0&C_{\phi}{}^{2}&0&0&In which, \(S_{\theta}=\sin\theta\), \(C_{\theta}=\cos\theta\).  
Consider the pendulum effect caused by gravity, according to the horizontal stiffness:  
\[k_{G}=\frac{mg}{(l-a)C_{\theta}} \tag{14}\]  
Thus, the comprehensive stiffness matrix of the antenna damping system can be expressed as:  
\[K=\left[ \begin{array}{cccc}k_{G}&0&0&0&0&0\\ 0&4{K}{S_{\theta}}^{2}+k_{G}&0&4k({a}{S_{\theta}}^{2}+b{S}_{\theta}C_{\theta} )&0&0\\ 0&0&4{K}{C_{\theta}}^{2}&0&0&0\\ 0&4k({a}{S_{\theta}}^{2}+b{S}_{\theta}C_{\theta})&0&4k({a}{S_{\theta}}+b{C}_{ \theta})^{2}&0&0\\ 0&0&0&0&4k{L}^{2}{C_{\theta}}^{2}&0\\ 0&0&0&0&0&(4k{S_{\theta}}^{2}+{k}_{G}){L}^{2}\end{array} \right] \tag{15}\]  
In the same way, the same method can be used to define the damping characteristics of the damping element, and the damping matrix \(C\) of the damping system can be calculated using the same method in Equation (13). The mass matrix of the antenna is denoted as \(M\), and the six-degree-of-freedom displacement of the center of mass of the antenna is denoted as \(\mathbf{q}=\left[ \begin{array}{cccc}x&y&z&\alpha&\beta&\gamma\end{array} \right]^{ \mathrm{T}}\). The antenna vibration differential equation determined by the damping system is:  
\[\mathbf{M}\ddot{\mathbf{q}}+\mathbf{C}\dot{\mathbf{q}}+\mathbf{K}\mathbf{q}=0 \tag{16}\]  
The natural frequency of the antenna damping system in the radial direction of the antenna is recorded as \(f_{n}\), and the damping rate is recorded as \(\zeta\). The ratio of the interested vibration frequency \(f_{\mathrm{c}}\) to the natural frequency of the vibration isolation is recorded as \(\lambda=f_{\mathrm{c}}/f_{\mathrm{n}}\). According to the above description, the vibration transmission rate of the designed antenna damping system must be not greater than \(1/\gamma=0.01\) at frequency \(f_{\mathrm{c}}\), that is:  
\[T_{\lambda}=\sqrt{\frac{1+4\zeta^{2}\lambda^{2}}{\left(1-\lambda^{2}\right)^{ 2}+4\zeta^{2}\lambda^{2}}}\leq\frac{1}{\gamma} \tag{17}\]  
From Equation (17), it can be obtained that:  
\[\lambda\geq\lambda_{\mathrm{m}}=\sqrt{1+2\zeta^{2}(\gamma^{2}-1)+\sqrt{\gamma^ {2}-1+\left[1+2\zeta^{2}(\gamma^{2}-1)\right]^{2}}} \tag{18}\]  
This means that the natural frequency \(f_{n}\) of the damping system must satisfy Equation (19), so that the vibration transmission rate at the interested frequency point can be ensured to be lower than the expected value.