# Design and Experiment of Magnetic Antenna Vibration and Noise Reduction System
## 3 Analysis and Design of Damping System
### Design and Performance Analysis of Vibration Damping System for Silicone Rubber Strip  
For the antenna damping system composed of four silicone rubber strips, as shown in Figure 2, the length and cross-sectional area of the silicone rubber strip in the static equilibrium state are respectively 1 and s, and the angle between itself and the vertical longitudinal section is \(\theta\). In addition, the tensile stress of each silicone rubber strip is \(\sigma\), and the relationship between itself and the antenna mass \(m\) is as follows:  
\[\sigma=\frac{mg}{s\cos\theta}\leq\sigma_{\text{max}} \tag{25}\]  
The geometric parameters of the silicone rubber strip should also meet the following size and space constraints:  
\[[l\cos\theta-a-(D-D_{1})/2+\delta]^{2}+(b+l\sin\theta)^{2}\leq D^{2}/4 \tag{26}\]  
In addition, combined with Equation (20), the silicone rubber strip with tensile stress \(\sigma\) should also meet the dynamic stiffness condition:  
\[k_{\text{trans}}=\max(\frac{4E_{\text{d}}s}{l}\sin^{2}\theta+\frac{mg}{(l-a)C_ {\theta}},\frac{4E_{\text{d}}s}{l}\cos^{2}\theta)\leq\frac{4\pi^{2}lf_{\text{ c}}^{2}}{\lambda_{\text{m}}^{2}L^{2}} \tag{27}\]  
In which, \(E_{\text{d}}=E_{\text{d}}(\sigma)\) is the dynamic Young's modulus of the silicone rubber strip under tensile stress \(\sigma\), determined by Figure 7.  
In summary, the optimal design of the silicone rubber strip in the vibration damping system can be described as follows: under the constraints of size space and maximum stress, making the maximum radial stiffness of the vibration damping system to reach the minimum value, as shown in the Equation (28).  
\[\begin{array}{l}\text{min}k_{\text{trans}}=\max(\frac{4E_{\text{d}}s}{l}\sin ^{2}\theta+\frac{mg}{(l-a)C_{\theta}},\frac{4E_{\text{d}}s}{l}\cos^{2}\theta) \\ s.t.\Bigg{\{}\begin{array}{l}\sigma=\frac{mg}{s\cos\theta}\leq\sigma_{\text{ max}}\\ [l\cos\theta-a-(D-D_{1})/2+\delta]^{2}+(b+l\sin\theta)^{2}\leq D^{2}/4\end{array} \tag{28}\]  
Based on the above optimization method, the antenna damping system is designed as follows: the hardness of the silicone rubber strip is 40 HA, the cross-sectional area of the four pieces of silicone rubber strip in the free state is \(s_{0}=4.8\times 10^{-5}\) m\({}^{2}\), the free length is \(l_{0}=0.06\) m, and the distance of fixed damping point and the center of the antenna is  
Figure 6: The equivalent static Young’s modulus and dynamic Young’s modulus of a silicone rubber strip under different stress states.  
\(L=0.3\) m. In the state of static equilibrium, the cross-sectional area of the silicone rubber strip is \(s=3.2\times 10^{-5}\) m\({}^{2}\), the length is \(l=0.09\) m, the angle between the silicone rubber strip and the vertical longitudinal section is \(\theta=4.3^{\circ}\), and the tensile stress of the silicone rubber strip is \(\sigma=0.5\) MPa. Practice has proved that applying this silicone rubber strip configuration can meet the damping rate requirement \(\zeta\leq 0.05\).  
According to the above design, the structure of the antenna damping system is shown in Figure 7. The silicone rubber strips between the antenna and two annular polyoxymethylene (POM) structures serve as the vibration isolation system. The annular polyoxymethylene structures are assembled into a tubular fiberglass shell with radial clearance, and are axially clamped via a rubber gasket. The design value of the lateral natural frequency of the system is \(f_{y}=1.80\) Hz, the design value of the vertical natural frequency is \(f_{z}=3.32\) Hz, the design value of the natural frequency in the yaw direction is \(f_{rx}=2.31\) Hz, and the design value of the natural frequency in the pitch direction is \(f_{rx}=4.30\) Hz; all the natural frequencies mentioned above meet the requirement \(f_{n}\leq 4.7\) Hz.  
According to the above-mentioned design scheme, the actual antenna damping system is shown in Figure 8.