# Design and Experiment of Magnetic Antenna Vibration and Noise Reduction System
## 3 Analysis and Design of Damping System
### Identification of Mechanical Characteristic Parameters of Silicone Rubber Elastic Elements
We adjusted the position of the fastening point to set the initial effective length under the stress-free state as \(l_{0}=0.2\) m, and ensured that each silicone rubber strip always remained vertical under the load state. We changed the load mass \(m_{T}\) applied to a single piece of silicone rubber strip sample in turn, and tested the length \(l\) of each piece of silicone rubber strip sample in a static equilibrium state to obtain the elongation change value \(\Delta l=l-l_{0}\) of the sample. Then, we tested the vertical first-order natural vibration frequency \(f_{T}\) of the load near its static equilibrium position when a single piece of silicone rubber strip sample was suspended with different load masses \(m_{T}\). The strain of the silicone rubber strip sample under load \(m_{T}\) is:  
\[\varepsilon=\Delta l/l_{0} \tag{21}\]  
After being stretched, the cross-sectional area of the silicone rubber strip will decrease correspondingly, which conforms to the characteristics of super-elastic materials within a certain stress range. Therefore, according to the change of elongation, the actual cross-sectional area of the silicone rubber strip sample under different strains can be calculated. Then the axial tensile stress of the silicone rubber strip sample under different load conditions can be calculated, as shown in the following Equation (21):  
\[\sigma=\frac{m_{T}g}{s_{0}l_{0}/(\Delta l+l_{0})}=\frac{m_{T}g(1+\varepsilon) }{s_{0}} \tag{22}\]  
The stress-strain characteristics of this silicone rubber strip can be expressed as follows using the Ogden model [21]:  
\[\sigma=\sum_{i=1}^{N}\frac{2\mu_{i}}{\alpha_{i}}(\epsilon^{\alpha_{i}-1}- \varepsilon^{-\frac{1}{2}\alpha_{i}-1}) \tag{23}\]  
Using the above-mentioned silicone rubber strip samples to do actual tests, after obtaining the stress and strain under different loads, the fitting was performed according to the Ogden model described in Equation (23). We obtained the fitting parameters \(\mu_{1}=-2.72\), \(\mu_{2}=-0.31\), \(\mu_{3}=3.88\) and \(\alpha_{1}=-10.64\), \(\alpha_{2}=6.00\), \(\alpha_{3}=-10.65\). The comparison of the actually measured stress-strain curve of the silicone rubber strip and the fitting result is shown in Figure 4. The results show that the fitting results can accurately characterize the mechanical properties of the material, and according to that, the detailed design of the silicone rubber strip required can actually be carried out. According to the actual load-deformation measurement results of the silicone rubber strip sample, the static stiffness-load mass curve can be obtained through different calculations. According to the actual measurement results of the vibration frequency of the resonant system under different load masses, and the conversion according to the following formula, the dynamic stiffness-load mass curve of the silicone rubber strip sample under the corresponding load can be obtained. \[k_{d}=4\pi^{2}f_{\mathrm{T}}^{2}m_{\mathrm{T}} \tag{24}\]  
The actual measurement results are shown in Figure 5, which shows that the static and dynamic stiffness of this silicone rubber strip exhibit strong nonlinear characteristics, and the dynamic stiffness is much higher than its static stiffness. When this material is applied to the antenna damping system described in this article, the length and cross-sectional dimensions of the silicone rubber strip are undetermined parameters. Therefore, it is necessary to use the equivalent Young's modulus independent of the length and cross-sectional size of the silicone rubber strip to determine its characteristics, which will facilitate subsequent design calculations. Figure 6 shows the equivalent static  
Figure 4: The measured stress–strain curve and fitting result of the silicone rubber strip sample. Figure 5: The relationship between the static and dynamic stiffness of the silicone rubber strip and the load mass.