Design and Experiment of Magnetic Antenna Vibration and Noise Reduction System

Featured Application

The proposed magnetic antenna damping system based on silicone rubber elastic element can effectively isolate the ambient vibration.

Abstract

The magnetic antenna can receive extremely weak underwater magnetic signals, and induce the magnetic field intensity which can reach the femto-tesla (fT) level, making it widely used in underwater electromagnetic signal detection. However, even the slightest vibration in the environment transmitted to the antenna will make it cut the geomagnetic field movement; as a result, electromagnetic-induced noise would be generated, and affect the performance of the magnetic antenna. The best method to suppress the motion induced noise is to isolate the magnetic antenna effectively from the ambient vibration. In this paper, the vibration reduction requirements of magnetic antenna are identified, and the principle of vibration absorber system of the magnetic antenna model is established. A magnetic antenna damping system, mainly focusing on low frequency, is proposed by using a silicon rubber elastic element. Based on the proposed method, a magnetic antenna damping system with an attenuation rate up to 100 times is designed, and some performance experiments are scheduled. The experiment results show that the performance of the magnetic antenna damping system meets the design requirements.

1. Introduction

In recent years, with the development of deep-sea exploration technology, the influence of the ocean on electromagnetic waves has become the focus of attention for many researchers [1]. The low-frequency electromagnetic wave has the advantages of low attenuation and strong anti-interference ability in seawater [2], as a reliable means of underwater electromagnetic signal detection. Compared with traditional underwater detection methods, magnetic antenna has many advantages in detection depth, direction, size and use mode, making it gradually become the main means of deep-sea detection. A magnetic antenna is essentially a device for receiving electromagnetic waves in space; by winding two separate coils around a magnetic bar, it can receive electromagnetic signals of femto-tesla level. However, once the weak vibration in the environment is transmitted to the antenna, it will cause the antenna to cut the geomagnetic field movement, resulting in electromagnetic induction noise, and then affect the performance of the magnetic antenna. For example, when an underwater vehicle’s engine is working or encounters the impact of ocean current, the magnetic antenna will produce large electromagnetic noise. Therefore, the suppression of this kind of motion induced noise becomes the key to the design and application of magnetic antenna.

In order to study the source of motion-induced noise and find the methods to suppress motion-induced noise, researchers have carried out a lot of research and experiments. In 1971, Manning [3] analyzed the vibration characteristics and noise of flexible ring antenna and pointed out that under the action of random fluctuating stress of seawater, the transverse vibration, longitudinal vibration and torsional vibration generated by magnetic antenna directly related to the performance of magnetic antenna by cutting the geomagnetic field and producing electromagnetic induction noise. Three years later, Burrows [4,5] proved that transverse vibration is the main cause of electromagnetic induction noise in antenna background noise by analyzing the motion-induced noise of magnetic antenna. Zhu et al. [6] studied the motion-induced noise and magnetostrictive noise of underwater antenna, analyzed the generation mechanism and calculation method of the motion-induced noise of antenna by electrodes, established the calculation formula of random pulsating pressure spectrum density of fluid, and obtained the estimation formula of electromagnetic noise of magnetic antenna. Lin et al. [7] investigated vibration-induced noise in extremely low frequency magnetic receiving antennas. The results show that the received useful signal would be easily lost within the antenna noise due to vibration of the antenna. Once proper vibration reduction technology is adopted, the antenna electromagnetic noise caused by vibration can been alleviated effectively. Many studies have reported the improvement of vibration isolation performance [8,9,10,11]. Quasi-zero stiffness techniques by using nonlinear X-shape structures [12], cam-roller-spring mechanisms [13], and bio-inspired structures [14,15] were proposed to acquire ultra-low vibration transmissibility in various applications. Despite the wide applications, most of them achieve quasi-zero stiffness by contacting-form structures, and suffer from friction, backlash, and space consuming. Magnetic negative stiffness mechanisms were proposed to counteract the positive stiffness so as to obtain quasi-zero stiffness without disturbances such as friction and backlash [16,17]. However, they introduce significant magnetic disturbance which will deteriorate the performance of the magnetic antenna. A vibration isolation system with ultra-low vibration transmissibility and zero magnetic disturbance is urgently required in the application of magnetic antenna.

In this paper, the vibration reduction requirements of magnetic antenna are identified, and the principle of vibration absorber system of the magnetic antenna model is established. A magnetic antenna damping system, mainly focused on low frequency, is proposed by using a silicon rubber elastic element. The prototype of the damping system was then verified by dynamic simulation and function comparison experiments. The experiment results prove that the damping method of magnetic antenna meets the design requirements.

2. Demand Analysis of Vibration Reduction

In essence, the induced noise of magnetic antenna motion is caused by the change of magnetic flux caused by the vibration of coil in the geomagnetic field, which mainly comes from fluid excitation and transmission vibration. Not long ago, Lin et al. [7] built a magnetic antenna test platform and simulated its working environment in underwater electromagnetic signal detection. By observing the measured spectrum of vibration acceleration signal, the signal spectrum at the concerned frequency f = 60 Hz in vibration state is about −65 dB, which translates to an acceleration of about 562 μg.

The structure of the magnetic antenna is shown in Figure 1. It is a rod-shaped structure, which is internally wrapped with magnetic cores and coils, with total length l_m = 0.8 m and weight m = 6.5 kg. Due to the strength of geomagnetic field H_earth ≈ 0.3 guass, according to the formula of electromagnetism B = μ × H_earth, the magnetic induction intensity B is 30 T. As the sensitivity of the magnetic antenna receiving signal acquisition equipment is 0.8 V/nT, and the effective signal amplitude is 200 fT [7], multiplication of the receiver sensitivity by the effective signal amplitude can obtain the corresponding induced voltage of 0.00016 V. In order to collect a valid signal, the induced electromotive force generated by the coil vibration must be less than 0.00016 V. The relevant parameters are listed in

Table 1. The parameters of the antenna.

Physical Quantity	Symbol	Unit	Value
Length of the antenna coil	l	m	0.512
Magnetic core permeability	μ	/	1 × 10 6
Number of turns	n	/	32,000
Cross-sectional area	S	m2	2.52 × 10−3
Strength of geomagnetic field	Hearth	guass	0.3
Concerned frequency	f	Hz	60
Magnetic field strength	B	T	30
Maximum induced electromotive force	εmax	V	0.00016
Maximum vibration acceleration	Amax	g	to be determined

.

Assume that the relative vibration acceleration and velocity of the two ends of the magnetic antenna with time are:

$$a=A\cdot \sin (2\pi ft)$$

(1)

$$v=-\frac{A}{2\pi f}\cdot \cos (2\pi ft)$$

(2)

Therefore, the maximum angular velocity of antenna rotation is given as:

$${\omega }_{\max }=\frac{V_{\max }}{l}=\frac{A_{\max }}{2\pi fl}$$

(3)

where l is the length of the antenna coil, Substitution into the induced electromotive force equation [18] yields the correlation formula of maximum induced electromotive force and maximum acceleration:

$${\epsilon }_{\max }=nBS{\omega }_{\max }=nBS\cdot \frac{V_{\max }}{l}=nBS\cdot \frac{A_{\max }}{2\pi fl}$$

(4)

Substitution of the parameters of antenna into Equation (4) can obtain the maximum vibration acceleration of magnetic antenna is A_max ≈ 5 × 10⁻⁵ m/s².

Therefore, at a frequency of 60 Hz, the vibration reduction system of the magnetic antenna must reduce the vibration acceleration transmitted to the magnetic antenna by the environment where the antenna is located to below 5 × 10⁻⁵ m/s², to 5 μg. According to the above, the vibration acceleration of the simulated magnetic antenna underwater detection is about 562 μg, so the vibration transfer rate of the vibration reduction system should be less than 0.01 (−40 dB).

3. Analysis and Design of Damping System

3.1. 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 δ ≥ 0.02 m in the static balance state. The relevant parameters are listed in

Table 2. The structure and mechanical parameters of the antenna and its damping system.

Physical Quantity	Symbol	Unit	Value
Antenna quality	m	kg	6.50
Moment of inertia of antenna	Iyy = I	kgm2	0.348
Diameter of antenna	D1	m	0.06
length of antenna	L1	m	0.80
The vertical distance between the damping fixed point and the center of the antenna	a	m	0.018
The lateral distance between the damping fixed point and the center of the antenna	b	m	0.036
The axial distance between the damping fixed point and the center of the antenna	L	m	$0.3\le L\le 0.35$
Diameter of the distribution circle of the vibration damping fixed point on the mounting base	D	m	0.15
The minimum distance between the antenna and the inner wall of the base in the static balance state	δ	m	$\delta \ge 0.02$
The height difference between the antenna and the center of the base in the static balance state	h	m	$(D-D_1)/2-\delta$
The angle between the elastic element and the vertical profile in the static balance state	θ	deg	to be determined
The length of the elastic element in static balance state	l	m	$\le (D/2+c)/cos\theta$
The stiffness of the elastic element	k	N/m	to be determined

.

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:

$$\{\begin{array}{l}({\overline{x}}_1,{\overline{y}}_1,{\overline{z}}_1)=(L,-b,-a)\\ ({\overline{x}}_2,{\overline{y}}_2,{\overline{z}}_2)=(L,b,-a)\\ ({\overline{x}}_3,{\overline{y}}_3,{\overline{z}}_3)=(-L,-b,-a)\\ ({\overline{x}}_4,{\overline{y}}_4,{\overline{z}}_4)=(-L,b,-a)\end{array}$$

(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:

$${\phi }_1={\phi }_3=-{\phi }_2=-{\phi }_4=\theta$$

(6)

The translational stiffness matrix of the elastic element in its own local coordinate system is denoted as:

$${\mathit{K}}_{\mathrm{T}}=\mathrm{diag}(0,0,k)$$

(7)

Then, the six-degree-of-freedom stiffness matrix of the elastic element in its own local coordinate system is:

$${\mathit{K}}_0=\mathrm{diag}({\mathit{K}}_{\mathrm{T}},0)=\mathrm{diag}(0,0,k,0,0,0)$$

(8)

The directional cosine matrix [19] from the global coordinate system to the local coordinate system of each elastic element is:

$${\mathit{A}}_i=\mathit{A}({\phi }_i,0,0)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos {\phi }_i& \sin {\phi }_i\\ 0& -\sin {\phi }_i& \cos {\phi }_i\end{array}\right]$$

(9)

According to literature [20], there is a rotation matrix:

$${\mathit{R}}_i=\left[\begin{array}{cc}{\mathit{A}}_i& 0\\ 0& {\mathit{A}}_i\end{array}\right]$$

(10)

In addition, there is a transformation matrix [20]:

$${\mathit{T}}_i=\left[\begin{array}{cc}\mathit{I}& {\mathit{U}}_i\\ 0& \mathit{I}\end{array}\right]$$

(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:

$${\mathit{U}}_i=\mathit{U}({\overline{x}}_i,{\overline{y}}_i,{\overline{z}}_i)=\left[\begin{array}{ccc}0& {\overline{z}}_i& -{\overline{y}}_i\\ -{\overline{z}}_i& 0& {\overline{x}}_i\\ {\overline{y}}_i& -{\overline{x}}_i& 0\end{array}\right]$$

(12)

According to Equation (13), the four elastic elements can be equivalent to the stiffness matrix relative to the antenna centroid coordinate system [20]:

$$\begin{array}{ll}{\mathit{K}}_s& ={\displaystyle \sum _{i=1}^4{\mathit{T}}_i^{\mathrm{T}}{\mathit{R}}_i^{\mathrm{T}}{\mathit{K}}_0{\mathit{R}}_i{\mathit{T}}_i}={\displaystyle \sum _{i=1}^4\left[\begin{array}{cc}{\mathit{A}}_i^{\mathrm{T}}{\mathit{K}}_{\mathrm{T}}{\mathit{A}}_i& {\mathit{A}}_i^{\mathrm{T}}{\mathit{K}}_{\mathrm{T}}{\mathit{A}}_i{\mathit{U}}_i\\ {\mathit{U}}_i^{\mathrm{T}}{\mathit{A}}_i^{\mathrm{T}}{\mathit{K}}_{\mathrm{T}}{\mathit{A}}_i& {\mathit{U}}_i^{\mathrm{T}}{\mathit{A}}_i^{\mathrm{T}}{\mathit{K}}_{\mathrm{T}}{\mathit{A}}_i{\mathit{U}}_i\end{array}\right]}\\ & =4k\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& S_{\theta }{}^2& 0& a{S}_{\theta }{}^2+b{S}_{\theta }{C}_{\theta }& 0& 0\\ 0& 0& C_{\theta }{}^2& 0& 0& 0\\ 0& a{S}_{\theta }{}^2+b{S}_{\theta }{C}_{\theta }& 0& {(a{S}_{\theta }+b{C}_{\theta })}^2& 0& 0\\ 0& 0& 0& 0& L^2{C}_{\theta }{}^2& 0\\ 0& 0& 0& 0& 0& L^2{S}_{\theta }{}^2\end{array}\right]\end{array}$$

(13)

In which, $S_{\theta }=\sin \theta$, $C_{\theta }=\cos \theta$.

Consider the pendulum effect caused by gravity, according to the horizontal stiffness:

$$k_G=\frac{mg}{(l-a)C_{\theta }}$$

(14)

Thus, the comprehensive stiffness matrix of the antenna damping system can be expressed as:

$$\mathit{K}=\left[\begin{array}{cccccc}{k}_G& 0& 0& 0& 0& 0\\ 0& 4k{S}_{\theta }{}^2+k_G& 0& 4k(a{S}_{\theta }{}^2+b{S}_{\theta }{C}_{\theta })& 0& 0\\ 0& 0& 4k{C}_{\theta }{}^2& 0& 0& 0\\ 0& 4k(a{S}_{\theta }{}^2+b{S}_{\theta }{C}_{\theta })& 0& 4k{(a{S}_{\theta }+b{C}_{\theta })}^2& 0& 0\\ 0& 0& 0& 0& 4k{L}^2{C}_{\theta }{}^2& 0\\ 0& 0& 0& 0& 0& (4k{S}_{\theta }{}^2+k_G)L^2\end{array}\right]$$

(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 $\mathit{q}={[\begin{array}{cccccc}x& y& z& \alpha & \beta & \gamma \end{array}]}^{\mathrm{T}}$. The antenna vibration differential equation determined by the damping system is:

$$\mathit{M}\ddot{\mathit{q}}+\mathit{C}\dot{\mathit{q}}+\mathit{K}\mathit{q}=0$$

(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 ζ. The ratio of the interested vibration frequency f_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_c, that is:

$$T_{\lambda }=\sqrt{\frac{1+4{\zeta }^2{\lambda }^2}{{(1-{\lambda }^2)}^2+4{\zeta }^2{\lambda }^2}}\le \frac{1}{\gamma }$$

(17)

From Equation (17), it can be obtained that:

$$\lambda \ge {\lambda }_{\mathrm{m}}=\sqrt{1+2{\zeta }^2({\gamma }^2-1)+\sqrt{{\gamma }^2-1+{[1+2{\zeta }^2({\gamma }^2-1)]}^2}}$$

(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.

$$f_{\mathrm{n}}\le f_{\mathrm{c}}/{\lambda }_{\mathrm{m}}$$

(19)

Considering the high requirements for damping performance in the concerned frequency band, based on engineering practice experience, it is proposed to design the antenna damping system as a small damping rate system with $\zeta \le 0.05$. Combined with $\gamma =100$, it can be concluded that ${\lambda }_{\mathrm{m}}\ge 12.78$. Therefore, the natural frequencies of the radial translation and pitch and yaw modes of the antenna damping system must satisfy that $f_{\mathrm{n}}\le 4.7$ Hz.

Combining Equation (15), it can be seen that in order to achieve the desired damping performance, the maximum radial stiffness of the antenna damping system must meet the following conditions:

$$k_{\mathrm{trans}}=\max (4k{S}_{\theta }{}^2+k_G,4k{C}_{\theta }{}^2)\le \min (\frac{4{\pi }^{2}m{f}_{\mathrm{c}}^2}{{\lambda }_{\mathrm{m}}^2},\frac{4{\pi }^{2}I{f}_{\mathrm{c}}^2}{{\lambda }_{\mathrm{m}}^2{L}^2})=\frac{4{\pi }^{2}I{f}_{\mathrm{c}}^2}{{\lambda }_{\mathrm{m}}^2{L}^2}$$

(20)

According to the above relationship, the radial integrated stiffness-damping rate matching curve of the antenna damping system can be drawn, as shown in Figure 3. When the stiffness and damping rate parameters for matching are selected in the lower left field of the curve, the vibration damping performance can be guaranteed to meet the index requirements.

Obviously, the stiffness k of an elastic element is closely related to its length l, section characteristic dimensions (such as cross-sectional area s), and material mechanical properties (such as Young’s modulus E). In this scene, the inclination angle θ of the elastic element not only affects its length l, but also affects its stiffness distribution in the vertical and horizontal directions, which is the key optimization parameter for the detailed design of the next step.

3.2. Identification of Mechanical Characteristic Parameters of Silicone Rubber Elastic Elements

In order to prevent the magnetic field in the space near the antenna of the designed damping system from changing caused by vibration, non-magnetic materials must be selected to design the antenna damping system, including elastic elements and corresponding transition structures. In order to make sure the stiffness of the damping system is low enough in a limited space, it is necessary to take a silicone rubber strip with small cross-sectional size and relatively small hardness as an elastic element. Considering that rubber and non-metal antennas and mounting bases cannot be vulcanized and fixed, if the pressure is used to make the connection fixed, when the pressure is excessive, the local stress would be too large to cause defects. When the pressure is too small, due to the creep of the rubber material, the fixed points of the damping system would be slack and unstable after operating for a long time.

Therefore, in this paper, the silicone rubber strip is designed into a loop, and the two ends are respectively suspended on the antenna and the suspension rod of the mounting base, it can keep balance naturally under the effect of gravity, without excessive pressing force for fastening. When this scheme is adopted to design a silicone rubber strip, the following factors must be taken into consideration:

• The silicone rubber strip must have sufficient load-bearing capacity. In other words, the tensile stress cannot be too large, which means that the cross-sectional size of the silicone rubber strip cannot be too small;
• There is a nonlinear relationship between the deformation of the silicone rubber strip and the stress it is subjected to. Knowing that the antenna and the mounting base satisfy minimum spacing constraint in the static equilibrium state, the stress-strain relationship is required to reverse the initial free length and section size of the silicone rubber strip;
• The static stiffness and dynamic stiffness of the silicone rubber strip are quite different, while the damping performance depends on its dynamic stiffness. Therefore, the precise design of the vibration damping performance needs to be based on the relationship between its dynamic stiffness and stress.

Based on the above analysis, in order to make the antenna damping system using a silicone rubber strip meet the space layout constraints and the damping performance indicators, it is necessary to accurately obtain the static and dynamic mechanical properties of the selected silicone rubber strip material. For this reason, first, for the silicon rubber strip material to be used, a sample of a certain length and cross-sectional size is selected, and its static stress–strain relationship is obtained through a static loading test. Then, a number of the sample suspension masses are used to test the vibration isolation natural frequency to obtain the dynamic stiffness characteristics of the silicone rubber strip samples under different loads. Further, data processing is used to obtain material dynamic mechanical characteristic parameters that are not related to geometric dimensions, which is called dynamic Young’s modulus-stress characteristic curve, providing a basis for the subsequent new-size silicone rubber driving stiffness estimation and antenna vibration damping performance evaluation.

We chose a sample of a rectangular cross-section silicone rubber strip with a Shore hardness of 40 HA, a density of $\rho =1.1\times {10}^3{\mathrm{kg}/\mathrm{m}}^3$, and the cross-sectional area in the free state was $s_0=1.05\times {10}^{-4}{\mathrm{m}}^2$. We took four pieces of the aforementioned silicone rubber strip samples of a certain length. One end of each sample was fixed on the same rigid base, and the other end was fixed to the four corners of the rectangular plate-shaped metal load, respectively. We adjusted the position of the fastening point to set the initial effective length under the stress-free state as $l_0=0.2\mathrm{m}$, 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:

$$\epsilon =\Delta l/l_0$$

(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):

$$\sigma =\frac{m_{\mathrm{T}}g}{s_0{l}_0/(\Delta l+l_0)}=\frac{m_{\mathrm{T}}g(1+\epsilon )}{s_0}$$

(22)

The stress–strain characteristics of this silicone rubber strip can be expressed as follows using the Ogden model [21]:

$$\sigma ={\displaystyle \sum _{i=1}^N\frac{2{\mu }_i}{{\alpha }_i}}({\epsilon }^{{\alpha }_i-1}-{\epsilon }^{-\frac{1}{2}{\alpha }_i-1})$$

(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 μ₁ = −2.72, μ₂ = −0.31, μ₃ = 3.88 and α₁ = −10.64, α₂ = 6.00, α₃ = −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.

$$k_d=4{\pi }^2{f}_{\mathrm{T}}^2{m}_{\mathrm{T}}$$

(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 Young’s modulus and equivalent dynamic Young’s modulus of the aforementioned silicone rubber strip material under different axial stress states.

3.3. 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 l and s, and the angle between itself and the vertical longitudinal section is θ. In addition, the tensile stress of each silicone rubber strip is σ, and the relationship between itself and the antenna mass m is as follows:

$$\sigma =\frac{mg}{s\cos \theta }\le {\sigma }_{\max }$$

(25)

The geometric parameters of the silicone rubber strip should also meet the following size and space constraints:

$${[l\cos \theta -a-(D-D_1)/2+\delta ]}^2+{(b+l\sin \theta )}^2\le D^2/4$$

(26)

In addition, combined with Equation (20), the silicone rubber strip with tensile stress σ should also meet the dynamic stiffness condition:

$$k_{\mathrm{trans}}=\max (\frac{4E_{\mathrm{d}}s}{l}{\sin }^2\theta +\frac{mg}{(l-a)C_{\theta }},\frac{4E_{\mathrm{d}}s}{l}{\cos }^2\theta )\le \frac{4{\pi }^{2}I{f}_{\mathrm{c}}^2}{{\lambda }_{\mathrm{m}}^2{L}^2}$$

(27)

In which, $E_{\mathrm{d}}=E_{\mathrm{d}}(\sigma )$ is the dynamic Young’s modulus of the silicone rubber strip under tensile stress σ, 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).

$$\begin{array}{l}\min k_{\mathrm{trans}}=\max (\frac{4E_{\mathrm{d}}s}{l}{\sin }^2\theta +\frac{mg}{(l-a)C_{\theta }},\frac{4E_{\mathrm{d}}s}{l}{\cos }^2\theta )\\ s.t.\{\begin{array}{l}\sigma =\frac{mg}{s\cos \theta }\le {\sigma }_{\max }\\ {[l\cos \theta -a-(D-D_1)/2+\delta ]}^2+{(b+l\sin \theta )}^2\le D^2/4\end{array}\end{array}$$

(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 $s_0=4.8\times {10}^{-5}{\mathrm{m}}^2$, the free length is $l_0=0.06\mathrm{m}$, and the distance of fixed damping point and the center of the antenna is $L=0.3\mathrm{m}$. In the state of static equilibrium, the cross-sectional area of the silicone rubber strip is $s=3.2\times {10}^{-5}{\mathrm{m}}^2$, the length is $l=0.09\mathrm{m}$, the angle between the silicone rubber strip and the vertical longitudinal section is $\theta =4.3°$, and the tensile stress of the silicone rubber strip is $\sigma =0.5\mathrm{MPa}$. Practice has proved that applying this silicone rubber strip configuration can meet the damping rate requirement $\zeta \le 0.05$.

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 $f_y=1.80\mathrm{Hz}$, the design value of the vertical natural frequency is $f_z=3.32\mathrm{Hz}$, the design value of the natural frequency in the yaw direction is $f_{rx}=2.31\mathrm{Hz}$, and the design value of the natural frequency in the pitch direction is $f_{rx}=4.30\mathrm{Hz}$; all the natural frequencies mentioned above meet the requirement $f_{\mathrm{n}}\le 4.7\mathrm{Hz}$.

According to the above-mentioned design scheme, the actual antenna damping system is shown in Figure 8.

4. Test Verification and Analysis

Two experiments are designed to verify the performance of the magnetic antenna damping system. The dynamic simulation test comprises of a vibration test on an electric shaking table to verify the vibration isolation performance of the magnetic antenna damping system. The electromagnetic noise induced by the electric shaking table is much higher than that induced by the magnetic antenna itself. A function comparison test is carried out on a dedicatedly designed vibration excitation apparatus to avoid electromagnetic noise, so as to verify whether the electromagnetic noise induced by the magnetic antenna can be significantly reduced via the proposed vibration isolation design.

4.1. Dynamic Simulation Test

In order to verify the vibration isolation performance of the magnetic antenna damping system, the project team conducted a simulation test on an electric shaking table. The test method is as follows:

• Install the magnetic antenna damping system on the shaking table, and use the pressure plate to compact the four supports;
• Set the test conditions according to the vibration level of the magnetic antenna installation platform, the range of vibration frequency is from 5 Hz to 200 Hz, the acceleration is 1 g, and the vibration mode is random vibration;
• Arrange an acceleration sensor on the antenna end, and obtain the vibration response parameters on the antenna in real time through a signal analyzer.

The installation diagram of the shaking table simulation test is shown in Figure 9.

In this vibration test, two positions were selected during the layout of the acceleration sensor: the antenna and the shell. The vibration test was carried out in the Z direction, and we obtained the vibration response curve of the antenna end through the dynamic signal analyzer, and calculated the attenuation rate of magnetic antenna damping device under a given vibration stress. The test results are shown in Figure 10.

Through the test data, we can easily find that, in the Z horizontal direction, under the condition of 1 g, the vibration attenuation of the antenna at 5–200 Hz is 0.01 times, the vibration reduction effect is excellent, and the attenuation exceeds 40 dB at the interested frequency 60 Hz, which well satisfies the vibration reduction need. It can be seen that the vibration transmissibility spiked up to near 0.1 (−20 dB) when the frequency was near 100 Hz. This is due to the natural mode of the connection structures between the silicone rubber strip and the electric shaking table, mainly attributed to flexibility of radial clearance and axial rubber gasket between the annular polyoxymethylene structures and the tubular fiberglass shell. This can be improved in final application by filling glue into the clearance between the annular polyoxymethylene structures and the tubular fiberglass shell.

4.2. Function Comparison Test

After the dynamic simulation test, we have a clear understanding of the performance of the vibration damping system. Now, through the functional test of receiving electromagnetic wave signals, the suppression effect of the damping device on the induction noise of magnetic antenna motion can be verified. The experiment is carried out on a dedicatedly designed vibration excitation apparatus to avoid electromagnetic noise, so as to verify whether the electromagnetic noise induced by the magnetic antenna can be significantly reduced via the proposed vibration isolation design. The vibration base platform is made up of concrete. A temporarily designed pneumatic vibration exciter with a remote air pump is adopted to avoid electromagnetic noise. As mentioned before, the clearance between the annular polyoxymethylene structures and the tubular fiberglass shell is filled with glue so as to suppress resonant vibration of the vibration isolation system near 100 Hz.

The specific steps are as follows:

• Install a single magnetic antenna and a magnetic antenna with a vibration damping system on two base surfaces with good vibration consistency, and compare the ambient electromagnetic noise received by the two antennas without vibration;
• The acceleration sensors are respectively arranged on the shell of the magnetic antenna damping system and antenna, then use the vibration shaker to apply the same size of excitation to the two base mesa, respectively. Analyze and process acceleration sensor feedback data, and calculate the attenuation rate of vibration from the housing to the antenna;
• Compare the ambient electromagnetic noise received by the two magnetic antennas with and without vibration reduction to verify whether the vibration reduction system meets the requirements. The test layout is shown in Figure 11.

The shell-antenna vibration transmission curve is shown in Figure 12. The experimental result coincides roughly with that of Figure 10, because they both show overall trends of vibration attenuation of 40 dB within the frequency domain above 60 Hz. There are slight differences in the low frequency domain and some high frequency domains. Resonant vibration of the vibration isolation system near 100 Hz has been suppressed in comparison with Figure 10, which is mainly the result of the structural improvement by filling up the clearance between the annular polyoxymethylene structures and the tubular fiberglass shell. Another reason might be the difference of the vibration characteristics between the temporarily designed pneumatic vibration exciter and the standard electric shaking table. Since the key point of this experiment was to verify whether the electromagnetic noise induced by the magnetic antenna can be significantly reduced, the authors think that the vibration transmissibility test error is temporarily acceptable before a more precise vibration exciter without electromagnetic noise is developed.

Function comparison test results are shown in Figure 13.

According to the above comparison test results, when there is no excitation, the electromagnetic noise induced by the magnetic antenna with and without vibration damping is below 200 fT, which proves that the antenna has a good reception effect in a quiet environment. When the vibration excitation is applied, the electromagnetic noise of the magnetic antenna without vibration damping increases obviously and exceeds 200 fT, which will affect the normal reception. However, the electromagnetic noise of the magnetic antenna with vibration damping basically maintains the same as that of the magnetic antenna without vibration excitation, which does not rise significantly and remains below 200 fT. The vibration transmission rate of the vibration damping system is basically about 0.01 (−40 dB) at the concerned frequency of 60 Hz, which meets the needs of vibration damping. The peak value of the vibration transmission curve is about 5 Hz, which is the natural frequency of the vibration reduction system. The vibration at this frequency point is amplified by nearly 20 dB, so the vibration reduction system must avoid this frequency band when working.

The magnetic antenna is a newly developed underwater low-frequency communication antenna, and it is easily affected by motion-induced noise. Therefore, no additional electromagnetic interference can be added to the design of vibration isolation measures. In the early stage, we only used an ordinary rubber vibration isolation pad to reduce vibration; with limited vibration isolation effect, the motion induced noise of the antenna is about 400 fT. Now, by adopting the new vibration isolation system introduced in this paper, the vibration in the communication frequency band is attenuated by more than 40 dB, and the antenna motion-induced noise is reduced to less than 60 fT, which significantly improved the communication performance.

5. Conclusions

In this paper, the research background and application of magnetic antenna are described. By analyzing the environmental vibration of magnetic antenna during underwater detection, we know from the vibration reduction requirement that the radial vibration transmissivity of the antenna should be less than 0.01 (−40 dB). On this basis, the principle model of the magnetic antenna damping system was established, and the main parameters to be optimized were defined. A magnetic antenna damping system based on silicone rubber elastic element was proposed, and its mechanical parameters were analyzed. Finally, the prototype design and performance analysis of the vibration damping system were completed. Dynamic simulation tests are carried out to verify that the performance of the prototype meets the requirements of vibration reduction, and the function comparison test confirms that the magnetic antenna equipped with the vibration absorption system can receive electromagnetic wave signal normally.