Introduction

As human exploration and utilization of marine resources continue to expand, underwater communication plays an increasingly vital role in marine scientific research, resource development, underwater operations, and military applications1,2. Due to the slower attenuation rate of low-frequency electromagnetic waves in seawater and their ability to travel long distances with strong resistance to interference, cross-medium electromagnetic communication in special environments such as seawater typically employs low-frequency electromagnetic communication3,4. However, traditional low-frequency electromagnetic communication systems also face challenges. Since the antenna size is related to the wavelength, the antenna requires a large area. To achieve long-distance communication, the transmission power of the system often reaches several megawatts, leading to increased energy consumption5.

To achieve the miniaturization of low-frequency communication systems, in 2017, mechanical antennas that generate low-frequency electromagnetic waves through mechanical motion were proposed6,7,8,9,10. The working principle of these mechanical antennas involves the generation of low-frequency electromagnetic waves through mechanical motion. It is possible to convert mechanical energy to electromagnetic energy by designing special mechanical structures. The radiation quality factor proposed by Chu and Harrington is considered the lower bound of the performance of traditional antennas11,12. However, mechanical antennas use innovative operating mechanisms and have been shown to break the radiation factor limit of traditional antennas in the low-frequency range6,13,14. It implies that mechanical antennas can efficiently emit low-frequency electromagnetic waves in a smaller size, thereby facilitating the miniaturization of low-frequency communication systems.

Many researchers have used motors as a source of motion to drive the rotation of permanent magnets15,16,17,18,19 or electrostatically charged materials (electret)20,21,22 to generate low-frequency electromagnetic signals (ULF/VLF). However, this approach faces several challenges in practical applications. Firstly, to produce low-frequency electromagnetic waves at the corresponding frequencies, the rotating system needs to achieve rotational speeds of tens of thousands of revolutions per minute, which places exceptionally high demands on the motor and mechanical structure. Secondly, since permanent magnets and electrets need to rotate at high speeds, the driving system consumes significant mechanical energy, reducing system efficiency and hindering miniaturization. Moreover, for mechanical antennas employing electrets, changes in environmental conditions such as temperature and humidity can decrease the electret’s charge stability, thereby affecting the mechanical antenna’s performance.

To overcome the issues above, Srinivas Prasad M N and others have proposed the concept of the magnetic pendulum mechanical antenna23,24,25,26,27. The magnetic pendulum mechanical antenna utilizes magnetic forces provided by a permanent magnet array and an excitation coil to drive the oscillation of the permanent magnet, thereby achieving the emission of low-frequency electromagnetic waves. This design eliminates the need for auxiliary devices such as motors, simplifies the system structure, and reduces energy consumption. Moreover, as the magnetic pendulum mechanical antenna is no longer dependent on a motor drive’s high-speed rotation, the system’s reliability and durability are significantly improved.

The rotor of the magnetic pendulum mechanical antenna oscillates under the combined action of the coil magnetic field and an external magnetic field. It utilizes the resonance principle to increase the amplitude of rotor oscillation and enhance radiation power. To calculate the resonance frequency of the magnetic pendulum mechanical antenna, Prasad et al. simplified the complex external magnetic field and established a dynamic model of the rotor permanent magnet using a uniform magnetic field model24. However, the magnetic forces acting on the rotor permanent magnet due to the stator permanent magnet and the driving coil are very complex, and the uniform magnetic field model needs to be revised to analyze the rotor’s motion trajectory accurately. Therefore, an accurate dynamic model is required to describe the rotor’s motion.

This paper proposes a new analytical method that utilizes the concept of virtual charges to establish dynamic equations for the rotor permanent magnet under the action of the stator permanent magnet’s static magnetic field and the driving coil’s alternating magnetic field. This method allows for the analysis of the rotor permanent magnet’s time-domain characteristics and its frequency-domain characteristics. Based on this, the magnetic pendulum mechanical antenna’s magnetic field model is established, and a prototype of the magnetic pendulum mechanical antenna is designed using 3D printing technology. The experimental results show good agreement with the theoretical results, verifying the accuracy of the magnetic field model and providing a theoretical basis for further research on magnetic pendulum mechanical antennas.

Method and model

Dynamic analysis

Figure 1a illustrates a structural scheme of a magnetic pendulum mechanical antenna. The central component of the mechanical antenna is a radially magnetized cylindrical permanent magnet rotor, which is responsible for the movement of the mechanical antenna. Symmetrically mounted on either side of the rotor are rectangular permanent magnets that act as the stator of the mechanical antenna, with their magnetization directions aligned with those of the rotor. The stator and rotor are surrounded by an outer excitation coil.

Fig. 1
figure 1

Structure of the mechanical antenna. (a) The isometric drawing of the mechanical antenna structure. (b) The cross-section drawn of the mechanical antenna structure.

Full size image

During the operation of the mechanical antenna, the excitation coil is filled with alternating current. At the same time, the stator remains fixed, and the rotor performs a one-degree-of-freedom oscillation around its axis. The primary function of the coil is to generate an alternating magnetic field in three-dimensional space, thereby providing the rotor with the necessary driving torque to deviate from its equilibrium position. Concurrently, the background magnetic field produced by the stator provides a restoring torque that brings the rotor back to its equilibrium position. This interaction ensures that the rotor oscillates within a small offset range, thereby generating an alternating electromagnetic field through its motion, realizing the emission of low-frequency electromagnetic waves.

When current is passed through the coil, a magnetic field \(\vec {H_c}=H_{cx}\vec {e_x}+H_{cy}\vec {e_y}+H_{cz}\vec {e_z}\) is generated in space, as shown in Fig. 1b. When the rotor is placed along the Y-axis, as depicted in Fig. 1a, the magnetic field component along the Y-axis, \(H_{cy}\) , does not produce torque on the rotor. However, the magnetic field components perpendicular to the Y-axis, \(H_{cx}\) and \(H_{cz}\) , will produce torques on the rotor in opposite directions, denoted as \(T_{cx}\) and \(T_{cz}\), respectively. The stator generates a background magnetic field \(\vec {H_s}=H_{sx}\vec {e_x}+H_{sy}\vec {e_y}+H_{sz}\vec {e_z}\) in space, with the magnetic field component parallel to the Y-axis not producing a torque on the rotor. The magnetic field components perpendicular to the Y-axis, \(H_{sx}\) and \(H_{sz}\), will produce torques on the rotor in opposite directions, denoted as \(T_{sx}\) and \(T_{sz}\) respectively. At the initial state, the rotor has an initial oscillation angle, \(\theta _r (t=0)\), such that the net torque acting on the rotor at this angle is \(T(\theta _r)=0\).

When the coil is excited with alternating current, the magnetic field \(\overrightarrow{H_c}\) will periodically vary, and the torques \(T_{cx}\) and \(T_{cz}\) produced by the coil on the rotor will also periodically vary. As a result, the net torque acting on the rotor is not zero, and the rotor is driven to oscillate periodically by the net torque. By utilizing the oscillation of the rotor, an alternating electromagnetic field is generated in space, which allows for the transmission of information outward. However, the motion trajectory of the rotor is influenced by various factors, including the geometric dimensions of the stator, rotor, and coil, the amplitude of the current through the coil, and the excitation frequency. Since the waveform and spectral characteristics of the electromagnetic field are related to the motion trajectory of the rotor, it is necessary to establish the equations of motion for the rotor, analyze its motion trajectory, and further conduct the magnetic field characteristic analysis of the magnetic pendulum mechanical antenna.

Torque produced by the stator’s magnetic field

One of the core components of the magnetic pendulum mechanical antenna is a cylindrical permanent magnet rotor placed along the Y-axis, with a radius of \(R_r\) and a length of \(H_r\). The rotor’s magnetization intensity is \(M_r\), which is magnetized radially in the X-axis direction, generating a magnetic field in space. The rotor’s motion is a one-degree-of-freedom oscillation, with the oscillation angle at time t denoted as \(\theta _r(t)\). The stator, an essential part of the mechanical antenna, has dimensions of \(2L_s\), \(2D_s\) and \(2H_s\) along the X-axis, Y-axis, and Z-axis, respectively. The two stators are symmetrically placed along the X-axis, with a centre-to-centre distance of \(2D_{ss}\). The stators are magnetized with an intensity of \(M_s\) in the X-axis direction, and they will produce a stable background magnetic field in the vicinity of the rotor, as shown in Fig. 1b.

Under the assumption of uniform magnetization, the rotor’s volume magnetic charge density \(\sigma _V = -\nabla \cdot \vec {M}_r = 0\), and the rotor’s surface magnetic charge density \(\sigma _S=\vec {n} \cdot \vec {M}_r\), where \(\vec {n}\) is the unit normal vector onto the rotor’s surface. Therefore, the surface magnetic charge density \(\sigma _S=0\) on its two circular surfaces. The rotor’s magnetic charge is primarily distributed on the cylindrical surface, and the surface magnetic charge density can be represented as \(\sigma _S=M_r\cos \varphi _r\), where \(\varphi _r\) is the angle between the area element and the rotor’s magnetization direction, as shown in Fig. 2a.

Fig. 2
figure 2

The forces acting on the rotor of a magnetic pendulum mechanical antenna. (a) the forces acting on the rotor. (b) the forces acting on an equivalent magnetic dipole.

Full size image

Considering a symmetric area element \(\textrm{d} S = R_r \textrm{d} \varphi _r \textrm{d} h\) on the rotor’s cylindrical surface, the magnetic charges of the area element are as follows:

$$\begin{aligned} \textrm{d} q_{m}= \pm M_r \cos \varphi _{r} \textrm{d} S = \pm M_{r} R_{r} \cos \varphi _{r} \textrm{d} \varphi _{r} \textrm{d} h \end{aligned}$$
(1)

In the equation, the magnetic charge \(\textrm{d} q_{m}\) is in units of A m. The positive and negative magnetic charges form a magnetic dipole pair, as shown in Fig. 2b. The magnitude of the magnetic dipole moment is:

$$\begin{aligned} \textrm{d} m=2 M_{r} R_{r}^{2} \cos \varphi _{r} \textrm{d} \varphi _{r} \textrm{d} h \end{aligned}$$
(2)

The unit of the magnetic dipole moment is \(\textrm{A}\,\textrm{m}^2\), and its direction is from the negative magnetic charge to the positive magnetic charge, \(\vec {e}_m=\cos (\theta _r+\varphi _{r}) \vec {e}_x+\sin (\theta _r+\varphi _{r}) \vec {e}_z\).

The angle between the stator magnetic field component \(H_{sx}\) and the magnetic dipole moment dm is \(\theta _r+\varphi _{r}\), and the magnetic dipole experiences a torque of \(dT_{sx}\); the angle between the stator magnetic field component \(H_{sz}\) and the magnetic dipole moment dm is \(\pi /2-\theta _r-\varphi _{r}\), and the magnetic dipole experiences a torque of \(\textrm{d} T_{sz}\).

$$\begin{aligned} \begin{aligned} \textrm{d} T_{sx}&=\mu _{0} \cdot \textrm{d} \vec {m} \times \vec {H}_{sx} =2\mu _{0}M_{r}R_{r}^{2}H_{sx}\sin (\theta _{r}+\varphi _{r})\cos \varphi _{r}\textrm{d} \varphi _{r}\textrm{d} h\\ \textrm{d} T_{sz}&=\mu _{0} \cdot \textrm{d} \vec {m} \times \vec {H}_{sz} =2\mu _{0}M_{r}R_{r}^{2}H_{sz}\cos (\theta _{r}+\varphi _{r})\cos \varphi _{r}\textrm{d} \varphi _{r}\textrm{d} h \end{aligned} \end{aligned}$$
(3)

In the equation, \(\mu _{0} = 4\pi \times 10^{-7} \mathrm {N/A^2}\) represents the magnetic permeability of the vacuum, and the positive direction of the torque \(\textrm{d} T_{sx}\) is along the clockwise direction. In contrast, the positive direction of the torque \(\textrm{d} T_{sz}\) is along the counterclockwise direction.

The background magnetic field generated in space by the stator structure within the mechanical antenna is28:

$$\begin{aligned} \begin{aligned} H_{sx}(x,y,z) &=\frac{M_{s}}{4\pi } [f(x-D_{ss},y,z,L_s,D_s,H_s)-f(x-D_{ss},y,z,-L_s,D_s,H_s)\\&\quad + f(x+D_{ss},y,z,L_s,D_s,H_s)-f(x+D_{ss},y,z,-L_s,D_s,H_s)] \\ H_{sy}(x,y,z) &=\frac{M_{s}}{4\pi } [g(x-D_{ss},y,z,L_s,D_s,H_s)-g(x-D_{ss},y,z,-L_s,D_s,H_s)\\&\quad + g(x+D_{ss},y,z,L_s,D_s,H_s)-g(x+D_{ss},y,z,-L_s,D_s,H_s)] \\ H_{sz}(x,y,z) &=\frac{M_{s}}{4\pi } [g(x-D_{ss},z,y,L_s,D_s,H_s)-g(x-D_{ss},z,y,-L_s,D_s,H_s)\\&\quad + g(x+D_{ss},z,y,L_s,D_s,H_s)-g(x+D_{ss},z,y,-L_s,D_s,H_s)] \end{aligned} \end{aligned}$$
(4)

The auxiliary functions \(f(x,y,z,L_s,D_s,H_s)\) and \(g(x,y,z,L_s,D_s,H_s)\) are defined as follows:

$$\begin{aligned} \begin{aligned} f(x,y,z,L_s,D_s,H_s)&=\Gamma (x,y,z,L_s,D_s,H_s)-\Gamma (x,y,z,L_s,D_s,-H_s) +\Gamma (x,y,z,L_s,-D_s,-H_s)-\Gamma (x,y,z,L_s,-D_s,H_s) \\ g(x,y,z,L_s,D_s,H_s)&=\Phi (x,y,z,L_s,D_s,H_s)-\Phi (x,y,z,L_s,D_s,-H_s)+\Phi (x,y,z,L_s,-D_s,-H_s)-\Phi (x,y,z,L_s,-D_s,H_s) \end{aligned} \end{aligned}$$
(5)

The auxiliary functions \(\Gamma (x,y,z,L_s,D_s,H_s)\) and \(\Phi (x,y,z,L_s,D_s,H_s)\) are defined as follows:

$$\begin{aligned} \begin{aligned}&\Gamma (x,y,z,L_{s},D_{s},H_{s})=\operatorname {sgn}(x-L_{s})\cdot \arcsin (\frac{(y-D_{s})(z-H_{s})}{\sqrt{\left( x-L_{s}\right) ^{2}+\left( y-D_{s}\right) ^{2}}\sqrt{\left( x-L_{s}\right) ^{2}+\left( z-H_{s}\right) ^{2}}})\\&\Phi (x,y,z,L_{s},D_{s},H_{s})=\ln [z-H_{s}+\sqrt{\left( x-L_{s}\right) ^{2}+\left( y-D_{s}\right) ^{2}+\left( z-H_{s}\right) ^{2}} ] \end{aligned} \end{aligned}$$
(6)

The function \(\operatorname {sgn}\) is the sign function. Combining Eqs. (3) and (4), the torque experienced by the rotor in the stator magnetic field component \(H_{sx}\) when the rotor swings at an angle \(\theta _{r}\) is:

$$\begin{aligned} T_{sx}(t)=\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\pi }^{\pi }\textrm{d} T_{sx}(t)=T_{sxs}\sin \theta _{r}(t)+T_{sxc}\cos \theta _{r}(t) \end{aligned}$$
(7)

where the parameters \(T_{sxs}\) and \(T_{sxc}\) are:

$$\begin{aligned} \begin{aligned}&T_{sxs} =T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}} \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{sx}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin ^{2}\xi \textrm{d} \xi \textrm{d} h \\&T_{sxc} =T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}} \int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{sx}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin \xi \cos \xi \textrm{d} \xi \textrm{d} h \end{aligned} \end{aligned}$$
(8)

The parameter \(T_0 = 2\mu _{0}M_{r}R_{r}^{2}\), \(\xi =\theta _{r}+\varphi _{r}\), and the purpose of the parameter transformation is to separate the variable \(\theta _{r}\) from the integral in Eq. (7), thereby facilitating subsequent integration processing. At this point, the parameters \(T_{sxs}\) and \(T_{sxc}\) in Eq. (7) are independent of the time variable and are solely related to the mechanical antenna structure. Since the magnetic field \(H_{sx}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\) is an even function of the variable \(\xi\), the parameter \(T_{sxc}=0\), and the torque in Eq. (7) is further simplified to:

$$\begin{aligned} T_{sx}(t)=T_{sxs}\sin \theta _{r}(t) \end{aligned}$$
(9)

Similarly, when the rotor swings at an angle \(\theta _{r}\), the torque experienced by the rotor in the stator magnetic field \(H_{sz}\) is:

$$\begin{aligned} T_{sz}(t)=\int _{-\frac{H_r}{2}}^{\frac{H_r}{2}}\int _{-\pi }^{\pi }\textrm{d} T_{sz}(t)=T_{szs}\sin \theta _r(t)+T_{szc}\cos \theta _r(t) \end{aligned}$$
(10)

where the parameters \(T_{szs}\) and \(T_{szc}\) are:

$$\begin{aligned} \begin{aligned}&T_{szs}=T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{sz}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin \xi \cos \xi \textrm{d} \xi \textrm{d} h \\&T_{szc}=T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{sz}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\cos ^{2}\xi \textrm{d} \xi \textrm{d} h \end{aligned} \end{aligned}$$
(11)

Given that the magnetic field \(H_{sz}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\) is an odd function of the variable \(\xi\), with parameter \(T_{szc}=0\), the torque (10) is further simplified to:

$$\begin{aligned} T_{sz}(t)=T_{szs}\sin \theta _r(t) \end{aligned}$$
(12)

Torque produced by the coil’s magnetic field

In order to simplify the model, the article neglects the effects of the height and thickness of the coil, assumes that the coil is concentrated within the XOY plane with dimensions of \(-L_{c} \le x \le L_{c}\) and \(-D_{c} \le y \le D_{c}\), and has a total number of turns represented by \(N_c\) as shown in Fig. 3.

Fig. 3
figure 3

Mechanical antenna structure in the XOY plane.

Full size image

A sinusoidal current \(I_c\) flows through the coil with a frequency \(f_c\). According to Biot-Savart’s law, the magnetic field \(\textbf{H}_c\) produced by the coil at point P(x, y, z) in space is:

$$\begin{aligned} \textbf{H}_c(t)=\frac{I_cN_c}{4\pi }\oint \frac{\textrm{d} \vec {l}\times \vec {e}_r}{r^2}\sin (2\pi f_ct) \end{aligned}$$
(13)

In the equation, \(\textrm{d} \vec {l}\) represents the current element, and \(\vec {r}\) is the distance from the current element to point P. Ignoring the effects of the coil’s rounded corners, the coil’s magnetic field can be considered as the combined magnetic field of four straight conductors.

$$\begin{aligned} \begin{aligned} \vec {H}_{cx}(t){=}H_{cxm}\sin (2\pi f_ct)\vec {e}_{x} =&-\frac{N_{c}I_{c}}{4\pi }[\frac{z}{\alpha _{+}}(\frac{y_{+}}{\rho _{x_{+},y_{+}}}-\frac{y_{-}}{\rho _{x_{+},y_{-}}})-\frac{z}{\alpha _{-}}(\frac{y_{+}}{\rho _{x_{-},y_{+}}}-\frac{y_{-}}{\rho _{x_{-},y_{-}}})]\sin (2\pi f_ct)\vec {e}_{x} \\ \vec {H}_{cy}(t){=}H_{cym}\sin (2\pi f_ct)\vec {e}_{y} =&-\frac{N_{c}I_{c}}{4\pi }[\frac{z}{\beta _{+}}(\frac{x_{+}}{\rho _{x_{+},y_{+}}}-\frac{x_{-}}{\rho _{x_{-},y_{+}}})-\frac{z}{\beta _{-}}(\frac{x_{+}}{\rho _{x_{+},y_{-}}}-\frac{x_{-}}{\rho _{x_{-},y_{-}}})]\sin (2\pi f_ct)\vec {e}_{y} \\ \vec {H}_{cz}(t){=}H_{czm}\sin (2\pi f_ct)\vec {e}_{z} =&\frac{N_{c}I_{c}}{4\pi }[\frac{x_{+}}{\alpha _{+}}(\frac{y_{+}}{\rho _{x_{+},y_{+}}}-\frac{y_{-}}{\rho _{x_{+},y_{-}}})-\frac{x_{-}}{\alpha _{-}}(\frac{y_{+}}{\rho _{x_{-},y_{+}}}-\frac{y_{-}}{\rho _{x_{-},y_{-}}})\\&+\frac{y_{+}}{\beta _{+}}(\frac{x_{+}}{\rho _{x_{+},y_{+}}}-\frac{x_{-}}{\rho _{x_{-},y_{+}}}) -\frac{y_{-}}{\beta _{-}}(\frac{x_{+}}{\rho _{x_{+},y_{-}}}-\frac{x_{-}}{\rho _{x_{-},y_{-}}})]\sin (2\pi f_ct)\vec {e}_{z} \end{aligned} \end{aligned}$$
(14)

In the equation, \(H_{cxm}\), \(H_{cym}\) and \(H_{czm}\) represent the maximum values of the X-axis, Y-axis, and Z-axis components of the magnetic field produced by the coil. The parameters \(x_\pm , y_\pm , \alpha _\pm , \beta _\pm , \rho _{x_\pm ,y_\pm }\) are:

$$\begin{aligned} \begin{aligned}&x_{\pm }=x\pm L_{c} \\&y_{\pm }=y\pm D_{c} \\&\alpha _{\pm }=x_{\pm }^{2}+z^{2} \\&\beta _{\pm }={y_{\pm }}^{2}+z^{2} \\&\rho _{x_{\pm },y_{\pm }}=\sqrt{x_{\pm } ^{2}+y_{\pm } ^{2}+z^{2}} \end{aligned} \end{aligned}$$
(15)

For the magnetic dipole in Fig. 2b, the magnetic field \(H_{cx}\) produced by the coil is at an angle \(\theta _{r}+\varphi _{r}\) to the magnetic dipole moment \(\textrm{d} m\), and the magnetic dipole experiences a torque of \(\textrm{d} T_{cx}\); the magnetic field \(H_{cz}\) produced by the coil is at an angle \(\pi /2-\theta _{r}-\varphi _{r}\) to the magnetic dipole moment \(\textrm{d} m\), and the magnetic dipole experiences a torque of \(\textrm{d} T_{cz}\).

$$\begin{aligned} \begin{aligned}&\textrm{d} T_{cx}=T_0 H_{cxm}\sin (2\pi f_ct)\sin (\theta _{r}+\varphi _{r}) \cos \varphi _{r}\textrm{d} \varphi _{r}\textrm{d} h\\&\textrm{d} T_{cz}=T_0 H_{czm}\sin (2\pi f_ct)\cos (\theta _{r}+\varphi _{r}) \cos \varphi _{r}\textrm{d} \varphi _{r}\textrm{d} h \end{aligned} \end{aligned}$$
(16)

In the equation, the positive direction of torque \(\textrm{d} T_{cx}\) is clockwise, and the positive direction of torque \(\textrm{d} T_{cz}\) is counterclockwise.

When the rotor is at an angle \(\theta _{r}\), the torque experienced by the rotor in the magnetic field \(\vec {H}_{cx}\) is:

$$\begin{aligned} T_{cx}(t)=[ T_{cxs}\sin \theta _{r}(t)+T_{cxc}\cos \theta _{r}(t) ] \sin (2\pi f_ct) \end{aligned}$$
(17)

where the parameters \(T_{cxs}\) and \(T_{cxc}\) are:

$$\begin{aligned} \begin{aligned}&T_{cxs}=T_0 \int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{cxm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin ^{2}\xi \textrm{d} \xi \textrm{d} h\\&T_{cxc}=T_0 \int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{cxm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin \xi \cos \xi \textrm{d} \xi \textrm{d} h \end{aligned} \end{aligned}$$
(18)

where \(\xi =\theta _{r}+\varphi _{r}\), since the magnetic field \(H_{cxm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\) is an odd function of the variable \(\xi\), the parameter \(T_{cxs}=0\), and the torque (17) is further simplified to:

$$\begin{aligned} T_{cx}(t)=T_{cxc}\cos \theta _{r}(t)\sin (2\pi f_ct) \end{aligned}$$
(19)

Similarly, the torque experienced by the rotor in the magnetic field \(H_{cz}\) is:

$$\begin{aligned} T_{cz}(t)=[T_{czs}\sin \theta _{r}(t)+T_{czc}\cos \theta _{r}(t)]\sin (2\pi f_ct) \end{aligned}$$
(20)

where the parameters \(T_{czs}\) and \(T_{czc}\) are:

$$\begin{aligned} \begin{aligned}&T_{czs}=T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{czm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\sin \xi \cos \xi \textrm{d} \xi \textrm{d} h\\&T_{czc}=T_0\int _{-\frac{H_{r}}{2}}^{\frac{H_{r}}{2}}\int _{-\frac{\pi }{2}}^{\frac{\pi }{2}}H_{czm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\cos ^{2}\xi \textrm{d} \xi \textrm{d} h \end{aligned} \end{aligned}$$
(21)

Since the magnetic field \(H_{czm}(R_{r}\cos \xi ,h,R_{r}\sin \xi )\) is an even function of the variable \(\xi\), the integration \(T_{czs}=0\), the torque (20) is further simplified to:

$$\begin{aligned} T_{cz}(t)=T_{czc}\cos \theta _{r}(t)\sin (2\pi f_ct) \end{aligned}$$
(22)

Friction torque

During the operation of mechanical antennas, friction torques caused by the bearings act upon them. These friction torques serve as a damping force, suppressing the oscillation of the rotor and influencing both the amplitude of the rotor’s oscillation and the characteristic frequencies of the system’s oscillation. Consequently, analyzing the impact of friction torque on rotor motion is paramount. The rotor is supported at both ends by bearings, and as the rotor rotates, friction forces are generated within the bearings. The friction torque in the bearings is defined as follows:

$$\begin{aligned} T_{b}=\mu F R_r \end{aligned}$$
(23)

In the equation, \(\mu\) represents the friction coefficient of the bearing, F denotes the load applied to the bearing, \(F=m_r g\), \(m_r\) is the mass of the rotor.

Dynamic differential equations

The rotor is subjected to torques \(T_{sx}, T_{sz}, T_{cx}\), \(T_{cz}\), and \(T_{b}\), and the rotor moves under the action of the resultant torque. The differential equation of the rotor’s force is given by:

$$\begin{aligned} \ddot{\theta }_r(t)+k\sin \theta _r(t)+T_f\operatorname {sgn}({\dot{\theta }}_r(t))=F\cos \theta _r(t)\sin (2\pi f_ct) \end{aligned}$$
(24)

In the equation, \(k=(T_{sxs}-T_{szs})/I_r\), represents the equivalent stiffness, \(T_f = T_b/I_r\) is the equivalent friction torque, \(F=(T_{czc}-T_{cxc})/I_r\) is the equivalent excitation force, \(I_r=m_rR_r^{2}/2\) is the rotor’s moment of inertia.

Affected by resonance, the rotor will exhibit different time-domain and frequency-domain characteristics at different frequencies. In this section, the motion characteristics of the antenna were analyzed through numerical calculations. Specifically, the process commences with the utilization of a numerical integration method to accurately determine the parameters of the stator magnetic torque, coil magnetic torque, and the friction torque acting on the rotor. Subsequently, the equivalent stiffness, equivalent friction torque, and equivalent excitation force of the rotor within the dynamic equation are meticulously calculated. The final stage involves employing the Runge-Kutta method to solve the dynamic equation, thereby yielding the numerical solution for the rotor trajectory.

The stator dimensions along the X-axis, Y-axis, and Z-axis are 5 mm, 25 mm, and 15 mm, respectively; the diameter and length of the rotor are 3.9 mm and 30 mm; the centre-to-centre distance between the stator and rotor is 19.3 mm, with the shortest distance from the stator to the rotor’s cylindrical surface being 14.9 mm; the equivalent length and width of the coil are 70 mm and 50 mm, respectively, with 100 turns in the coil, and the effective intensity of the current in the coil is 0.1A. The stator and rotor are NdFeB, with residual magnetization \(B_r=1.25\textrm{T}\) and magnetic permeability \(\mu _r=1.09\)29. Therefore, the torque parameter \(T_{sxs}=0.0185\), \(T_{szs}=1.88 \times 10^{-4}\), \(T_{cxc}=6.46 \times 10^{-8}\), \(T_{czc} = 1.1 \times 10^{-4}\), \(T_b = 5.06 \times 10^{-6}\), the equivalent parameter \(k=3.66 \times 10^6\), \(T_f = 1.01 \times 10^3\), \(F = 2.20 \times 10^4\). The natural frequency of the mechanical antenna can be calculated using the equivalent stiffness, \(f_n = \sqrt{k}/(2\pi )=304\,\textrm{Hz}\).

Figures 4 and 5 depict the time-domain and frequency-domain plots of the rotor’s motion excited at 200 Hz, 300 Hz, and 400 Hz. The rotor’s angular displacement in the time domain exhibits apparent harmonic motion, corresponding to the peaks in the frequency domain plots, which align with the excitation frequencies. The amplitude of the rotor’s angular displacement is influenced by the excitation frequency, with significant increases in both angular displacement and angular velocity amplitude as the frequency approaches the rotor’s resonance frequency. It suggests leveraging the rotor’s resonance characteristics can increase its oscillation amplitude. Thus, studying the rotor’s resonance frequency is a crucial parameter in the design of magnetic pendulum mechanical antennas.

Fig. 4
figure 4

Time-domain plot of the rotor’s motion under different excitation frequencies (a) 200 Hz, (b) 300 Hz, (c) 400 Hz.

Full size image
Fig. 5
figure 5

Frequency-domain plot of the rotor’s motion under different excitation frequencies (a) 200 Hz, (b) 300 Hz, (c) 400 Hz.

Full size image

Figure 6 illustrates the influence of the rotor radius \(R_r\), height \(H_r\), stator dimensions along the axes, \(L_s, D_s, H_s\), and the minimum spacing \(D_{rs}\) between the stator and rotor on the mechanical antenna’s natural frequency. It can be observed from the figure that as the rotor radius \(R_r\), height \(H_r\), and the minimum spacing \(D_{rs}\) increase, the natural frequency decreases, with the natural frequency being more sensitive to \(R_r\) and \(D_{rs}\). On the other hand, as the stator size increases, the natural frequency increases, and the trend of increase becomes gradually slower. It indicates that the natural frequency is significantly affected by the dimensions of both the rotor and the stator, and thus, in the design process, the dimensions of the stator and rotor should be optimized based on the desired resonance frequency.

Fig. 6
figure 6

Influence of geometric parameters on the mechanical antenna’s natural frequency.

Full size image

Figure 7 demonstrates the relationship between the rotor’s motion amplitude near the resonant frequency and the resonance frequency. The operating frequency of the antenna is \(f_c = 0.9 f_n\). It can be observed that as the resonance frequency increases, the amplitude of the rotor’s motion decreases. This phenomenon indicates that resonance frequencies in the low-frequency range are more conducive to achieving larger rotor motion amplitudes for magnetic pendulum mechanical antennas. Therefore, when designing magnetic pendulum mechanical antennas, conditions suitable for the low-frequency range should be primarily considered to achieve effective energy transmission and signal reception.

Fig. 7
figure 7

Relationship between the rotor’s motion amplitude near the resonant frequency and the resonance frequency.

Full size image

Mechanical antenna’s magnetic field model

When the rotor is moving, it generates alternating electromagnetic field in space. The superposition of rotor’s magnetic field and excitation coil’s magnetic field constitutes the mechanical antenna’s magnetic field and transmits signals outward. Therefore, studying the electromagnetic field model of rotor motion is particularly important. The article has constructed a magnetic field model for the rotating body based on the magnetic dipole’s field under motion. As shown in Fig. 8, a coordinate system is established where the rotor’s axis is aligned along the Y-axis. Two magnetic charge, \(\pm q_{m}\), are selected at symmetrical positions on a cylindrical surface of height h to form a magnetic dipole. The angle between the dipole and the Y-axis is \(\phi _r\), the angle between the dipole and the direction of magnetization is \(\varphi _r\), and the angle between the rotor and the X-axis is \(\theta _r\).

Fig. 8
figure 8

Position relationship between the moving magnetic dipole and the field point.

Full size image

The transformation relationship from the Cartesian coordinate system (Fig. 4) to the spherical coordinate system is as follows:

$$\begin{aligned} \begin{aligned}&x = R \sin \phi \cos \theta \\&y = R \cos \phi \\&z = R \sin \phi \sin \theta \end{aligned} \end{aligned}$$
(25)

In the equation, R represents the distance in the spherical coordinate system. The magnetic charge of the rotor is concentrated in the cylindrical surface in the form of a magnetic dipole. As the rotor moves, the magnetic dipole will move similarly. The moving magnetic dipole will generate a changing magnetic field in space30,31:

$$\begin{aligned} \vec {B}_m=\frac{\mu _0}{4\pi }\{\frac{3\vec {n}(\vec {n}\cdot \vec {m})-\vec {m}}{R^3}+\frac{R}{c}\frac{\textrm{d}}{\textrm{d}t}[\frac{3\vec {n}(\vec {n}\cdot \vec {m})-\vec {m}}{R^3}] -\frac{\textrm{d}}{\textrm{d}t}[\frac{3\vec {n}(\vec {n}\cdot \vec {m})-\vec {m}}{c^2R^2}\frac{\textrm{d}R}{\textrm{d}t}]+\frac{1}{c^2}\frac{\textrm{d}^2}{\textrm{d}t^2}[\frac{\vec {n}\times (\vec {n}\times \vec {m})}{KR}]\}_\tau \end{aligned}$$
(26)

In the equation, the subscript \(\tau\) denotes that the parameters represent the time of the magnetic dipole’s motion, which is when the electromagnetic wave is generated, \(\tau =t-R/c\). t is the time the electromagnetic wave reaches the field point, which is influenced by the reception distance, causing a lag phenomenon at the receiving point. \(c=3 \times 10^8 \mathrm {m/s}\) represents the speed of light; \(\vec {R}=\vec {r}-\vec {r}^{\prime }(\tau )\) is the distance vector from the magnetic dipole to the field point; \(\vec {r}\) is the position of the field point; \(\vec {r}^{\prime }(\tau )\) is the position of the center of the magnetic dipole; \(\vec {n}\) is the unit direction vector of the distance from the magnetic dipole to the field point; the parameter \(K=1-\vec {n} \cdot \vec {v}/c\); \(\vec {v}\) is the velocity of the magnetic dipole at time \(\tau\).

Considering the actual situation, the article simplifies Eq. (26) as follows:

  1. (1)

    Since electromagnetic waves propagate at the speed of light, the time delay caused by the distance between the receiver and the transmitter can be ignored, \(\tau =t\);

  2. (2)

    The rotor’s movement speed is much slower than the speed of light, \(v \ll c\), therefore the parameter \(K=1-\vec {n} \cdot \vec {v}/c \approx 1\).

  3. (3)

    The size of the rotor is much smaller than the distance to the field point, hence the effect of the rotor’s size can be neglected, and \(R=r\) as well as \(n=\vec {e}_r\).

At this point, Eq. (26) will be simplified to:

$$\begin{aligned} \vec {B}_m=\frac{\mu _0}{4\pi }\{\frac{3(\vec {e}_r\cdot \vec {m})\vec {e}_r-\vec {m}}{r^3}+\frac{3(\vec {e}_r\cdot \vec {m})\vec {e}_r-\vec {m}}{cr^2}+\frac{\vec {e}_r\times (\vec {e}_r\times \vec {m})}{c^2r}\} \end{aligned}$$
(27)

In Eq. (27), the first term of the magnetic field is inversely proportional to \(r^3\) and predominates in the near-field region; the second term is inversely proportional to \(r^2\) and predominates in the intermediate region; the third term is inversely proportional to r and predominates in the far-field region.

In the near-field region, only the first term of magnetic field needs to be considered, and the moving magnetic dipole’s magnetic field can be simplified to:

$$\begin{aligned} { \vec {B}_m=\frac{\mu _0}{4\pi }\cdot \frac{3(\textbf{e}_r\cdot \textbf{m})\textbf{e}_r-\textbf{m}}{r^3} } \end{aligned}$$
(28)

As shown in Fig. 8, at the symmetric position on the cylindrical surface, the area element \(\textrm{d} S=R_r \textrm{d} \varphi _r \textrm{d}h\) has a magnetic charge quantity \(\textrm{d} q_m = M_r R_r \cos \varphi _r \textrm{d} \varphi _r \textrm{d}h\), forming a pair of magnetic dipoles. The azimuth angle of the dipole \(\theta = \theta _r + \varphi _r\) and the magnetic dipole moment and its derivative are:

$$\begin{aligned} \textrm{d} \vec {m}=2M_rR_r^2[\cos (\theta _r+\varphi _r)\vec {e}_x+\sin (\theta _r+\varphi _r)\vec {e}_z]\cos \varphi _r \textrm{d} \varphi _r \textrm{d}h \end{aligned}$$
(29)

To facilitate integration operations, we expand the magnetic field (28) in the Cartesian coordinate system, assuming the field point position is P(x, y, z) and the direction vector is \(\vec {e}_r=(x_p\vec {e}_x+y_p\vec {e}_y+z_p\vec {e}_z)/r\), where \(r=\sqrt{x_p^2+y_p^2+z_p^2}\). By substituting Eq. (29) into the magnetic field Eq. (28), we obtain the magnetic field of the dipole in the near-field range:

$$\begin{aligned} \textrm{d} \vec {B}_r =&\frac{2\mu _{0}M_rR_r^2 \cos \varphi _r \textrm{d} \varphi _r \textrm{d}h}{4\pi } \{ \frac{3[x_{p}\cos (\theta _r+\varphi _r)+z_{p}\sin (\theta _{r}+\varphi _r)]}{r^{5}} \nonumber \\&\times (x_{p}\vec {e}_{x}+y_{p}\vec {e}_{y}+z_{p}\vec {e}_{z})-\frac{\cos (\theta _r+\varphi _r)\vec {e}_x+\sin (\theta _r+\varphi _r)\vec {e}_z}{r^3} \} \end{aligned}$$
(30)

Performing the integration of Eq. (30), we can obtain the magnetic field of the rotor in the near-field range:

$$\begin{aligned} \vec {B}_r=&\frac{\mu _{0}M_{r}V_{r}}{4\pi r^{5}}\{(3x_{p}^{2}\cos \theta _{r}+3x_{p}z_{p}\sin \theta _{r}-r^{2}\cos \theta _{r})\vec {e}_{x}+3(x_{p}y_{p}\cos \theta _{r}+y_{p}z_{p}\sin \theta _{r})\vec {e}_{y} \nonumber \\&\quad +(3z_{p}^{2}\sin \theta _{r}+3x_{p}z_{p}\cos \theta _{r}-r^{2}\sin \theta _{r})\vec {e}_{z}\} \end{aligned}$$
(31)

From Eq. (31), it can be observed that the rotor’s magnetic field is influenced by several factors, including the rotor’s magnetization intensity, rotor volume, position of the field point, and the angle of oscillation.

According to Eqs. (13) and (31), we can obtain the magnetic field of the mechanical antenna in the near-field range:

$$\begin{aligned} { \vec {B} = \vec {B}_r + \mu _{0}\vec {H}_c } \end{aligned}$$
(32)

Experimental results and discussion

An prototype of the magnetic pendulum mechanical antenna was designed to accurately measure the magnetic pendulum mechanical antenna’s magnetic field characteristics. Firstly, 3D printing technology was employed to fabricate the mechanical antenna frame, with PLA (polylactic acid) selected as the printing material. PLA is a biodegradable material that offers good mechanical strength and does not interfere with the magnetic field distribution. Using 3D printing technology, it was possible to manufacture complex-shaped mechanical antenna frames with high precision and stability, meeting specific design requirements. Secondly, at the ends of the rotor, ceramic bearings were used for support to ensure the stability of the rotor. The inner diameter of the bearing (MR74) is 4 mm, the outer diameter is 7 mm, and the width is 2.5 mm. A specific installation clearance is set from the rotor to facilitate the installation of the bearing, as shown in Fig. 9.

Fig. 9
figure 9

Prototype of the magnetic pendulum mechanical antenna.

Full size image

The excitation system consisted of a RIGOL DG4062 waveform generator (frequency range: 1 \(\mu\)Hz–60 MHz, amplitude range: 1 mVpp–10 Vpp) driving a custom-wound solenoid coil. Magnetic field measurements were performed using a calibrated Schwarzbeck FESP 5133-F loop antenna (frequency range 10 Hz–200 kHz) positioned 50 cm above the antenna centroid. The measured data was displayed and recorded by Tektronix MD32 oscilloscope(bandwidth 1 GHz, vertical resolution 8 bits, sampling rate 5 GS/s)

During the measurement process, a RIGOL DG4062 signal generator was configured to output sinusoidal signals with frequencies sweeping from 100 to 500 Hz in 10 Hz increments. Each frequency was maintained for 60 s to ensure steady-state oscillation. The resulting magnetic field components were captured at 50 cm above the antenna centroid using a Schwarzbeck FESP 5133-F loop antenna, with signals acquired through a Tektronix MDO32 oscilloscope configured for 1 \(\mu s\) sampling interval (1 MS/s rate), 1-s acquisition windows. Following signal acquisition, raw data underwent post-processing with Butterworth bandpass filter to remove the influence of clutter.

Figure 10 shows the magnetic field signal captured by the magnetic field receiving device at a height of 50 cm when a 200 Hz alternating current is applied to the excitation coil. The spatial magnetic field of the mechanical antenna is the superposition of the magnetic field generated by the rotor and the coil’s magnetic field. From the time-domain curve, the mechanical antenna’s magnetic field exhibits sinusoidal behaviour. The theoretically calculated signal frequency and the experimentally tested frequency are 200 Hz from the frequency-domain curve. The theoretically calculated magnetic field of mechanical antenna is 0.067 A/m, while the experimentally tested magnetic field is 0.069 A/m, with an error of 2.9% between the experimental and theoretical results, indicating good consistency. Based on this experimental outcome, it is possible to alter the magnetic field’s frequency by adjusting the excitation coil’s frequency, thereby enabling the extraction of signals at the receiving end for information transmission.

Fig. 10
figure 10

Comparison of magnetic field signals between experiment and theory (a) time-domain plot (b) frequency-domain plot.

Full size image

Figure 11 compares the mechanical antenna’s experimentally measured magnitude-frequency response curve at a height of 50 cm and the theoretically calculated results. The experimental findings indicate that the mechanical antenna’s magnitude-frequency response reaches a peak at a frequency of 299 Hz, whereas mechanical antenna’s natural frequency is determined to be 304 Hz. It suggests the mechanical antenna enters its optimal radiation efficiency before approaching its natural frequency. Below the frequency of 299 Hz, the mechanical antenna’s magnetic field increases with the frequency, which can be attributed to resonance causing an amplification of the rotor’s movement and enhancing the rotor’s magnetic field. The closer the mechanical antenna’s frequency is to the resonance frequency, the stronger the magnetic field. However, particularly when the frequency exceeds the mechanical antenna’s natural frequency, the magnetic field decreases abruptly. It is because, beyond the natural frequency, the direction of the rotor’s magnetic field is opposite, leading to a cancellation effect between the rotor’s magnetic field and the coil’s magnetic field, as depicted in Fig. 12. Furthermore, as shown in Fig. 11, within the resonance frequency range of 290–310 Hz, the deviation between the theoretical and experimental results increases. This is due to the fact that in the vicinity of the resonance frequency, the rotor undertakes large-amplitude oscillations. At this time, the angular velocity of the rotor increases. Under such a state, extremely high requirements are imposed on the mechanical structure and the dynamic balance of the rotor, resulting in a decrease in the stability of the prototype and an increase in the experimental error. Excluding this unstable resonance frequency range, within the frequency range of 100–500 Hz, the average error between the model and the experimental results amounts to 5.1%, which can be employed to measure the accuracy level of the theoretical model within the stable range.

Fig. 11
figure 11

Comparison of amplitude-frequency characteristics between experimental and theoretical calculations.

Full size image
Fig. 12
figure 12

Phase-frequency curve.

Full size image

The intensity of the received magnetic field is not only influenced by frequency variations but also by the position of the receiver. Analyzing the impact of position on magnetic field intensity is essential for subsequent signal communication. Figure 13 illustrates the relationship between magnetic field intensity and the receiver’s position on a spherical surface with a radius of 1 meter, operating at 200 Hz. Some measuring points are selected to validate the magnetic field characteristic results. According to the magnetic field characteristics of the magnetic pendulum mechanical antenna, on the sphere with a radius of 1 m, the magnetic field \(H_x\) reaches its maximum value in the XOZ plane, with an angle of \(45^{\circ }\) relative to the Z axis. The theoretical result is \(6.4 \times 10^{-3} \textrm{A}\,\textrm{m}\), and the experimental test result is \(6.05 \times 10^{-3} \textrm{A}\,\textrm{m}\), with an error of 5.49%. The magnetic field \(H_y\) reaches its maximum value in the YOZ plane, with an angle of \(45^{\circ }\) relative to the Z axis. The theoretical result is \(6.3 \times 10^{-3} \textrm{A}\,\textrm{m}\), and the experimental test result is \(5.95 \times 10^{-3} \textrm{A}\,\textrm{m}\), with an error of 5.51%. The magnetic field \(H_z\) reaches the maximum value at the Z-axis. The theoretical result is \(8.46 \times 10^{-3} \textrm{A}\,\textrm{m}\), and the experimental test result is \(8.53 \times 10^{-3} \textrm{A}\,\textrm{m}\), with an error of 0.84%. It proves that the experimental and theoretical results have a good agreement. Therefore, in the design process of the mechanical antenna, the influence of receiver position should be considered to ensure the optimal magnetic field and signal quality.

Fig. 13
figure 13

Magnetic field of the mechanical antenna.

Full size image

Conclusion

This paper rigorously analyzes the coupled interactions among the rotor permanent magnet, stator permanent magnets, and driving coil through torque-based modeling, and establishes a dynamic model for the magnetic pendulum mechanical antenna. This model accurately describes the swinging behaviour of the rotor’s permanent magnet under the combined action of an alternating magnetic field and an external magnetic field, providing a theoretical foundation for analyzing the motion laws of the magnetic pendulum mechanical antenna. Furthermore, based on the dynamic model, we have developed a magnetic field model of the magnetic pendulum mechanical antenna, which can accurately predict the magnetic field characteristics of the mechanical antenna. In order to verify the accuracy of the magnetic field model, a prototype of the magnetic pendulum mechanical antenna was designed and fabricated using 3D printing technology, and the experimental results were compared and analyzed with the theoretical predictions. The results indicate that the error between the theoretical calculations and the experimental measurements is 5.1%, demonstrating good consistency. It thus proves the accuracy and reliability of the magnetic field model.

The proposed model provides a theoretical foundation for optimizing the design of magnetic pendulum mechanical antennas in practical applications. For instance, in underwater communication systems, this model can adjust the resonant frequency of the antenna to the required communication frequency band by optimizing the rotor-stator geometric parameters, and utilize the resonance effect to increase the radiation power and the communication distance. Future implementations may integrate this model with adaptive control algorithms to dynamically adjust excitation frequencies based on environmental variations.