Levitation and nonlinearity characteristics of a diamagnetic levitation structure

Abstract

To better understand and make use of its characteristics, a diamagnetic levitation structure, which consists of NdFeB permanent magnets and pyrolytic graphite sheets, is studied in this report. Finite element analysis is performed to analyze the influence of floating magnets with different structures and sizes on the equilibrium spacing and movement space of a floating magnet. The key factors that affect the equilibrium spacing and movement space are obtained, and the effectiveness of the simulation is verified experimentally. Compared to the experimental results, the maximum simulation error is less than 10%. A forced vibration model with a single degree of freedom is used to denote the diamagnetic levitation structure, and the natural frequency is analyzed to be 4.87 Hz for the structure in this study. The nonlinear characteristics of the system are studied to obtain a better understanding of the vibration of the nonlinear system. The diamagnetic levitation structure has great potential in the application of vibration energy harvesters or noncontact sensors and actuators.

Keywords

Diamagnetic levitation magnet pyrolytic graphite nonlinear movement space equilibrium position

1. Introduction

Recently, diamagnetic levitation structures have attracted considerable interest because of their potential application in micro/nano-electro-mechanical systems (MNEMS). In 1842, Earnshaw stated that an object could not be stably levitated when subjected to forces that varied by the inverse square of the distance between them [1]. However, it was later determined that this theorem did not apply to diamagnetic materials. In 1939, small pieces of bismuth and graphite were levitated in a non-uniform strong electromagnetic field by Braunbek [2]. Furthermore, Simon and Geim [3] used diamagnetic materials to levitate a living frog and stably suspend a magnet.

In recent years, a few studies [4, 5] on diamagnetic levitation have been reported, such as the magnetic levitation of superconductors based on the Meissner effect [6]. However, this levitation [7] requires superconductors in the superconducting state. Fortunately, with the improvement of rare-earth permanent magnets and diamagnetic levitation technology, diamagnetically stable levitation at room temperature is becoming more common. In 2001, a magnetic levitation device was achieved at room temperature by Simon et al. [8], and this device could work without any superconductors or external energy input. Then, Palagummi et al. [9, 10] proposed a vibration energy harvester based on this structure. The diamagnetic levitation technology has begun to gradually move towards engineering applications. Abadie et al. [11] used a NdFeB permanent magnet and pyrolytic graphite sheet to create a nano force sensor, and the resolution of this sensor could reach 5 nN. Hilbert and Jakoby [12] designed a magnetic membrane actuator composed of NdFeB permanent magnets, pyrolytic graphite plate, and coils, with a floater magnet integrated on a polymer composite film. Clara et al. [13] studied an advanced viscosity and density sensor based on diamagnetically stabilized levitation. The diamagnetic levitation system has certain characteristics, such as simple structure, small volume and lightweight, and it can be widely used in sensors [13], micro rotors [14, 15] energy harvesters [9, 10, 16], micro gyroscopes [17] and biomedical fields.

Currently, scholars have introduced several diamagnetic levitation structures, and the most common diamagnetic levitation system typically consists of NdFeB permanent magnets and pyrolytic graphite. To provide a thorough understanding of the diamagnetic levitation structure, a set of exploratory experiments is designed to systematically characterize the suspension features and nonlinear behavior of the diamagnetic levitation structure. This report can serve as a reference for the study of diamagnetic levitation systems.

Figure 1.

Structural model of the diamagnetic levitation system.

Figure 2.

The boundary and its normal vector of the floating magnet.

2. Structure and principle of a diamagnetic levitation system

The structure of the diamagnetic levitation system is illustrated in Fig. 1, which consists of a lifting magnet, two pyrolytic graphite sheets, a floating magnet and a non-magnetic shell. The magnetization of the two magnets is along their axes. In the vertical direction, the floating magnet is pulled by the lifting magnet. The attractive force is exerted on the floating magnet by the lifting magnet, and the gravity of the floating magnet is balanced by the attractive force. Simultaneously, the floating magnet is subjected to the diamagnetic forces by the upper and lower pyrolytic graphite sheets. A stable levitation of the floating magnet is achieved between the two graphite sheets. Theoretically, the equilibrium position of the floating magnet is exactly at the midpoint between the two graphite sheets in the vertical direction. The floating magnet is expected to return to its equilibrium automatically with the resultant force when it deviates from the position of equilibrium in the horizontal or vertical directions.

3. Analysis of the levitation space

3.1 Calculation of the magnetic force

The interaction force between two magnets can be obtained by integrating the Cauchy stress tensor on the boundary of a magnet. As indicated in Fig. 2, ${n}_{1}$ is the boundary normal vector pointing out from the magnet surface. ${T}_{1}$ and ${T}_{2}$ are the stress tensors in the floating magnet and air, respectively.

Because the stress tensor is continuous at the common boundary of the floating magnet and air, the relationship between ${T}_{1}$ and ${T}_{2}$ can be expressed as follows [18]:

$\displaystyle{n}_{1}({T}_{2}-{T}_{1})=0$ (1)

Under the action of the magnetic field of the lifting magnet, the stress tensor of air can be expressed as follows:

$\displaystyle{T}_{2}=-pI-\left({\frac{1}{2\mu_{0}}{B}_{1}\cdot{B}_{1}}\right){% I}+\frac{1}{\mu_{0}}{B}_{1}{B}_{1}^{T}$ (2)

where $p$ is the air pressure; $I$ is the identity matrix; $\mu_{0}$ is the permeability of vacuum; and $B_{1}$ is the magnetic induction intensity of the lifting magnet at the location of the floating magnet. If the air is approximated to be a vacuum, then $p=$ 0 and $B_{1}=\mu_{0}H$ , and they can be substituted into Eq. (2). Then, the stress tensor is projected on the boundary, and the expression for the stress tensor can be given as follows:

$\displaystyle{n}_{1}T_{2}={-}\frac{1}{2}\left({{H}\cdot{B}_{1}}\right){n}_{1}+% \left({{n}_{1}\cdot{H}}\right){B}_{1}^{T}$ (3)

Using integral calculus, the magnetic force $F_{1}$ exerted by the lifting magnet on the floating magnet can be expressed as follows:

$\displaystyle{F}_{1}=\oint\limits_{\partial\Omega_{1}}{{n}_{1}}{T}_{2}\text{dS}$ (4)

where S is the surface area of the floating magnet.

3.2 Calculation of the diamagnetic force

A highly oriented pyrolytic graphite (HOPG) is a type of anisotropic material, and the force per unit volume exerted on the graphite sheet by the floating magnet can be written as follows [19]:

$\displaystyle d{{F}}_{2}={{M}}_{m}\cdot\left({\nabla\cdot{{B}}_{2}}\right)d{v}$ (5)

where $M_{m}$ and $B_{2}$ denote the magnetization and the magnetic induction strength at the location of the diamagnetic material in the magnetic field of the floating magnet, respectively. Because the susceptibility of the diamagnetic material $\chi_{m}$ is extremely small, $M_{m}$ can be written as follows:

$\displaystyle{{M}}_{{m}}=\frac{{{\chi}}_{m}}{\mu_{0}}{{B}}_{2}=\frac{1}{\mu_{0% }}\left[{{\begin{array}[]{*{20}c}{\chi_{{m}x}}\hfill&0\hfill&0\hfill\\ 0\hfill&{\chi_{{m}y}}\hfill&0\hfill\\ 0\hfill&0\hfill&{\chi_{{m}z}}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{{B}_{{2x}}}\hfill\\ {{B}_{{2y}}}\hfill\\ {{B}_{{2z}}}\hfill\\ \end{array}}}\right]$ (6)

By substituting Eq. (6) into Eq. (5), the integral can be calculated across the entire diamagnetic material. The diamagnetic force between the diamagnet and the magnet can be evaluated as follows:

$\displaystyle{F}_{2x}=\frac{\chi_{m{x}}}{\mu_{0}}\int\!\!\!\int\!\!\!\int_{v}{% B_{2x}\frac{\partial B_{2x}}{\partial x}}\cdot dv=\frac{\chi_{m{x}}}{2\mu_{0}}% \int\!\!\!\int\!\!\!\int_{v}\left({\frac{\partial\left\|{{{B}}_{2}}\right\|^{2% }}{\partial x}}\right)\cdot dv$ (7) $\displaystyle{F}_{{2y}}=\frac{\chi_{m{y}}}{\mu_{0}}\int\!\!\!\int\!\!\!\int_{v% }{B_{2y}\frac{\partial B_{2y}}{\partial y}}\cdot dv=\frac{\chi_{m{y}}}{2\mu_{0% }}\int\!\!\!\int\!\!\!\int_{v}\left({\frac{\partial\left\|{{{B}}_{2}}\right\|^% {2}}{\partial y}}\right)\cdot dv$ (8) $\displaystyle{F}_{{2z}}=\frac{\chi_{m{z}}}{\mu_{0}}\int\!\!\!\int\!\!\!\int_{v% }{B_{2z}\frac{\partial B_{2z}}{\partial z}}\cdot dv=\frac{\chi_{m{z}}}{2\mu_{0% }}\int\!\!\!\int\!\!\!\int_{v}\left({\frac{\partial\left\|{{{B}}_{2}}\right\|^% {2}}{\partial z}}\right)\cdot dv$ (9)

Figure 3.

Structure size and force analysis of the floating magnet.

where $V$ denotes the volume of the diamagnetic material; and $F_{2x}$ , $F_{2y}$ and $F_{2z}$ are the components of the diamagnetic force in the $x$ , $y$ and $z$ directions, respectively. Based on the Gauss divergence law, Eqs (7)–(9) can be rewritten as follows:

$\displaystyle{F}_{{2x}}=\frac{\chi_{{mx}}}{2\mu_{0}}\mathop{{\int\!\!\!\!\!% \int}\mskip-21.0mu \bigcirc}\nolimits_{s}{\left\|{B_{2}}\right\|}^{2}n_{x}% \cdot ds$ (10) $\displaystyle{F}_{{2y}}=\frac{\chi_{{my}}}{2\mu_{0}}\mathop{{\int\!\!\!\!\!% \int}\mskip-21.0mu \bigcirc}\nolimits_{s}{\left\|{B_{2}}\right\|}^{2}n_{y}% \cdot ds$ (11) $\displaystyle{F}_{{2z}}=\frac{\chi_{{mz}}}{2\mu_{0}}\mathop{{\int\!\!\!\!\!% \int}\mskip-21.0mu \bigcirc}\nolimits_{s}{\left\|{B_{2}}\right\|}^{2}n_{z}% \cdot ds$ (12)

where $d s$ represents the surface area unit of the diamagnetic material; and $n_{x}$ , $n_{y}$ and $n_{z}$ denote the surface normal vector components of the diamagnetic material in the $x$ , $y$ and $z$ directions, respectively. When the magnetic field of the floating magnet is calculated, the acting force between the diamagnetic material and the floating magnet can be obtained using Eqs (10)–(12).

3.3 Maximum levitation space

Let us assume that $G$ represents the gravity of the floating magnet, $F_{21}$ and $F_{22}$ represent the diamagnetic force exerted on the floating magnet by the upper and lower pyrolytic graphite sheets, respectively, $F_{1}$ represents the magnetic attractive force exerted on the floating magnet by the lifting magnet, and the upward direction is positive, which is illustrated in Fig. 3. Then, the restoring force exerted on the floating magnet can be given as follows:

$\displaystyle F_{R}=F_{1}-G+F_{22}-F_{21}$ (13)

When the floating magnet moves upwards from its equilibrium position, the distance between the two permanent magnets decreases, which causes the magnetic attractive force to increase. The increment of the magnetic force, $\Delta F_{1}$ , can be expressed as follows:

$\displaystyle\Delta F_{1}=F_{1}-G$ (14)

Meanwhile, the distance $L_{1}$ between the floating magnet and the upper pyrolytic graphite sheet decreases, the diamagnetic force exerted on the floating magnet by the upper pyrolytic graphite increases, and the diamagnetic force from the lower pyrolytic graphite sheet decreases. The increment of the diamagnetic force, $\Delta F_{2}$ , can be written as follows:

$\displaystyle\Delta F_{2}=F_{21}-F_{22}$ (15)

The floating magnet can return to the equilibrium position only when $\Delta F_{2}$ is always greater than $\Delta F_{1}$ . The magnetic force $F_{1}$ and the diamagnetic forces $F_{21}$ and $F_{22}$ can be calculated in COMSOL 4.4. Correspondingly, the maximum rising displacement of the floating magnet, $L_{\text{1max}}$ , can be obtained. Similarly, $L_{\text{2max}}$ , which is the maximum declining displacement of the floating magnet, can be calculated when the floating magnet moves downwards from the equilibrium position. The distance between the upper and lower pyrolytic graphite sheets can be given as follows:

$\displaystyle D=L_{1\max}+L_{2\max}+d$ (16)

where $d$ is the thickness of the floating magnet. Because the diamagnetic force is smaller than the gravity of the floating magnet and the change rate of the magnetic force versus the distance $Z$ is relatively large, the movement range of the floating magnet around the equilibrium position is small. In the small movement range, the magnetic force change with the distance can be regarded as linear. In other words, the increment and decrement of the magnetic force around the equilibrium position are considered to be approximately equivalent. Furthermore, the diamagnetic force increment is equivalent to the diamagnetic force decrement when the floating magnet is located in the middle of the upper and lower pyrolytic graphite sheets. Correspondingly, the maximum rising distance of the floating magnet is considered equal to its maximum declining distance, i.e., $L_{1\max}\approx L_{2\max}$ . Then, Eq. (16) can be rewritten as follows:

$\displaystyle D=2L_{1\max}+d$ (17)

Specifically, the maximum movement space of the floating magnet can be expressed by the maximum rising distance.

The maximum levitation space of the diamagnetic levitation device primarily depends on the relationship between the magnetic force variation and the diamagnetic force variation of the floating magnet. Based on Eq. (4), the magnetic force and the magnetic field intensity of a permanent magnet are related to the structure of the magnet. Using Eqs (10)–(12), the diamagnetic forces between a diamagnet and a magnet are associated with the magnetic field intensity, structure of the magnet, susceptibility and structure of the diamagnet. Thus, the maximum movement space of the diamagnetic levitation device can be adjusted by changing the structure type of the floating magnet.

4. Analysis of the static levitation characteristics

In this study, a method [20] for decreasing the error in the magnetic and diamagnetic forces was introduced to ensure the accuracy of the FEA (finite element analysis) simulation result. The magnetic force between magnets and the diamagnetic force between the magnet and the diamagnet are calculated in the FEA software COMSOL Multiphysics, and the movement space can then be obtained. Consequently, the influence of the floating magnet with different structural parameters on the movement space of the floating magnet is analyzed, and the experimental results prove that the simulation is accurate. In this study, the thickness and diameter of the graphite pyrolytic sheets are 1 mm and 10 mm, respectively.

In COMSOL Multiphysics, a stationary study is used to calculate the magnetic and diamagnetic forces. The free-meshing algorithm using tetrahedral elements is applied to all domains. The maximum element size of the magnets and the pyrolytic graphite sheets is set at 0.2 mm, and the meshing scale of the air domain is set to “Extremely fine” with a 6.6 mm element size. The simulation model is depicted in Fig. 4, in which there are approximately 71,124 tetrahedral elements in the two meshed magnets. The elements of air surrounding the two magnets are refined to be consistent with those of the two magnets. A simulation model consisting of a lifting magnet ( $\Phi$ 4.7625 mm $\times$ 3.175 mm) and a floating magnet ( $\Phi$ 3.175 mm $\times$ 1.5875 mm) is built, in which the distance between the two magnets is 26.89 mm. On an Intel ${}^{\@setsize{\scriptsize}{9.5pt}{\viiipt}{\@viiipt}\textregistered}$ 4 GHz i7 4790 K and 32 GB RAM computer, the solution time of the model is 23 s, and the magnetic force between the two magnets is 923.3 $\mu$ N.

Figure 4.

FEA model for a floating magnet and a lifting magnet (surrounded by blue lines).

4.1 Influence of the thickness of the cylinder floating magnet on the movement space

For the same lifting magnet, ten different floating magnets (with the same diameter) are used in this study. The detailed parameters of these permanent magnets are listed in Table 1. The magnetic forces and the diamagnetic forces are calculated using FEA software.

Table 1
Magnetic parameters and sizes of the lifting magnet and the floating magnet

		Structure	Size (mm)	Grade	Br (T)
LIfting magnet		cylinder	$\Phi$ 4.7625 $\times$ 3.175	N52	1.45
Floating magnet	group 1	cylinder	$\Phi$ 3.175 $\times$ 0.4	N42	1.3
	group 2	cylinder	$\Phi$ 3.175 $\times$ 0.7	N42	1.3
	group 3	cylinder	$\Phi$ 3.175 $\times$ 1	N42	1.3
	group 4	cylinder	$\Phi$ 3.175 $\times$ 1.3	N42	1.3
	group 5	cylinder	$\Phi$ 3.175 $\times$ 1.5875	N42	1.3
	group 6	cylinder	$\Phi$ 3.175 $\times$ 1.9	N42	1.3
	group 7	cylinder	$\Phi$ 3.175 $\times$ 2.2	N42	1.3
	group 8	cylinder	$\Phi$ 3.175 $\times$ 2.5	N42	1.3
	group 9	cylinder	$\Phi$ 3.175 $\times$ 2.8	N42	1.3
	group 10	cylinder	$\Phi$ 3.175 $\times$ 3.175	N42	1.3

The relationship of the magnetic force and the distance between the lifting magnet and the floating magnet for the fifth group is depicted in Fig. 5.

Figure 5.

Relationship of the magnetic force and the distance between the two magnets.

From Fig. 5, it can be seen that the magnetic force varies linearly with the distance between the two magnets in the range of 25 mm to 28 mm, which verifies the rationality of $L_{1\max}\approx L_{2\max}$ . A linear fitting is performed using MATLAB ${}^{\text{TM}}$ to determine the relationship of the magnetic force versus the distance Z between the two magnets, and the fitting equation can be given as follows:

$\displaystyle F_{1}=-141.4Z+4725.2$ (18)

where $F_{1}$ and Z are in units of $\mu$ N and mm, respectively.

If we set $F_{1}=G=$ 923.3 $\mu$ N, then we can obtain $Z=$ 26.89 mm. Specifically, when the distance between the two magnets is 26.89 mm, the floating magnet achieves a force balance. Therefore, 26.89 mm is considered to be the equilibrium spacing of the fifth group.

The magnitude of the magnetic force acting on the floating magnet is related to the distance between the two magnets. When $Z=$ 26.89 mm, $L_{1\max}$ is equal to 0.25 mm, 0.6 mm, 0.8 mm, 0.9 mm and 1 mm, and the relations of $\Delta F_{1}$ and $\Delta F_{2}$ versus $L_{1\max}$ are studied, as indicated in Fig. 6.

Figure 6.

Relationship of the magnetic force variation $\Delta F_{1}$ and diamagnetic force variation $\Delta F_{2}$ with the displacement of the floating magnet.

Figure 7.

Equilibrium spacing and movement space of the floating magnet versus thickness of the floating magnet.

In Fig. 6, for a decreasing $L_{1\max}$ , $\Delta F_{2}$ increases. When $L_{1\max}=L_{2\max}=$ 0.9 mm, $\Delta F_{2}$ is always larger than $\Delta F_{1}$ , which indicates that the resultant force $F_{R}$ can ensure that the floating magnet returns to its equilibrium position automatically. Thus, $L_{1\max}=$ 0.9 mm is regarded as the maximum rising distance of the fifth group of floating magnets, and it is used to represent the maximum movement space of the group.

Similarly, the equilibrium spacing and the maximum movement space of the other groups are obtained, as indicated in Fig. 7.

As seen from Fig. 7, when the diameter of the floating magnet is 3.175 mm, the equilibrium spacing and the maximum movement space of the floating magnet are nearly constant as the thickness of the floating magnet increases from 0.4 mm to 3.175 mm gradually. Therefore, it is infeasible to increase the maximum movement space of the diamagnetic levitation system by changing the thickness of the floating magnet.

4.2 Influence of the volume of the cylindrical floating magnet on the movement space

The parameters of the magnets in this study are provided in Table 2, and the thickness-diameter ratio of the floating magnet is maintained at 0.5 for the same lifting magnet. Furthermore, the volume of the floating magnet, equilibrium spacing and maximum movement space of the floating magnet are varied using FEA software, and the simulation results are illustrated in Fig. 8.

Table 2
Magnetic parameters and sizes of the lifting magnet and the floating magnet

		Structure	Size (mm)	Grade	Br (T)
Lifting magnet		cylinder	$\Phi$ 4.7625 $\times$ 3.175	N52	1.45
Floating magnet	group 1	cylinder	$\Phi$ 1 $\times$ 0.5	N42	1.3
	group 2	cylinder	$\Phi$ 2 $\times$ 1	N42	1.3
	group 3	cylinder	$\Phi$ 3.175 $\times$ 1.5875	N42	1.3
	group 4	cylinder	$\Phi$ 4 $\times$ 2	N42	1.3
	group 5	cylinder	$\Phi$ 5 $\times$ 2.5	N42	1.3
	group 6	cylinder	$\Phi$ 6 $\times$ 3	N42	1.3
	group 7	cylinder	$\Phi$ 7 $\times$ 3.5	N42	1.3
	group 8	cylinder	$\Phi$ 8 $\times$ 4	N42	1.3
	group 9	cylinder	$\Phi$ 9 $\times$ 4.5	N42	1.3
	group 10	cylinder	$\Phi$ 10 $\times$ 5	N42	1.3

Figure 8.

Trend of the equilibrium spacing and movement space of the floating magnet versus volume of the cylindrical floating magnet.

As seen from Fig. 8, when the thickness-diameter ratio is constant, the equilibrium spacing has limited relation to the volume of the floating magnet; however, the maximum movement space first increases then decreases with an increment in the volume of the floating magnet, and the maximum movement space reaches a maximum value when the volume of the lifting magnet is approximately equal to that of the floating magnet. When the volume of the floating magnet is smaller than that of the lifting magnet, the variation of the magnetic force and the diamagnetic force of the floating magnet are small on both sides of the equilibrium position. In this situation, with an increase in the volume of the floating magnet, the increasing rate of the diamagnetic force is faster than that of the magnetic force; hence, the maximum movement space tends to become larger. However, when the volume of the floating magnet is greater than that of the lifting magnet, the increasing rate of the diamagnetic force variation is slower than that of the magnetic force variation; therefore, the maximum movement space decreases.

Table 3

Magnetic parameters and sizes of the lifting magnet and the floating magnet

		Structure	Size (mm)	Grade	Br (T)
Lifting magnet		cylinder	$\Phi$ 4.7625 $\times$ 3.175	N52	1.45
Floating magnet	group 1	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 0.5 $\times$ 2.5	N42	1.3
	group 2	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 1 $\times$ 2.5	N42	1.3
	group 3	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 1.5 $\times$ 2.5	N42	1.3
	group 4	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 2 $\times$ 2.5	N42	1.3
	group 5	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 2.5 $\times$ 2.5	N42	1.3
	group 6	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 3 $\times$ 2.5	N42	1.3
	group 7	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 3.5 $\times$ 2.5	N42	1.3
	group 8	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 4 $\times$ 2.5	N42	1.3
	group 9	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	5 $\times$ 4.5 $\times$ 2.5	N42	1.3

Figure 9.

Trend of the equilibrium spacing and movement space of the floating magnet versus inner diameter of the annular magnet.

4.3 Influence of the floating magnet with an annular structure on the movement space

Based on the floating magnet of the 5th group in Section 4.2, the cylindrical structure is replaced by an annular structure without changing the thickness $h$ or diameter $d_{o}$ of the floating magnet. The detailed parameters are listed in Table 3. To explore the effects of the internal diameter of the annular magnets on the movement space of the diamagnetic levitation device, the equilibrium spacing and maximum movement space of the floating magnet are calculated using FEA software, as indicated in Fig. 9.

As seen from Fig. 9, when the internal diameter of the annular magnet increases gradually from 0.5 mm to 4.5 mm, the equilibrium spacing decreases gradually from 26.84 mm to 26.28 mm, and the maximum movement space decreases from 0.8 mm to 0.4 mm. The maximum movement space is reduced by 50%. The reason is that the rate of decrease of the magnetic force variation is less than that of the diamagnetic force variation when the inner diameter of the annular magnet increases. Consequently, the maximum movement space of the floating magnet becomes increasingly smaller.

4.4 Influence of the floating magnet with a cylinder-ring composite structure on the movement space

As indicated in Table 4, the lifting magnet remains unchanged. Based on the floating magnet of the 2nd group (d ${}_{\text{o}}$ $\times$ d ${}_{\text{i}}$ $\times$ h $=$ 5 $\times$ 1 $\times$ 2.5 mm) in Section 4.3, two cylindrical magnets are fixed at both ends of the ring magnet, forming a hollow floating magnet. Additionally, the diameter of the cylindrical magnet is equal to the outer diameter of the annular magnet, and all of the magnets are axially magnetized. Hollow magnets of different sizes can be obtained by changing the thickness of the cylindrical magnet. The effects of the thickness of the cylindrical magnets on the movement space of the diamagnetic levitation device are studied by calculating the equilibrium spacing and the maximum movement space of the hollow floating magnet using FEA software, and the simulation results are indicated in Fig. 10.

Table 4
Magnetic parameters and sizes of the lifting magnet and the floating magnet

		Structure	Size (mm)	Grade	Br (T)
Lifting magnet		cylinder	$\Phi$ 4.7625 $\times$ 3.175	N52	1.45
Floating magnet	group 1	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 0.5	N42	1.3
	group 2	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 1.	N42	1.3
	group 3	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 1.5	N42	1.3
	group 4	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 2	N42	1.3
	group 5	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 2.5	N42	1.3
	group 6	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 3	N42	1.3
	group 7	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 3.5	N42	1.3
	group 8	ring magnet sandwiched between two cylinders (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h ${}_{1}+$ d ${}_{\text{o}}\times$ h ${}_{2}$ )	5 $\times$ 1 $\times$ 2.5 $+$ 5 $\times$ 4	N42	1.3

As can be seen in Fig. 10, with an increase in the thickness of the cylinder magnets on both sides of the floating magnet, the equilibrium spacing of the floating magnet increases gradually but the maximum movement space becomes increasingly smaller. The maximum movement space reduces from 0.7 mm to 0.3 mm whereas the thickness of the cylindrical magnets increases from 0.5 mm to 4 mm. For the two cylindrical magnets, $\Delta F_{1}$ rapidly increases; however, $\Delta F_{2}$ increases slower than $\Delta F_{1}$ , resulting in decreasing trend for the movement space.

Table 5

Magnetic parameters and sizes of the lifting magnet and the floating magnet

		Structure	Size (mm)	Grade	Br (T)
Floating magnet	cylinder	$\Phi$ 3.175 $\times$ 1.5875	N42	1.3
Lifting magnet	group 1	cylinder	$\Phi$ 2.54 $\times$ 0.79375	N42	1.3
	group 2	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	3.175 $\times$ 1.5875 $\times$ 1.5875	N42	1.3
	group 3	cylinder	$\Phi$ 3.175 $\times$ 1.5875	N42	1.3
	group 4	cylinder	$\Phi$ 3.175 $\times$ 3.175	N52	1.45
	group 5	outer ring inner cylinder	6.35 $\times$ 3.175 $\times$ 3.175 $+$ 3.175 $\times$ 3.175	N42	1.3
		(d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h $+$ d ${}_{\text{i}}\times$ h)
	group 6	cylinder	$\Phi$ 4.7625 $\times$ 3.175	N52	1.45
	group 7	ring (d ${}_{\text{o}}\times$ d ${}_{\text{i}}\times$ h)	6.35 $\times$ 3.175 $\times$ 3.175	N42	1.3

Figure 10.

Trend of the equilibrium spacing and movement space of the floating magnet versus thickness of the cylindrical magnets on both sides of the floating magnet.

Figure 11.

Experimental setup of the floating magnet equilibrium spacing.

Figure 12.

Experimental results, relative error and simulation results of the equilibrium spacing.

4.5 Experimental validation

In this experiment, the floating magnet remains unchanged, and seven groups of different lifting magnets are used. The floating magnet can be located at the equilibrium of the forces by adjusting the distance between the two magnets. In this adjustment process, the axes of the two magnets will automatically overlap. When the floating magnet is located at its equilibrium, the experimental equilibrium spacing can be obtained. The differences between the experimental equilibrium spacing and the theoretical equilibrium spacing represent the simulation errors of the magnetic force. The parameters of the floating magnet and the lifting magnet are provided in Table 5, and the annular magnet and the cylindrical magnet in the lifting magnet of the 5th group are magnetized in opposite directions.

Static stable levitation is achieved for the floating magnet under the combined effects of the two magnets, as indicated in Fig. 11. For the equilibrium spacing, the simulation results, experimental values and relative error are illustrated in Fig. 12.

As depicted in Fig. 12, the simulation results can reflect the changing trend of the experimental values: the simulation error increases with an increase in the equilibrium spacing, and the maximum error is less than 10%.

5. Vibration model and nonlinear analysis

Under vertical excitation, the diamagnetic levitation device is simplified as a forced vibration model of a single degree of freedom system consisting of a mass, spring and damper, as indicated in Fig. 13.

Figure 13.

Physical model of the diamagnetic levitation structure.

Figure 14.

Restoring force versus displacement of the floating magnet in the vertical direction.

As seen from Fig. 6, for the lifting magnet ( $\Phi$ 4.7625 mm $\times$ 3.175 mm) of the 6th group in Section 4.5 and the floating magnet ( $\Phi$ 3.175 mm $\times$ 1.5875 mm), the diamagnetic force is nonlinear when the displacement varies. Based on Eq. (13), the restoring force $F_{R}$ of the floating magnet is a nonlinear force along the vertical direction; hence, the vibration of the floating magnet includes nonlinear characteristics, and the nonlinear vibration primarily depends on the nonlinearity of the diamagnetic force. The nonlinearity of the diamagnetic force can be effectively reduced by decreasing the vibration range of the floating magnet. For example, the linear fitting errors of the diamagnetic force are 14.7 $\mu$ N, 31.82 $\mu$ N and 100 $\mu$ N for the vibration spaces of 0.25 mm, 0.6 mm and 0.9 mm, respectively. When the vibration range of the floating magnet is 0.9 mm, a linear fitting and nonlinear fitting are implemented using MATLAB ${}^{\text{TM}}$ to determine the relationship between the total force exerted on the floating magnet and the displacement. The fitting results are illustrated in Fig. 14. The linear equation can be given as follows:

$\displaystyle F_{{R1}}{=}0.0884088{z}$ (19)

where $F_{R1}$ and $z$ are in units of $\mu$ N and mm, respectively.

For a linear vibration system, the nature frequency of the system can be computed with the equation as follows [21]:

$\displaystyle f_{n}=\frac{\omega_{n}}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{K}{m}}$ (20)

By substituting $K=$ 0.0884088 N/m and $m=$ 9.45 $\times$ 10 ${}^{-5}$ kg into Eq. (20), the vertical resonance frequency of the linear vibration system $f_{n1}$ can be calculated as 4.87 Hz.

Similarly, the nonlinear fitting equation can be given as follows:

$\displaystyle F_{R2}=0.1223808z^{5}-0.0374239z^{3}+0.0262614z$ (21)

where $F_{R2}$ and $z$ are in units of $\mu$ N and mm, respectively.

Then, the nonlinear vibration Eq. (22) of the floating magnet can be given as follows:

$\displaystyle{m\ddot{z}}+{f}_{m}\left({{\ddot{z},\dot{z}}}\right)+{f}_{k}\left% ({{z,\dot{z}}}\right)={m\ddot{z}}+F_{R2}=\textit{Fsin}\omega{t}$ (22)

where ${f}_{m}\left({\ddot{z},\dot{z}}\right)$ represents the synthetic expression of the nonlinear inertial force and nonlinear damping force; and ${f}_{k}\left({z,\dot{z}}\right)$ represents the synthetic expression of the nonlinear damping force and nonlinear elasticity force.

When the motion is in its initial state, $z=$ 0, the external excitation is $y(t)=\textit{Ysin}\omega t$ , and the equivalent equation for the nonlinear vibration equation can be written as follows:

$\displaystyle{m}_{e}{\ddot{z}}+c_{e}{\dot{z}}+{k}_{e}z=m\omega^{2}Y\sin\omega t$ (23)

where $m_{e}$ is the equivalent mass: $c_{e}$ is the equivalent damping coefficient; and $k_{e}$ is the equivalent stiffness coefficient.

Because of the damping, the free vibration will disappear after a certain time. By assuming that the stationary solution of Eq. (23) is $z=\textit{Zsin}(\omega t-\theta)=\textit{Zsin}\varphi$ , then the equivalent linearized amplitude can be given as $Z=m\omega^{2}\textit{Ycos}\theta/(k_{e}-m_{e}\omega^{2})$ , and the phase angle difference is $\theta=\textit{arctanc}_{e}/(k_{e}-m_{e}\omega^{2})$ based on the linear vibration theory.

The equivalent natural frequency of the nonlinear system can be written as follows:

$\displaystyle f_{n}=\frac{1}{2\pi}\sqrt{\frac{k_{e}}{m_{e}}}$ (24)

where ${m}_{e}=m-\left[{\frac{1}{\pi\omega^{2}Z}\int\limits_{0}^{2\pi}{f_{m}\left({% \ddot{z},\dot{z}}\right)}\cos\varphi d\varphi}\right]$ , ${k}_{e}=\frac{1}{\pi Z}\int\limits_{0}^{2\pi}{f_{k}}\left({z,\dot{z}}\right)% \sin\varphi d\varphi$ .

The resonance frequency was calculated using Eqs (22) and (24) and can be written as follows:

$\displaystyle{f}_{n2}=\sqrt{7.039-7.523Z^{2}+20.502Z^{4}}$ (25)

The natural frequency of the nonlinear system depends primarily on the amplitude of the floating magnet. In other words, it depends on the physical characteristics of the system as well as the displacement and frequency of the external excitation. Accordingly, the natural frequency of the diamagnetic levitation system can be regulated by adjusting the size of each part of the diamagnetic levitation device, residual magnetization of the two magnets and amplitude and frequency of the external excitation. When the external excitation source is fixed, the diamagnetic levitation device can work in a certain frequency range, and the ideal vibration is achieved by reasonably configuring the residual magnetization, size of the two magnets and distance between the two pyrolytic graphite sheets.

6. Conclusions

In summary, we have demonstrated that a diamagnetic levitation structure can be used for the stable levitation of a floating magnet. For a floating magnet of different shapes, the influence of its thickness, volume, inner diameter and structure on the maximum movement space and equilibrium spacing has been studied in detail. The maximum movement space is less affected by the thickness of the cylindrical floating magnet. However, increasing the volume and inner diameter of the floating magnet will result in a significant reduction in the maximum movement space. Additionally, the above-mentioned factors have limited effects on the equilibrium spacing; it is impracticable to reduce the volume of the diamagnetic levitation structure by changing these parameters. The validity of the simulation analysis is verified by the experimental results, and the maximum error of the finite element analysis is within 10%. The diamagnetic levitation device exhibits a nonlinear characteristic, which is beneficial for expanding the frequency response range. The natural frequency of the diamagnetic levitation structure is low, i.e., it is primarily concentrated in the vicinity of 4.87 Hz. Furthermore, the resonance frequency can be tuned by adjusting the size of the components of the structure, residual magnetization of the lifting magnet and floating magnet or the external excitation. The diamagnetic levitation structure presented in this study is highly suitable for building a vibration energy harvester or contactless sensing elements due to its stability and nonlinearity.

Footnotes

Acknowledgments

This study is supported by the National Natural Science Foundation of China under grant no. 51475436 and the Henan Province Key Project in Science and Technologies under grant no. 152102210042.

References

Earnshaw

, On the nature of the molecular forces which regulate the constitution of the luminiferous ether, Trans. Camb. Phil. Soc 7 (1842), 97–112.

Braunbek

, Freies schweben diamagnetischer körper im magnetfeld, Zeitschrift für Physik 112 (1939), 764–769.

Simon

and Geim

, Diamagnetic levitation: flying frogs and floating magnets, Journal of Applied Physics 87 (2000), 6200–6204.

Nemiroski

Kumar

A.A.

Soh

Harburg

D.V.

H.-D.

and Whitesides

G.M.

, High-sensitivity measurement of density by magnetic levitation, Analytical Chemistry 88 (2016), 2666–2674.

Kamal

K.Y.

Herranz

van Loon

J.J.

Christianen

P.C.

and Medina

F.J.

, Evaluation of simulated microgravity environments induced by diamagnetic levitation of plant cell suspension cultures, Microgravity Science and Technology 28 (2016), 309–317.

Davis

Logothetis

and Soltis

, Stability of magnets levitated above superconductors, Journal of Applied Physics 64 (1988), 4212–4218.

Murakami

Kobayashi

Gyoda

Ando

and Yamada

, Development of magnetic levitation synchronous motor with high-Tc superconducting bearing, International Journal of Applied Electromagnetics and Mechanics 39 (2012), 97–103.

Simon

Heflinger

and Geim

, Diamagnetically stabilized magnet levitation, American Journal of Physics 69 (2001), 702–713.

Palagummi

and Yuan

, An optimal design of a mono-stable vertical diamagnetic levitation based electromagnetic vibration energy harvester, Journal of Sound and Vibration 342 (2015), 330–345.

10.

Palagummi

Zou

and Yuan

, A horizontal diamagnetic levitation based low frequency vibration energy harvester, Journal of Vibration and Acoustics 137 (2015), 061004.

11.

Abadie

Piat

Oster

and Boukallel

, Modeling and experimentation of a passive low frequency nanoforce sensor based on diamagnetic levitation, Sensors and Actuators A: Physical 173 (2012), 227–237.

12.

Hilber

and Jakoby

, A magnetic membrane actuator in composite technology utilizing diamagnetic levitation, IEEE Sensors Journal 13 (2013), 2786–2791.

13.

Clara

Antlinger

Abdallah

Reichel

Hilber

and Jakoby

, An advanced viscosity and density sensor based on diamagnetically stabilized levitation, Sensors and Actuators A: Physical 248 (2016), 46–53.

14.

Khanna

J.N.

Irwen

and Wang

W.C.

, Diamagnetically levitating three phase motor with optical feedback control, International Journal of Optomechatronics 4 (2010), 424–448.

15.

Xiao

and Takahata

, Analytical and experimental study of micromachined graphite rotor based on diamagnetic levitation, in: Nano/Micro Engineered and Molecular Systems (NEMS), 2015 IEEE 10th International Conference on, IEEE, Xi’an, China, 2015, pp. 116–119.

16.

Duan

Takahata

and Su

, Analysis of multi-direction low-frequency vibration energy harvester using diamagnetic levitation, International Journal of Applied Electromagnetics and Mechanics 47 (2015), 847–860.

17.

Liu

Zhang

Liu

Chen

Cui

and Li

, An innovative micro-diamagnetic levitation system with coils applied in micro-gyroscope, Microsystem Technologies 16 (2010), 431–439.

18.

Wangsness

R.K.

, Electromagnetic fields, Wiley, New York, 1986.

19.

Chen

, Study on diamagnetic bearings for MEMS, Ph.D. Dissertation, Shanghai Jiao Tong University School, 2009.

20.

Takahata

and Su

, FEA analysis and working performance of micro vibration energy harvester based on diamagnetic levitation, Microsystem Technologies 21 (2015), 903–909.

21.

Shao

, Mechanical System Dynamics (in Chinese), Machinery Industry Press, Beijing, 2005.

22.

Wen

, Theory of mechanical vibration and its application (in Chinese), Higher education Press, Higher education Press, 2009.