A Novel Nonlinear Eddy Current Damping Design Method for a Triple-magnet Magnetic Suspension Dynamic Vibration Absorber

Abstract

A novel Nonlinear Eddy Current Damping (NL-ECD) design method for a Triple-magnet Magnetic Suspension Dynamic Vibration Absorber (TMSDVA) is proposed, modelled, and tested. The implementation of the NL-ECD design method is completely passive and does not require an external energy supply. First, the mathematical model of the TMSDVA vibration reduction system is established. Additionally, the NL-ECD calculation model is also derived. The NL-ECD coefficient can be designed and adjusted by changing the dimensional parameters of the damping tube. Then the correctness of the NL-ECD derivation process is verified through simulations and experiments. It is found that the NL-ECD coefficient is strongly related to the position of the moving magnet. And the simulation results show the same trend as the experimental results. Next, the dynamic responses of the TMSDVA with NL-ECD are studied. Using NL-ECD can not only eliminate higher branch responses in the TMSDVA vibration reduction system, but also retain non-linear dynamics response characteristics of the system. As a result, a better vibration reduction effect of the TMSDVA is obtained. Moreover, the NL-ECD design method proposed in this paper can be either applicable to the TMSDVA or applicable to other types of magnetic DVAs using cylindrical magnets or ring magnets.

Keywords

nonlinear eddy current damping triple-magnet magnetic dynamic vibration absorber higher branch response nonlinear dynamics passive vibration reduction

Highlights

A novel Nonlinear Eddy Current Damping (NL-ECD) design method is proposed, modelled, and tested.

The realization of the method is completely passive and does not require an external energy supply.

The NL-ECD coefficient of the Triple-magnet Magnetic Suspension Dynamic Vibration Absorber (TMSDVA) can be designed and adjusted by changing the dimensional parameters of the damping tube.

Using NL-ECD can both eliminate higher branch responses and retain non-linear response characteristics in the TMSDVA vibration reduction system.

The NL-ECD design method can also be applicable to other types of magnetic DVAs utilizing cylindrical magnets or ring magnets.

Introduction

Dynamic Vibration Absorber (DVA), a passive vibration reduction device, was first proposed by Frahm in 1909 (Xu, 2015). DVAs have been widely used in the vibration control of engineering structures. Linear Dynamic Vibration Absorber, which is also called Tuned Mass Damper (TMD), is usually composed of a linear stiffness element, a mass block, and a damping component (Liu et al., 2007). Thanks to its large damping, TMD can have a broad vibration reduction frequency band. However, due to the high demand for the damping component, TMD usually requires more installation space. And TMD may lead to the occurrence of a significant resonance frequency offset of the vibration reduction object (Den Hartog, 1947; Ormondroyd and Den Hartog, 1928). Roberson (Roberson, 1952) indicated that introducing nonlinear stiffnesses to DVAs can also expand the vibration reduction frequency band and enhance the robustness of DVAs. And the nonlinear system will not exhibit an obvious resonance peak when the Nonlinear Dynamic Vibration Absorber (NLDVA) performs a good vibration reduction effect. There are many kinds of NLDVAs, including cubic stiffness NLDVAs (Oueini et al., 1999), piecewise stiffness NLDVAs (Pun and Liu, 2000) and tuned stiffness NLDVAs (Walsh and Lamancusa, 1992), et al. Nonlinear Energy Sink (NES) is a kind of NLDVA that can effectively transfer vibration energy from the primary structure to itself and dissipate it through its damping. NES was first proposed and named by Vakakis (Vakakis, 2001). NESs differ from other NLDVAs because of their targeted energy transfer (Zhang et al., 2019), which is fast, unidirectional, and irreversible (Guo et al., 2015). As a result, NESs have received a lot of attention from researchers in recent years. Furthermore, the related theories have gradually matured (Bab et al., 2017; Li et al., 2019; Tsiatas and Karatzia, 2020).

Besides, in the studies of NLDVAs, the researchers have also paid attention to DVAs with magnetic properties. The nonlinear stiffnesses of magnetic DVAs are generated by the interaction between the magnets (Sun et al., 2021; Zhang et al., 2017). Compared with NLDVAs constructed from non-magnetic components, magnetic DVAs greatly reduce the difficulty of obtaining nonlinear stiffness. Many types of magnetic DVAs have been proposed and studied, including those with rods and ring magnet sets (Benacchio et al., 2016; Yamakawa et al., 1977), as well as those with rectangular magnets (Chen et al., 2020). However, existing magnetic DVAs typically require numerous magnets, leading to more complex structures. Furthermore, due to this requirement, magnetic DVAs, which have high installation space requirements, are not conducive to miniaturization.

Although NLDVAs typically possess wide vibration reduction frequency bands, the strong nonlinear stiffnesses in NLDVAs can lead to complex dynamics in the vibration reduction system, resulting in behaviours such as bifurcation and chaotic motion (Chen et al., 2018; Zang and Chen, 2017). Starosvetsky et al. (Starosvetsky and Gendelman, 2009) have investigated bifurcation control of a NLDVA using nonlinear damping. This type of nonlinear damping characteristic is feasible in hydraulic dampers based on the same principle of orifice flow, with the addition of semi-active control (Golafshani and Mirdamadi, 2009). Such damping can be realized in hydraulic dampers containing several on/off orifice valves in a moving piston. These pieces of evidence mean that the realization of the nonlinear damping needs external energy supply. However, for magnetic DVAs, due to the magnetic properties of the magnets, employing Eddy Current Damping (ECD) (Liu et al., 2020b; Lu et al., 2017; Tom et al., 2016) may be a convenient and effective method for eliminating bifurcation behaviours in the system.

Eddy currents can be induced in a good non-magnetic conductor, such as aluminium, because of a moving permanent magnetic source (e.g., a permanent magnet) in the vicinity of the conductor (Bae et al., 2009; Kriezis et al., 1992). The interaction between the induced magnetic field resulting from eddy currents and the external magnetic field generates a braking force, referred to as the ECD force, against the motion of the permanent magnetic source (Uhlig et al., 2012). This force is proportional to the velocity of the permanent magnetic source within the range relevant to engineering applications and behaves like a viscous force (Yan et al., 2021). Compared to traditional dampers (such as hydraulic or rubber dampers), eddy current dampers offer advantages of non-contact, frictionless, and high reliability (Kriezis et al., 1992; Yan et al., 2021). Moreover, eddy current dampers do not degrade in performance over time. Additionally, due to their non-contact characteristics, eddy current dampers do not exhibit additional stiffnesses like traditional viscoelastic damping, which arises from material shear deformation (Ebrahimi et al., 2008). The ECD has been widely used in the linear TMD design (Bae et al., 2018; Liu et al., 2020a, b; Lu et al., 2017; Niu et al., 2018; Tom et al., 2016), which can effectively dissipate the vibration energy of the vibration reduction object. However, currently, few relevant studies on the application of the ECD in NLDVAs have been reported. The existing damping adjustment methods for NLDVAs typically rely on active or semi-active control, which requires an external energy supply.

Based on the aforementioned background, a novel Nonlinear Eddy Current Damping (NL-ECD) design method for a Triple-magnet Magnetic Suspension Dynamic Vibration Absorber (TMSDVA) (Chen et al., 2023a, 2023b, 2023c, 2023d) is proposed, modelled, and tested in this work. This NL-ECD design method can provide a new idea for the bifurcation behaviour control of the TMSDVA and make the vibration reduction performance of the TMSDVA more stable. The implementation of the NL-ECD design method is completely passive and does not require an external energy supply. The NL-ECD coefficient can be designed and adjusted by changing the dimensional parameters of the damping tube, which is both simple and convenient. The TMSDVA consists solely of three cylindrical permanent magnets and a non-magnetic conductive metal tube (damping tube), making its structure much simpler than that of other existing magnetic DVAs. Using NL-ECD can not only eliminate higher branch responses in the TMSDVA vibration reduction system, but also retain non-linear dynamics response characteristics of the system, resulting in a better vibration reduction effect of the TMSDVA. Moreover, the NL-ECD design method proposed in this paper is also applicable to other types of magnetic DVAs by using cylindrical magnets or ring magnets.

The paper is organized as follows: In Section 2, the structure of the TMSDVA is introduced first. Then the mathematical model of the TMSDVA vibration reduction system and the calculation model of the NL-ECD design method are established. In Section 3, simulations and experiments are conducted to verify the correctness of the derivation process described in Section 2.2. In Section 4, the dynamic responses of the TMSDVA with NL-ECD are analysed to determine how NL-ECD affects the vibration reduction performance of the TMSDVA. Section 5 concludes the paper.

Structure and Mathematical Model

Triple-Magnet Magnetic Suspension Dynamic Vibration Absorber (TMSDVA)

Figure 1(a) shows the schematic diagram of the TMSDVA. The TMSDVA is composed of three cylindrical permanent magnets A, B, C, and a non-magnetic conductive metal tube (damping tube) D1. Both the magnets B and C share the same specifications, which are fixed with the damping tube D1. The position of the magnet B and the magnet C is symmetrical about the centre of the damping tube D1. The magnet A repels not only magnet B, but also magnet C (the magnetic pole arrangements are shown in Figure 1(a)). When the magnet A moves relative to the damping tube D1, eddy currents can be generated on the surface of the tube, which provides linear ECD for the TMSDVA. The ECD coefficient of the TMSDVA can be designed and adjusted by changing the dimensional parameters of the damping tube D1. E is a non-magnetic insulating mass block (such as a rubber block, etc.) attached to the magnet A. The mass block E and magnet A are collectively referred to as a suspension oscillator. The mass of the mass block E can be changed to adjust the overall mass of the suspension oscillator while keeping magnet A unchanged. This facilitates the optimisation of the vibration reduction effect of the TMSDVA. l is the distance between the two fixed magnets B and C (from the upper surface of magnet B to the lower surface of magnet C). s is the distance between the magnet B and the suspension oscillator (from the upper surface of magnet B to the lower surface of the suspension oscillator). When the suspension oscillator is excited to move, s changes accordingly. The material of the tube D1 in this paper is selected as aluminium, and the inner surface of tube D1 is smoothed. $l_{T}$ is the total length of tube D1. $r_{n}$ and $r_{w}$ are the inner radius and the outer radius of the tube D1, respectively.

Figure 1.

(a) Schematic diagram of the TMSDVA. (b) Equivalent dynamics model of the system with the TMSDVA.

The TMSDVA reported in Figure 1(a) is attached to a vibration reduction object (the primary structure). The equivalent dynamics model of the system is depicted in Figure 1(b). According to Newton's second law, the dynamic equations can be established as:

{\begin{aligned} - k_{p} z_{p} - c_{p} {\dot{z}}_{p} + F_{mg} - c_{m} ({\dot{z}}_{p} - {\dot{z}}_{m}) + m_{p} a = m_{p} {\ddot{z}}_{p} \\ - F_{mg} - c_{m} ({\dot{z}}_{m} - {\dot{z}}_{p}) = m_{m} {\ddot{z}}_{m} \end{aligned}

(1)

In Eq. (1),

k_{p}

and

c_{p}

are the equivalent stiffness coefficient and the equivalent damping coefficient of the primary structure, respectively.

F_{mg}

is the resultant force of the nonlinear magnetic suspension force

F_{m}

and the gravity acting on the suspension oscillator

m_{m} g

c_{m}

is the ECD coefficient of the TMSDVA, and

m_{m}

is the mass of the suspension oscillator.

z_{p}

and

z_{m}

are the displacements of the primary structure and the suspension oscillator, respectively. a is the excitation acceleration applied to the primary structure.

It is worth noting that the two fixed magnets and tube D1 cannot move relative to the primary structure. Thus, $m_{p}$ is calculated as:

\begin{matrix} m_{p} = m_{1} + m_{t} + 2 m_{fm} \end{matrix}

(2)

In Eq. (2),

m_{1}

is the equivalent mass of the primary structure.

m_{t}

is the mass of tube D1.

m_{fm}

is the mass of fixed magnet B or C. The calculation model of the nonlinear magnetic suspension force

F_{m}

was established based on the equivalent magnetizing current theory (Sun et al., 2021; Zhang et al., 2017). The detailed derivation of the nonlinear magnetic force calculation model can be found in Appendix A. The magnetic induction

B

generated by magnet B can be written as:

\begin{aligned} B = & \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{(z - z_{1}) r_{B} \cos θ}{{r^{'}}^{3}} d θ i \\ + \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{(z - z_{1}) r_{B} \sin θ}{{r^{'}}^{3}} d θ j # \\ + \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{r_{B} (r_{B} - x \cos θ - y \sin θ)}{{r^{'}}^{3}} d θ k \end{aligned}

(3)

In Eq. (3),

r^{'} = ({(x - r_{B} \cos θ)}^{2} + {(y - r_{B} \sin θ)}^{2} -

{(z - z_{1})}^{2})^{1 / 2}

l_{B}

and

r_{B}

are the height and radius of magnet B, respectively.

μ_{0}

is the vacuum magnetic permeability.

M_{B}

is the magnetization of magnet B. The magnetic induction of magnet C is the same as that of magnet B. Let i , j , and k be the unit vectors in the x, y, and

z

directions, respectively. The magnetic induction

B

is represented as a vector:

\begin{matrix} B = B_{i} i + B_{j} j + B_{k} k \end{matrix}

(4)

The nonlinear magnetic force

F_{m}

on the primary structure can be expressed as:

\begin{aligned} F_{m} & = M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{i} (x + r_{A} \cos φ, y + r_{A} \sin φ, z_{a} + z_{2}) \cos φ r_{A} d φ \\ + M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{j} (x + r_{A} \cos φ, y + r_{A} \sin φ, z_{a} + z_{2}) \sin φ r_{A} d φ \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{i} (x + r_{A} \cos φ, y + r_{A} \sin φ, z_{b} + z_{2}) \cos φ r_{A} d φ \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{j} (x + r_{A} \cos φ, y + r_{A} \sin φ, z_{b} + z_{2}) \sin φ r_{A} d φ \end{aligned}

(5)

where

z_{a} = s + \frac{l_{A}}{2} + \frac{l_{B}}{2}

z_{b} = l - s - \frac{l_{A}}{2} + \frac{l_{B}}{2}

l_{A}

and

r_{A}

are the height and radius of magnet A, respectively.

M_{A}

is the magnetization of magnet A. The relationship among s,

z_{p}

, and

z_{m}

is given by:

\begin{matrix} s = \frac{l - l_{A}}{2} + (z_{m} - z_{p}) \end{matrix}

(6)

According to Newton's third law, the nonlinear magnetic suspension force on the suspension oscillator is

- F_{m}

. Due to the suspension oscillator's gravity, the actual nonlinear force

F_{mg}

on the primary structure is:

\begin{matrix} F_{mg} = - (- F_{m} - m_{m} g) = F_{m} + m_{m} g \end{matrix}

(7)

Nonlinear Eddy Current Damping

A theoretical model of ECD based on electromagnetic eddy current theory has been proposed in Ref. (Bae et al., 2009). To reduce the computational complexity, the authors approximated the radial magnetic induction of the cylindrical permanent magnet as a constant, which made the results inaccurate. Therefore, based on Faraday's law of electromagnetic induction, the NL-ECD calculation model considering the non-uniformly distributed radial magnetic induction of the permanent magnet is established in this section.

To establish the NL-ECD calculation model, a linear ECD calculation model must be established first. Figure 2 shows the schematic diagram of the linear ECD equivalent calculation model. It is assumed that the eddy current moves only along the conductor plane perpendicular to the axis of the damping tube D1. The magnet A can only move inside tube D1. In other words, the length of the damping tube should be larger than the height of magnet A. Besides, because of the unpredictable tilts of magnet A inside the damping tube, the tilts of magnet A are not considered in our calculation model. A Cartesian coordinate system ( $O - x y z$ ) at the centre of magnet A is established. Based on Eqs. (3) and (4), the magnetic induction in the y -direction produced by magnet A can be expressed as:

\begin{matrix} B_{j} = \frac{μ_{0} M_{A}}{4 π} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{1} \int_{0}^{2 π} \frac{(z - z_{1}) r_{A} \sin θ}{{({(x - r_{A} \cos θ)}^{2} + {(y - r_{A} \sin θ)}^{2} - {(z - z_{1})}^{2})}^{\frac{3}{2}}} d θ \end{matrix}

(8)

Figure 2.

Schematic diagram of the linear ECD equivalent calculation model.

From Eq. (3), the value of $B_{j}$ is related to the spatial coordinates ( $x$ , y, $z$ ).

As shown in Figure 2, magnet A moves along the z -axis. We assume that there is a closed loop (grey circle in the top view) in a micro-segment $Δ z$ of the damping tube. The width of the closed loop is $Δ r$ . And the distance between the closed loop and the centre axis of the damping tube is r. According to Faraday's law of electromagnetic induction, the induction electromotive force $Δ E$ in the closed loop can be written as:

Δ E = B_{j} L v

(9)

where v is the velocity of magnet A. L in Eq. (9) can be expressed as:

L = 2 π r

(10)

The equivalent resistance

Δ R_{Δ z}

of the closed loop is given by:

Δ R_{Δ z} = \frac{ρ L}{Δ r Δ z}

(11)

In Eq. (11),

ρ

is the resistivity of the damping tube made of aluminium, and the value is

2.85 \times 10^{- 8} Ω \cdot m

. Then the current

Δ i

in the closed loop is:

Δ i = \frac{Δ E}{Δ R_{Δ z}} = \frac{B_{j} v}{ρ} Δ r Δ z

(12)

Hence, the force on the closed loop is:

Δ F = B_{j} Δ i L = \frac{2 π r B_{j}^{2} v}{ρ} Δ r Δ z

(13)

Then the N-pole force of magnet A acting on a segment of damping tube can be expressed as:

F_{N}^{'} = \int \int Δ F = \int \int \frac{2 π r B_{j}^{2} v}{ρ} Δ r Δ z

(14)

We establish another Cartesian coordinate system (

O_{p} - x_{p} y_{p} z_{p}

) at the centre of the end of the damping tube and let the position of the centre of magnet A in the damping tube be (0, 0,

z_{p}

). Then the N-pole ECD force acting on magnet A is:

F_{N} (z_{p}) = - \frac{2 π v}{ρ} \int_{0}^{l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z

(15)

In Eq. (15),

r_{n}

and

r_{w}

are the inner and outer radii of the damping tube D1, respectively.

l_{N} (z_{p})

is the length of the damping tube subjected to the N-pole force of magnet A. Both

l_{N} (z_{p})

and

F_{N} (z_{p})

will change as the position

z_{p}

of magnet A changes.

F_{N} (z_{p})

is proportional to v, which indicates that the ECD force is a type of linear viscous damping force. The negative sign in Eq. (15) proves that the direction of the force acting on magnet A is opposite to that of the motion velocity v.

The equation of the S-pole force acting on magnet A has the same form as that of the N-pole force. However, the length of the damping tube subjected to the S-pole force of magnet A is equal to $l_{T} - l_{N} (z_{p})$ . Therefore, the S-pole ECD force acting on magnet A should be written as:

F_{S} (z_{p}) = - \frac{2 π v}{ρ} \int_{0}^{l_{T} - l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z

(16)

Hence, the ECD force acting on magnet A is given by:

F (z_{p}) = - \frac{2 π v}{ρ} (\int_{0}^{l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z + \int_{0}^{l_{T} - l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z)

(17)

According to the relationship between the damping force and the damping coefficient

c_{m}

\begin{matrix} F = - c_{m} v # 18 \end{matrix}

(18)

the linear ECD coefficient can be expressed as:

c_{m} (z_{p}) = \frac{2 π}{ρ} (\int_{0}^{l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z + \int_{0}^{l_{T} - l_{N} (z_{p})} \int_{r_{n}}^{r_{w}} r B_{j}^{2} Δ r Δ z)

(19)

From Eq. (19), the ECD coefficient

c_{m} (z_{p})

can be calculated when magnet A is at any position inside the damping tube D1. Based on the above derivation process, the NL-ECD calculation model can be derived and established. We keep the inner radius of the damping tube unchanged. The outer radius of the damping tube is changed as shown in Figure 3.

Figure 3.

Schematic diagram of the NL-ECD equivalent calculation model.

From the figure, when magnet A is at different positions inside the damping tube D2, the corresponding thickness of the tube D2 is different. The outer radius $r_{w}$ of the damping tube D2 should be expressed as a function of $z_{p}$ . Therefore, based on Eq. (19), the NL-ECD coefficient calculation model is given by:

c_{m} (z_{p}) = \frac{2 π}{ρ} (\int_{0}^{l_{N} (z_{p})} \int_{r_{n}}^{r_{w} (z_{p})} r B_{j}^{2} Δ r Δ z + \int_{0}^{l_{T} - l_{N} (z_{p})} \int_{r_{n}}^{r_{w} (z_{p})} r B_{j}^{2} Δ r Δ z)

(20)

Simulation and Experimental Verification

In this section, simulations and experiments are carried out to verify the correctness of the NL-ECD calculation model. First, the magnetic induction of the magnet A should be measured through experiments. And the experimental data need to be compared with simulation magnetic induction results to find the actual magnetization of magnet A. The height and radius of magnet A are 29.5 mm and 14.75 mm, respectively. Figure 4 shows the magnetization measurement system. The Gaussmeter was used to measure the magnetic induction of magnet A along the $z$ -axis. And the displacement knob in the system was used to adjust the distance between magnet A and the gaussmeter probe. We substitute the magnet parameters into Eqs. (3) and (4) to calculate the magnet A's axial magnetic induction $B_{k}$ . Figure 5 depicts simulation and experimental results of the magnetic induction of magnet A. From the figure, the axial magnetic induction $B_{k}$ of the magnet decreases gradually with increasing distance. The simulation results show that when the magnetization $M_{A}$ of magnet A is larger, the overall axial magnetic induction $B_{k}$ of magnet A is larger. Then, we compare the experimental data with the simulation results and find that the error between the experimental results and the simulation results is minimum when $M_{A} = 6.1 \times 10^{5}$ A/m. Therefore, the magnetization $M_{A}$ of magnet A is about $6.1 \times 10^{5}$ A/m.

Figure 4.

Magnetization measurement system.

Figure 5.

Magnetic induction of magnet A along the z -axis.

Figure 6 shows the physical map of the NL-ECD damping tube. The inner diameter of the damping tube is 42 mm. The total length of the damping tube is 600 mm. All three segments ( $l_{T 1}$ , $l_{T 2}$ , and $l_{T 3}$ ) of the damping tube share the same length of 200 mm. Both the $l_{T 1}$ and $l_{T 3}$ segments share the same outer diameters of 58 mm. The outer diameter of the $l_{T 2}$ segment is 50 mm. To avoid friction between magnet A and the NL-ECD damping tube as much as possible, a polytetrafluoroethylene (PTFE) tube was attached inside the wall of the damping tube. The outer diameter and inner diameter of the PTFE tube are 40 mm and 37 mm, respectively.

Figure 6.

Physical map of the NL-ECD damping tube.

The parameters of magnet A and NL-ECD damping tube are substituted into Eq. (20). Based on the above parameters and the direction of the magnet velocity v, it can be calculated that $z_{p}$ varies from −14.75 mm to −585.25 mm. The integral shown in Eq. (20) is calculated by MATLAB 2020b. Figure 7 displays the simulation results of NL-ECD coefficients when magnet A is at different positions inside the NL-ECD damping tube. We assume that the magnet A moves towards the $l_{T 3}$ segment from the $l_{T 1}$ segment. From the overall trend shown in Figure 7, the NL-ECD coefficient curve is symmetric. The symmetry centre of the curve is at the centre of the damping tube. When the magnet A moves inside the $l_{T 1}$ segment, the NL-ECD coefficient will increase first and then remain constant, and finally decrease. After that, when magnet A moves inside the $l_{T 2}$ segment, the NL-ECD coefficient will decrease first and then remain constant, and finally increase. When magnet A moves inside the $l_{T 3}$ segment, the NL-ECD coefficient will increase first and then remain constant. Finally, when magnet A moves close to the end of the $l_{T 3}$ segment, the NL-ECD coefficient will decrease.

Figure 7.

Simulation results of NL-ECD coefficients when magnet A is at different positions inside the damping tube.

Because of the structural features of the TMSDVA displayed in Figure 1(a), the movement of magnet A cannot exceed the two fixed magnets. It can also be said that magnet A usually moves near the centre of the damping tube. Hence, magnet A is mainly affected by the NL-ECD coefficients near the centre of the damping tube, such as the NL-ECD coefficients in the interval of [−0.15 m, −0.45 m]. Therefore, the NL-ECD coefficients shown in the interval are the focus of further study.

Since eddy currents can only be generated when magnet A and the damping tube move relatively, it is difficult to obtain the NL-ECD coefficients through direct experimental measurements. Therefore, we choose to indirectly verify the correctness of the NL-ECD force calculation model in Section 2.2 by measuring the motion state of magnet A under the joint action of gravity and the NL-ECD force. Figure 8 depicts the NL-ECD coefficient measurement system. The NL-ECD damping tube is first fixed vertically using a gripper. Then magnet A is placed vertically at the top of the damping tube so that magnet A can fall freely in the tube. An acceleration sensor was attached to magnet A to measure the falling acceleration of magnet A. Finally, the acceleration data can be input into a computer through a signal acquisition device. The sampling frequency of the signal acquisition device is 1.652 kHz.

Figure 8.

NL-ECD coefficient measurement system.

The dynamic equation for the motion of magnet A in the damping tube can be written as:

m_{A} g - c_{m} {\dot{z}}_{A} = m_{A} {\ddot{z}}_{A}

(21)

In Eq. (21),

{\dot{z}}_{A}

is the velocity of magnet A.

{\ddot{z}}_{A}

is the acceleration of magnet A.

m_{A}

is the mass of magnet A and acceleration sensor (the mass of the acceleration sensor is approximately 13.5 g). We have solved Eq. (20) using MATLAB and obtained the NL-ECD coefficient

c_{m}

as shown in Figure 7.

c_{m}

is essentially a function of the displacement

z_{p}

of magnet A. Based on this, by solving Eq. (21) using the fourth-order Runge-Kutta method, the simulation motion displacement

z_{A}

, velocity

{\dot{z}}_{A}

, and acceleration

{\ddot{z}}_{A}

of magnet A under the effect of gravity and NL-ECD force can be obtained. The correctness of the NL-ECD calculation model can be indirectly verified by comparing the simulation acceleration

{\ddot{z}}_{A}

with the actual acceleration of magnet A obtained from experimental measurements. This is because if the NL-ECD calculation model is correct, the simulation acceleration characteristics of magnet A and the experimental acceleration characteristics will be in agreement within the error allowance. Figure 9(a) plots the simulation results of the displacement

z_{A}

of magnet A. From the figure, magnet A reaches −0.15 m at about 0.553 s and −0.45 m at about 1.525 s. Moreover, magnet A reaches the bottom of the damping tube at about 2.098 s.

Figure 9.

(a) Simulation results of the displacement of magnet A. (b) Simulation results of the velocity of magnet A in the time range of [0.553 s, 1.525 s]. (c) Simulation and experimental results of the acceleration of magnet A in the time range of [0.553 s, 1.525 s]. All black lines in this figure correspond to simulation results of Eq. (21). Colour lines in Figure 9(c) correspond to experimental measured acceleration of magnet A.

The simulation results of the velocity ${\dot{z}}_{A}$ of magnet A in the time range of [0.553 s, 1.525 s] are displayed in Figure 9(b). During [0.553 s, 0.763 s], the velocity ${\dot{z}}_{A}$ of magnet A increases over time. In this case, magnet A moves from −0.15 m to −0.2 m, and from Figure 7, the NL-ECD coefficient continuously becomes smaller. After 0.763 s, the velocity ${\dot{z}}_{A}$ of the magnet A increases first and then remains constant. Magnet A moves from −0.2 m towards −0.4 m in this situation. From Figure 7, the NL-ECD coefficient decreases first and then kept constant. When magnet A moves close to −0.4 m, the velocity ${\dot{z}}_{A}$ of magnet A decreases owing to the NL-ECD coefficient starting to increase. During [1.349 s, 1.525 s], magnet A moves from −0.4 m to −0.45 m. The velocity ${\dot{z}}_{A}$ of magnet A continues to decrease due to the increased NL-ECD coefficients shown in Figure 7. Figure 9(c) presents the simulation and experimental results of the acceleration of magnet A in the time range of [0.553 s, 1.525 s]. The experimental results exhibit the same trend as the simulation result ${\ddot{z}}_{A}$ , which can verify the correctness of the derivation process in Section 2.2.

There are still some errors between the simulation results and the experimental results. The reasons for the errors are as follows:

Firstly, the end of the acceleration sensor is connected to the signal acquisition device by a long special wire for the sensor. As magnet A falls inside the damping tube, the mass of the wire acting on magnet A is increased, resulting in the mass $m_{A}$ in Eq. (21) continuously increasing in the actual situation. However, in simulations, the increasing mass of the wire acting on magnet A cannot be accurately calculated because of the flexibility of the wire. Therefore, the mass $m_{A}$ in Eq. (21) is kept constant in the simulations.

Secondly, in order to allow magnet A to fall freely, the inner diameter of the PTFE tube is larger than the diameter of magnet A. As a result, magnet A may have small tilts during its fall and may result in random slight collision between magnet A and the tube wall. However, these unpredictable tilts of magnet A are not considered in our calculation model.

Thirdly, although the surface of magnet A is very smooth and the friction coefficient of the PTFE tube is very small, the friction between magnet A and the PTFE tube wall is still unavoidable during the experiment. At the same time, the friction between the wire and the wall of the PTFE tube could also affect the experimental results.

Besides, in the actual situation, the distribution of eddy currents on the surface of the damping tube is non-uniform. The assumption of “eddy currents move only along the conductor plane perpendicular to the axis of the damping tube” in the model derivation process may also be a reason causing some errors.

Dynamic Responses of the TMSDVA with NL-ECD

In this part, dynamic responses of the TMSDVA with NL-ECD are analysed to see how NL-ECD affects the vibration reduction performance of the TMSDVA. The parameters of the magnets in the TMSDVA are shown in Table 1 of Appendix B. The magnet distance l is 46 mm. The parameters of the TMSDVA vibration reduction system are shown as follows:

{\begin{aligned} k_{p} = 340.74 N \cdot m^{- 1}, c_{p} = 0.1085 N \cdot s \cdot m^{- 1}, m_{p} = 0.0863 kg \\ m_{m} = 0.0042 kg \end{aligned}

(22)

Table 1.

Parameters of Magnets A, B, and C in Section 4.

Material	Parameter	Numerical Value
Magnet A material: $N d_{2} F e_{14} B$ (N35)	Density $ρ_{A}$ /(kg/m³)	7500
	Height $l_{A}$ /(mm)	10
	radius $r_{A}$ /(mm)	3
	Magnetization $M_{A}$ /(A/m)	6.4 × 10⁵
Magnet B and C material: $N d_{2} F e_{14} B$ (N35)	Density $ρ_{B}$ /(kg/m³)	7500
	Height $l_{B}$ /(mm)	4
	radius $r_{B}$ /(mm)	3
	Magnetization $M_{B}$ /(A/m)	6.4 × 10⁵
	Vacuum permeability $μ_{0}$ /(N/A²)	4π × 10⁻⁷

Figure 10(a) shows the magnetic suspension force $F_{mg}$ . Figure 10(b) displays the nonlinear stiffness of the magnetic force. Both figures indicate that the magnetic force is essentially a kind of nonlinear force. Therefore, the TMSDVA is a type of nonlinear dynamic vibration absorber. When the excitation acceleration is too large, the TMSDVA vibration reduction system will exhibit higher branch responses (Starosvetsky and Gendelman, 2009). This can cause the TMSDVA lose its vibration reduction effect. The NL-ECD can be used with the TMSDVA to eliminate the higher branch responses in the vibration reduction system.

Figure 10.

Diagram of (a) the nonlinear magnetic suspension force $F_{mg}$ and (b) the nonlinear stiffness when the magnet distance is 46 mm.

Because the magnet distance of the TMSDVA is 46 mm and the height of magnet A is 10 mm, the movement range of magnet A is [-18 mm, 18 mm]. For the TMSDVA shown in Figure 1(a), the inner radius of the linear ECD damping tube D1 is 4 mm, and the outer radius is 5 mm. The length of the damping tube is 60 mm. According to Eq. (19), the ECD coefficient of the TMSDVA in the magnet movement range is always kept constant. In other words, the TMSDVA has linear damping under this parameter condition. After calculation, the value of the linear ECD coefficient $c_{m}$ is about 0.08 Ns/m. The root-mean-square value of the excitation acceleration a is 0.15 g. The Runge-Kutta method is used to solve the dynamic equations of the system (Eq. (1)).

Figure 11(a) depicts the amplitude-frequency response curve of the TMSDVA vibration reduction system when the linear ECD coefficient is about 0.08 Ns/m. From the figure, a higher branch response occurs in the vibration reduction system. The vibration peak value of the primary structure without TMSDVA is approximately 26.3 mm. Due to the large vibration peak value 18.7 mm of the higher branch response in the vibration reduction system, the vibration amplitude of the primary structure is reduced by only 28.90%. In addition to the larger vibration peak of the higher branch response, another smaller vibration peak appears at the frequency of 10.1 Hz. Figure 11(b) plots the time-domain waveform of the system at the frequency of 10.1 Hz. The envelopes of the time-domain waveform shown in Figure 11(b) undergo obvious modulation, which illustrates that modulated responses appear in the vibration reduction system. The peak value of the time-domain waveform is about 4.7 mm.

Figure 11.

The root-mean-square value of the acceleration amplitude is 0.15 g. The damping of the TMSDVA is linear, and the ECD coefficient is about 0.08 Ns/m. (a) The amplitude-frequency response curve of the TMSDVA vibration reduction system. (b) The time-domain waveform of the TMSDVA vibration reduction system at the frequency of 10.1 Hz.

According to Ref. (Starosvetsky and Gendelman, 2009), increasing the damping coefficient of the TMSDVA is an effective way to eliminate higher branch responses of the system. Hence, we increase the outer radius of the linear ECD damping tube D1 to 7 mm and keep the inner radius unchanged. The linear ECD coefficient is increased to 0.15 Ns/m from 0.08 Ns/m. It should be noted that, to investigate the effect of damping variation on the vibration reduction effect of the TMSDVA, we ignore the variation in the mass of the damping tube. Figure 12(a) shows the amplitude-frequency response curve of the TMSDVA vibration reduction system when the linear ECD coefficient is 0.15 Ns/m. From the figure, no higher branch response occurs in the vibration reduction system anymore. The peak vibration of the response curve appears at the frequency of 9.8 Hz. Figure 12(b) displays the time-domain waveform of the system at the frequency of 9.8 Hz, which illustrates that the system undergoes steady periodic motion. It demonstrates that there is also no modulated response occurring in the vibration reduction system. In this case, the TMSDVA performs like a TMD (Lu et al., 2017). The peak value of the time-domain waveform is about 5.6 mm, which is significantly larger than that shown in Figure 11(b). The vibration amplitude of the primary structure is reduced by 78.71% under these conditions.

Figure 12.

The root-mean-square value of the acceleration amplitude is 0.15 g. The damping of the TMSDVA is linear, and the ECD coefficient is about 0.15 Ns/m. (a) The amplitude-frequency response curve of the TMSDVA vibration reduction system. (b) The time-domain waveform of the TMSDVA vibration reduction system at the frequency of 9.8 Hz.

Although increasing the linear ECD coefficient can eliminate higher branch responses in the vibration reduction system, the modulated responses in the system cannot be retained. According to Ref. (Starosvetsky and Gendelman, 2009), the vibration reduction effect of the TMSDVA can be further improved using NL-ECD instead of linear ECD. Figure 13 shows the schematic diagram of the TMSDVA with NL-ECD. The inner radius of the NL-ECD damping tube D2 is still 4 mm. All three segments ( $l_{T 1}$ , $l_{T 2}$ , and $l_{T 3}$ ) of the damping tube share the same length of 20 mm. Both the $l_{T 1}$ and $l_{T 3}$ segments share the same outer radius of 7 mm. The outer radius of the $l_{T 2}$ segment is 5 mm. The NL-ECD coefficient curve is shown in Figure 14.

Figure 13.

Schematic diagram of the TMSDVA with NL-ECD.

Figure 14.

ECD coefficient curve of the TMSDVA.

Figure 15(a) shows the amplitude-frequency response curve of the primary structure with the TMSDVA using NL-ECD. From Figure 15(a), no higher branch response occurs in the vibration reduction system. The peak vibration of the response curve appears at the frequency of 10.25 Hz. Figure 15(b) displays the time-domain waveform of the system at the frequency of 10.25 Hz, which illustrates that the envelopes of the time-domain waveform still undergo modulation in this case. The peak value of the time domain waveform is about 4.7 mm, which is equal to that shown in Figure 11(b). The vibration amplitude of the primary structure is reduced by 82.13%. Therefore, the vibration reduction effect of the TMSDVA with NL-ECD is better than that of the TMSDVA with linear ECD.

Figure 15.

The root-mean-square value of the acceleration amplitude is 0.15 g. The NL-ECD coefficients of the TMSDVA are shown in Figure 14. (a) The amplitude-frequency response curve of the TMSDVA vibration reduction system. (b) The time-domain waveform of the TMSDVA vibration reduction system at the frequency of 10.25 Hz.

The above phenomena demonstrate that using NL-ECD can not only eliminate higher branch responses in the TMSDVA vibration reduction system, but also retain the non-linear dynamics response characteristics of the system, leading to a better vibration reduction effect of the TMSDVA.

Conclusion

In this work, a novel Nonlinear Eddy Current Damping (NL-ECD) design method for a Triple-magnet Magnetic Suspension Dynamic Vibration Absorber (TMSDVA) is proposed, modelled, and tested. The NL-ECD coefficient can be designed and adjusted by changing the dimensional parameters of the damping tube. The realization of this method is completely passive and does not require an external energy supply.

From the analysis, the NL-ECD coefficient is strongly related to the position of the moving magnet. Using NL-ECD can not only eliminate higher branch responses in the TMSDVA vibration reduction system, but also retain the non-linear dynamics response characteristics of the system, leading to a better vibration reduction effect of the TMSDVA.

Moreover, the NL-ECD design method proposed in this paper can be applicable to either the TMSDVA or to other types of magnetic DVAs using cylindrical magnets or ring magnets (e.g., the magnetic DVAs in Refs. Benacchio et al., 2016, Yamakawa et al., 1977).

Footnotes

Author Contributions

Xiaoyu Chen: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing-Original Draft, Visualization.

Yonggang Leng: Conceptualization, Methodology, Supervision, Project administration, Funding acquisition, Writing-Original Draft.

Shengbo Fan: Conceptualization, Methodology, Investigation, Project administration.

Xukun Su: Conceptualization, Methodology, Investigation.

Junjie Xu: Conceptualization, Methodology.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant No. 52275122 and 12132010).

Declaration of Conflicting Interests

The authors declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “A Novel Nonlinear Eddy Current Damping Design Method for a Triple-magnet Magnetic Suspension Dynamic Vibration Absorber”.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Appendix A

The detailed derivation of Eqs. (3) to (6) is given below. According to Refs. (Sun et al., 2021; Zhang et al., 2017), for axially magnetised cylindrical permanent magnets, the surface equivalent magnetising current is distributed in a circular shape around the magnetisation axis of the magnet. We take the centre of the circular current loop as the origin and establish a coordinate system as shown in Figure A-1. Then the following relations can be obtained: (\rm A-1)

\begin{matrix} R = x i + y j + z k \end{matrix}

(\rm A-2)

\begin{matrix} l = l \cos θ i + l \sin θ j, d l = - l \sin θ d θ i + l \cos θ d θ j \end{matrix}

(\rm A-3)

\begin{matrix} r = R - l = (x - l \cos θ) i + (y - l \sin θ) j + z k \end{matrix}

(\rm A-4)

\begin{matrix} d l \times r = z l \cos θ d θ i + z l \sin θ d θ j + l (l - x \cos θ - y \sin θ) d θ k \end{matrix}

Biot-Savart's Law is expressed as: (\rm A-5)

\begin{matrix} B = \int_{L} \frac{μ_{0} I}{4 π} \frac{d l \times r}{r^{3}} \end{matrix}

In formula (A-5),

B

is the magnetic induction intensity generated by a permanent magnet. L is the integral path of the current.

μ_{0}

is the vacuum magnetic permeability. I can be expressed as: (\rm A-6)

\begin{matrix} I = \int_{t} K_{m} d t \end{matrix}

t

represents the width of the current in this section.

K_{m}

is the surface magnetizing current density: (\rm A-7)

\begin{matrix} K_{m} = M \times \hat{n} \end{matrix}

In formula (A-7),

M

is the magnetization intensity of the permanent magnet, and

\hat{n}

is the surface normal unit vector. Figure A-2 shows the positions of magnetizing currents on the surface of cylindrical magnets in the coordinate system. Different from Figure A-1, the origin of the coordinate system is moved to the geometric centre of magnet B. In Figure A-2, the magnetic induction intensity at an arbitrary point produced by magnet B can be expressed as: (\rm A-8)

\begin{aligned} B = & \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{(z - z_{1}) r_{B} \cos θ}{{r^{'}}^{3}} d θ i \\ + \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{(z - z_{1}) r_{Bsin} θ}{{r^{'}}^{3}} d θ j \\ + \frac{μ_{0} M_{B}}{4 π} \int_{- \frac{l_{B}}{2}}^{\frac{l_{B}}{2}} d z_{1} \int_{0}^{2 π} \frac{r_{B} (r_{B} - x \cos θ - y \sin θ)}{{r^{'}}^{3}} d θ k \end{aligned}

In Eq. (A-8),

r^{'} = ({(x - r_{B} \cos θ)}^{2} + {(y - r_{Bsin} θ)}^{2} -

{(z - z_{1})}^{2})^{1 / 2}

l_{B}

and

r_{B}

are the height and radius of magnet B, respectively.

M_{B}

is the magnetization of magnet B. The magnetic induction of magnet A is the same as that of magnet B. Let i , j, and k be the unit vectors in the x, y, and

z

directions, respectively. The magnetic induction

B

is represented as a vector: (\rm A-9)

B = B_{i} i + B_{j} j + B_{k} k

According to Ampere's law, the interacting magnetic force

F

between the two cylindrical permanent magnets A and B is derived as: (\rm A-10)

F = \underset{S}{\int \int} K_{m A} \times B d S = \underset{S}{\int \int} | \begin{matrix} i & j & k \\ - M_{A} \sin φ & M_{A} \cos φ & 0 \\ B_{i} & B_{j} & B_{k} \end{matrix} | d z_{2} r_{A} d φ

And

F

can also be written as: (\rm A-11)

\begin{aligned} F = & M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{k} (x + r_{A} c o s φ, y + r_{A} s i n φ, z + z_{2}) c o s φ r_{A} d φ i \\ + M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{k} (x + r_{A} c o s φ, y + r_{A} s i n φ, z + z_{2}) s i n φ r_{A} d φ j \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{i} (x + r_{A} c o s φ, y + r_{A} s i n φ, z + z_{2}) c o s φ r_{A} d φ k \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{j} (x + r_{A} c o s φ, y + r_{A} s i n φ, z + z_{2}) s i n φ r_{A} d φ k \end{aligned}

In formula (A-11),

z = s + \frac{l_{A}}{2} + \frac{l_{B}}{2}

l_{A}

and

r_{A}

are the height and radius of magnet A.

M_{A}

is the magnetization of magnet A. Based on the structural features of the TMSDVA, the nonlinear magnetic force

F_{m}

is derived as: (\rm A-12)

\begin{aligned} F_{m} = & M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{i} (x + r_{A} \cos φ, y + r_{Asin} φ, z_{a} + z_{2}) \cos φ r_{A} d φ \\ + M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{j} (x + r_{A} \cos φ, y + r_{Asin} φ, z_{a} + z_{2}) \sin φ r_{A} d φ \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{i} (x + r_{A} \cos φ, y + r_{Asin} φ, z_{b} + z_{2}) \cos φ r_{A} d φ \\ - M_{A} \int_{- \frac{l_{A}}{2}}^{\frac{l_{A}}{2}} d z_{2} \int_{0}^{2 π} B_{j} (x + r_{A} \cos φ, y + r_{Asin} φ, z_{b} + z_{2}) \sin φ r_{A} d φ \end{aligned}

where

z_{a} = s + \frac{l_{A}}{2} + \frac{l_{B}}{2}

z_{b} = l - s - \frac{l_{A}}{2} + \frac{l_{B}}{2}

. The relationship among s,

z_{p}

and

z_{m}

is given by: (\rm A-13)

\begin{matrix} s = \frac{l - l_{A}}{2} + (z_{m} - z_{p}) \end{matrix}

In this section, Eqs. (A-8), (A-9), (A-12) and (A-13) are corresponding to Eqs. (3) to (6) in Section 2.1, respectively.

Appendix B

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