Equivalent circuit model of eddy current damping regarding frequency-dependence with test validation

Abstract

This study proposes an equivalent circuit model to simulate the mechanical behavior and frequency-dependent characteristic of eddy current (EC) damping, with the validations from multi-physics finite element (FE) modeling and dynamic testing. The equivalent circuit model is first presented with a theoretical expression of the EC damping force. Then, the transient analysis with an ANSYS-based FE model of an EC damper is performed. The time-history forces from the FE model are compared with that from the proposed equivalent circuit model. The favorable agreement indicates that the proposed model can simulate the nonlinear behavior of EC damping under different excitation scenarios. A noncontact and friction-free planar EC damper is designed, and its dynamic behavior is measured by employing shake table testing. The experimental observations can be reproduced by the proposed equivalent circuit model with reasonable accuracy and reliability. The proposed equivalent circuit model is compared with the classical viscous model and the higher-order fractional model using a complex EC damper simulated in ANSYS to show the advantages of the proposed model regarding model simplicity and prediction accuracy. A single-degree-of-freedom (SDOF) structure with different EC damping models is further analyzed to illustrate the need for accurate EC damping modeling.

Keywords

Eddy current damping mechanical model frequency dependence dynamic behavior finite element model

Introduction

Eddy current (EC) damping is used to dissipate the structural kinetic energy into heat, thereby suppressing structural vibration (Sodano and Bae, 2004). Due to the relative motion between the magnet and the conductor, the ECs in the conductor and the EC damping force will be induced (Touzani and Rappaz, 2014). Then, the ECs would be dissipated into heat because of the resistance of the conductor. Physically, the energy is dissipated in a noncontact way, and the EC damping is free of mechanical contact with the structure, exhibiting favorable characteristics of long-term reliability, and does not require any electronic equipment or external power supply (Shi and Zhu, 2015). Hence, the EC damping mechanism and its applications have been broadly investigated for decades (Irazu and Elejabarrieta, 2018).

In terms of structural vibration suppression, EC damping has received significant interest in recent years. The EC damping is utilized to attenuate the vibration of the cantilever beam and has become one of the most active studies (Bae et al., 2014; Sodano and Inman, 2007). Moreover, different types and configurations of EC damper have been proposed (Li et al., 2019; Liu and Lui, 2020; Shi et al., 2017), such as the EC-embedded base isolation (Villaverde, 2017), tuned mass damper (TMD) (Wang et al., 2012, 2019), the electromagnetic damper with increased energy density (Zuo et al., 2011), and the EC damper with friction (Amjadian and Agrawal, 2018). To fully utilize the energy-dissipation efficiency of the EC damping, researchers have attempted to increase the damping density by optimizing the magnet configuration (Beek et al., 2016). For example, an EC damper with alternating magnetic field directions is proposed and presents a significantly increased damping capacity of three to five times (Zuo, et al., 2011).

Usually, the physics-oriented mechanical modeling of nonlinear behavior of energy dissipation mechanisms is the research focus for developing novel dampers. The feasible damping model can advance relevant energy dissipation techniques for structural vibration control and real-world implementation. For example, the phenomenological model has been successfully proposed for magnetorheological (MR) dampers (Spencer et al., 1997). Since then, the MR damper has been widely studied with numerous parametric models (Wang and Liao, 2011), which strongly supports the implementation of semi-active and smart control (Chang and Spencer, 2010; Kim et al., 2010; Li and Wang, 2011). The viscoelastic (VE) damping presents similar progress with different mechanical models developed for modeling the nonlinear behavior and characteristics (Xu et al., 2016).

In the existing studies on EC damping modeling, the EC force is usually assumed to be viscous-type and is linearly proportional to the relative velocity between the magnet and the conductor (Lu et al., 2017; Wang, et al., 2012). However, experimental results show that the EC damping may exhibit nonlinear properties (Pan et al., 2016; Zuo, et al., 2011), and such nonlinear behavior may be sensitive to the excitation and vibration frequency. As the excitation frequency increases, the widely used viscous model would be inevitably challenged due to the frequency-dependent characteristics with a significant decrease of the equivalent damping coefficient and a noticeable increase of stiffness contribution (Bae et al., 2009; Pan, et al., 2016; Zuo, et al., 2011). Although such frequency-dependent features have been observed and reported by relevant studies, reliable modeling of EC damping has not been fully developed. The physics equations of the EC damping are too complex for engineering, and the viscous model is overly simplified to simulate the frequency-dependent features of EC damping. Therefore, a feasible and reliable mechanical model, with a satisfactory balance between the model simplicity, prediction accuracy, and computational efficiency, may be desired to simulate the nonlinear behavior and inherent characteristic of EC damping, especially with the objectives of engineering design and practice.

In the previous efforts, the authors investigated the dynamic and nonlinear behavior of the EC damping mechanism by using shaking table tests and finite element modeling (FEM) (Shi et al., 2020), and presented the regressed functions of the excitation frequency for calculating the equivalent stiffness and damping coefficient. However, the regressed functions are mostly applicable in the single-frequency case. Then, the eddy-current force is approximated based on the magnetoquasistatic assumption by the authors (Loong et al., 2020). Experimental results showed that the approximated analysis could determine the eddy-current force when the eddy-current damper vibrated below 5.0 Hz with satisfactory accuracy. In the present study, a parametric model with higher adaptability would be proposed. The proposed model can simulate the EC damping characteristic in the multi-frequency scenarios.

In this paper, an equivalent circuit model is proposed to simulate the nonlinear behavior of EC damping. The transient analysis with multi-physics FEM is conducted. The FEM results are compared with the numerical prediction of the proposed equivalent model under various excitation scenarios. A noncontact and friction-free EC damper is then dynamically tested with accurate measurements to further investigate the prediction performance of the equivalent circuit model. Following that, a large-scale EC damper is numerically modeled in ANSYS to compare the modeling accuracy of the proposed circuit model, the higher-order fractional derivative model, and the viscous damping model. Furthermore, a single-degree-of-freedom (SDOF) structure with EC damping is analyzed, and the influence of different EC damping models is discussed.

Equivalent circuit model formulation

According to Faraday’s law of induction, EC can be induced from the relative motion between a magnet and a conductor. The induced ECs flow in closed loops within conductors, and the energy can be dissipated in the form of heat, which is shown in Figure 1(a). In the current study, the EC is simplified as a loop of electrical current. Assume that the EC flows in a closed-loop, the EC can be simplified as a single-coil moving in the constant magnetic field, which is shown in Figure 1(b). Then, the model can be transformed into an ideal resistor–inductor circuit (RL circuit) in Figure 1(c).

Figure 1.

Proposed equivalent circuit model of eddy current damper: (a) Schematic diagram of the eddy current damping; (b) simplified schematic diagram; (c) the ideal resistor–inductor circuit.

According to Faraday’s law, the induced electromotive voltage $E (t)$ can be expressed as

E (t) = \frac{d ϕ}{d t} = B l_{w} V (t)

(1)

where

ϕ

is the magnetic flux,

l_{w}

is the width of magnets, and

B

is the magnetic field through the surface. Under a constant gap between the magnet and the copper plate, the magnetic field

B

is almost constant and will not change during the relative displacement between the magnet and the copper plate.

V (t)

is the relative velocity between the magnet and the copper plate. According to Lenz’s law, the EC will exert a drag force on the moving magnet, which opposes its motion, and the force effect can be given as

F (t) = B l_{w} I (t)

(2)

where

I (t)

is the time-dependent current through the circuit.

Considering the actual nonuniform magnetic field and the difference between the real EC damper and the ideal resistor–inductor circuit model, the coefficient of correction $α$ is introduced, and the corrected force can be expressed as

F (t) = α B l_{w} I (t)

(3)

Based on Kirchhoff’s Voltage Law, the RL circuit shown in Figure 1(c) can be related by a first-order differential equation as

E (t) - I (t) R - L \frac{d I (t)}{d t} = 0

(4)

where

R

and

L

are the resistance and inductance of the equivalent circuit, respectively. Introducing the time constant

τ = L / R

and equation (1), the first-order differential equation of equation (4) can be written as

τ \frac{d I (t)}{d t} + I (t) = \frac{B l_{w}}{R} V (t)

(5)

We can set $v (t) = I (t) R / B l_{w}$ , then

I (t) = \frac{v (t) B l_{w}}{R}

(6)

Substituting equation (6) into equation (5) yields

τ \frac{d v (t)}{d t} + v (t) = V (t)

(7)

with equation (6), the EC damping force in equation (3) can be rewritten as

F (t) = C_{0} v (t) = C (t) V (t)

(8)

where

V (t)

is the relative velocity between the magnet and the conductor,

C_{0} = α B^{2} l_{w}^{2} / R

is termed as the pseudo-coefficient of damping, and

C (t) = C_{0} v (t) / V (t)

is the coefficient of EC damping. When

v (t) / V (t) = 1.0

, the pseudo-coefficient of damping

C_{0}

is accordingly equal to the coefficient of damping

C (t)

From equations (7) and (8), if the time constant $τ = 0$ , then $v (t) = V (t)$ , and the EC damping force is linearly proportional to the relative velocity between the magnet and the copper plate. In this particular case, the EC damper presents the characteristic of ideal viscous damping. When the moving speed is constant as $V (t) = V_{c}$ , and the initial condition is $v (t) = V_{0}$ , then the variable $v (t)$ can be solved as

v (t) = (V_{c} - V_{0}) (1 - e^{- \frac{t}{τ}}) + V_{0}

(9)

There is a decaying exponential transient item $e^{- \frac{t}{τ}}$ in equation (9). As the time variable increases, the transient item gradually approaches zero and the variable $v (t)$ becomes close to the $V_{c}$ eventually. In such case, the originally time-dependent coefficient of damping $C (t)$ approaches to a constant value $C_{0}$ when it reaches the steady-state.

When the model is subjected to a sinusoidal excitation with velocity amplitude of $V_{m}$ and a frequency of $ω$ , $V (t) = V_{m} \cos (ω t)$ , $v (t)$ can be determined from equation (7) as

v (t) = C_{1} e^{- \frac{t}{τ}} + \frac{\cos (ω t + θ)}{\sqrt{1 + ω^{2} τ^{2}}} V_{m}

(10)

θ = - \tan^{- 1} (τ ω)

(11)

As time increases, the term $C_{1} e^{- \frac{t}{τ}}$ in equation (10) tends to zero and can be neglected. Hence, the EC damping force under harmonic vibration can be written as

F (t) = \frac{C_{0}}{\sqrt{1 + ω^{2} τ^{2}}} V_{m} \cos (ω t + θ)

(12)

where a phase difference

θ

is observed between the EC damping force

F (t)

and the velocity excitation

V (t) = V_{m} \cos (ω t)

. Such time delay in the time domain may deviate the proposed damping model from the classical models including the viscous damping. Also, it can be observed from the equation (12) that the EC damping force is related to the excitation frequency

ω

and the time constant

τ

, and the EC damping force decreases along with the increase of the excitation frequency under the same peaks of the excitation velocity

V_{m}

Numerical simulation

To analyze the prediction performance of the proposed equivalent circuit model, the EC damping force under different excitations was numerically simulated at the ANSYS Electronics Desktop. The time-history predictions of the EC behaviors from the proposed equivalent model and the multi-physics FEM model were compared. The configuration of the planar EC FE model and the current density distribution within the copper plate are shown in Figure 2. The dimensions for the magnet and the copper plate are $50 mm \times 50 mm \times 20 mm$ and $400 mm \times 200 mm \times 10 mm$ , respectively. The parameters for the FE model are listed in Table 1. The prediction performance of the equivalent circuit model was investigated under four excitation scenarios, including the constant velocity, time-varying velocity, harmonic motion, and dual-frequency excitation.

Figure 2.

Current density configuration of the FE model. Note. FE: finite element.

Table 1.

Parameters of the finite element model and the equivalent circuit model.

Parameter		Default value
Copper	Length (mm)	400
	Width (mm)	200
	Thickness (mm)	10
	Conductivity (Siemens/m)	58,000,000
Magnet	Length (mm)	50
	Width (mm)	50
	Thickness (mm)	20
	Conductivity (Siemens/m)	625,000
	Magnetization (T)	1.1
Gap between copper plate and magnet	Gap (mm)	8

The time history of the EC damping force subjected to a constant velocity excitation is shown in Figure 3(c), and the input excitation with a constant velocity of 0.3 m/s is correspondingly illustrated in Figure 3(a). The pseudo-coefficient of damping $C_{0} = 14.58 N⋅s/m$ and the time constant $τ = 0.007 s$ are determined by the result obtained from the constant velocity excitation using a curve-fitting process. It is observed that the EC damping force increases gradually to a stable value with an initial value of zero. The force amplitude at the stable stage after 0.05 s is predicted to be 4.42 N by the proposed equivalent model, with a discrepancy of 0.60% as compared to the FEM result.

Figure 3.

Comparison between predicted and FEM results under constant and step velocity excitation: (a) Constant velocity; (b) time-varying velocity; (c) eddy current damping force under constant velocity; (d) eddy current damping force under time-varying velocity. Note. FEM: finite element modeling.

The EC damping behavior under a piecewise velocity excitation is shown in Figure 3(d) with the input velocity as illustrated in Figure 3(b). Similar to the constant case, the predicted damping force agrees favorably with the simulation outputs of the FEM model in terms of stabilized force amplitudes and gradual increase process. Noticeably in Figure 3, the EC damping force may not respond immediately following the instantaneous change of excitation velocity. According to the illustrations in Figure 3(c) and (d), the damping force may exhibit a so-called “response time” to adapt to the next stage smoothly. Such observation indicates that there may exist a time delay for the EC damping force. It is shown that the proposed equivalent circuit model can also simulate this phenomenon.

In order to investigate the frequency-dependent characteristic of the EC damping, the harmonic motion was assumed as the input for generating the EC damping force. The force–displacement plots of the EC damper under excitation frequencies of 1.0 Hz and 4.0 Hz are shown in Figure 4. The slope and the plump degree of the force curves are illustrated to be distinguished from each other. This may indicate that the hysteresis-oriented equivalent stiffness and damping coefficient may present frequency sensitivity. By comparing Figure 4(c) and (d), it is shown that the equivalent stiffness increases with the increase of excitation frequency. Overall, the equivalent circuit model can simulate the EC damping behavior under frequency-dependent excitations with the phenomenal stiffness changes, as shown in Figure 4.

Figure 4.

Comparisons between the proposed model and the FE model under harmonic motions: (a) and (c) are the time history of EC damping force and the force–displacement plot under 1.0 Hz harmonic motion, respectively; (b) and (d) are the time history of EC damping force and the force-displacement plot under 4.0 Hz harmonic motion, respectively. Note. EC: eddy current; FE: finite element.

In structural vibration analysis, the excitation input may be multi-frequency, so the prediction performance of the equivalent circuit model under multi-frequency excitation may be of great interest to be further studied. For simplicity, the dual-frequency excitation is adopted to generate the EC damping force and is expressed as

x (t) = A_{1} \sin (2 π f_{1} t) + A_{2} \sin (2 π f_{2} t)

(13)

where

[A_{1}, A_{2}]

and

[f_{1}, f_{2}]

are the pair of excitation amplitude and frequency, respectively, corresponding to the first and second frequency components in sequence. In this illustrative scenario,

A_{1}

and

A_{2}

are identically set to be 10.0 mm, while

f_{1}

and

f_{2}

are set at 1.0 Hz and 4.0 Hz, respectively.

The EC damping force and the corresponding force–displacement relationship, generated from the equivalent model and the detailed FE model, are compared in Figure 5. The normalized root-mean-square (RMS) discrepancy is 0.46% for the EC forces in Figure 5(a). We can see in Figure 5(b) that there exists a phenomenal stiffness within the illustrated force plot, and the proposed equivalent circuit model can portray such force–displacement behaviors satisfactorily. The estimation of the peak EC force is 4.573 N and 4.566 N for the equivalent circuit model and the FE model, respectively, with a 0.15% discrepancy.

Figure 5.

Comparison between the proposed model and the FE model under dual-frequency excitation: (a) Time history of the eddy current damping force; (b) force–displacement relationship. Note. FE: finite element.

Experimental validation

A planar prototype of the eddy current damper was designed and tested at the civil engineering laboratory of the Hong Kong University of Science and Technology to validate the equivalent circuit model. Figure 6 shows the experimental setups. Figure 6(a) shows a schematic view of the eddy current damper, including two pieces of permanent magnets with the opposite magnetization directions on both sides and one piece of copper plate in the middle. The details of the copper plate and the magnets are shown in Figure 6(b) and (c). The diagram of the setup is shown in Figure 6(d).

Figure 6.

Experimental setup of the eddy current damper: (a) Schematic view; (b) details of the copper plate; (c) details of the magnets; (d) diagram of setup.

The dimension of the copper plate is

160 mm \times 80 mm \times 10 mm

. The two permanent magnets are identical, and the geometric size is

100 mm \times 50 mm \times 20 mm

. The clear distance between the magnet and the copper plate is assigned as 8.0 mm during the dynamic testing. The fundamental parameters of the EC damper and the equivalent circuit model are summarized in Table 2. The pseudo-coefficient of damping

C_{0}

and time constant

τ

of the equivalent circuit model are set as

32 N⋅s/m

and 0.007 s, respectively, by employing the curve-fitting method.

Table 2.

Fundamental information of the eddy current damper and equivalent circuit model for the dynamic experiment.

Parameter		Value
Copper	Length (mm)	160
	Width (mm)	80
	Thickness (mm)	10
	Conductivity (Siemens/m)	58,000,000
Magnet	Length (mm)	100
	Width (mm)	50
	Thickness (mm)	20
	Conductivity (Siemens/m)	625,000
	Magnetization (T)	0.9
Gap between copper plate and magnet	Gap (mm)	8

The two magnets were fixed to the shaking table through a rigid wood frame. The copper plate, as shown in Figure 6(b), was connected to two load cells on both sides with a measurement capacity of 30 kg. These load cells are further connected to an aluminum frame that was mounted in the ground. Hence, the copper plate practically stands still during the experiment, and the effects of inertial force from the copper plate on the measurement are minimized. This experimental setup is a friction-free testing condition, which guarantees the accurate measurement of EC damping forces under different testing scenarios.

The uni-directional and harmonic motions were simulated by using the shake table with different frequencies and changing amplitudes. The shake table generates the movement of the magnets, which introduced the relative motion between the static copper and the moving magnets. The EC damping force was then generated by the interaction between the copper plate and the magnets and was measured by the load cells. Furthermore, the displacement of the shake table was measured by the laser displacement sensors. Table 3 lists the testing cases with assumed amplitudes and frequencies of the shaking table excitations. The excitation amplitude and frequency were selected based on the performance limits of the shake table on large velocities. In the testing, the first loop was inevitably disturbed by the abrupt and initial condition, and only the stationary testing segments were measured for further analysis.

Table 3.

Testing cases with different frequencies and amplitudes.

Case	Frequency (Hz)	Displacement amplitude (mm)	Case	Frequency (Hz)	Displacement amplitude (mm)
1	1.0	4.0	12	3.0	8.0
2	2.0	4.0	13	4.0	8.0
3	3.0	4.0	14	5.0	8.0
4	4.0	4.0	15	6.0	8.0
5	5.0	4.0	16	1.0	12.0
6	6.0	4.0	17	2.0	12.0
7	7.0	4.0	18	3.0	12.0
8	8.0	4.0	19	4.0	12.0
9	9.0	4.0	20	1.0	16.0
10	1.0	8.0	21	2.0	16.0
11	2.0	8.0	22	3.0	16.0

The excitation velocity was derived by differentiating the displacement measured from the laser displacement sensor. The obtained input velocities were then used to calculate the EC damping force by employing the proposed model with equations (7) and (8). The nine testing cases with designed amplitudes of 4.0 mm, as numbered as Case 1–9, are selected for illustration here. The relationship between the EC damping force and the excitation displacement for the chosen cases is shown in Figure 7. The experimental measurements and the numerical predictions are in favorable agreement regarding the shapes of the hysteresis loops. The width of the red line in Figure 7 represents the error of the different cycles of test. Hence, the proposed equivalent model may accurately simulate the physical behavior of the EC damping force. Moreover, similar to the observation in the numerical section, the frequency-dependent characteristic of EC damping may also be simulated with the phenomenon of excitation frequency-based slope increases of hysteresis force loops.

Figure 7.

Comparison of the eddy current damping force between the dynamic testing and the equivalent circuit model with an amplitude of 4.0 mm: (a) 1.0 Hz; (b) 2.0 Hz; (c) 3.0 Hz; (d) 4.0 Hz; (e) 5.0 Hz; (f) 6.0 Hz; (g) 7.0 Hz; (h) 8.0 Hz; (i) 9.0 Hz.

The EC force and the excitation velocity are normalized by the peak force and peak velocity, respectively. The time histories of the normalized results for the case with an amplitude of 4.0 mm and a frequency of 8.0 Hz are presented in Figure 8. Obviously, there is a response shift between the excitation velocity and the corresponding EC forces with a time delay of 0.006 s approximately. Subsequently, the velocity and the EC force may not simultaneously approach the peak values. For example, when the velocity is zero, the normalized EC force is 0.256. It is much different from the ideal viscous-type damping model with the directly proportional relationship between the damping force and velocity.

Figure 8.

Time history of the normalized eddy current force and velocity for the case with an amplitude of 4.0 mm and frequency of 8 Hz.

In order to quantify the prediction accuracy between the measured physical behaviors and the predicted numerical models, the normalized root-mean-square discrepancy (NRMSD) is adopted in the current study. The NRMSD is the root-mean-square discrepancy (RMSD) normalized by the range of observed values of a variable being predicted, which can be expressed as

NRMSD = \frac{RMSD}{y_{m a x} - y_{m i n}}

(14)

where

y_{m a x}

is the maximum value of the variable,

y_{m i n}

is the minimum value of the variable, and the lower value of this index indicates less residual variance and higher prediction precision.

The NRMSD of the EC force between the experimental measurements and the predicted results is indicated in Figure 9 for all the testing cases listed in Table 3 with the assigned four amplitudes of 4.0 mm, 8.0 mm, 12.0 mm, and 16.0 mm. Although the overall discrepancy of the EC force, for the cases with relatively higher amplitudes (8.0 mm, 12.0 mm, and 16.0 mm), is slightly larger than those with an amplitude of 4.0 mm, the normalized root-mean-square discrepancies are still relatively minor and acceptable. Such observation indicates that the equivalent circuit model can predict the experimental results reasonably well. Moreover, the NRMSD seems to be less sensitive to excitation amplitudes and frequency, indicating promising prediction reliability of the proposed equivalent circuit model.

Figure 9.

Normalized root-mean-square deviation of the predicted eddy current forces with varying excitation frequencies.

Comparison with existing damping models

The EC damper used in engineering practice may be more complicated. The optimal configurations of the EC damper are studied by different research groups (Ebrahimi et al., 2008; Zuo, et al., 2011), to increase the efficiency of the EC damping. To investigate the feasibility of the equivalent circuit model for modeling complex EC damper, an illustrative example of an EC damper with alternating magnetic fields was adopted and simulated at the ANSYS Electronics Desktop. The FEM simulation results are then compared with the predictions obtained from the equivalent circuit model, the higher-order fractional model, and the classical viscous model.

Figure 10(a) shows the schematic view of the EC damper, which consists of six layers of magnets, two layers of soft irons, and five layers of copper plates. Figure 10(b) shows the arrangement and the magnetization direction of the magnets. The adjacent magnets are arranged with alternating pole direction on the X–Y plane, and the adjacent magnets along the Z-direction are arranged with the same pole direction. The magnetization of the magnet is 1.45 T, and the relative permeability is 1.10. The dimensions of the magnet blocks are $100 mm \times 50 mm \times 25 mm$ , which is shown in Figure 10(c), while the size of the copper plate is $490 mm \times 270 mm \times 6 mm$ , as shown in Figure 10(d).

Figure 10.

Design of a magnetic eddy current damper: (a) Finite element model in ANSYS; (b) magnetization direction of the magnets; (c) detail of the magnet; (d) detail of the copper plate.

The higher-order fractional model (Xu, et al., 2016), which consists of a fractional Maxwell model and a fractional Kelvin model in parallel, is adopted for comparison in the current study. The higher-order fractional model shown can be written as

D^{β} [γ (t)] = \frac{D^{λ} [F_{1} (t)]}{μ_{1}} + \frac{F_{1} (t)}{η_{1}}

(15)

F_{2} (t) = η_{2} D^{k} [γ (t)] + μ_{2} γ (t)

(16)

F_{1} (t) + F_{2} (t) = F (t)

(17)

where

μ_{1}

and

μ_{2}

are the elastic moduli of the spring elements,

η_{1}

and

η_{2}

are the viscosity coefficients of the dashpots, and

λ

β

, and

k

are the orders of fractional derivatives.

γ (t)

is the displacement, and

F_{1} (t)

and

F_{2} (t)

are the forces of the fractional Maxwell model and the fractional Kelvin model, respectively.

The parameters $c_{1} = μ_{1} / η_{1}$ and $c_{2} = μ_{2} / η_{2}$ are adopted for convenience, so the model described by equations (15)–(17) can be further expressed as

\frac{1}{η_{2}} (D^{λ} [F (t)] + c_{1} F (t)) = D^{λ + k} [γ (t)] + c_{2} D^{λ} [γ (t)] + c_{1} D^{k} [γ (t)] + \frac{μ_{1}}{η_{2}} D^{β} [γ (t)] + c_{1} c_{2} γ (t)

(18)

Equation (18) can be numerically solved at the Simulink platform of MATLAB. The parameters of the higher-order fractional derivative model are determined by the genetic algorithm (Park et al., 2006), and the model parameters in the present study are $λ = 0.9212$ , $β = 0.9683$ , $k = 0.5322$ , $μ_{1} = 4762 kN/m$ , $μ_{2} = 8.435 \times 10^{- 6} kN/m$ , $η_{1} = 61.17 kN⋅s/m$ , and $η_{2} = 1.127 \times 10^{- 5} kN⋅s/m$ . The pseudo-coefficient of damping $C_{0}$ for the equivalent model is set as $53 .95 kN⋅s/m$ , and the time constant $τ$ is set as 0.0119 s. Furthermore, the viscous damping model is utilized for comparison, and the damping coefficient is regressed as $53 .95 kN⋅s/m$ . The comparisons between the predicted results of the three mechanical models and the FEM data under 5.0 Hz harmonic excitation are illustrated in Figure 11. The simulations have been conducted for two cycles for each case, and the normalized RMS deviation can be calculated by using equation (14).

Figure 11.

Comparison between FEM result and different models for the eddy current force under 5 Hz harmonic excitation: (a) Time histories of the eddy current damping force; (b) force–displacement. Note. FEM: finite element modeling.

Figure 11 shows that the equivalent circuit model and the fractional model can approximately simulate the dynamic behavior of the EC damping force. However, significant differences exist between the viscous model and the other two models at the initial condition. The viscous damping model cannot simulate the phenomenon where the EC force increases gradually at the initial state. Under this selected excitation with a frequency of 5.0 Hz, the peak values of damping forces for the equivalent circuit model, the fractional model, and the viscous damping model are 15.85, 15.93, and 16.96 $kN .s/m$ , respectively. As compared with the FEM result (15.75 $kN$ ), the amplitude discrepancies for the three damping models are 0.66%, 1.16%, and 7.73%, respectively. Moreover, the normalized RMS deviations of the second loop cycle under 5.0 Hz harmonic excitation for the equivalent circuit model, the fractional model, and the viscous model are 1.15%, 1.04%, and 26.35%, respectively.

The frequency-dependent EC damping may present the equivalent damping and stiffness relevant to the excitation frequency, which has been reported by some researchers (Bae, et al., 2009; Pan, et al., 2016; Zuo, et al., 2011). To quantify the characteristics of the frequency dependence, the EC damping force can be regressed by adopting the standard linear model as

F = c_{eq} \dot{x} + k_{eq} x

(19)

where

x

and

\dot{x}

denote the displacement and velocity between magnet and conductor, respectively;

c_{eq}

and

k_{eq}

represent the equivalent damping and equivalent stiffness coefficient, respectively, and can be estimated by linear regression techniques. The equivalent damping and stiffness fitting for the target force loops generated by the FE model and the two nonlinear models under varying excitation frequencies are calculated using equation (19), and the pairs of regressed parameters are presented in Figure 12.

Figure 12.

Equivalent damping and stiffness for FEM result and different models under varying excitation frequencies. Note. FEM: finite element modeling.

The damping coefficient of the FEM decreases with the increase of the excitation frequency, which is consistent with the results in Reference (Bae, et al., 2009; Zuo, et al., 2011). The equivalent circuit model and the fractional model can simulate the frequency-dependent phenomenon, and the damping coefficients for these two models decrease along with the increase of excitation frequency. For example, when the excitation frequency is 5.0 Hz, the damping coefficients for the equivalent circuit model and the fractional model are identified to be 47.27 $kN⋅s/m$ and 47.89 $kN⋅s/m$ , respectively. Compared with the FEM result of 46.61 $kN⋅s/m$ , the deviations for the two nonlinear models are 1.42%, and 2.75%, respectively, indicating favorable agreement.

The FEM results show the tendency of the simultaneous increase of phenomenally equivalent stiffness with the rise of the excitation frequency. Both the equivalent circuit model and fractional model can portray the trend. Noticeably, regarding the FEM curve in the black line as the objective, the proposed equivalent circuit model shows better, comparable, and worse agreement than the fractional model for the frequency ranges of below 3 Hz, 3–10 Hz, and above 10 Hz, respectively. The proposed mechanical model may be feasible and advantageous over the high-order fractional model, due to the presented properties of higher computational efficiency and fewer parameters to be determined.

Following the comparison of model accuracy, it would be interesting to further study the potential influence of adopting linear and nonlinear EC damping models on dynamic analysis of the vibration system. An SDOF vibration system is selected, as shown in Figure 13. The mass is 5000 kg, and the stiffness is 12,633 kN/m, with the vibration frequency of 8.0 Hz. The EC damper in Figure 10 provides the damping to the vibration system. The equivalent circuit model, the fractional model, and the viscous model are adopted to calculate the responses of the vibration system. Sinusoidal excitation with an amplitude of $1.0 m / s^{2}$ is adopted to excite the system. The acceleration amplification factor of the SDOF vibration system under varying excitation frequencies is shown in Figure 14.

Figure 13.

Single-degree-of-freedom vibration system with eddy current damping.

Figure 14.

Amplification factor of the SDOF vibration system with different EC damping models. Note. EC: eddy current; SDOF: Single-degree-of-freedom.

It is observed in Figure 14 that the systems with the equivalent circuit model and the fractional model exhibit the same resonant frequency of 8.4 Hz due to the equivalent stiffness of the EC damping, while the resonant frequency for the system with the viscous model is 7.92 Hz. The peak amplification factors for the vibration system with the equivalent circuit model, the fractional model, and the viscous model are 6.885, 6.689, and 4.783, respectively. Noticeably, there is a significant discrepancy of 28.49% between the viscous model and the fractional model. Overall, the viscous damping model may introduce a slight overestimation and a significant underestimation of the system response with the excitation frequency scenarios smaller than 7.7 Hz and larger than 7.7 Hz, respectively.

Conclusion

An equivalent circuit model is proposed and systematically investigated in the current study, to simulate the nonlinear behavior and frequency-dependent characteristic of EC damping. The ECs are simplified as an ideal resistor–inductor circuit model with the derivation of time-varying damping coefficient. It is numerically and experimentally demonstrated that the proposed equivalent model could simulate and predict the EC damping force satisfactorily and efficiently.

An FE model of EC damping is modeled in ANSYS. The nonlinear behavior of EC damping can be favorably simulated by the proposed equivalent circuit model under various excitation scenarios. Then, a noncontact and friction-free EC damper is utilized for testing dynamic behavior, and the equivalent circuit model can reproduce the experimental measurements and observations. The time-delay phenomenon between the normalized EC damping force and the normalized velocity is discussed, explaining the observed stiffness phenomenon. The characteristic of complicated EC damper modeled in ANSYS can be simulated by the proposed equivalent circuit model, comparing the higher-order fractional model and the viscous damping model. The proposed model may be feasible with relatively better accuracy, higher computational efficiency, and fewer parameters to be determined. An SDOF structure with EC damping is analyzed to study the potential simulation influence by adopting different EC damping models. The result indicates that the proposed equivalent circuit model and the high-order fractional model show similar results, and the viscous model may underestimate the system responses in some cases. In the next study, the applicability of the theoretical model to other configurations of the EC damper will be researched.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is sponsored by the National Natural Science Foundation of China (Grant No: 51878483), the Key Laboratory of Performance Evolution and Control for Engineering Structures (Tongji University), Ministry of Education (No. 2019KF-6), the Shanghai Qi Zhi Institute (Grant No. SYXF0120020109), and the Peak Discipline Construction Project of Shanghai (No. 2021-CE-03).

ORCID iD

Jiazeng Shan

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