Improved Eddy Current Damping Model for Transverse Vibrations

Henry A. Sodanoa, Jae-Sung Baeb, Daniel J. Inmana and W. Keith Belvinc

a Center for Intelligent Material Systems and Structures

Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0261, USA

b Wind Power/Fluid Machinery Research Center

New & Renewable Energy Research Department Korea Institute of Energy Research

Daejeon, 305-343, Korea

c Structural Dynamics Branch NASA Langley Research Center Hampton, VA 23681-0001, USA

ABSTRACT Eddy Currents are formed in a conductive material when it is subjected to a time changing magnetic flux. The currents circulate inside of the conductor in such a way that a new magnetic field is generated with a polarity that varies with the change in the applied magnetic flux. The interaction between the applied magnetic field and the field due to the eddy currents causes the generation of a force that opposes the change in flux. However, due to the internal resistance of the conductor the eddy currents will dissipate into heat, causing a removal of energy from the system. This dissipation of energy allows magnet and conductor to form a damper that may be used to suppress the vibration of a structure. This magnetic damping system is of non-contacting nature, allowing it to be easily applied and add significant damping to the structure without changing dynamic response. In a previous study, the concept and theoretical model was developed for one eddy current damping system that was shown to be effective in the suppression of transverse beam vibrations. The study developed a mathematical model of the system and showed that it accurately predicted the damping induced on a beam. However, while this model was very accurate when the magnet was far from the beam, it was less accurate for the case that the gap between the magnet and beam was small. In the present study, an improved theoretical model of the previously developed system will be formulated using the image method allowing the eddy current density to be more accurately computed.

1. INTRODUCTION

In the event that a nonmagnetic conductive metal is subjected to a time varying magnetic field, eddy currents will be generated. These eddy currents circulate inside the conductor in a way that a new magnetic field with a polarity dependent on the direction of change in applied field, thus causing a force that resists the change in field. However, due to the electrical resistance of the conductive metal, the induced currents will dissipated into heat at the rate of I2R and the force will disappear. In the case of a dynamic system, the conductive metal is continuously moving in the magnetic field and experiences a continuous change in flux that induces an electromotive force (emf) allowing the induced currents to regenerate. The process of the eddy currents being generated and dissipated causes a repulsive damping force that is proportional to the velocity of the change in magnetic flux. This system acts as a viscous damper because of the removal of energy from the system in the form of the dissipated eddy currents. The use of eddy currents for damping of dynamic systems has been known for decades and its application to magnetic braking systems [1-4] and lateral vibration control of rotating machinery [5-6] has been thoroughly investigated.

While the theory and applications of rotary magnetic braking systems have been well documented, there are

many more applications of eddy current dampers. Schmid and Varga [7] studied a vibration-reducing system with eddy current dampers (ECDs) for high resolution and nanotechnology devices such as an STM (Scanning tunneling microscope). Lee [8] considered the dynamic stability of conducting beam-plates in transverse magnetic fields. The research showed that three regions of stability existed, damped stable oscillation, static asymptotic stability and static divergence instability. The buckling field was also found to exhibit a linear dependence on the geometry of the ratio of the thickness and length of the beam-plate. Kobayashi and Aida [9] explored the use of a Houde damper (a type of damped vibration absorber) using an eddy current damper as the energy dissipation mechanism. The eddy current damper consisted of a conducting plated moving between two permanent magnets. The study found the Houde damper could increase the damping ratio by 2% and suppress the displacement of the pipe by a factor of eight to ten. Kienholtz et al. [10] investigated the use of a magnetic tuned mass damper for vibration suppression of a spacecraft solar array and a magnetically damped isolation mount for the payload inside of a space shuttle. The magnetic tuned mass damper system targeted two modes of the solar array (1st torsion at 0.153Hz and 1st out of plane bending of 0.222Hz) and increased the damping by 30 dB and 28 dB respectively, while the higher frequency untargeted modes 0.4-0.8 Hz were damped in the range of 11-16 dB.

While the aforementioned studies have investigated a variety of applications, several studies have been

performed to determine the effect of magnetic fields on the vibration of beam. Morisue [11] investigated the effect of an applied magnetic field on a conducting cantilever beam and analyzed the beam response. The response was predicted using finite difference methods and the results were found to compare well with experiments performed at Argonne National Laboratory. In a similar study Takagi et al. [12] studied the deflection of a thin copper plate subjected to magnetic fields both analytically and experimentally. They used an electromagnet with very high current (several hundred Amperes) to generate the magnetic field then analyzed the response of the plate to the applied field. Matsuzaki et al. [13] proposed the concept of a new vibration control system in which the vibration of a beam periodically magnetized along the span is suppressed using electromagnetic forces generated by a current passing between the magnetized sections. To confirm the vibration suppression capabilities of their proposed system, they performed a theoretical analysis of a thin beam with two magnetized segments subjected to an impulsive force and showed the concept to be viable. Following the proposal of the previous concept, Matsuzaki et al. [14] performed an experimental study to show the effectiveness of this new concept. However, a partially magnetized beam was not available to the authors, so a thin beam with a current carrying wire bonded to its surface along with a permanent magnet was used. The system was then used to determine if the electromagnetic force generated by the wire was sufficient to suppress the vibration of the beam. The results of their study showed that indeed the force is capable of damping the beam’s first few modes of vibration.

More recently, Kwak et al. [15] investigated the effects of an eddy current damper on the vibration of a

cantilever beam and their experimental results showed that the eddy current damper can be an effective device for vibration suppression. The authors ECD used a fixed copper conducting plate and flexible linkage attached to the tip of the beam in order to utilize the axial magnetic flux and generate eddy current damping forces. Bae et al. [16] modified and developed the theoretical model of the eddy current damper constructed by Kwak et al [15]. Using this new model, the authors investigated the damping characteristics of the ECD and simulated the vibration suppression capabilities of a cantilever beam with an attached ECD numerically.

When using eddy currents, the typical method of inducing an emf in the conductor is to place the metal

directly between two oppositely poled magnets with the metal moving perpendicular to the magnet’s poling axis, a schematic of this process is shown in Fig. 1, and has been studied in Ref. [1-7], [9-10], and [15-16]. This configuration is optimal because of the concentrated magnetic field between the two magnets. While this configuration is effective for magnetic braking, in certain applications it is not possible. The research presented by Sodano et al. [17] proposed and developed a theoretical model of one such system in which the transverse vibrations of a cantilever beam were damped by a permanent magnet fixed to a location perpendicular to the beams motion and the magnet’s radial flux was used to generate the damping force. It was shown through experiments that the theoretical model very accurately estimated the damping when the magnet was greater than 5mm from the beam, but was less accurate for smaller gaps. The present study will revisit the theoretical model of the system developed by Sodano et al. [17] and provide improvements to its accuracy over the whole range of magnet locations.

Fig. 1: Schematic of conductive material passing through a magnetic field and the generation of eddy currents.

In this manuscript the magnetic flux generated by a cylindrical permanent magnet will be calculated, allowing the eddy current density to be determined. After finding the eddy current density, the image method described by Lee and Park [18] will be used to enforce the necessary boundary condition of zero eddy current density at the edges of the conductor. By doing this, the accuracy of the predicted damping force will be improved over the tested range of distances between the magnet and beam. The modifications of the model will be applied to the eddy current damper previously developed by Sodano et al. [17] and the increased accuracy due to these improvements will be shown through a comparison of both models with experimental data.

2. EDDY CURRENT DAMPING MODEL

A schematic showing the layout of the eddy current damping system is shown in Fig. 2, which consists of a cantilever beam with a copper plated located in the magnetic filed generated by a cylindrical permanent magnet. The permanent magnet generates a magnetic field in the vertical (z) and horizontal or radial (y or R) axes. When the conducting sheet of thickness δ, conductivity σ and distance lg from the circular magnet is set into motion with a velocity v relative to the surface of the permanent magnet, an electric field is generated in the conducting sheet. Since the deflection of beam is in the vertical direction, the vertical component of the magnetic field does not contribute to the generation of eddy currents. Hence, the electric field on the conductor is dependent on the horizontal component By of the magnetic field. As shown in Fig. 3, the eddy currents circulate on the conducting sheet in the x-y plane, causing a magnetic field to be generated.

Fig. 2: Cantilever beam in magnetic field generated by permanent magnet.

Magnet

x

y

z

y

v

×v B

gl

L

b-b

Fig. 3: Magnetic field and the eddy currents induced in the cantilever beam.

Due to the symmetry of the circular permanent magnet the surface charges are ignored, allowing the eddy

current density J induced in the conducting sheet moving in the vertical direction to be written as

( )BvJ ×= σ (1) where the ×v B term is the cross product of the velocity v of the conductor and the magnetic flux density B defining the electromotive force driving the eddy currents J. The velocity and magnetic flux can be written as follows

kjiv zv++= 00 (2)

kjiB zyx BBB ++= (3)

where the velocity of the conductor is only in the z-direction. After substitution of equations (2) and (3) into equation (1), the eddy current density is defined by

( ) ( )jiBvJ xyz BBv +−=×= σσ (4) The above equation confirms that the magnetic flux in the z direction has no effect on the induced eddy currents and that the induced currents are solely dependent on the x- and y-components of the magnetic flux or the flux tangential to the face of the conducting sheet.

The magnetic flux density due to a circular magnetized strip, shown in Fig. 4 can be written by [23]

φπ

µ πd

RdMd ∫×

=2

0 31

100

4RlB (5)

where µ0 and M0 are the permeability and the magnetization per unit length, respectively. The vector R1 is defined by the distance between the differential element on the circular strip and the point on the y-z plane as shown in Fig. 4 and defined as

rRR −=1 (6)

where

kjR zy += (7)

jir φφ sincos bb += (8) The length vector dl of the infinitesimal strip is

jil φφφφ dbdbd cossin +−= (9)

where b is the radius of the circular magnet.

φ

R

1R

rθ

( ), ,P R zθ

y

z

x

dlb

b

Fig. 4: Schematic of the Circular magnetized strip depicting the variable used in the analysis.

Substituting equations (6), (8)and (9) into equation (5), the magnetic flux density due to the circular magnetized strip is

( )( )zybIbzMd

ybzyb

bzMBy ,,4sin2

sin4 1

002

0 23222

00

πµφ

φ

φπ

µ π=

−++= ∫ (10)

( )( )zybIbMd

ybzyb

ybbMBz ,,4sin2

sin4 2

002

0 23222

00

πµφ

φ

φπ

µ π=

−++

−= ∫ (11)

where 1I and 2I include the elliptic integrals and are shown in Appendix A. Hence, the magnetic flux density due to the circular magnet of length L are written by

( ) ( ) ( )∫− ′′−′−=0

100 ,,

4,

Ly zdzzybIzzbMzyBπ

µ (12)

( ) ( )∫− ′′−=0

200 ,,

4,

Lz zdzzybIbMzyBπ

µ (13)

where 'z and L are the distance in the z direction from the center of a magnetized infinitesimal strip and the length of the circular magnet, respectively.

Since the velocity of the conducting sheet is in the z direction, the magnetic flux density Bz does not contribute to the damping force. Using equations (4), (12), and (13), the damping force due to the eddy current is defined by

( )( )∫

∫ ∫∫

−=

−=

×=

c

c

r

gy

g

r

y

V

dylyyBv

dydlyyBv

dV

0

2

2

0 0

2

,2

,

πσδ

φσδπ

k

k

BJF

(14)

where δ and v are the thickness and the vertical velocity of the conducting sheet, respectively, rc is the equivalent radius of the conductor that preserves its surface area and lg is the distance between the conducting sheet and the bottom on magnet as shown in Fig. 3. 2.2 Application of the Image method to Improve Model Accuracy

The derivation performed in section 2.1 is valid for the case of an infinite conducting sheet, meaning that the edge effects of the conductor are unaccounted for. Neglecting the edge effects in the model will cause the predicted damping to be greater than actually present, because the eddy current density is not required to be zero at the edges. In order to account for the edge effects, the image method [18] can be used to satisfy the boundary condition of zero eddy current density at the conducting plate’s boundaries, a schematic showing how the image method is used can be seen in Fig. 5. By introducing an imaginary eddy current density the net eddy current in the radial direction J ′ can be written as

( ))2()1(yy JJ −=′J (15)

where y is the radial direction and the imaginary eddy current density is written as

)2()( )1()2( yAJyJ yy −= (16) where ( )1

yJ is the predicted eddy current density and the dimension A corresponds to half the length of the conducting plate as shown in Fig. 6. Only one imaginary eddy current is needed because the conductor is modeled as a circular plate with the same area as the original conductor as shown in Fig. 6. This assumption is made to simplify the integration of equations (12)-(14). Substituting equations (15) and (16) into equation (14) the damping force accounting for the imaginary eddy currents is defined as

( ) ( )

( ) ( )

−−−=

−−−=

×′=

∫∫

∫ ∫∫ ∫

∫

cc

cc

r

gy

r

gy

g

r

yg

r

y

V

dylyAyBdylyyBv

dydlyAyBdydlyyBv

dV

0

2

0

2

2

0 0

22

0 0

2

,2,2

,2,

πσδ

φφσδππ

k

k

BJF

(17)

Because it is too difficult to integrate equations (12) and (13) analytically, a numerical integration method is used to obtain the damping force in equation (17).

Fig. 5: Schematic demonstrating the effect of the imaginary eddy currents.

Fig. 6: schematic showing the variables associated with the conducting plate.

2.3 Modeling of Cantilever Beam

The dynamic response of the beam can be formulated using the assumed modes method applied to an Euler-Bernoulli beam. This method assumes that the response can be modeled as a summation of the mode shapes of the system and a temporal coordinate as

( ) ( ) ( ) ( ) ( )trxtrxtxuN

iii φφ == ∑

=1

, (18)

where )(xiφ is the assumed mode shapes of the structure that can be set to satisfy any combination of boundary conditions, r(t) is the temporal coordinate of the displacement and N is the number of modes to be included in the analysis. The kinetic energy T, potential energy V, non-conservative forces D, and external forces Q for the beam are defined by

( ) ( )∫

∂∂

=L

dxttxuxT

0

2,21 ρ (19)

( ) ( )∫

∂

∂=

Ldx

xtxuxEIU

0

2

2

2 ,21

(20)

( ) 2,

∂∂

−=ttxucD e (21)

( ) ( ) ( )[ ] ( )∫ −+=L

ji dxtxuxxtFtxfQ0

,, δ (22)

where u(x,t) is the displacement of the beam, ρ is the density per unit area, V is the volume of the beam, F is a concentrated force acting on the beam, f(x,t) is a distributed force acting on the beam, E is the modulus of elasticity, I(x) is the moment of inertial of the beam and ce is the viscous damping force from the eddy currents. Using the assumed series solution of equation (26), the kinetic energy, potential energy and external forces can be rewritten as

( ) ( ) ( ) ( ) ( )

= ∫∑∑

= =

L

ji

N

i

N

jji dxxxxtrtrT

01 12

1 φφρ&& (23)

( ) ( ) ( ) ( ) ( )

= ∫∑∑

= =

L jiN

i

N

jji dx

dtxd

dtxdxEItrtrU

0 2

2

2

2

1 121 φφ

(24)

( ) ( ) ( ) ( )[ ]xxctrtrD jie

N

i

N

jji φφ∑∑

= =

=1 12

1&& (25)

( ) ( ) ( ) ( ) ( )∑ ∫ ∑= =

+=

m

kk

L p

iikik trxtFdxxtxfQ

10

1

, φφ (26)

To obtain the equations of motion for the Euler-Bernoulli beam, Lagrange’s equation defined by

jjjjj

QrV

rD

rT

rT

dtd

=∂∂

+∂∂

+∂∂

−

∂∂

&& (27)

Using equations (23)-(26), equation (27) can be solved to obtain the equation of motion defined by

( ) ( ) ( ) ( ) ( ) ( ) ( )∫ ∑=

+=++L p

iii xtFdxxtxftrtrtr

01

, φφKCM &&& (28)

where the mass matrix M, the damping matrix C, and the stiffness matrix K are defined by

( ) ( ) ( )∫==L T

ij dxxxxm0

φφρM (29)

( ) ( ) ( )∫==L T

ij dxxxxEIk0

φφ &&&&K (30)

( ) ( )eeT

eij xcxc φφ==C (31) where ( )exφ is the magnitude of the mode shape at the location of the eddy current damper. The above equation of motion defines the interaction between the beam and the passive eddy current damper.

3. EXPERIMENTAL SETUP

Following the development of an improved model, it is necessary to demonstrate its effectiveness for predicting the response of the system. To do so, experiments were performed on an aluminum cantilever beam with dimensions shown in Fig. 7. For the experiments performed in this study, a neodymium-iron-boron permanent magnet with radius and length of 6.35 mm and 12.7 mm, respectively, was used. The other physical properties of the beam, conductor and magnet are listed in Table 1. To accurately model the system being tested it was necessary to include the eddy current generated in the aluminum beam as well as the copper plate.

Fig. 7: Schematic showing the dimensions of the beam.

Table 1: Physical properties of the beam, conductor and magnet.

Property Value Young’s modulus of beam 75 GPa Density of beam 2700 kg/m3 Conductivity of beam 3.82 x 107 Thickness of copper conductor 0.62 mm Conductivity of copper conductor 5.80 x 107 Permanent magnet composition NdFeB 35 Residual magnetic Flux of magnet 1.21 kGauss

The goal of the experiments was to measure the damping of the beam as a function of the gap lg between the copper conducting plate and the surface of the permanent magnet. To do this both the response to an initial displacement and the frequency response were measured. From these two measurements the damping of the beam can be calculated by determining the log decrement of the initial condition response and the unified matrix polynomial approach (UMPA) can be applied to the frequency response. It was necessary to find the damping using both of these methods because the eddy currents add significant damping when the magnet is placed in close proximity to the beam, making the damping measurement difficult.

In order to accurately measure the damping of the aluminum beam using an initial displacement and the log decrement method, the initial condition must be consistent throughout all tests. This is necessitated further due to the need to measure the damping for numerous different gap distances lg for the magnet and conducting plate. Therefore to ensure that the initial displacement was consistent throughout every test a very thin steel plate was attached to the beam and an electromagnet was positioned at a fixed distance from the beam and steel plate. A small switch was constructed to allow the magnet to be activated and thus pull the steel plate into contact with the surface of the electromagnet and provide a fixed initial displacement. When the switch was turned off the electromagnet releases the beam allowing its vibration to damp out, this system can be seen in Fig. 8. Once the beam is set into motion, a Polytec laser vibrometer was used to measure the displacement of the tip of the beam.

Fig. 8: Experimental setup of the aluminum beam and eddy current damper.

To measure the frequency response of the aluminum beam, a piezoelectric patch was attached to the root of the beam and excited using a swept sine signal through the frequency range of interest. The displacement response of the beam was then measured using a Polytec laser vibrometer and recorded using a SigLab FFT analyzer. With the two excitation systems developed, the next step was to construct an accurate method of positioning the permanent magnet a fixed distance from the conducting plate. To do this, the permanent magnet was bonded to a wooden block that was fixed to a fine threaded lead screw, as shown in Fig. 8, this lead screw allowed the position of the magnet to be accurately varied. A wooden block was used so that the magnetic field was not distorted due to high permeability materials in close proximity to the magnet.

4. RESULTS OF MODEL AND EXPERIMENTS 4.1 Numerical Calculation of the Magnetic Flux

Once the experimental setup had been designed the model could be used to predict the response of the

system. However, before the damping force generated by the eddy currents and the beam response could be found the magnetic field of the permanent magnet had to be calculated. Because equations (12) and (13) are too difficult to solve analytically, they were numerically integrated. The resulting magnetic flux B of this integration is shown in Fig. 9, for the case of a cylindrical permanent magnet with length L and whose surface is located at z=0. The contours in Fig. 9 indicate the radial component By of the magnetic flux. Since the conductor moves in the z direction, the z component Bz of the magnetic flux does not contribute to the generation of eddy currents in the conductive material indicating that only the radial component By, affects the strength of the eddy current flowing through the conducting sheet.

Electromagnet

Magnet y

z

Fig. 9: Magnetic flux and contour of yB for a single cylindrical magnet.

After determining the magnetic flux generated by the permanent magnet, the induced eddy currents could be calculated and the damping force found. The eddy current density when the magnet is located at a distance of 1 mm from the beam is shown in Fig. 10. This Figure also shows the imaginary eddy current density which was calculated using the image method and the resulting eddy current density after the image method is applied to satisfy the electrical boundary conditions of the conductor. The use of the image method allowed our theoretical model’s accuracy to be greatly increased.

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

Radial Distance from Center of Conductor (mm)

Eddy

Cur

rent

Den

sity

Eddy Current Density without Edge EffectsImaginary Eddy Current DensityEddy Current Density with Edge Effects

Fig. 10: Eddy current density before and after the image method is applied.

4.2 Validation of Model through Experiments

Once the magnetic flux and eddy current density had been numerically calculated, the accuracy of the theoretical model could be validated using the experimental setup previously detailed. However, first the effectiveness of the damping concept developed by Sodano et al. [17] will be shown. The performance of the eddy current damper can be seen by looking at the frequency response shown in Fig. 11. As can be seen, the magnetic damper adds significant damping to the cantilever beam, with the first through third mode experiencing a reduction of 42.4 dB, 21.9 dB and 14.3 dB, respectively. Furthermore, the Figure demonstrates that the dynamic response of the system is unaffected by the addition of the damper into the system. This occurs because the damper is of non-contact nature, meaning that it does not induce and type of mass loading or added

stiffness to the host structure. This point is a crucial one for systems that have been designed with a specific dynamic response, yet require additional damping subsequent to their design. Other methods of damping would alter the desired performance of the system, while the eddy current damper developed in the present study would not.

0 50 100 150 200 250-60

-50

-40

-30

-20

-10

0

10

20

Frequency (Hz)

Mag

nitu

de (d

B)

DampedUndamped

Fig. 11: Experimentally measured damped and undamped frequency response of the beam.

The predicted and measured damping ratio of the beam was used to demonstrate the model’s accuracy. As mentioned, the theoretical model developed in this study provides increased accuracy over the model previously developed by Sodano et al. [17]. To demonstrate the improvement in accuracy that is obtained through the use of the image method, Fig. 12 shows the experimental and predicted damping ratio for the eddy current damping system with the model developed by Sodano et al. [17] as well as the predicted damping ratio when the image method is used. From the Figure, it can be seen that the improved mathematical model developed in this manuscript provides a significant increase in accuracy when the magnet is in close proximity to the conductor. To further show the improvement in accuracy when the image method is used, Fig. 13 shows the experimental and predicted response to an initial displacement.

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Magnet Gap (mm)

Dam

ping

Rat

io

Experimental DataFinte Conductor without Edge EffectsFinte Conductor with Edge Effects

Fig. 12: Experimental and predicted damping ratio of the beam’s first mode for the system used in Sodano et al. [17] and the damping ratio predicted by the improved model developed in this paper.

1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5-1

-0.5

0

0.5

1

1.5

2

2.5

3

Time (sec)

Dis

plac

emen

t (m

m)

Experimental DataFinite Conductor without Edge EffectsFinite Conductor with Edge Effects

Fig 13: Experimental, predicted without edge effects and predicted with edge effects initial condition response for the case of a magnet gap of 1mm.

CONCLUSIONS The research presented in this manuscript has utilized the eddy currents induced in a moving conductor for passive vibration damping. Because the damper is of passive nature it is robust to parameter changes, requires no additional energy, and is easy to apply to the structure. Furthermore, the damper developed in this paper is a non-contact damper, thus allowing significant damping to be added while permiting the other properties and dynamics of the structure to be unaffected by its addition to the system. This point is of significant importance to the many structures that were designed with specific key parameters in place, through which the addition of a damping scheme such as constrained layer damping would change. In a previous study, a mathematical model of the eddy current damping system was developed. However, while this model was very accurate for gaps between the conductor and magnet greater than 5mm, it was less accurate for smaller gaps. In the present study, the image method was utilized to increase the accuracy of the predicted eddy current density. By incorporating this method into the theoretical model of the system, the predicted damping ratio was significantly increased aver the whole range of magnet gaps. Using damping systems of this type present many advantages over traditionally used dampers and with the accuracy of the modeling techniques provided in this manuscript, the system can be easily designed to fit any needed application.

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[17] Sodano, H.A., Bae, J.S., Inman, D.J. and Belvin, W.K., 2004, “Concept and Model of Eddy Current Damper for Vibration Suppression of a Beam,” Journal of Sound and Vibration, Submitted.

[18] Lee, K. and Park, K., 2002, “Modeling Eddy Currents with Boundary Conditions by Using Coulombs Law and the Method of Images,” IEEE Transactions on Magnetics, Vol. 38, No. 2, pp. 1333-1340.

[19] Cheng, D. K., 1992, Field and Wave Electromagnetics, Addison-Wesley Publishing Company.

APPENDIX A The integration I1 in equation (10) is

( )

−

+

−

−

−

+

−

=

−+= ∫

22222

21212

2

2

0 3222

1

4,4

34,4

4,4

34,4

1

sin2

sin

nybE

nybEp

nybE

nybEm

bynp

dybzb

I

ππππ

φφ

φπ

(A1)

where

2222 zybm ++= (A2)

( ) 222 zybn +−= (A3)

( ) 22 zybp ++= (A4)

The elliptic integrals of equation (A1) are

( ) ( )∫ −==φ

θθφ0

2121 sin1, dmmE (A5)

( ) ( )∫−

−==φ

θθφ0

2122 sin1, dmmE (A6)

The integration I2 in equation (11) is

( )

−

+

−

+

−

+

−

=

−+

−= ∫

22222

21212

2

0 3222

2

4,4

34,4

4,4

34,4

1

sin2

sin

nybE

nybEp

nybE

nybEs

bnp

dybzb

ybI

ππππ

φφ

φπ

(A7)

where

2222 zybm ++= (A8)

( ) 222 zybn +−= (A9)

( ) 22 zybp ++= (A10) 222 zybs −−= (A11)

The elliptic integrals of equation (A7) are

( ) ( )∫ −==φ

θθφ0

2121 sin1, dmmE (A12)

( ) ( )∫−

−==φ

θθφ0

2122 sin1, dmmE (A13)