structural optimization

Progress In Electromagnetics Research B, Vol. 49, 130, 2013

MODELING THE DYNAMIC ELECTROMECHANICALSUSPENSION BEHAVIOR OF AN ELECTRODYNAMICEDDY CURRENT MAGLEV DEVICE

Nirmal Paudel* and Jonathan Z. Bird

Department of Electrical and Computer Engineering, Universityof North Carolina at Charlotte, 9201 University City Boulevard,Charlotte, NC 29223, USA

AbstractA 2-D analytic based eddy-current transient model for aconducting plate is derived that is capable of accounting for continuouschanges in the input conditions. Only the source field on the surfaceof the conducting plate needs to be known. In addition, a 2-Dsteady-state analytic based eddy-current model that is capable ofaccounting for frequency and velocity changes in two directions isderived. Both analytic based models have been validated using finiteelement code. The transient and steady-state models are integratedinto an electromechanical system where the magnetic source is aHalbach rotor. The accuracy of both calculation methods is compared.The stiffness and damping coefficients are derived using the steady-state model.

1. INTRODUCTION

Most magnetic suspension (maglev) systems create suspension forcesby either electromagnetic [1] or electrodynamic [2, 3] means. Both suchmethods typically involve translationally moving a magnetic sourcerapidly over a conductive surface. However, this motion gives rise toa large drag force. Various (costly) methods are used to reduce thisdrag force. An alternative is to try to use this drag force to createpropulsion. This can be achieved by rotating a magnetic source ratherthan simply translationally moving it, as illustrated in Figure 1. Themagnetic thrust force creation is analogous to an automobile wheelusing frictional forces to create propulsion. The rotation of the magnetscan create both a propulsion force in addition to suspension force [4, 5].

Received 11 December 2012, Accepted 5 February 2013, Scheduled 12 February 2013* Corresponding author: Nirmal Paudel ([email protected]).

2 Paudel and Bird

Figure 1. Finite element analysis model ofa four pole-pair Halbach rotor rotating andtranslationally moving above an aluminumplate guideway. The field created by theHalbach rotor is shown as well as the inducedguideway currents within the aluminumguideway plate.

Figure 2. Field linescreated by a four pole-pair Halbach rotor.

In order to create a large lift force a flux-focusing Halbach rotor [68],as shown in Figure 2, can be used.

If the relative velocity of the rotor, as seen by the conductingplate, is greater than the translational velocity, propulsion forces result,whereas if the rotor is rotated slower than the translational velocitybraking forces are created [4, 5]. The use of a flat aluminum plateenables the normal forces to be used for suspension purposes. Theslip, sl, between the translational and rotational velocity is given by

sl(t) = m(t)ro vx(t) (1)where m is the mechanical angular velocity, ro the outer radius of therotor, and vx the translational velocity. The circumferential velocityvc of the rotor is defined as

vc(t) = m(t)ro (2)

The device shown in Figure 1 has been termed an electrodynamic wheel(EDW).

It is well known that electrodynamic suspension systems are highlyunderdamped [9]. Davis and Wilkie [10] analytically studied thedynamics of a long wire moving above a thin continuously uniformnonferromagnetic conducting plate while Baiko et al. [11] studiedthe dynamics using a rectangular coil. Both authors calculated thatpositive vertical damping is present at low-speeds while at high-speedthe vertical damping can become negative. In contrast, Yoshida andTakakura [12] and Urankar [13] calculated that vertical damping is

Progress In Electromagnetics Research B, Vol. 49, 2013 3

always positive when a conducting coil is translationally moved abovea conducting sheet. However, the damping values greatly reduce athigh speed. Davis and Wilkie [10] and Yoshida and Takakura [12]calculated the forces utilizing a thin-sheet approximation approachin which the current is assumed to be constant throughout the platethickness while Baiko et al. [11] and Urankar [13] accounted for currentvariation throughout the plate thickness.

Yamada et al. [14], Iwamoto et al. [15], Fujiwara [16], and Higashiet al. [17] calculated damping between translationally moving coils ona vehicle and stationary coils on a guideway. Iwamoto et al. used alumped parameter based analysis while Fujiwara used a field basedapproach. They all concluded that the vertical magnetic dampingbecomes negative at medium and high speed, while Ooi [18], Takanoand Ogiwara [19], Kratki and Oberretl [20], and He and Coffey [21]concluded the opposite. They used lumped parameter models to showthat vertical magnetic damping was always positive but it decreasedto very low values at high translational speeds.

The damping responses from experimental laboratory studieshave been equally contradictory [9]. For instance, Zhu et al. [22]and Yamada et al. [14] performed vibration experiments using arotating drum and a cantilevered magnet. They concluded thatnegative vertical damping occurred at high-speed, while Fujiwara [16]experimental results using superconducting magnets over guidewaycoils showed no negative damping at high speeds.

All researchers concluded that the inherent magnetic dampingwas insufficient and therefore active control of an electrodynamicmaglev system is essential [9]. In this paper the dynamic responsewhen a magnetic source has translational, vertical and rotationalmotion will be considered. The modeling characteristics when usingtwo different formulation techniques for simulating the dynamics inan electromechanical eddy current device that utilizes a continuouslinear finite thickness conducting plate will be presented. Exactelectrodynamic stiffness and damping equations are derived.

2. FIELD BASED MODELING

The applicable quasi-static Maxwells equations to model the problemshown in Figure 1 are [23, 24]

E = dBdt

(3)

B = oJ (4) B = 0 (5)

4 Paudel and Bird

J = E (6)B = A (7)E = dA

dtV (8)

where E = electric field intensity (Vm1), B = magnetic flux density(Wbm2), 0 = permeability of free space (Hm1), J = current density(Am2), A = magnetic vector potential (Wbm1), = conductivity(Sm1) and V = electric scalar potential (V). Using the Coulomb gauge

A = 0 (9)and assuming the conductive plate region is linear and simplyconnected then the governing transient eddy current field equationwithin a conductive plate can be described by [23]

2A = 0dAdt

(10)

In general the vector potential can be a function of both position andtime such that A(x(t), y(t), z(t), t). In this case the chain-rule can beapplied to the time derivative term in (10) such that [25]

dAdt

=At

+ (v )A (11)where v is a velocity vector. Substituting (11) into (10) gives

2A = 0At

+ 0(v )A (12)In this case the last term on the right side of (12) models the

fields spatial change due to the position of the magnetic source whilethe first term on the right models changes in source field itself withrespect to time.

Transient eddy current models are capable of accuratelysimulating the motion of complicated moving source fields. However,transient eddy-current simulations that utilize (10) or (12) can be verytime intensive when incorporated into a dynamic mechanical system.This is especially so when there is continuous motion of the source fieldand there is a need for constant feedback between the eddy currentforce and the mechanical motion. In contrast, steady-state field basedsolutions can be solved much more quickly since the time variation isassumed to be [26]

A(x, y, z, t) = A(x, y, z)ejet (13)

and therefore (12) reduces down to

2A = 0(jeA+ (v )A) (14)


Equation (14) can be calculated with relative ease when comparedwith the transient solution. The vast majority of steady-state modelsincorporate only one velocity term in the conducting region. Invariablythis velocity term is in the direction of motion [24, 26, 27]. In thispaper, the modeling accuracy will be considered when incorporatingboth a translational velocity, vx, and a heave velocity, vy, into anelectromechanical transient simulation [28]. This study will be limitedto a 2-D based analysis. However, the results and conclusions presentedin this paper can be extended to 3-D based problems. For 2-D problems(10) and (14) reduce down to [5, 29, 30]

2Az = 0Azt

(15)

2Az = 0(jeAz + vx

Azx

+ vyAzy

)(16)

A 2-D solution is approximately accurate as long as the plateis sufficiently wide and the plate overhang width is significantlygreat [24]. Two different dynamic electromechanical simulations willbe compared. The first model utilizes the coupled electromechanicalsystem summarized in Figure 3 in which an analytic based transienteddy current formulation is coupled to a transient mechanical system.The electromagnetic forces, Fx, Fy are calculated using a transientelectromagnetic model. In the second model, shown in Figure 4, asteady-state based eddy current model in which translational velocity,vx, heave velocity, vy, as well as rotational motion, e, is accounted foris coupled to the transient mechanical model. The transient changesin vertical and horizontal position of the source are accounted forby feeding back the position and velocity terms determined from themechanical model at each time step.

An analytic solution to (16) is derived in Section 3. This modelis an extension of the 2-D model presented in [26]. In addition,the transient model for a step change developed in [31] is extendedso as to be capable of accounting for continuous variations in thesource conditions as derived in Section 4. Both the steady-state

Figure 3. The formulation for transient model.

6 Paudel and Bird

Figure 4. The formulation for a steady-state model.

Figure 5. Illustration of the conductive (2) and non-conductive(1,3) regions and boundaries used by the analytic based model.The conductive plate has a thickness of b.

and transient eddy current models are incorporated into the samemechanical model so that the accuracy of the steady-state modelwhen coupled to a transient mechanical system can be assessed. Theaccuracy is compared by using the EDW source.

3. STEADY STATE MODEL

The 2-D problem region is illustrated in Figure 5. It consists of twonon-conducting regions 1, 3 and a conducting region 2. Theconducting plate is of finite thickness, b and it is assumed to havea length that is significantly longer than the source field. A magneticsource (not shown) is assumed to be located only in region 1.

3.1. Governing Equations and Boundary Conditions

The governing equation within the conducting region is given by (16).The source field, Bs, and the reflected field, Br, due to the inducedcurrent in the non-conducting region 1 and 3 is

B(x, y) = Bs(x, y) +Br(x, y) (17)The reflected field can be further written in terms of the scalarpotential, n, defined as

Br = 0n, in n (18)


where n = 1, 3 for region 1 and region 3. Taking the divergence of (17)and since Bs = 0 it can be noted that the non-conducting regioncan be modeled using

2nx2

+2ny2

= 0 in n. (19)

By utilizing the scalar potential in 1, 3 the source term only needs tobe accounted for on the conductive boundary [26, 32]. The boundaryconditions for the electromagnetic fields at the top boundary interfacebetween the non-conducting and conductive guideway regions, 12, are

nnc (B1 B2) = 0, on 12 (20)nnc (H1 H2) = 0, on 12 (21)

Since the permeability of the non-conducting and conducting regionsis the same, the boundary condition (21) can be written as

nnc (B1 B2) = 0, on 12 (22)The field in the non-conducting region 1 is composed of a source field,Bs and an eddy current reflected field Br defined as

B1 = Bs +Br (23)

The reflected field can be defined in terms of the scalar potential

Br = 01 (24)The field within the conductive region 2, the transmitted field, can beexpressed in terms of a vector potential defined by

B2 = Assz (25)Substituting (23) and (25) the boundary conditions on 12 can bewritten in terms of the vector and scalar field values such that [26, 33]

01x

+Bsx(x, b) =Asszy

, on 12 (26)

01y

Bsy(x, b) =Asszx

, on 12 (27)

Similarly, when there is no source in region 3 the boundary conditionson 12 are

03x

=Asszy

, on 23 (28)

03y

= Assz

x, on 23 (29)

where Bsx(x, b), Bsy(x, b) are the magnetic source terms. The source

field is centered at x = 0. On the outer non-conducting boundaries

8 Paudel and Bird

1 = 0 on 1 and 3 = 0 on 3. In this paper, the source is assumed tobe only located in 1 therefore it is only present in boundary condition(26) and (27). However, this method can be used in a situation whena source is presented on both the top and the bottom surfaces. Inthat case, boundary conditions (28) and (29) will have an extra sourceterms similar to (26) and (27).

3.2. Fourier Solution of Governing Equations

The governing Equations (16) and (19) must satisfy the boundaryconditions (26)(29) and outer boundary requirements. The solution ofthis problem has been obtained by using the spatial Fourier transformtechnique [34] in which the Fourier transform for the vector and scalarpotential regions with respect to the x-axis are

Assz (, y) =

Assz (x, y)ejxdx (30)

n(, y) =

n(x, y)ejxdx. (31)

By utilizing (30), (16) reduces to

2Assz (, y)y2

2Assz (, y)y

2Assz (, y) = 0, in 2 (32)

where

2 = 2 + 0s0 (33)so = j(eo + vxo) (34)

=vy0

2(35)

Solving (32) gives the general solution in 2 as

Assz (, y) =[M()ey +N()ey

]ey (36)

where2 = 2 + 2 (37)

and M() and N() are unknowns. The Fourier transform of (19) is

2n(, y)y2

= 2n(, y), in n (38)


where n = 1 and 3. Solving (38) and noting that when moving awayfrom the plate along the y-axis in 1 and 3 the field must reduce tozero one obtains the solutions

1(, y) = X1()ey, in 1 (39)3(, y) = X3()ey, in 3 (40)

where X1() and X3() are unknowns. The boundary conditions (26)(29) are Fourier transformed with respect to x and (36), (39) and (40)are substituted into the transformed boundary conditions. Solving forthe unknowns M() and N() enables the magnetic vector potentialto be derived as

Assz (, y) = Tss(, y)Bs(, b) (41)

where

T ss(, y) =

[[ ( + )]ey [ ( )]ey] e(yb)eb[2 ( + )2] eb[2 ( )2] (42)

can be interpreted as the transmission function for an arbitrary sourcefield, Bs(, b), imparted on the plate surface, 12. The Bs(, b) sourcefield is

Bs(, y) = Bsx(, y) + jBsy(, y) (43)

The reflected field can be determined by solving for X1(), from whichit is determined that [31, 33]

Bry(, y) = jBrx(, y) (44)

whereBry(, y) =

[Bty(, b)Bsy(, b)

]e(by) (45)

and the transmitted field Bty(, y) is

Bty(, y) = Assz (, y)

x= jT ss(, y)Bs(, b) (46)

The transmitted, reflected and source fields at the boundary, y = b,are therefore

Bty(, b) = Bry(, y) +B

sy(, b) (47)

Btx(, b) = Brx(, y) +B

sx(, b) (48)

3.3. Force Calculation

The forces are calculated by evaluating the stress tensor equation on12 (y = b). Due to Parsevals theorem the force integration can be

10 Paudel and Bird

evaluated in the Fourier domain thereby avoiding the need to firstobtain the inverse transform [29, 35]. The thrust and lift forces are

Fx =w

4pi0Re

Btx Btyd (49)

Fy =w

8pi0Re

(Bty B

ty Btx Btx

)d (50)

where star-superscript denotes complex conjugation. After substitut-ing (47), (48) into (49), (50) it can be shown that for an arbitrarysource the force equations are given by [31, 33]

F ss =w

8pi0

[2Az(, b)Bs(, b) |Bs(, b)|2]d, on 12 (51)

where the normal, Fy, and tangential force, Fx, on the rotor are

F ss = Fy + jFx (52)

Therefore

Fx = Im[F ss ] (53)Fy = Re[F ss ] (54)

Based on convention lift force is defined as a positive force and thereforethe negative signs in (53) and (54) ensures that both lift and thrustforce are the force acting on the rotor source. The force equationgiven by (51) can be further simplified. Substituting (41) into (51)and rearranging gives

F ss =w

8pi0

(, b)|Bs(, b)|2d, on 12 (55)

where

(, b) = 2T ss(, b) 1 (56)=

0[so y]22 + 0so + 2 coth(b)

(57)

4. TRANSIENT EDDY CURRENT MODEL FOR ANARBITRARILY CHANGING SOURCE

In [31, 33] the 2-D transient eddy current forces due to a step changein angular velocity and/or translational velocity of a source field above


a linear conductive plate were derived. The model was validated bycomparing it with finite element analysis (FEA) code. Unlike withthe steady-state model, the motional effects are accounted for bymoving the source field. The model is capable of predicting transientforce changes given an initial steady-state condition. In order to becoupled to a dynamic electromechanical model the transient eddy-current model must be capable of predicting forces for continuouschanges in operating inputs from non-steady state initial conditions. Inthis section the model presented in [31] is extended to model continuouschanges in the source input. The Fourier and Laplace transformedsolution for the vector potential in 2, for the case when Az(x, y) = 0at t = 0 is [31]

Atrz (, y, s) = Ttr(, y, s)Bs(, b, s) (58)

The superscript tr stands for transient. The transmission function forthe transient case is

T tr(, y, s) =(+ )ey + ( )eyeb(+ )2 eb( )2 (59)

and2 = 2 + 0s. (60)

4.1. Vector Potential Step and Impulse Response

If the source field is a unit-step

Bs(, b, t) =1s

(61)

then by using (58) and following the derivation method given in [31]the vector potential field at y = b is

Astepz (, b, t) = A1u(t) +9

n=0

(Ant e

snt t +Anc esnc t)

(62)

where u(t) denotes the unit step function. In (62) only the first 10roots of cot and tan are evaluated numerically and the root index isdenoted by the superscript n = 0, 1, . . . 9. The other variables in (62)are

A1 = T (, b, 0) = 1/(2) (63)

Ant = 8knt

ob32(2knt cot(2k

nt ) + b)

snt (knt + cot(knt )) (+ sec2(knt ))(64)

Anc =8knc

ob32(2knc cot(2k

nc ) + b)

snc (+ csc2(knc )) (knc + tan(knc ))(65)

12 Paudel and Bird

and the time constants are

snq = [(

2knqb

)2+ 2

]1o

(66)

where the subscript q = t or c denotes the root solutions for the tanand cot terms. The impulse response can be obtained from the stepresponse solution. Laplace transforming (62) and multiplying throughby s gives

Aimpz (, b, s) = A1 +9

n=0

(Ant s

s snt+

Anc s

s snc

)(67)

inverse Laplace transforming (67) one obtains

Aimpz (, b, t) = A1(t)+9

n=0

(Ant +Anc ) (t)+

9n=0

(Ant s

nt e

snt t +Anc snc e

snc t)

(68)where (t) is the unit impulse function.

The transient response due to an arbitrary source changeBs(, b, ) at any point in time can be obtained by utilizing theconvolution integral of the impulse response [36]

Az(, b, t) =

t0

Bs(, b, )Aimpz (, b, t )d (69)

substituting (68) into (69) gives

Atrz (, b, t) =

t0

Bs(, b, )

[(A1 +

9m=0

(Ant +Anc )

)(t )

+9

n=0

(Ant s

nt e

snt (t) +Anc snc e

snc (t))]

d. (70)

The x and y-component flux density transient response can be derivedfrom the derivative of (70) [33].

4.2. Force Calculations

The transient forces can be computed using [31, 33]

F tr(t) =w

8pi0

[2Az(, b, t)Bs(, b, t) |Bs(, b, t)|2]d, on 12

(71)


such that the normal and tangential force will be

F tr(t) = Fy + jFx (72)

The accuracy of this transient eddy current model was validated withan FEA model [33]. The validation is provided in the Appendix.

5. HALBACH ROTOR SOURCE FIELD

The field results derived in Sections 3 and 4 can be utilized with anysource term. Only the Fourier transformed field value on the surface ofthe conducting plate must be specified. In this analysis the source fieldis assumed to be a Halbach rotor that can simultaneously rotate andmove in the x, y plane above the conductive plate. The coordinatesystem for such a Halbach rotor is shown in Figure 6. The centerof the Halbach rotor is located at (xo, yo). The rotation frequencyis assumed to be at a single frequency, e(t), but its value howevercan change with time. The electrical and mechanical angular velocity,m(t), are related by e(t) = m(t)P where P is the number of pole-pairs. An airgap, g, between the rotor and conducting plate is alwaysassumed.

Figure 6. Coordinate system for Halbach rotor position above aconductive plate.

14 Paudel and Bird

5.1. Transient Source Equation

The 2-D Halbach rotor field in air was derived in [37]. It can beexpressed in vector potential form as

Asz(r, , t) =C

P

ejP

rPejPmt (73)

where

C =(2BrPP + 1

)(1 + r)r2Po (r

P+1o rP+1i )

(1 r)2r2Pi (1 + r)2r2Po(74)

Br = magnet remanence, ri = inner rotor radius and r = relativepermeability. The magnet eddy-current losses are neglected in theanalysis but as the Halbach magnets are highly segmented, these losseswill be relatively low [38]. The Halbach rotors coordinate axis islocated at (xo, yo) where

yo = b+ g + ro (75)

Using the complex analysis conversion [26, 33]

ejP

rP=

1(xo jyo)P (76)

and converting (73) to the conducting plate Cartesian coordinatereference frame gives

Asz(x, y, t) =CejPmt

P [(x xo) j(y yo)]P (77)

The translational source velocity term, vx can also be included into(77). After including this, the source field flux density is given by

Bsy(x, y, m, vx, t) = Azx

=C

[(xvxtxo)j(yyo)]P+1 ejPmt (78)

Bsx(x, y, t) = jBsy(x, y, t) (79)

Equations (78), (79) are utilized in the transient solution. The Fouriertransformed transient source solution at y = b is given by [26]

Bsx(, b, t) = (j)P2P !CpiP e(g+ro+jxo)ej(Pmvx)tu() (80)

Bsy(, b, t) = jBsx(, b, t) (81)and the source equation defined by (43) for a Halbach rotor is thengiven by

Bs(, b, t) = 2(j)P 2P !CpiP e(g+ro+jxo)ej(Pmvx)tu() (82)


5.2. Steady-state Source Equation

In the steady-state formulation the oscillating source frequency, e, aswell as both the translational and heave velocity terms are accountedfor within the conducting plate solution given by (41). Therefore, thesteady-state source equation used in (41) is [26, 33]

Bsy(x, y) = Azx

=C

[x xo j(y yo)]P+1 (83)Bsx(x, y) = jB

sy(x, y) (84)

Substituting (83) and (84) into (43) and taking the Fourier transformgives the steady-state source solution at y = b as

Bs(, b) =4piCP

P !e(ro+g+jxo)u(). (85)

Equation (85) was used in (41).

6. TWO DEGREE OF FREEDOM VEHICLESIMULATION

In order to understand the impact on the electromechanical dynamiccharacteristics when the eddy current forces are calculated using thesteady-state force model, a 2-degree of freedom electromechanicalmodel comparison has been made. An electromechanical model usingfour EDWs has been created in the Matlab-Simulink environment.Each EDW is connected to the vehicle through a drive shaft. Thetraction motors have not been modeled. The torque is directly appliedto the drive shafts. The basic configuration of the vehicle is shownin Figure 7 and the block diagram for the integration of the wheeland vehicle model is shown in Figure 8. The parameters used bythis model are given in Table 1. The selection between the transientand steady state eddy-current model is achieved by changing the

Figure 7. The maglev vehicle used for simulation where Bv, Hv andLv are the breadth, height and length of the vehicle.

16 Paudel and Bird

Figure 8. Block diagram showing the electrodynamic wheel and(mechanical) vehicle coupling. The steady state model includes vywhile the transient model includes the time.

position of the switch (in Figure 8). Only the variation in the vehicleheight, yo, in the y-axis, and translational position, xo along the x-axis is considered. The vehicles x and y-axis motion acts like anelectromechanical nonlinear spring-mass system [39]. The governingmechanical equations are

md2y(t)dt2

= Fy(t) Fg(t) (86)

md2x(t)dt2

= Fx(t) Fd(t) (87)where Fg = gravitational force and m = mass of vehicle and the rotormagnets. Fx(t) and Fy(t) are the thrust and lift force respectively. Theaerodynamic drag force, Fd(t), is given by [40]

Fd(t) = 0.5CdAvx(t)2 (88)

where, = density of air, Cd = aerodynamic drag coefficient, A =frontal area of the vehicle. No aerodynamic damping in y-directionis included. The values were chosen to match an experimental setup.The vehicle was started with initial conditions: 10ms1 translationalvelocity, 0ms1 heave velocity, airgap g = 10mm, m = 400 rads1.These initial conditions result in a positive slip sl = 10ms1. Theresponse when using these initial conditions with the steady-statemodel (given in Section 3) and transient model (Section 4) arecompared in Figure 9 and Figure 10. Both use the same mechanicalmodel. The large initial transient is due to the positive slip of thevehicle creating a thrust and lift force and consequently accelerationin the x and y direction at t = 0 s. One can note that the steady-statemodel tracks the transient response quite closely. This is because ofthe presence of the vy term in (16). The sudden change in lift force

Progress In Electromagnetics Research B, Vol. 49, 2013 17 W

eigh

t (N

)Th

rust

,F

(N

) Li

ft fo

rce,

F

(N

)Ai

rgap

, g

(mm)

H

eave

, v

(m

s )

Time (s)(a)

xy

y

W

eigh

t (N

)Th

rust

,F

(N

) Li

ft fo

rce,

F

(N

)Ai

rgap

, g

(mm)

H

eave

, v

(m

s )

Time (s)(b)

xy

y

1

1

Figure 9. Simulation response when a step change in maglev vehicleweight of 50N occurs at t = 5 s. The resulting response when (a) thetransient eddy current model is used and (b) the steady-state eddycurrent model is used is shown. The thrust, lift, air-gap variation andheave velocity have been plotted.

Tr

ansl

atio

nal

velo

city

, v (m

s )

x1

Time (s)

Dra

g (N

)

Figure 10. Translational velocity, vx and aerodynamic drag force fora step change in weight.

Ai

rgap

Erro

r (%)

Time (s)

Figure 11. Percentage error between the steady state and transienteddy current electromechanical system for the airgap as a function oftime.

18 Paudel and Bird

Table 1. Dynamic simulation parameters.

Description Value Units

Vehicle

Length of the vehicle, LV 40 cmBreadth of the vehicle, BV 20 cmHeight of the vehicle, HV 10 cmThickness of vehicle, TV 4 cm

Frontal area of the vehicle, A 0.0476 m2

Density of iron, Fe 7.93 g/cm3

Length of drive shaft, LDS 4 cmRadius of drive shaft, RDS 1 cmTotal mass of vehicle, m 21.38 kg

Aerodynamic drag coefficient, Cd 0.25 kgs1

Halbachrotor

Outer radius, ro 50 mmInner radius, ri 34.20 mm

Width, w 50 mmMagnet (NdFeB), Br 1.42 T

Magnet relative permeability, r 1.08 -Pole-pairs, P 4 -

Conductingplate

Conductivity (Al), 2.459 107 Sm1Single sheet width, 50 mm

Thickness, b 10 mmAir-gap between rotor and plate, g 10 mm

creates a mechanical acceleration and consequently this is capturedby the vy term, without the feedback created by vy the steady-statemodel cannot account for the dynamic variation in the airgap, g.The resulting error in airgap estimation is relatively small, as shownin Figure 11. Electromagnetic damping is clearly present. The liftand thrust forces are highly coupled. The translational velocity issmoothly increasing because the conducting plate is assumed to beinfinitely uniform in the x-axis. At time t = 5 s a step change inmass occurs and this results in a second transient phase; again theelectromechanical system with steady-state forces closely tracks thetransient eddy-current electromechanical model.

A comparison for a step change in angular velocity is shownin Figure 12 and Figure 13. The model starts in a steady-statecondition and then a step change in m from 400 to 600 rads1


occurs at t = 1 s. This results in an increased slip and consequentlyan increase in translational velocity. As the velocity is greater thenew steady-state airgap value increases. The steady-state coupledelectromechanical model again closely tracks the eddy-current basedtransient electromechanical model.

Time (s)

v (m

s )

x1

v (m

s )

y1

Airg

ap (m

m)Li

ft (N

)Th

rust

(N)

(m

s )

m1

Aer

odyn

amic

D

rag

(N)

Figure 12. Electromechanical simulation results for a step change inangular velocity when using the stead-state and transient eddy currentmodel.

Ai

rgap

Erro

r (%)

Time (s)

Figure 13. Percentage error between the steady state and transienteddy current electromechanical system for the airgap as a function oftime.

20 Paudel and Bird

7. STIFFNESS AND DAMPING ANALYSIS

The two degree freedom vehicle simulation results indicate thatthe eddy current damping and stiffness characteristics of theelectromechanical system can be relatively accurately predicted byusing (16). Therefore, as the steady-state equations are significantlysimpler to understand and greatly faster to compute they have beenused to study the stiffness and damping characteristics for this system.The magnetic damping coefficients are dependent on the transmissionfunctions. The damping coefficient is defined as the negative derivativeof force with respect to the velocity [41]. Differentiating (55) withrespect to velocities in the x and y directions, one obtains[

Dxx DxyDyx Dyy

]=

[ dFxdvx dFxdvydFydvx

dFydvy

]=

Im[dF ss

dvx

]Im[dF ss

dvy

]Re[dF ss

dvx

]Re[dF ss

dvy

] (89)

The velocities, vx and vy in (89), are the velocities of the rotor and thedamping forces are computed on the vehicle rather than the conductiveplate. The derivative terms in (89) can be further written as

dF ss

dvx=

w

8pi0

(, b)vx

|Bs(, b)|2d (90)dF ss

dvy=

w

8pi0

(, b)vy

|Bs(, b)|2d (91)

The derivatives of the transmission function in (90) and (91) areevaluated analytically and are determined to be

(, b)vx

= j2(2 + 2( ) + 2) coth(b)+2( ) + b(2 + 2 2)csch(b)2

[2 coth(b) + 2 + 2]2(92)

(, b)vy

=

(2(2 ) + 2) coth(b) + (2 + 2)+b(2 + 2 2)csch(b)2

[2 coth(b) + 2 + 2]2. (93)

The stiffness coefficient is defined as the negative derivative of forceswith respect to the displacement [41]. The stiffness matrix for 2Dmodel can be obtained by taking the derivative of force with respectto the x and y-axis displacements as given by[

kxx kxykyx kyy

]=

[ dFxdxo dFxdyodFydxo

dFydyo

](94)


Using (52), (94) can be written as[kxx kxykyx kyy

]=

Im[dF ss

dxo

]Im[dF ss

dyo

]Re[dF ss

dxo

]Re[dF ss

dyo

] (95)

Assuming that the complex force function (52) has a derivative over allspace then it can be said that (52) is analytic and therefore the Cauchy-Riemann equation is applicable. The Cauchy-Riemann equation statesthat [42]:

Fyxo

=Fxdyo

(96)

Fyyo

= Fxdxo

(97)

Equation (97) is the 2-D form of Earnshaws theorem [4345].Substituting (96) and (97) into (95), the stiffness matrix becomes[

kxx kxykyx kyy

]=

[ Re[dF ssdyo ] Im[dF ssdyo ]Im[dF

ss

dyo] Re[dF

ss

dyo]

](98)

When the source field is given by (85) it can be noted that the airgap,g, will be changing therefore

dF ss

dyo=F ss

g

g

yo=F ss

g(99)

Substituting (85) into (55) and evaluating (99) gives

dF ss

dyo=

w

4pi0

(, b) |Bs(, b)|2 d, (100)

and from (96), (97) and (100) one obtains

dF ss

dxo=

jw

4pi0

(, b) |Bs(, b)|2 d, (101)

Using the parameters given in Table 1 the lift and drag force onthe EDW as a function of translational velocity for the case whene = 0 and vy = 0 is shown in Figure 14. The stiffness coefficientsobtained using (95) is shown in Figure 15. The stiffness coefficientskyy is positive for increase in translational velocity. This stiffnesscoefficient is acting similar to the mechanical spring; when the rotor ispushed close to the conductive plate, it will be pushed back because

22 Paudel and BirdD

rag

forc

e, F

(N

)y

Lift

forc

e, F

(N)

y

Lift force

Drag force

Translational velocity, v (m/s)x

Figure 14. Electrodynamic lift,Fy, and drag force, Fx, as afunction of translational velocityat (e, vy) = (0, 0). As e = 0 thetangential force is a drag force.


k , |k |xxyy

k , k yxxy

k

an

d k

(N

/mm)

yxxy

k

an

d |k

| (N

/mm)

xxyy

Figure 15. The electrodynamicstiffness coefficients as a functionof translational velocity at g =10mm and e = 0 rads1, vy =0ms1.

Lift force

Drag force

Dra

g fo

rce,

F

(N)

x

Lift

forc

e, F

(N

)y

Vertical velocity, v (m/s)y

Figure 16. Lift, Fy and dragforce, Fx versus EDW heave ve-locity, for (e, vx) = (0, 10ms1).


D

(Ns/m

)yx

D

(Ns/m

)xx

Dyx

Dxx

Figure 17. The electrodynamicdamping terms, Dxx and Dyx .

of a positive stiffness, a necessary condition for stability. The stabilityexists in the direction of positive stiffness if the reaction force acts tooppose perturbatory displacements [46]. The negative stiffness, kxy, isa consequence of the drag force decreasing with height.

Figure 16 shows that for small changes in the EDW heavevelocity the variation in the lift and drag force is linear. Figure 17and Figure 18 show the EDW vertical and horizontal dampingcharacteristics calculating using (89). The damping coefficient Dxxis positive at velocities below the peak of the drag force and thenbecomes negative with further increase in translational velocity. Interms of energy, the positive damping means taking away energy fromthe system whereas negative damping refers to adding energy to thesystem [47]. The damping coefficient Dyx is always negative and peaksat a low translational speed. Since the lift force increases with increasein vx (see Figure 14), energy is being added to the system, hence, thedamping coefficient Dyx is negative.


Both drag and lift force are decreasing with an increase inthe EDW heave velocity (see Figure 16), the energy is being takenaway from the system. Therefore, the damping coefficients Dxyand Dyy are both positive. The vertical damping coefficient, Dyydecreases and becomes almost zero with an increase in translationalvelocity. However, the damping coefficient Dxy reaches a maximumvalue at peak drag force and then decreases with further increases intranslational speed. These damping characteristics shown in Figure 18agree with the calculations performed by Yoshida and Takakura [12],Urankar [13] in which no negative vertical dampingDyy was calculated.

For the case when e 6= 0 a slip will be present as defined by (1).Depending on the slip value the tangential force can be either a thrustor a drag force as shown in Figure 19. The lift and tangential forceas function of slip and translational speed is shown in Figure 20while Figure 21 shows the stiffness contour plots. The same stiffnessrelationships given by (95), (96) and (97) apply when e 6= 0. Thecross coupled stiffness terms kxy, kyx become positive for positiveslip values and therefore this should improve stability. The damping


D

(Ns/m

)yy

D

(Ns/m

)xy

Dxy

Dyy

Figure 18. The electrodynamicdamping terms, Dxy and Dyy.

Dra

g/Th

rust

For

ce, F

(N)

x

Lift

Forc

e,F

(N)

y

Slip, s (m/s)l

Thrust/Drag

Lift

Figure 19. Thrust (tangential)force and lift (normal) force asa function of slip when vx =20ms1.

(a)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

-50

0

50

(N)

(b)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

50

100

150

(N)

Figure 20. (a) Thrust force and (b) lift force as a function of slip andtranslational velocity at g = 10mm and vy = 0m/s.

24 Paudel and Bird

(a)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

-10

5

10

(N/mm)

(b)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

5

15

25

(N/mm)

-5

0

20

10

Figure 21. (a) The stiffness coefficients kxy and kyx, (b) the stiffnesscoefficients kyy and |kxx| as a function of slip and translational velocityat g = 10mm and vy = 0m/s.

(a)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

15

20

(Ns/m)

(b)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

5

15

(Ns/m)

5

10

20

10

(c)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

-10

5

10

(Ns/m)

(d)

Tr

ansla

tiona

l vel

oci

ty (m

s

1 )

Slip (ms 1) -20 -10 0 10 20 30 40 500

5

10

15

20

25

30

-10

0

10

(Ns/m)

-5

0

5

-5

0

Figure 22. (a) Damping coefficient Dyy, (b) damping coefficient Dxx,(c) damping coefficient Dxy and (d) damping coefficient Dyx all as afunction of slip and translational velocity at g = 10mm and vy = 0m/s.

relationships are shown in Figure 22. It can be noted that the verticaldamping, Dyy, is always positive but decreases with slip value. Unlikein Figure 17, the horizontal damping coefficient, Dxx, becomes positivewhen both the translational and rotational speed are included. Themagnitude of Dxx however decreases with increase in slip value asshown in Figure 22(b). The off-diagonal damping term Dyx is positivefor positive slip values. Whereas the other off-diagonal damping termDyx is negative when operating with thrust and therefore this term is


Figure 23. Lift, thrust and airgap plot comparison between the FEAtransient model and analytical transient model when integrated withthe Matlab-Simmechanics vehicle model.

likely to create instabilities. The decrease of the magnetic dampingvalues at high slip values suggests that the inherent magnetic dampingis insufficient and therefore active control of an electrodynamic maglevsystem is essential.

8. CONCLUSIONS

A 2-D analytic based steady-state eddy current model thatincorporates heave and translational velocity as well as rotationalmotion has been derived. In addition, a dynamic eddy-current model

26 Paudel and Bird

capable of reacting to continuous changes in input conditions has alsobeen presented. The steady-state and transient eddy-current modelswere both incorporated into an electromechanical system in order toassess the calculation accuracy of the steady-state model when heavevelocity is included. An electrodynamic wheel (Halbach rotor) wasused as the source field. The simulation results indicate that theinclusion of the heave velocity, vy, into a steady-state model creates ameans for feedback in the electromechanical system thereby enablingthe steady-state based force calculations to quite accurately track thedynamic behavior. The electromechanical simulation time is greatlyreduced when the eddy current forces are computed from steady-stateequations. Using the concept of reflected and transmitted fields thetangential and normal force equations were derived in a simplifiedform this enabled the exact damping and stiffness equations to beanalytically derived.

9. ACKNOWLEDGMENT

This material is based upon work supported by the U.S. NationalScience Foundation under Grant No. 0925941.

APPENDIX A.

The analytic based electromechanical transient eddy current modelwith continuous time varying capabilities presented in Section 4 andSection 5 was validated by comparing it with a transient 2D FEAmodel that was also integrated into the electromechanical systemdescribed in Section 5 [33]. The transient FEA model utilized afictitious current sheet [5] approach to model the Halbach rotor. TheFEA model was developed in COMSOL v3.5 and integrated intothe Matlab SimMechanicsTM model environment utilizing Matlab s-functions. The comparison is made by using the parameters givenin Table 1 except that an equivalent time-varying current sheet valueJz = 1.1814106Am1 was used to model the source [5]. The vehiclewas started with initial conditions: airgap go = 10mm, translationalvelocity, vxo = 10ms1 and angular velocity, mo = 400 rads1. Theseinitial conditions result in a positive slip sl = 10ms1. The comparisonbetween the lift force, thrust force and the air-gap are illustratedin Figure 23. An excellent agreement between the FEA model andthe analytic based transient model for a continuously changing inputcondition was obtained. The integrated simulation approach developedbetween the transient FEA model and SimMechanicsTM vehicle modelwas extremely time intensive. For instance, to obtain the result shown


in Figure 23 took approximately 2 weeks. However, the computationaltime using the analytic based transient eddy current model could becompleted within a few minutes.

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structural optimization

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