AD-A259 178 ANL-92/21

Materials and Components Dynamic Stability ofTechnology Division

Materials and Components Maglev SystemsTechnology Division

Materials and ComponentsTechnology Division by Y. Cai, S. S. Chen,

T. M. Mulcahy, and D. M. Rote

DTICS ELECTF

m

JN 5 1993

SArgonne National Laboratory, Argonne, Illinois 60439

operated by The University of Chicagoi for the United States Department of Energy under Contract W-31-109-Eng-38

92-33092

92 12 29 064

Argonne National Laboratory, with facilities in the states of Illinois and Idaho, isowned by the United States government, and operated by The University of Chicagounder the provisions of a contract with the Department of Energy.

DISCLAIMERThis report was prepared as an account of work sponsored by an agency ofthe United States Government. Neither the United States Government norany agency thereof, nor any of their employees, makes any warranty, expressor implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or pro-cess disclosed, or represents that its use would not infringe privately ownedrights. Reference herein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, or otherwise, does notnecessarily constitute or imply its endorsement, recommendation, orfavoring by the United States Government or any agency thereof. The viewsand opinions of authors expressed herein do not necessarily state or reflectthose of the United States Government or any agency thereof.

Reproduced from the best available copy.

Available to DOE and DOE contractors from theOffice of Scientific and Technical Information

P.O. Box 62Oak Ridge, TN 37831

Prices available from (615) 576-8401

Available to the public from theNational Technical Information Service

U.S. Department of Commerce5285 Port Royal Road

Springfield, VA 22161

ARGONNE NATIONAL LABORATORY ANL-92/219700 South Cass Avenue, Argonne, Illinois 60439

DistributionCategory:

All Transportation SystemsReports (UC-330)

Dynamic Stability of Maglev Systems

Y. Cai, S. S. Chen, and T. M. Mulcahy

Materials and Components Technology Division

D. M. Rote

Center for Transportation Research

April 1992

Work supported by Aeasm ,-

U.S. DEPARTMENT OF ENERGY i0 W,-,

Office of Transportation Technologies Just 0

z'Dist r p1 0

I 'M q n IL J special

Contents

Abstract ....................................................... 1

1 Introduction .................................................. 1

1.1 Theoretical Studies ......................................... 21.2 Experimental Studies ...................................... 21.3 Experimental/Analytical Studies .................................................... 2

2 Motion-Dependent Magnetic Forces ....................................................... 3

2.1 Motion-Dependent Magnetic-Force Coefficients ................................ 32.2 Experimental Methods to Measure Motion-Dependent

Magnetic-Force Coefficients ........................................................... 52.3 Quasistatic Motion-Dependent Magnetic-Force Coefficients of

Maglev System with L-Shaped Guideway ......................................... 6

3 Stability of M aglev System s .................................................................. 12

4 Simplified Vehicle Models for Dynamic Instability ................................. 18

4.1 Two-Degree-of-Freedom Vehicle ...................................................... 194.2 Three-Degree-of-Freedom Vehicle ................................................ :.. 224.3 Six-Degree-of-Freedom Vehicle ..................................................... 264.4 Vehicle on Double L-Shaped Aluminum Sheet Guideway ................. 26

5 C losing R em arks ............................................................................... 36

A cknow ledgm ents ................................................................................. 39

R eferen ces ............................................................................................ 40

iii

Figures

1 Displacement components of a maglev system ................................. 4

2 Experimental apparatus for magnetic force measurement ................ 7

3 Schematic diagram of the apparatus used to measure magneticforces on an L-shaped aluminum sheet guideway ............................ 7

4 Measured lift and guidance magnetic forces .................................... 8

5 Measured lift and guidance magnetic stiffness ............................... 13

6 Magnetic forces divided by image force .......................................... 20

7 Natural frequency as a function of levitation height ........................ 22

8 Three-degree-of-freedom vehicle ..................................................... 23

9 Maglev system with a vehicle operating on double L-shapedaluminum sheet guideway ......................................................... 27

10 Displacement components of three-degree-of-freedom vehicle ............ 2D

11 Maglev-system eigenvalues vs. vehicle levitation height, withY* = 12.7 m m ............................................................................. 33

12 Modal shapes of three-degree-of-freedom maglev system withY* fi 12.7 m m ............................................................................. 34

13 Maglev-system eigenvalues vs. vehicle levitation height, withY * = 5 M m ................................................................................. 35

14 Maglev-system eigenvalues vs. lateral location of vehicle, withh = 12.7 mm and go = 25 mm ....................................................... 37

15 Maglev-system eigenvalues vs. lateral location of vehicle, withh = 7 mm and go = 25 mm ........................................................... 38

16 Real part of maglev-system eigenvalues vs. lateral location ofvehicle, with h = 7 mm and go = 10, 15, 20, and 25 mm ........... 39

iv

Table

Eigenvectors of vehicle motion ........................................................ 32

V

Dynamic Stability of Maglev Systems

by

Y. Cai, S. S. Chen, T. M. Mulcahy, and D. M. Rote

Abstract

Because dynamic instability is not acceptable for any commercial maglevsystems, it is important to consider this phenomenon in the development of allmaglev systems. This study considers the stability of maglev systems based onexperimental data, scoping calculations, and simple mathematical models.Divergence and flutter are obtained for coupled vibration of a three-degree-of-freedom maglev vehicle on a guideway consisting of double L-shaped aluminumsegments attached to a rotating wheel. The theory and analysis developed in thisstudy identifies basic stability characteristics and future research needs of maglevsystems.

1 Introduction

The dynamic response of maglev systems is important in several respects:safety and ride quality, guideway design, and system costs. Ride quality isdetermined by vehicle response and by environmental factors such as humidityand noise. The dynamic response of vehicles is the key element in thedetermination of ride quality, and vehicle stability is one of the importantelements relative to safety. To design a proper guideway that provides acceptableride quality in the stable region, the vehicle dynamics must be understood.Furthermore, the trade-off between guideway smoothness and the levitation andcontrol systems must be considered if maglev systems are to be economicallyfeasible. The link between the guideway and other maglev components is vehicledynamics. For a commercial maglev system, vehicle dynamics must be analyzedand tested in detail.

For safety, maglev systems should be stable. Magnetic forces are basicallyposition-dependent, although some are also velocity-dependent. These motion-dependent magnetic forces can induce various types of instability. In addition,the periodic structure of the motion-dependent magnetic forces may in some casesalso induce parametric and combination resonances.

2

Some analytical and experimental studies have been performed tounderstand the stability characteristics of maglev systems. Several examples aresummarized briefly as follows:

1.1 Theoretical Studies

"* Davis and Wilkie (1971) studied a magnetic coil moving over aconducting track and concluded that negative damping occurs atvelocities greater than the characteristic velocity based on thin-tracktheory.

"* Ohno et al. (1973) studied the pulsating lift forces in a linearsynchronous motor. These forces may cause parametric andcombination resonances, as well as heave and pitch oscillations.

"* Baiko et al. (cited in Chu and Moon 1983) considered the interactionsof induced eddy currents with on-board superconducting magnetsand found possible heave instabilities.

1.2 Experimental Studies

"* An experimental vehicle floating above a large rotating wheel wasfound by Moon (1974) to have sway-yaw instabilities.

"* Experiments performed at MIT on a test track showed pitch-heaveinstability (Moon 1975).

1.3 ExperimentallAnalytical Studies

" A conducting guideway, consisting of L-shaped aluminum segmentsattached to a rotating wheel to simulate the Japanese full-scaleguideway at Miyazaki, was studied experimentally and analyticallyby Chu and Moon (1983). Divergence and flutter were obtained forcoupled yaw-lateral vibration; the divergence leads to two stableequilibrium yaw positions, and the flutter instability leads to a limitcycle of coupled yaw and lateral motions in the neighborhood of thedrag peak.

"* Variation of the magnetic lift force due to variation of the levitatedheight corresponding to the sinusoidal guideway roughness wasstudied by Yabuno et al. (1989). Parametric resonance of heaving andpitching motions is possible.

3

Based on these published analytical results and experimental data, it isobvious that different types of dynamic instabilities can occur in maglev systems.Because dynamic instability is not acceptable for any commercial maglevsystems, it is important to consider this phenomenon in the development of allmaglev systems.

This study considers the stability of maglev systems and is based onexperimental data, scoping calculations, and simple mathematical models. Theobjective is to provide some basic stability characteristics and to identify futureresearch needs.

2 Motion-Dependent Magnetic Forces

2.1 Motion-Dependent Magnetic-Force Coefficients

Magnetic forces are needed for any vehicle dynamics analysis, guidewaystructural design, fastening design, and prediction of ride quality. These forcecomponents are considered from the standpoint of vehicle stability.

As an example, consider a vehicle with six degrees of freedom, threetranslations, ux, uy, uz, and three rotations, oa), coy, coz, as shown in Fig. 1. Let Ube the vector consisting of the six motion components, i.e.,

u2 Uy

U u3= Uz (1)u4 Ox

u 5 COY

u 6 ) (OZ

Velocity and acceleration are given by

Sau

and (2)

a2U

The motion-dependent magnetic forces can be written

4

U

z

uy

1Y

Fig. 1. Displacement components of amaglev system

6fi -- (miiij + Cjjj + kiuj), (3)

j= 1

Iwhere mij, cij, and kij are magnetic mass, damping, and stiffness coefficients.These coefficients can be obtained analytically, numerically, or experimentallyand are functions of the system parameters.

" Analytical Studies: Analyses for simple cases can be performed todetermine the characteristics of the coefficients. For example, ananalytical method may be used to identify the coefficients that can beneglected under specific conditions.

"* Numerical Methods: For the general case with complicatedgeometries, analytical methods may not be appropriate andnumerical methods will be more useful. Numerical methods (finite-

5

element method and boundary-element method) can be used tocalculate the values of all coefficients under specific conditions.

Experimental Techniques: Measurements of magnetic forces willgive the information required to calculate magnetic-force coefficients.

2.2 Experimental Methods to Measure Motion-DependentMagnetic-Force Coefficients

Quasistatic Motion Theory. The magnetic forces acting on an oscillatingvehicle are equal, at any instant in time, to those of the same vehicle moving witha constant velocity and with specific clearances equal to the actual instantaneousvalues. The magnetic lorces depend on the deviation from a reference state ofspeed and clearance, i.e., the motion-dependent magnetic forces depend only onup, but not uj and Up so that

6fi= bkjuj. (4)

In this case, the magnetic forces are determnned uniquely by vehicle position. Allelements of magnetic stiffness kij can be obtained. To determine kij, themagnetic-force component fi is measured as a function of uj. Stiffness is given by

kb = _j. (5)

In general, kij is a function of U.

Unsteady-Motion Theory. The magnetic forces acting on an oscillatingvehicle will depend on U, U, and UJ. The magnetic force based on the unsteady-motion theory can be obtained by measuring the magnetic force acting on thevehicle oscillating in the magnetic field. For example, if the displacementcomponent uj is excited, its displacement is given by

Uj = Uj exp(V 1 cot). (6)

The motion-dependent magnetic force of the component fi acting on the vehicle isgiven by

6

= [ab cos(vi) + N/i ab sin(Wb)IUj exp(4 :ii ct), (7)

where ai3 is the magnetic force amplitude and Vij is the phase angle between themagnetic force and the vehicle displacement uj. These values are measuredexperimentally.

Using Eqs. 3 and 6, we can also write the motion-dependent magnetic forcecomponent as

fi= k]Lbo+ -l o cj3 + kb )di exp(1 7 i (04) 8

A comparison of Eqs. 7 and 8 yields

ci = aij sin(bij) / co,

(9)mij =[1ki - aij cos(Wij)] / o02.

Based on Eqs. 5 and 9, all motion-dependent magnetic-force matrices can bedetermined from two experiments: quasistatic motion and unsteady motion.

If mij and ci are of no concern, the experiment using quasisteady motion issufficient to determine kij.

2.3 Quasistatic Motion-Dependent Magnetic-Force Coefficients ofMaglev System with L-Shaped Guideway

An experiment m ,s conducted recently at Argonne National Laboratory toinvestigate the lift, drag, and guidance magnetic forces on an NdFeB permanentmagnet moving over an -luminum (6061-T6) L-shaped ring mounted on the topsurface of a 1.2-m diameter rotating wheel (shown in Fig. 2). For a given rotatingspeed of the wheel, the lift and guidance magnetic forces were measured as theguidance gap Y* and lift height h were varied. A schematic diagram of themeasurement approach is shown in Fig. 3. Figure 4 shows those measuredforces as a function of h, with Y* fixed (Y* = 5 mm and 12.7 mm), or as a functionof Y* with h fixed (h = 7 mm and 12.7 mm) when the surface velocity of the lateralleg of the guideway is 36.1 m/s, the highest velocity tested.

During testing, the long side of the 25.4 x 50.8 x 6.35 mm rectangular magnetwas oriented parallel to the direction of motion of the L-shaped guideway and was

7

Fig. 2. Experimental apparatus for magnetic force measurement

.2

-J• Guidance direction

Y* 25.4 mm 25.4 x 50.8 x 6.35 mm

Magnet

....................... .. . . .

78.2 mm

Fig. 3. Schematic diagram of the apparatus used to measuremagnetic forces on an L-shaped aluminum sheetguideway

8

1200 a

Y* 5 mm

1000

Cl 800 Y-ML0MP + ... ~ MS~xa + M9-

S\ I MO 2485.2072086Ml -370.73754751

0o 600 M2 25.591416346

M3 -0.93776713995SM4 0.01o7442074721

400 M51 -0.00012885295531

R 0.99997403029

200

10 20 30 40 50

HEIGHT, mm(a)

1200 ... . . .

Y* = 12.7 mm

1000

S 800So Y-MO + Mlx +...Mex 8 + M9x9

0MO 2623.4089363""c Ml -410.50344932LA 60 M2 30.527419731

M31 -1.2607254486

M41 0.0306501 7351400 M5 -0.00038829515856

M6 2.0139656841e-06200 R 0.99998792531

0 1 . . , . . . , . . , . . . I .. . . . .

0 10 20 30 40 50 60

HEIGHT, mm(b)

Fig. 4. Measured lift and guidance magnetic forces

9

1000 . . . .

h=7mm

800

u 600 Y- __+MIx +.,._M8"x8_+_ Mx9

cc' MO 973.046717850 Ml -52.650400005

0M2 9.0560858017

M3 -0.72869010929

M4 0.031159677776

M5 -0.00071232713434

200 M6 7.9983147733e-06M7 -3.42286166266-08

0.99995749344

0 .. . ... .. l . " I . ". . .... .I . . " . I . ..

0 10 20 30 40 50 60

GAP, mm(c)500 . . . . p ' . . . . . . . p . . . " . . . . . .

h 12.7 mm

400

LLF 300

0 Y-M0+MlIx +...M Ox ++M Ox9UL. 200 MO 379.25275759_ 200 Ml 7.2100505188

-I M2 -1.0256181846

M3 0.059211127824

100 M4 -0.0014649957218M5 1.18140861280-05

R 0.99997139853

0 . . . " .. I - " " _" I . . . .. . I .. ...

0 10 20 30 40 50 60

GAP, mm(d)

Fig. 4. Cont'd

10

150 • • • . . . I . . . . . I . .

Y*= 12.7 mm

125

•m100

12

UIn 7 5 Y. M0O+MIx + ... MS"x8 ÷M'xg

Ot

C.)0

Z MO 25.974283328

SMl 16.871081909

5 50 M2 -1.2259988206

. M3 0.045480108193

M4 -0.00099175122258

2 5 M5 1.1891440675e-05

Me -5.9728308401.-08

R 0.99990427993

0 1 . . .I .. .. .I . . . I . . I . ... I . . .

O 10 20 30 40 50 60

HEIGHT, mm(e)

400 . ... ,

m 35Y* =5 Min

350

ui 30000 Y MO + M°x +..." M8*x8 + M9-x9

wn 250 MO 271.59560866U Ml 22.783510041

R M2 -2.1892659951

5 200 M3 0.10217075249

0 M4 -0.0027202422056

Ms 3.95738906849-05

150 Me -2.4621859254e-07

R 0.99991037086.

100 .

0 10 20 30 40 50 60

HEIGHT, mm(f)

Fig. 4. Cont'd

505

400 MO 860.15112485Ml -152.231601522M2 13.108312509

LI M3 -0.650461 26263

0MS -0.00032045122602

wU MO 2.74293078239-060 M7 -9.1 542572530.-0gz___4 200 R 0.999979617314

0

100

0 . 1 0 . . . .2 10 . . . .3 0 4 10 5 10 6 '0

GAP, mm

600 - * * 3

V.'MO+Ml~x +....M8*x8 +M9-x

50MO 809.86578864550Ml -125.40640562S

CD M2 7.3649603558urM3 -0.0975526064122

400 R M41 -0.008645585270210 MS 0.0004349S888168

m6 -7.5891941276e-06S300 M7 4.6309597314e-08

Z Rl 0.9999061183S

S200

100

h 7 mm

0 10 20 30 40 50 60

GAP, mm(h)

Fig. 4. Cont'd

12

held stationary by a two-component force transducer that comprised two BLHC2G1 load cells connected in series to measure the lift and guidance forcessimultaneously. Laboratory weights were used to calibrate the transducer and toassess crosstalk (which was found to be <2%). The base of the load cell assemblywas mounted on motorized stages that provided accurate positioning (±0.05 mm).Out-of-roundness of the L-shaped guideway ring varied, but was always less than±0.15 mm for the lateral leg and ±0.35 mm for the vertical leg. Ability to exactlyposition the magnet with respect to the guideway dominated our experimentalerror, estimated at ±5%.

The qualitative trends in the lift and guidance force data taken at lowervelocities for the L-shaped guideway are the same as shown in Fig. 4 for thehighest velocity tested. Two interesting features are evident. First, a maximumoccurs in the guidance force variation with respect to height variations at a fixedgap (as shown in Figs. 4e and 4f) that is caused by the corner region of theguideway. Second, a minimum occurs in the guidance force variation withrespect to gap variations at a fixed height (as shown in Figs. 4g and 4h) that iscaused by the edge of the lateral leg of the guideway. As will be shown in Section4.4, the first feature is associated with a flutter and the second with a divergenceinstability.

Based on the magnetic force data shown in Fig. 4, we can calculate thequasistatic motion-dependent magnetic-force coefficients with Eq. 5. All elementsof magnetic stiffness ke*, kgg, kgj, and kgg, were calculated and are shown inFig. 5 with various Y* and h.

The curve fits to both magnetic forces and stiffnesses were derived withpolynomial expressions (results are given in Figs. 4 and 5) and input into acomputer code to simulate coupled vibrations of the maglev vehicle.

3 Stability of Maglev SystemsWithout motion-dependent magnetic forces, the equation of motion for the

vehicle can be written as

[Mv]{U} + [CvgJU} + [Kv]{U} = {Q}, (10)

where Mv is the vehicle mass matrix, Cv is the vehicle damping matrix, Kv is thevehicle stiffness matrix, and Q is the generalized excitation force.

The motion-dependent magnetic forces are given in Eq. 3. With motion-dependent magnetic forces, Eq. 10 becomes

13

50 ... . .

SY* =5 mm

0

> -50_________ __

,Y-MO+Mlx +... M8"x8 +M9.x 9

"°l M-398.29603506

* -1o00 Ml 66.087837168UM2 -4.9034465682!S3 0.19558208619I

IiL M41 -0.004055206234 IS-1 5 0 M5 3.4066913034e-05 I

Z R 0.99938479873 I

-200 ....

0 10 20 30 40 50 60

HEIGHT, mm(a)

50 . 5 . 3 ... .. . • .. .. I .

SY*= 12.7 mm0

• -50> YMO+Ml*x +...MSx +M9gx9

W MO -427.85547152Ml 73.062098255u, -100

LU M2 -5.6154704794cc M3 0.2410212257800. M4 -0.0059675485976

- -150 M5 7.9577308981-.05Me -4.41547793168-07

IR 0.99889272488

-200 .0 10 20 30 40 50 60

HEIGHT, mm(b)

Fig. 5. Measured lift and guidance magnetic stiffness

14

10 . . . S . . .. I . . . . I .. .. .

E h=7mmE 0

>- -10

> ~Y- MO+ Mlx .. M~xs + M9 x

,, -20 MO 31.691508933a ml -14.298777941

w M2 2.6273152741

cc 3 0 M3 -0.238536497510 M4 0.011665956742

.L. M5 -0.00031084553385

S-4 0 M6 4.19953375219-06M7 -2.23942848939-08

R 0.99959180031

-50

0 10 20 30 40 50 60GAP, mm

(C)

1 0 " " " . . . S . I . . . . I . . .

E h =12.7 mm

M 0

Y MO+ Ml~x *... M + .Mg9

a MO 17.003607649a Ml -8.1060930153

WL -20 M2 1.4073249093

I M3 -0.12101504973o M4 0.0057048790699US

MS -0.000149823027393Me 2.0363327448e-06

M7 -1.1100374606e-08

R 0.99922553236

-4 0 . .. . . .. . . , .. . .; 0 . . . . ..

0 10 20 30 40 50 60GAP, mm

(d)

Fig. 5. Cont'd

15

8.0-AE . Y* =5 mm

6.0> Y-MO+M1~x +....Msx+M9"x9

MO 22.407288918> 4.0 MI -4.6465429478

M2 0.35828966062LU M3 -0.0142969892090

S2.0 M4 0.00029563595274. M5 -2.5101972190e-06

SR 0.98855894203

0"-. 0.0

zS-2.0

°4.0

0 10 20 30 40 50 60

HEIGHT, mm(e)

E 8.0 . . . . ' . 3 . ' . ' " -- -- "

E Y* =12.7mm4mm

LLF 6.0> Y0 M +.o ÷lM x + ... Mx 8 +M9.x 9

M mO 18.245024183Ml -3.157897297

L 4.0 M2 0.23337312217a M3 -0.010331754178

LU M4 0.00027806591002ccMs -4.09898885790-06, 2.0 Me 2.50503165220-08

.R 0.99562601734LU-Uz( 0.0

-2.0.•• ... •. . .... .............

0 10 20 30 40 50 60

HEIGHT, mm(f)

Fig. 5. Cont'd

16

20

E

T0

Im-

LU

a Mo -13.2951 0085IM, l 26.223336466

S-40 M2 -1.78124311940 M3 0.055500351o638

LUM4 -0.00069222848428IL M5 3.2308307559e-06

z .6 0 M6 -8.6841873449e-08M7 1.2995260345e-09

R 0.998948691 32

80 .... . . .. †,.. .††. .†... .††. .

0 10 20 30 40 50 60GAP, mm

(g)

E 2 0 -a. . a . " " ' . ' . . " -

E h =12.7mmcm 10

S -10

-U Y- No + .,! x 20.,.3,,+,,,,Ml

u 2 0 "s o / S2.o,2,,oo,.20 ml 28.455386695SM2 -2.3425643442

0 10 2030 M350 6

o 30. 0.10824325908

Ml !M4 M -0.0030092353317U -40 MS1 S.1 704867968e-05z U'me -5.27327607379-07

M7[ 2.48336594720-0950R 0.99950977088

.60w......................0 10 20 30 40 50 60

GAP, mm(h)

Fig. 5. Cont'd

17

[M+ +MmI{U} +[C' + Cm]{} + [K +Kml{U} = {Q}, (11)

where Mm is the magnetic mass matrix, Cm is the magnetic damping matrix,and Km is magnetic stiffness; their elements are mij, cij, and kij.

Once the magnetic-force coefficients are known, analysis of vehicle stabilityis straightforward. Equation 11 may be written as

[M0{} + [C]{U} + [K]{U} = {Q}. (12)

In general, M, C, and K are functions of U, U, and 0; therefore, a completesolution is rather difficult to obtain. In many practical situations, one can ignoreall nonlinear terms, so that M, C, and K are independent of vehicle motion.

By premultiplying by UI jT and forming the symmetric and antisymmetriccomponents of the matrices

[Mi] I=([M] + [M]T), [M2] I([M]- [MIT),

IC1]= .([C] + [CIT), [C2 ]= •([C]- [C]T), (13)

[KE] = [K] + [K]T), [K21 =([K]-[K]Tý

the terms may be separated, giving

{uj} T [M 1]{U} + {} T [C 2]{IJ} + {Uj} T [Kd]{U}

= -({Ij}T [M2 1{U} + {Uj} T [CI]{j} + {j}T [iK2 WU}) (14)

Equation 14 equates rates of work. The terms on the right-hand side of theequation produce a net work-resultant when integrated over a closed paththrough the space {U), the magnitude depending on the path taken. The forcescorresponding to the matrices M2, C1, and K2, appearing on the right-hand side,are thus, by definition, the nonconservative parts of the forces represented by M,C, and K. The terms on the left-hand side similarly can be shown to give rise to azero work-resultant over any closed path, and therefore together are the sum ofthe rates of work from the potential forces and the rate of change of kinetic energy.

18

Different types of instability can be classified according to the dominant terms inEq. 14:

" Magnetic-Damping-Controlled Instability (single-mode flutter): Thedominant terms are associated with the symmetric damping matrix[C 1]. Flutter arises because the magnetic damping forces create"negative damping," that is, a magnetic force that acts in phase withthe vehicle velocity.

" Magnetic-Stiffness-Controlled Instability (coupled-mode flutter): Thedominant terms are associated with the antisymmetric stiffnessmatrix MK2]. It is called coupled-mode flutter because at least twomodes are required to produce it.

Corresponding to the single- and coupled-mode flutter, parametric andcombination resonances may exist if the motion-dependent magnetic forces are aperiodic function of time.

" Parametric Resonance: When the period of a motion-dependentmagnetic force is a multiple of one of the natural frequencies of thevehicle, the vehicle may be dynamically unstable.

" Combination Resonance: When the period of the motion-dependentmagnetic forces is equal to the sum or difference divided by aninteger of the natural frequencies of the vehicle, the vehicle may alsobe subjected to dynamic instability.

In practical cases, two or more mechanisms may interact with one another, andEq. 12 is applicable for general cases.

It is noted that maglev systems are subjected to several groups of forces,including magnetic forces, aerodynamic forces, and guideway perturbation. Thetheory presented in this paper is applicable to maglev systems when they aresubjected to other types of forces. In particular, the aerodynamic effects can bedescribed exactly the same way as those given in Eqs. 1-14 and the dynamicresponse characteristics to aerodynamic forces are similar to magnetic forces; seeChen (1987) for details.

4 Simplified Vehicle Models for Dynamic Instability

Four different vehicles are considered, in order to provide an understandingof stability characteristics.

19

4. Two-Degree-of-Freedom Vehicle

A maglev vehicle is supported by magnetic forces; the resultant lift and dragforces of the coil above the continuous-sheet track can be representedapproximately by (Sinha 1987)

FL(V,zt) = ELF(t),

FD(v,z,t) = EDF(t),

F(t) = p012/4irz,(15)

EL = 1- 1(1 + v 2 /w 2 )n,

ED = (w/v)[1 - 1/(1 + v2/w2)n],

w = 2/poch,

where v is the forward velocity, z is the steady-state height of the coil above thetrack, t is the time, I is the constant coil current, and w is the characteristic speedand is related to track thickness h, conductivity c, and permeability g±o. The valueof n is 1 for a single conductor and varies from about 1/5 to 1/3 for coils (Rhodesand Mulhall 1981). Note that F(t) represents the repulsive force between the coiland its image coil. The force ratios, EL and ED, are given in Fig. 6 for n = 1, 1/3,and 1/5.

Assume that the vehicle is traveling at a velocity vo at an equilibrium heightzo. The instantaneous position and height of the vehicle are x and z respectively;therefore,

x(t) = vot + X(t),(16)

z(t) = zo + Z(t).

The equations of motion for the vehicle moving at a velocity v(t) with a levitationheight z(t) can be written as

mi(t) = -mg + FL(v,z,t),(17)

mit(t) = Fp - FD(v,z,t) - Fa,

where m is the mass of the vehicle, Fp is the propulsion force, and Fa is the aero-dynamic force. The aerodynamic force is given by

2D

'l ..... ...... .• ........ .t---- --:-- .-V - . .--- ...- .....---..- .---- .'-- .--- ---. " ........ ........-....:--.-.."-------- - '-' -- .....S...... . ..... u ..- .... .• . .•... 4... ....• !: ` ... . . -.. ... •!,,,. • -,. .,.

•..... . ... ..... .... . ....... ..... t----:; • , ?• : ÷ .: i....... .......---. . . ... --. . :• -

o . o -~ ~.I ... :........... .L .' - Ji...i.... ......................... L.... '.... -. .......... d.. ... ,.•...-..,..... . :... ..... ......•. .. .

I - - - - : '' : :

- ...... .. , ..L .... ...... ...... ...... .....•. ... - ..• .1 ........ ...... .. ...... ... 44 4 Z- - ," -o?-I

S..... .• .•. .:•i• ..•..z~ •...L••i . ..•..•.L. •. L.• .• .........L~ .... .

0.6 d1 -UF' I : : : :i L " : :

• I ' i , . , i i , . .." ..... .................... .- ... ... ............. .. ..... .-.- .-- ..... ......., ..... ...... , ..-....- ,--. ----. -..... ...........-. ...... ... ,... ... ..... ....

.t. ...... .•...L .. ... .. i .. ... .. J ...... J..... ... ... i.... : ....... .... L.... ...... ..... L.. ...... ..... . ....0 . 1 : i .• :+: : ... ..... ............ .- .....

0.4.

. ... ...... ..... ....... ..- ..i..-.-..- -....-.. .- -- -.- --. i-.-............:...... . ...... ....---- ......... ... .--.---- i..- ...

.. ... ... ... . .. ... .... ........ ... .. .... ... ............... .... .... 4.....4 .. . .. .....

o ,." .. ... ..... ....... ..... ......... ............................. ..... ..... . ....... .....

4 1 4R~ FI.................. .. ...... .......... 1" ' •

0 10 20 30 40 s0

V#W

Fig. 6. Magnetic forces divided by image force

Fa =Kav2' (18)

Ka = 0.5 CDLAp,

where L and A are the length and cross-sectional area of the vehicle, p is airdensity, CD is the drag coefficient (with its values varying from 0.2 to 0.3,depending on vehicle shape).

The equilibrium point of the vehicle vo and zo is defined by

FL(vo,zo) = mg, (19)

FD(vo,zo) = Fp - Fa.

Using Eqs. 16-19 and neglecting the nonlinear terms, we obtain the followingequations of motion for the vehicle X(t) and Z(t):

m2(t) - Czx(v,z,t)X(t) + Kzz(V,z,t)Z(t) = 0, (20)

mX(t) + Cxx(v,z,t)X(t) - Kxz(v,z,t)Z(t) = 0.

21

Czx(v,z,t) and Cxx(v,z,t) are magnetic damping, and Kzz(v,z,t) and Kxz(v,z,t) aremagnetic stiffness; these are given by

Czx(v,z,t) = iaFL(V,z,t)/iv I (vozo)'

Cxx(v,z,t) = i)FD(v,z,t)/iv I (vo,zo) + 2 Kavo, (21)

Kzz(v,z,t) = - aFL(v,z,t)/Mz I (vo,zo),

Kxz(v,z,t) = --aFD(V,Z,t)/iZ I (Vo,Z o )

Their values are given as follows:

Czx = (mg/eL)2nvo/[w 2(1 + v2/w2)n+l],

Cxx = wmg/v2 [w 2 - (2n - 1)v2/(w2 + v2)] + 2 Kavo,0 0 0(22)

Kzz = mg/zo,

Kxz = wmg/vozo.

For high speed vehicles, the values of Czx and Kxz are approximately zero.Therefore, the motions in the vertical direction and forward direction areuncoupled at high speeds.

Equations 20 can be analyzed; let

Z(t) = a exp(icot),(23)

X(t) = b exp(iot).

Substituting Eqs. 23 into Eqs. 20 gives the following frequency equation:

mi = .(24)--Kxz -mC02 + i(OxxJ-b

The natural frequencies can be determined from the determinant of the coefficientmatrix given in Eq. 24. At high speeds, the off-diagonal terms may be neglected.The natural frequency of the vertical motion fv is

fv = (g/zo)0.5/2x. (25)

22

. .... .... .... .... .... ....

9 -• -. -•-- -• -- -•- .-• - .. .. ..... .... ... -- .• - .- .-- ... ...... • .... .... ... .. ....-.'.-- • .-- --.. -- ..... ... .--i -- --J --. ----. ....

...... i.. . ......... .---.. ..4

,.•... ...... .... .... • .... .... .... ---..--........ .......... .......... •-. ........... i.--...•-------,-. ... ....

S........... .......... ...... . ..........

+ . ....... .... ......4.... . .............. .... 4.. . ....... i .......... "... •.... • .......... * -.. i--... i .... ... .... ... .. i .... ... i........ .. -. .... .... i... .... i.... s.. ...... •....... .... .... 4

.... ... ........ . . ----. .... .... ]......... ... .... ....... - .- .- -- -. - -. " .....• --i --• - '. - -4

. .... ... ............ . 4... .....

1 ..... 4...

.-.... .... ...... ..... .. *..S.... ...".' " "' " ' " " " ' ........... - -- - - --- ......... ... .... ......... .... .... • ' ' " • " ' !" " " !" ' ' ' ' ' '"...... .

0...... .... .......- - 1-

0 10 20 30

LEVITATION HEIGHT, cm

Fig. 7. Natural frequency as a function of levitation height

The natural frequency in the vertical direction depends on the levitation heightonly. Figure 7 shows the natural frequency as a function of the levitation height.At high speeds, oscillations in the vertical direction are stable.

In the forward direction, the motion is given by

X(t) = C1 + C2exp(st). (26)

For high-speed vehicles, the exponent s is approximately given by

s = (2n - 1)wg/v2 - 2Kavo. (27)

Note that the vehicle may be unstable if n = 1, and K. is zero. At high speeds, thesecond term given in Eq. 27 is larger than the first term regardless of the values ofn; therefore, s is negative and the system is stable.

4.2 Three-Degree-of-Freedom Vehicle

Figure 8 shows a three-degree-of-freedom vehicle traveling at a velocity vo atan equilibrium height zo. For a symmetric vehicle, the instantaneous positionand height of the vehicle are x(t), zi(t), and z2(t); therefore,

23

z

Z2 _I

FD2 FD

Fa Fp

FL2 FLU

Fig. 8. Three-degree-of-freedom vehicle

x(t) = vO(t) + X(t),

zl(t) = Zo + Zl(t), (28)

z2(t) = zo + Z2(t).

The equations of motion for the vehide moving at a velocity v(t) with levitation

height zi(t) and z2(t) can be written as

nm(t) = Fp - FDl(Zl,V,t) - FD2(z2,v,t) - Fa,

2[il(t) + i 2 (t)] = -mg + FLl(zl,v,t) + FL2(z 2,v,t), (29)

I. - 2(t)] = .1tL[FL1(zl,v,t) - FL2(Z2,v,t)].

where m is the vehicle mass, Io is the rotational moment of inertia about the

vehicle's center of mass, Fp is the propulsion force, and Fa is the aerodynamic

force that is assumed to act at the center of mass of the vehicle and is given by

Eq. 18.

24

For a symmetric vehicle with two identical levitation systems at the two ends,the equilibrium point of the vehicle vo, zio, and z2o, as well as the magnetic forces,is defined as

FLI(Vo,zlO) + FL2(vo,z20) = mg,

FD1(Vo,zlO) + FD2(Vo,Z20) = Fp - Fa,

Z1o = Z20 = zO, (30)

FL1(vo,zlO) =FL2(Vo,Z20),

FD1(Vo,z20)- FD2(Vo,z20).

Using Eqs. 28, 29, 30, and 18 and neglecting the nonlinear terms, we obtain thefollowing equations of motion of the vehicle, X(t), Zi(t), and Z2(t):

In(21 + Z2 ) + (Czxl + Czx2 )X + KzzlZ1 + Kzz2Z2 = 0

-- (l - i2) + (Czx1 - C 2 )i + (KzzlZ1-Kzz2Z2 = 0 (31)

mn + (Cxx1+ Cxx2 + 2kavo)]k + KxziZi + Kxz2Z2 = 0

where

Czxi --- (vo,zo)

Cxxi = i- ( oo

(32)Kzzi =M )F

KxMzi =-i"Vo,zo)

i = 1,2.

25

The magnetic damping coefficients Cxi and Cxxi, and the magnetic stiffnesscoefficients Kzzi and Kxi can be calculated from the magnetic lift and drag forcesgiven in Eq. 15.

At high speeds, Kzxi and Cxzi are approximately zero. The equations ofmotion become

In(21 + 22 ) + Kzzl + Kzz2Z2 = 0,

2 (2 1 - Z2 ) + KzzZ 1 - Kzz2Z2 = 0, (33)

mX + (Cx. 1 + Cx 2 + 2Kavo)X = 0.

In this case, the vehicle is stable at high speeds; this is similar to the two-degree-of-freedom vehicle. The natural frequency of vertical oscillations is the same asthat in the two-degree-of-freedom system given in Eq. 25. The natural frequency ofpitching oscillations is

f1(L~mg)0"5fp = -I -9o . (34)

For a square vehicle with length L and height h, the natural frequency ofpitching oscillations is J 0.5

1 (JgO.5(3 -(5fP= (35)z

The natural frequency of pitching oscillations is larger than vertical oscillations.For a long vehicle (h << L), fp is equal to about 1.7 f.. For a square vehicle, fp = 1.4fv. At high speeds, heaving and pitching oscillations are stable for the magneticlevitation described by Eq. 15. In the forward direction, the result is the same asthat for the two-degree-of-freedom vehicle.

26

4.3 Six-Degree-of-Freedom Vehicle

For the six-degree-of-freedom vehicle shown in Fig. 1, stability can be studiedfrom Eqs. 11 or 12. Once the coefficients of magnetic forces mi, ci1, and kb areknown, Eqs. 11 or 12 can be evaluated for Q = 0. Let the displacement of aparticular component be

uj(t) = ajexp(X + ico)t. (36)

Substituting Eq. 36 into Eq. 12 with Q = 0 yields

{(X + i(o) 2[M] + (X% + iwo)[C] + [K]) {A) = 10)}. (37)

The values of X and o) can be calculated based on Eq. 37 by setting the determinantof the coefficient matrix equal to zero.

Vehicle stability is determined by X, which is a function of v. If X < 0, vehiclemotion is damped; if X > 0, vehicle displacement increases with time untilnonlinear effects become important.

To solve this problem, all motion-dependent magnetic-force coefficients mustbe known. At this time, it appears that limited analytical, numerical, orexperimental data are available. For any future maglev systems, it will benecessary to investigate the characteristics of motion-dependent magnetic forcesto avoid dynamic instabilities.

4.4 Vehicle on Double L-Shaped Aluminum Sheet Guideway

Figure 9 shows a cross section of a vehicle on a double L-shaped aluminumsheet guideway. Assume that the vehicle travels at a constant velocity along xdirection. Two permanent magnets are attached to the bottom of vehicle andprovide lift and guidance force FL1, FL2, FG1, and FG2 (see Fig. 9). Assume at theinitial state that h1 = h 2 = h0 and g, = g2 = go; thus, the vehicle and guidewaygeometries can be expressed as follows:

Li = L2 = S = 76.2 (mm)

W 152.4 + S - 2g0 (mm)

H =0.9 W (mm)

z

8

T

-Z! 0W

S]F

mgS L• 2. L S PIS. L,

Fig. 9. Maglev system with a vehicle operating on double L-

shaped aluminum sheet guideway

a= 0.5 H (mm)

b = 0.5(W - 25.4) (mm).

Equations of motion for this three-degree-of-freedom maglev system can bewritten as

M'+ C' = FL1 +FL2 -mg

IO+ EO = (FGI + FG2 •)a +(FG1 + FG2 )b (38)

my + Dy = FG1 + FG2,

where m is the mass of the vehicle, C and D are damping ratios; I is the momentof inertia about the center of mass inertia moment of the vehicle [I =(m/12)(H 2 + W2)]. FL1, FL2, FG1, and FG2 are lift and guidance forces and arefunctions of y and z. At the equilibrium position, they are FL10(yo,zo), FL20(yo,zo),FGlo(yo,zo), and FG20(yo,zo). Apply them to Eqs. 38:

28

FLIo =

FL 1o + FI• = mg (39)

FGjo = -FG2o

Therefore

M = FL10 + FL2 2FL(ho) (40)g g

Let

1 (u+ u2)

y=u3 (41)

e-=(U 1 -U 2 )/2b

where u1, U2 , and u3 are shown in Fig. 10. Equations 38 can be rewritten as

m(Ql+ t 2) + c(ul- i 2)= 2FL, + FL2 - mg)

+(- 2 ) E(u1-f12) = 2a(FG1 + FG2 ) +2b(FG1 - FG2 ) (42)

ml + Du3 = FG1 + FG2 •

Note the reduced dependence of the forces on the new displacements of Eq. 41:

FL, = FL1 (ul,u3)

FI2 = FL4 (u2,u3 )(43)

FG1 = FG1(ul,u3)

F = = FG2(u2,u3 ).

29

z

ly

U 2

3

U 1

Fig. 10. Displacement components of three-degree-of-

freedom vehicle

Let

ui Uio + vi i = 1, 2 (44)

The linear approximation of lift and guidance forces can be expressed as

FL1 = FL10 +-VLl v Fv3

CFL +1Fiv3

FL2 = FL20 + aFL2 oFL 2'V2 2 v3

(45)FGl=fFGlo+•lv1 + 3 v3

FG2 , FGo + G- v2 +'FGv3

2 3

Using Eqs. 39 and 45, we can rewrite Eq. 42 as

30

C. C. 2RL, 2DFL 1 (DFG, aFG

Vl+V 2 +-Vl+-v 2 -- vlv---2 2 V 2 -- I' 2 1D, V3 =0In In In In~ In2 3 )V 3 )

E E( +(cb FL 1 + 2b 2 FGl'Xl

+ rŽb FL2 + 2b2 ____)2 46

1 C-)V210- 2b F+[1abf + aFL2 b! -oGI+aF2)V

m3 -3- -~ m v1 - -m,ýv2 mI(ý 3- aV 3 =0

or

[MI(V) + [Clivý) + [Kljvl =0 (47)

w h e r e, - - C 2 2 0o 1k 1 l k 1 2 k 1 3 ]

M -110 C C2 22 0 K= k2 l k22 k1231 (48)iJ0l-- 0 c23- J k3 l k32 k33 J

and

CCi 1 = Ci 2 = I

_EC2 1 C= 2 2.

Dc3= m

2l= I FL1-IV, (49)

31

2 aFL2

m OV

kl=2ab aFGj 2b2 cIFL2

I ~V 2 IF

(49)

k23 = 2abF( - FL2' (Cont'd)1 v 3 aIV 3 1 ClV 3 i)V

1 MGQ

m v

k32 ___ aFG 2

k3= --L (aFG +av )where

-FL ki 1(h), iaFG1 - kgi(h),

(50)

M = k1 g(g), aFGI = kg(g)."V3 O'V3

32

ktt(h), keg(g), kge(h), and kgg(g) are motion-dependent magnetic-force coefficients(see Fig. 5).

If we assume that the damping effects can be neglected, the eigenvalues of

Eq. 47 can be obtained from

[K] (vA = X [M] (Av (51)

where X = (oR + i (ol.

With magnetic forces and stiffnesses measured by the experiments (seeFigs. 4 and 5), the eigenvalues and eigenvectors of a maglev vehicle on a doubleL-shaped guideway were calculated with the theoretical model developed in thissection. Some very interesting results were obtained from those calculations.

Figure 11 shows that eigenvalues of vehicle motion versus levitation heightvary when guidance gaps are fixed (gl = g2 = Y* = 12.7 mm). The first mode 0)1shows an uncoupled heave motion; its imaginary part of the eigenvalue is zero,while the second and third modes are coupled roll-sway motions. Within therange of height h = 19.0 to 35 mm, the imaginary parts of the eigenvalues appearnot to be zero. This indicates that within this range, flutter does exist for thesecoupled roll-sway vibrations. Table 1 and Fig. 12 give eigenvectors and modalshapes of these three modes of vehicle motion, respectively. When the guidancegaps are fixed at g, = g2 = Y* = 5 mm, the same results are obtained, as shown inFig. 13; there is a flutter for coupled roll-sway modes.

Table 1. Eigenvectors of vehicle motion (Y* = 12.7 mm)

h = 15.0 mm h=25mm h = 37 mm

Mode V1 v2 v3 v1 v2 v3 Vi v2 v3Uncoupled 1 1 0 1 1 0 1 1 0heave mode(01

Coupled 1 -1 -0.009 0.586 -0.586 -0.332 -1 1 -0.205roll-swaymode w2

Coupled -0.545 0.545 1 -0.810 0.810 0.060 1 -1 0.448roll-swaymode o)3

33

80

Y*- 12.7 mm

- 0 O: 60 1

---. O)> 24 •-.-.--- - CO033

400

CL

• 20cc

0

0 10 20 30 40 50Height, mm

1 0 , , I , I , , I * , , ,

SY* 12.7 mm

C 5CD

00,

Z._

ccCu•& -5 CCuE 2

E CO3

-10 1

0 10 20 30 40 50Height, mm

Fig. 11. Maglev-system eigenvalues vs. vehicle levitation height,with Y* = 12.7 mm

34

r - E

E> crE

030

> >I

03 U'N

00tlw.

0"

I E0 E 'O

)S% Cu T

C) CY

35

12 0 , , I , , , I , ,

100 Y* 5 mm

00

> 80C

600

C. 40

d: 2020

0 10 20 30 40Height, mm

8 , , , , I . . . I . . . . I i

* 5 mm 0

> 4 02

0 0

coCL

.S -4

E

8 '

0 10 20 30 40

Height, mm

Fig. 13. Maglev-system eigenvalues vs. vehicle levitation height,with Y* = 5 mm

36

Figures 14 and 15 show eigenvalues of vehicle motion versus lateral locationof vehicle when g, = 92 = g0 = 25 mm and levitation heights h = 12.7 mm and h =7 mm, respectively. We notice that for the third mode, which presents thetransversal motion of vehicle, the real part is zero and imaginary part is not zerowithin a certain region. This indicates that the divergence is subjected to thelateral motion of the vehicle, given those vehicle and guideway parameters.Figure 16 shows the real part of the third mode versus lateral location of vehiclewhen the parameter-equilibrium guidance gap varies as g, = g2 = go = 10 mm,15 mm, 20 mm, and 25 mm. We found that divergence appears only in the case ofgo = 25 mm.

We must point out that the measured and calculated data for motion-dependent magnetic-forces and force coefficients are very limited and thatdamping effects were not considered in the above analysis. Even thoughdivergence and flutter appear in the eigenvalue results, we still have difficulty incompletely predicting the dynamic instability of this three-degree-of-freedommaglev vehicle model. Further research is needed in modeling to gain anunderstanding of dynamic instability in maglev systems.

5 Closing Remarks

" Motion-dependent magnetic forces are the key elements in modelingand understanding dynamic instabilities of maglev systems. At thistime, it appears that very limited data are available for motion-dependent magnetic forces. Efforts will be made to compileanalytical results and experimental data for motion-dependentmagnetic forces. When this work is completed, recommendationswill be presented on research needs on magnetic forces. In addition,specific methods to obtain motion-dependent magnetic forces will bedescribed in detail.

" Various options can be used to stabilize a maglev system: passiveelectrodynamic primary suspension damping, active electrodynamicprimary suspension damping, passive mechanical secondarysuspension, and active mechanical secondary suspension. With abetter understanding of vehicle stability characteristics, a bettercontrol law can be adopted to ensure a high level of ride comfort andsafety.

37

40-

00

C0

0

as 10 - - 0()2M

0

-30 -20 -10 0 10 20 30Lateral location of vehicle, mm

og =25 mmCh h-12.7 mm

> 10 -6-O

CL

0cc

-5-3 2 1 01 03

Laea octo fveilm

Fi.1.Cue-ytmegnale s aea oaino

0eilwt 2. madg 5m

38

0040

"-a 3090 25 mm/

CD/

20 h=7mm

0.0 10 C• oC 2

3cc 0

- 1 0 --- , . . . . . .. .

-30 -20 -10 0 10 20 30Lateral location of vehicle, mm

1 5 . . . . .. . ... I . .. ,

90 -25 mm -- co10 h 2

CL

0 -9- a(aS5

CU 0

"-5 - ' I I ' I * ' ' I ' I 9

-30 -20 -10 0 10 20 30Lateral location of vehicle, mm

Fig. 15. Maglev-system eigenvalues vs. lateral location ofvehicle, with h = 7 mm and go = 25 mm

39

4 5 i i i i . . . I . . . . I . . . . . . I o I i

-- go =10 mm d go =20 mm

35 - go =15 mm -- g =25 mm

i h =7mm

S25

0

t: 15asCL-

CC 5

-5-30 -20 -10 0 10 20 30

Lateral location of vehicle, mm

Fig. 16. Real part of maglev-system eigenvalues vs. laterallocation of vehicle, with h = 7 mm and go = 10, 15,20, and 25 mm

"* Computer programs are needed to screen new system concepts,evaluate various designs, and predict vehicle response. It appearsthat the stability characteristics of maglev vehicles under differentconditions have not been studied in detail in existing computer codes.When information on motion-dependent magnetic forces becomesavailable, the existing computer codes can be significantly improved.

"* Instabilities of maglev-system models have been observed at Argonneand other organizations. An integrated experimental/analyticalstudy of stability characteristics is an important aspect of maglevresearch.

Acknowledgments

This work was performed under the sponsorship of the U.S. Army Corps ofEngineers and the Federal Railroad Administration through interagencyagreements with the U.S. Department of Energy.

Additional thanks to S. Winkelman for performing tests on magnetic forces.

40

References

Chen, S. S. (1987), Flow-Induced Vibration of Circular Cylindrical Structures,Hemisphere Publishing Co., New York.

Chu, D., and Moon, F. C. (1983), Dynamic Instabilities in Magnetically LevitatedModels, J. Appl. Phys. 54(3), pp. 1619-1625.

Davis, L. C., and Wilkie, D. F. (1971), Analysis of Motion of Magnetic LevitationSystems: Implications, J. Appl. Phys. 42(12), pp. 4779-4793.

Moon, F. C. (1974), Laboratory Studies of Magnetic Levitation in the Thin TrackLimit, IEEE Trans. on Magnetics, MAG-10, No. 3, pp. 439-442 (September 1974).

Moon, F. C. (1975), "Vibration Problems in Magnetic Levitation and Propulsion,"Transport Without Wheels, ed. by E. R. Laithwaite, Elek Science, London,pp. 122-161.

Ohno, E., Iwamoto, M., and Yamada, T. (1973), Characteristic of SuperconductiveMagnetic Suspension and Propulsion for High-Speed Trains, Proc. IEEE 61(5), pp.579-586.

Rhodes, R. G., and Mulhall, B. E. (1981), Magnetic Levitation for Rail Transport,Oxford University Press, New York.

Sinha, P. K. (1987), Electromagnetic Suspension, Dynamics and Control, PeterPeregrinus Ltd, London, United Kingdom.

Yabuno, H., Takabayashi, Y., Yoshizawa, M., and Tsujioka, Y. (1989), Boundingand Pitching Oscillations of a Magnetically Levitated Vehicle Caused byGuideway Roughness, Int. Conference Maglev 89, pp. 405-410.

41

Distribution for ANL-92/21

Internal

Y. Cai (10) M. W. Wambsganss (3)S. S. Chen (10) Z. WangJ. L. He R. W. WeeksL. R. Johnson S. ZhuC. A. Malefyt (2) ANL Patent Dept.T. M. Mulcahy (5) ANL Contract FileD. M. Rote (5) TIS Files (3)R. A. Valentin

External

DOE-OSTI for distribution per UC-330 (91)ANL-TIS Libraries (2)Manager, Chicago Operations Office, DOEDirector, Technology Management Div., DOE-CHD. L. Bray, DOE-CHA. L. Taboas, DOE-CHMaterials and Components Technology Division Review Committee:

H. K. Birnbaum, University of Illinois at Urbana-Champaign, UrbanaR. C. Buchanan, University of Illinois at Urbana-Champaign, UrbanaM. S. Dresselhaus, Massachusetts Institute of Technology, Cambridge, MAB. G. Jones, University of Illinois at Urbana-Champaign, UrbanaC.-Y. Li, Cornell University, Ithaca, NYS.-N. Liu, Electric Power Research Institute, Palo Alto, CAR. E. Smith, Engineering Applied Sciences, Inc., Trafford, PA

0. P. Manley, DOE, Washington, DCJ. S. Coleman, DOE, Washington, DCD. Frederick, DOE, Washington, DCF. C. Moon, Cornell University, Ithaca, NY

is.i

U- 0

tow UUif- "

o -4

C,: 0o T . V Z)

s W n C14Exx

z OD 07- r-' E-n0 r.- o c

0 o Cl •-d' 4M U;

SUPPLEMENTARY

INFORMATION-

ElRRATA

ANL-92/21

Dynamic Stability of Maglev Systems

by

Y. Cai, S. S. Chen, T. M. Mulcahy, and D. M. Rote

Argonne National Laboratory, Argonne, Illinois 60439

Pages 27 to 31 are replaced by the attached new ones to correct a series oftypographical errors.

JT

z

0I y

W

wdNo

U- U.. t

mgL L2 I_ S I L I

Fig. 9. Maglev system with a vehicle operating on double L-shaped aluminum sheet guideway

a=0.5H (mm)

b = 0.5(W - 25.4) (mm).

Equations of motion for this three-degree-of-freedom maglev system can bewritten as

mz+Cz = FL1 +FL 2 -Mg

I0 + E6 = a(FG1 + FG2 )+ b(FL1 - FL2 ) (38)

my- + Dy = FG1 + FG2,

where m is the mass of the vehicle, C and D are damping ratios; I is the momentof inertia about the center of mass of the vehicle [I = (m/12)(H2 + W2)]. FL1, FL2,FG 1, and FG2 are lift and guidance forces and are functions of y and z. At theequilibrium position, they are FLlo(yo,zo), FL20(yo,zo), FG1 o(yo,zo), and FG20(Yo,Zo).Apply them to Eqs. 38:

28

FL10 = F,2

FL1 o + FL2o= mg (39)

FGjo = -FG20

Therefore

m= FFL10 + FL20 = 2FL(ho)g g (40)

Let

"= -(u 1 +u 2)2

Y = u 3 (41)

0 = (ul-u 2 )/2b

where ul, u2 , and u3 are shown in Fig. 10. Equations 38 can be rewritten as

m(uil + ui2)+C(ul + u2 ) 2(FL1 +FL 2 -mg)

" (t- ii2)+ E (il- ( 12) =2a(FGI + FG2 )+ 2b(FLI - FL2) (42)

mui3 4 DM3 = FG, + FG2.

Note the reduced dependence of the forces on the new displacements of Eq. 41:

FL1 = FLI(Ul,U3 )

FL2 = FL2(U2,U3)

(43)FG1 = FGI(Ul,U3)

FG2 = FG2(u2,U3 ).

)29

z

uU 3 Ul

Fig. 10. Displacement components of three-degree-of-

freedom vehicle

Let

ui Uio + vi i = 1, 2, 3 (44)

The linear approximation of lift and guidance forces can be expressed as

= +FL1 +FLvFLFL fao+ V+ v 3

)v 1 Dv3

€)FL2 + FL2 v

FL2 = FL2 0 + R22 + v 3

(45)=FG 1 + FG1 v3FG1 =FG1 0+ •,v+

%)V i ov3

FG2 =FG2o+-G2 v2 +-2V 3

Using Eqs. 39 and 45, we can rewrite Eq. 42 as

30

C. C. 2aFL1 2 aFL2 2 (aFL1 + RL2 '*

vl 2 vlV Vi ___a_m In I may1 mnay 2 (I~ 5y3 aV3 )

-Vl E. +E.' +(2ab aFG, 2b2 pLIV

+r(abaFG 2 2b2 DFL2 )2 (46)

[2ab(aFGI aFG2 ')2b2 (aFLI aFL2 ]3=

I aV3 aV3 ) I aV3 aiV3

D. aFG1 V 1 aFG2 1 aG 1 aGV3V V V2 (a- I + a iG23 =0m m ay1 may52 m~ aV3 CVj

or

[M]IBý)+[CIIfv}+[KlfvI 0 (47)

where

1= 1 01 [c11 c90 0 ~ k1 l k12 k13]

m -1I 1 0, C=1c91 C'20 K= k21 k22 k23,(8L0 0i Lo 0 0c 2 3j Lk31 k32 k33j

and

Cci=C11 =.L -

EC2 1 = -C22=

DC3 3 -=I

k= 2 iaFL1 (49)m v

I' - 31

m av2

21 =-ý(aFL1 aFL2

k = 2ab aFG1 2b2 aFL1

k22ab aFG 2b2 aFL2I G Iv2 v

(49)

k23 ab____I aFG2 +2b 2 D'FL 1 aFL2 (otd

I av av3 I iav3 iav3

m v

k3.- 1 (aFG aF

k3 _~aG,+ aiG

where

-!12 -= k a1(h), aFG2 = k g h ,

(50)aFL1 ~~) aFG _

-573 ayV ggg)

FL = klg(g), aFG2 =kgg(g).5ay3 v