DEMO 2010/1

ΑΡΙΘΜΗΤΙΚΗ ΜΕΛΕΤΗ ΕΝΟΣ ΗΛ/ΜΑΓΝΗΤΙΚΟΥ ΕΚΤΟΞΕΥΤΗΡΑ

Θ. Ε. ΡΑΠΤΗΣ

NNNNumerical umerical umerical umerical Study of anStudy of anStudy of anStudy of an E/M E/M E/M E/M LauncherLauncherLauncherLauncher. . . .

T. E. RAPTIS

ΕΚΕΦΕ «∆ΗΜΟΚΡΙΤΟΣ»

153 10, T.Θ. 60228, ΑΓ. ΠΑΡΑΣΚΕΥΗ ΑΤΤΙΚΗΣ, ΕΛΛΑΣ

“DEMOKRITOS”

National Centre for Scientific Research

153 10, PO Box 60228, AG. PARASKEVI, ATTIKI, GREECE

AΡΙΘΜΗΤΙΚΗ ΜΕΛΕΤΗ ΕΝΟΣ H/M ΕΚΤΟΞΕΥΤΗΡΑ

Θ. Ε. Ράπτης

ΕΚΕΦΕ «∆ΗΜΟΚΡΙΤΟΣ»

∆ιεύθυνση Τεχνολογικών Εφαρµογών (rtheo@dat.demokritos.gr)

NUMERICAL STUDY OF AN E/M LAUNCHER

T. E. Raptis

NCSR “DEMOKRITOS” Division of Applied Technology

(rtheo@dat.demokritos.gr)

Aθήνα, 22, Ιανουαρίου, 2010

Athens, 22, January, 2010

DEMO 2010/1

Numerical Study of an E/M Launcher.

T. E. Raptis

National Centre for Scientific Research “Demokritos”

Division of Applied Technologies

rtheo@dat.demokritos.gr

Abstract: We report on a new design for an Electromagnetic Launcher (EML)

which combines both magnetic levitation and propulsion characteristics based on an

asymmetric combination of dipoles superposed on a simple magnetic rail. We

provide both a pole method and a Finite Element analysis of a planar section of the

overall configuration of a single accelerating stage and derive the force

characteristics. We also describe the limitations of this system with respect to actual

applications

1. Introduction

The subject of Electromagnetic Launchers (EML) has recently been introduced

as an alternative method for propulsion as well as for defense applications due to its

simplicity and its capability to abolish chemical propellants.

While a lot of work has been done which can be found in a number of

conference proceedings on the subject [4 - 6] the main point of view is towards the

rail-gun and coil-gun concepts. These are more or less directly based on the current

interaction in order to produce a subsequent Lorentz force as the main propulsion

mechanism. A review of the various types of configuration of coilguns can be found

in [10] with a direct comparison of their properties.

Lately, NASA developed an alternative technology (ELAM) [11] based on the

combination of a Maglift rail similar to the one used in the magnetic train

(MAGLEV) for levitation purposes associated with a linear induction motor for

acceleration with promising results. In this report we examine a slightly different

mechanism which provides for both simplicity of construction and for simultaneous

levitation and propulsion forces.

Although this technology is not yet mature enough for a full scale Mass Driver

capable of directly setting a vehicle into Earth’s orbit it has other potential

applications. Its main advantage would be that of reducing the cost by cutting the

propellant weight in a two-stage assisted launch system. Other applications include

replacement of old steam based aircraft boosters used in carrier ships at sea

operations.

In section 2 the main idea is presented with the aid of a simplified model based

on permanent magnets forming a magnetic catapult. It is shown through numerical

simulation that it is in principle possible to form the shape of the resulting potential

obtained with the method of poles through the introduction of an idealized

asymmetry in the rail magnet poles. In section 3 a realistic model based on

electromagnets is presented with a magnetic driving head with two separate

electromagnets carrying strongly asymmetric currents. A full FEM study of a planar

configuration based on the model of previous section is performed and the force

curve is derived by sliding the driving head across the magnetic rail. An approximate

equation of motion for a vehicle is also deduced and the limitations of the device

functioning in open air are explored.

2. A simple magnetic catapult

A basic model can be introduced using permanent magnets that form a two-

part system composed of a 1) driving head with a two pole bar magnet shaped so that

the two poles both face downward and 2) a magnetic rail with a number of dipole

moments equally distributed and directed towards the normal to the ground plane.

This configuration is shown in Fig. 1

A first analysis of a “slingshot” effect in a linear arrangement of permanent

magnets has been given by D. Roper [3]. To this aim the method of poles has been

employed in order to obtain an approximate potential describing the forces between

the poles. Assuming an equally distributed amount of magnetic charge across the

surface of each pole in the form wm / where w is the length of the pole surface one

can write the potential energy of the interaction between two separate magnetic poles

as

( )[ ]∫ =+−′

=L

a

w

mm

yxx

dxV

0

2/122

,β

αβ µµ (1)

With recourse to Fig. 1, where g is the gap between the rail magnets, w and h is the

length and height respectively of each identical rail magnet, l is the length of the

driving head while the length of each head pole is also w, one gets the total potential

function which upon integration yields

1

{ }M

NN NS SN SS

n

V V V V V=

= − − +∑ (2a)

2 2

2 2ln

np np ij

ij n

nm nm ij

x x yV

x x yµ

+ + =

+ + (2b)

where

( 1) /2, , ( , )nm np nm ijx n g nw x l x x w y y h y= − + − = + = −m

A calculation with w = 2, h = 0.5, g = 1, l = 5.2, y = 1.5 and M = 50 results in

the potential of Fig. 2. We take initially µn = µ0 = 1 in arbitrary units (the log term

does not contribute to the dimensions). There it is shown that the system has an

average slope towards the positive direction of the z-axis.

We next try to improve this characteristic by introducing a modulation of the

rail dipole moments in the form of a “frozen” stationary wave. To this aim we

introduce the modulated terms

+=

)(

2sin0

gwM

knn

πµµ

(3a)

+

+=)(

2sin10

gwM

knn

πµµ

(3b)

where M is the number of rail dipoles and k is an effective magnetization wavelength

for the frozen wave across the rail length. The new configuration and the associated

potentials for the two cases above are shown in Fig. 3-5 respectively for the values w

= 8, h = 0.4, g = 2, l = 10.4, y = 2, M = 6, L = 60, k = L/3 = 20, µ0 = 1.

Further testing with the above model showed that it is possible to shape the

potential in an almost linear smooth form by taking simply a linear variation of the

rail dipoles and an asymmetric distribution of magnetization on the head poles. An

important conclusion of this toy model is that such a catapult must be activated only

when the magnetically assisted launch vehicle is inside a certain useful range

between the barriers of the potential. This shows that a proper application requires

the use of electromagnets and an appropriate automatic control system (ACS). The

rail must then be constructed by a number of independent accelerating stages

activated with the aid of a mechanism similar to a photocell with the emitter located

on the vehicle and a receiver at each end of a separate such stage in order to activate

and deactivate the currents locally.

3. FEM modeling of the EML system

We propose here a simple and effective design of a full EML system

composed of a special driving head and a linear rail of electromagnets. In the

analysis that follows we used the finite element software FEMM4.2 by David

Meeker (www.femm.info) that is capable of handling a 2-dimensional planar

configurations.

In order to achieve an enhanced performance we design the driving head in a

special way. The head is now composed of two separate iron core gamma-shaped

electromagnets in order to introduce an asymmetry in the magnetic flux during the

interaction with the rail. A planar section of this configuration with the resulting

fields is shown in Fig. 6. The rail is composed of a series of iron core electromagnets

of which the current values fall linearly from a maximum to a minimum values like

InII ∆−= max .

Two simulations were performed under the condition that the rail current

limits are lesser or equal to those of the driving head. Under this condition both the

horizontal and vertical forces were found to be positive resulting into a simultaneous

levitation and propulsion effect. The forces were computed from the Maxwell stress

tensor as

[ ]nBHnHBnBH )()()(2

1•−•+•= ∫dCf (3)

using a square contour surrounding the driving head where n stands for the unit

vector normal to the contour while the driving head was moving from left to right

along the rail.

In both simulations 15 electromagnets were used for the rail elements with the

same indicative values for the currents of the driving head. These were 100 A for the

left part and 1000 A for the right one. The total head length and height were 160mm

and 80mm respectively with a 20mm inner diameter. The head coils had a winding

radius of 20mm and length of 80mm.

In the first simulation we used Imax = 500 A and Imin = 173.3 A with a step of

23.3A for the rail elements. The gap between the rail elements was 15mm and the

vertical distance from the head was 10mm. The coil length and height was chosen to

be 6mm and 50mm while the iron core length and height were 10mm and 60mm. In

Fig. 7 we show both the vertical and horizontal force components which were

sampled at steps of 5mm. The continuous line represents the horizontal propulsion

component while the discontinuous curve is the vertical repulsive component. In the

second simulation only the rail geometry and currents are changed. We chose Imax =

1000 A and Imin = 253.3 A with a step of 53.3A. The gap is also taken to be 5mm and

the vertical head distance to be 20mm. The rail coils length and height are 10mm and

50mm and the iron core length and height are 30mm and 60mm respectively. In Fig.

8 we again show the resulting force components.

It is observed that while the propulsive term has an almost smooth piecewise

linear behavior the repulsive levitation component has a strongly oscillating

characteristic which can be a source of instability. This can be compensated by the

addition of a separate magnetic levitation line which does not add to the propulsion

effect but acts in combination with the ACS to regularize the behavior of the total

repulsive term. A compensator can be part of the ACS in order to fill in the

oscillating terms of the accelerating line by a complementary force so that their total

would remain almost constant.

The propulsion term can be simulated relatively well by a piecewise parabolic

potential like the one of Fig 9 which is obtained by interpolating the force

components with two straight lines and integrating. The slope is the same on both

intervals but with opposite sign and equal to 240 for the data of Fig. 7. For a real

launcher it would be necessary to incorporate several accelerating stages which

correspond to a periodic repetition of the potential of Fig 9.

Assuming that the model can be scaled up to a realistic vehicle with an

accelerating stage of 300m one can estimate the efficiency of a multi-stage system by

studying the motion in open air. Including air resistance and a small lifting angle for

the launcher ramp and taking into account a small gap between accelerating stages as

an arbitrary parameter we may write an equation of motion in the form

φρ sin2

1 2 Mgdz

dVvSK

dt

dvM d −−−= (4)

where ρ is the air density ~1.2kg/m3

at 25o C, S is the cross section area of the

vehicle towards the air, Kd the drag coefficient ~0.05 – 0.1 for aerodynamically

shaped objects, V(z) is a scaled up periodic version of the potential of Fig 9

including a gap parameter and φ is the ramp angle.

The above was solved numerically for a mass of 1Kg, S = 0.1 m2, φ = 30

o and

a gap of 10 meters between subsequent accelerating stages. It is also assumed an

arrow shaped object with a minimum drag coefficient 0.05 which gives a total of

0.006 in open air for the velocity coefficient in (4). The result is shown in Fig. 10

where the solid line represents position and the dotted one the velocity. It is shown

that the system has a maximum limiting velocity of ~2km/sec which can be reached

in less than 0.3 sec. At this point the arrow has reached almost 600 m which proves

that it is inefficient to use more than two accelerating stages in open air.

It might be possible to reduce the air resistance using additional aerodynamic

effects like the well known “groung effect” first used in the Russian ekranoplane and

do the installation inside a tunnel. Study of the overall behavior in such a system is

more complicated than our simplified model used here and lies beyond the scope of

the present report.

4. Conclusions

In this report we show that it is possible to construct electromagnet based

systems that can be used for propulsion with purely magnetic forces. We designed a

system more complicated than existing coilguns that can work on open air in a way

similar to magnetic trains. It was shown that such a system can comprise an amount

of upward levitating forces and horizontal propulsion forces. Based on this we gave a

simplified model of motion which shows that the system can achieve strong

accelerations for a limited length after which it becomes inefficient due to increased

air resistance. Possible improvements are proposed that require a more detailed 3-

dimensional simulation in a tunnel.

References

[1] J. D. Jackson, “Classical Electrodynamics” (1975) Wiley

[2] P. Hammond, J. K. Sykulski, “Engineeering Electromagnetism” (1994)

Oxford Univ. Press

[3] D. Roper, http://www.roperld.com/Science/LinearMotor.pdf

[4] IEEE Trans. Magnetics Special Issue 27 1991

[5] IEEE Trans. Magnetics Special Issue 29 (1) 1993

[6] IEEE Trans. Magnetics Special Issue 39 (1) 2003

[7] A R Hoyle et al 2000 J. Phys. D: Appl. Phys. 33 120-126

[8] Ian R. McNab, IEEE Transactions on Magnetics, Vol. 39, No. 1, January

2003.

[9] D. Wetz et al., Acta Phys. Pol. A, 115 (6) 2009.

[10] T. G. Engel et al., Power Modulator Symposium, 2006. Issue , 14-18 May

2006 405 – 410

[11] AIAA, SAE, ASME, and ASEE, Joint Propulsion Conference, 26th,

Orlando, FL, July 16-18, 1990, NASA Tech. Report Server (NTRS)

FIGURES

Fig. 1

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Fig. 9

Fig. 10