tromcondbly.with

lsive f

[1] bu

One of the important problems in developing the railgun is the design of the rail and the related elements. The current inthe rails and associated magnetic eld extends from the breech to the armature location. The temporal changes occur as thecurrent provided by the pulsed power system varies during the inbore cycle. The rail is subjected to a hostile load environ-ment. So, the rail should have high ability against these complicated loads [7,8]. The dynamic interaction between armatures

0307-904X/$ - see front matter 2011 Elsevier Inc. All rights reserved.

Corresponding author. Tel.: +86 335 8060195; fax: +86 335 8074783.E-mail address: xlz@ysu.edu.cn (L. Xu).

Applied Mathematical Modelling 36 (2012) 14651476

Contents lists available at SciVerse ScienceDirect

Applied Mathematical Modellingdoi:10.1016/j.apm.2011.09.0365.9 km/s were obtained using an arc as the driving armature. The results lead people to believe that higher velocities can beobtained and a lot of studies on the railgun were done. Fitch and Rose [2] investigated the effects of the current modes on thetransfer efciency, and found that the transfer efciency can be increased effectively when the current becomes zero at theshot exit. Thus, the rapid rise-constant-slow set current mode of the operating current was proposed. For the mode, the ef-ciency of the railgun system is relatively high, and a relatively large muzzle velocity of the projectile can be obtained for agiven rail length that is a proper operating current mode. For reducing energy loss, some electric sources were developed andthe distributed-current-feed and distributed-energy-store railguns were presented [36]. Using these electric sources, thebreech electric energy and the resistance energy loss are reduced, so the rail length can be increased. For these novel electricsources, the rapid rise-constant-slow set current mode is accepted.1. Introduction

For many years, the utility of elec(see Fig. 1). It consists of two parallelintegral part of the projectile assembetween the rails. This eld interactsjectile and produces a mutually repubecome an attractive approach.

In 1978, Rashleigh and Marshallagnetic guns has been explored. The railgun is a kind of the electromagnetic gunuctors rails across which an armature makes electrical contact. The armature is anWhen current ows in the circuit, a magnetic eld is established in the spacethe current to produce the Lorentz or J B force, which both accelerates the pro-orce on the rails. Velocities up to 6000 m/s have been reported, making the railgun

ilt an inductively driven rail-gun macroparticle accelerator in which velocities ofForces of rails for electromagnetic railguns

Lizhong Xu , Yanbo GengMechanical Engineering Institute, Yanshan University, China

a r t i c l e i n f o

Article history:Received 6 October 2010Received in revised form 28 August 2011Accepted 1 September 2011Available online 8 September 2011

Keywords:RailgunForceStressRailArmature

a b s t r a c t

In this paper, using electromagnetic theory, the equations for the total magnetic force onthe rails are deduced. Besides this, the equations for the changes of the magnetic forcealong with time and the running position of the armature under jointed pulse operatingcurrent are presented. Then, the equations for changes of the bending stress and shearstress in the rail along with time and the running position of the armature are given.Changes of the magnetic force distribution between the two rails along with the runningposition of the armature and other parameters are investigated. The total electromagneticforces on the rail under jointed pulse operating current are analyzed. Distribution andchanges of the bending stresses in the rail are studied. A number of results are obtained.The results are useful for design and application of the railgun system.

2011 Elsevier Inc. All rights reserved.

journal homepage: www.elsevier .com/locate /apm

Lewis [13]. Liudas et al. [14] developed a 2D plane stress nite element model resting upon discrete elastic supports and

1466 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476performed the transient analysis for a set of constant loading velocities (6001600 m/s) and for the experimentally derivedtransient-loading prole.

However, the electromagnetic loads on the rails were considered to be uniformly distributed in above-mentioned studies.Actually, the electromagnetic loads on the rails are not distributed uniformly, which causes a large peak load on the rails. Itinuences on the operating behavior of the railgun signicantly.

In this paper, using electromagnetic theory, the equations for the magnetic force between the two rails are deduced.The equations for the magnetic force applied to the rail by the armature are given as well. From them, the equationsfor the total magnetic force on the rails are obtained. Here, the non-uniform distribution of the electromagnetic loadson the rails is considered. Based on these equations, the equations for the changes of the magnetic force along with timeand the running position of the armature under jointed pulse operating current are given. Then, the rails are considered asan elastic foundation beam and the equations for the changes of the internal forces and the displacement of the rail alongwith time and the running position of the armature are obtained. Besides this, the equations for the bending stress and theshear stress in the rail are given. Using these equations, the changes of the magnetic force distribution between the tworails along with the running position of the armature and other parameters are investigated. Distribution of electromag-netic forces to the rail applied by armature is investigated. Then, the total electromagnetic forces on the rail under jointedpulse operating current are investigated. The distribution and changes of the bending stresses in the rail along with sys-tem parameters are analyzed. A number of results are obtained. The results are useful for design and application of therailgun system.

2. Magnetic forces on rails and armature

As shown in Fig. 1, two copper rails are arranged in parallel. The rails are copper strips h w and of lengths L. Thedistance between the two copper strips is b. The armature together with the projectile can run along the rails. The currentowing in the rails produces a magnetic ux density B between the rails, and this magnetic eld interacts with the currentowing in the armature. The resulting Lorentz force F accelerates the armature together with the projectile along the rails.and rails at this contact interface presents a great challenge to all scientists working in this interdisciplinary research area.The armature performance is sensitive to variations of the normal forces at the contact interface. Therefore, the evaluation ofthe loads on rails is an important task. An analytical approach to investigation of the dynamic response of laboratory rail-guns, including projectile movement, was developed in a number of works by Tzeng [9] and Tzeng and Sun [10]. The tran-sient elastic waves in electromagnetic launchers and their inuence on armature contact pressure were studied by Johnsonand Moon [11,12]. The transient resonance at critical velocities of the projectile was numerically studied by Nechitailo and

Fig. 1. Schematic of railgun and its magnetic eld (a) schematic of railgun (b) geometric parameters and coordinate system.Besides this, the two rails interacts with each other as well because of the currents owing in them and the correspondingmagnetic eld.

For the simple rails, the current in the rails is known to localize near their surfaces. It is known as skin effect. Because ofthe effects, a large current is required for a given launch velocity. Meanwhile, the heating loss of the electric resistance for therail is quite large, and the electrical source for producing large current is also difcult to make. Hence, a laminated rail isproposed. The laminated rail consists of several thin rails with the insulating layers between them. For the rails, the currentrequired is only 1/N of the current for simple rails under the same launch velocity (here, N is the number of the thin rails). Itcan reduce the heating loss of the electric resistance for the rail, and the uniform current density is obtained in each thin rail.So, the laminated rail is widely used for the railgun. The studies are for the railgun with laminated rail, and the current in therail is considered to be distributed uniformly.

The magnetic force between the two rails are analyzed as shown in Fig. 2. Let P0(z0,y0) denote one point on the left rail. Thecurrent element at the point can be expressed as dI Ihw dr0 Ihw dz

0dy0. Let P(z,y) denote one point on the right rail. Thus, thecomponents of the magnetic eld vector at the point P(z,y) caused by current element dI at the point P0(z0,y0) are given asbelow

whereFro

given.The ot

whereTh

currenper un

whereFro

Fig. 2.(c) mag

L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476 1467dBy l0Izz0

4phwyy0 2zz0 2 x

yy0 2zz02x2p xl

yy02zz0 2xl2p

dr0;

>>>:1

l0 is the permeability, I is the current intensity in the rail, l is the running position of the armature.m Eq. (1), the components of the magnetic eld vector at the point P(z,y) caused by the current I in the left rail can bedBz l0Iyy0

4phwyy0 2zz0 2 x

yy0 2zz02x2p xl

yy02zz0 2xl2p

dr0;

8>>>

4ph w D D y y z z y y0 z z0 x2 y y0 z z0 x l

Letthe pocan be

whereOw

the to

2ph wc X D0 y y0 z z0 0 2 0 2 2 0 2 0 2 2>: >;

where

2 3

rail. It

Kq

3. Ma

1468 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476Three modes of the operating current were proposed. The rst is the constant current mode, the second is the rapid rise-slow set current mode, and the third is the rapid rise-constant-slow set current mode. For the rst mode, a large muzzlegnetic forces under jointed pulse operating current zy y02 x x02 z2

q z b2wy y02 x x02 z b 2w2

q64 75dy0dx0dydz:y y0 z z0 x2 y y0 z z0 x l

l04ph2wc

Z b2ww

Z h=2h=2

Z c0

Z h=2h=2

x x0y y02 x x022 34ph2w2 b h=2 0 h=2 y y02 z z02

x2 2

q x l2 2 2

q264

375dy0dz0dydzl0Z bw Z h=2 Z w Z h=2 z z0wherecan be calculated as

q qrr qar Krr KarI2 KqI2; 8Thus, the total magnetic forces applied to each rail include the magnetic force from the armature and one from anothery y02 x x02 z2 y y02 x x02 z b2

zq z bq64 75dy0dx0dydz:qar KarI2; 7

Kar l04ph2wc

Z bwb

Z h=2h=2

Z c0

Z h=2h=2

x x0y y02 x x02y y z z x y y z z x l

In a same manner, the magnetic forces applied to the rail by the armature can be calculated as well. It isKra l02Z Z z z0

2 2

xq x lq64 75dr0>< >=

dv :where 2 38 9P(z,y) denote one point on the armature (see Fig. 2(c)). The equation for the component Byof themagnetic eld vector atint P(z,y) caused by current I in the left rail is identical to Eq. (2). The current element at the point P(z,y) on the armatureexpressed as dI Ihc dxdydz. Thus, the magnetic forces applied to the armature by the left rail can be obtained

F left l0I2

4ph2wc

ZX

ZD0

z z0y y02 z z02

xyy02 z z02x2

q x ly y02 zz02 x l2

q264

375dr0

8>:

9>=>;dv ; 5

the symbol X denotes the zone: (l c) 6 x 6 l, (h/2) 6 y 6 (h/2) and 0 6 z 6 b; dv = dxdydz.ing to the symmetry, the magnetic force applied to the armature by the right rail is equal to one by the left rail. Thus,tal magnetic force applied to the armature by the two rails can be given as

Fa F left Fright 2F left KraI2; 6where

Krr l02 2Z Z

0

z z00 2 0 2

x2 2

q x l2 2 2

q264

375dr0dr:

veloci

projectivelyopera

No

I I0e t2 6 t 6 t3;

8

whereFro

2 2 28

L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476 1469Fa1 Kq1I1 Kq1I0 sin xt 0>>:

10

where

K l0Z bw Z h=2 Z w Z h=2 z z0ThTh

wherethe toI0 is a constant current, s is the time constant, x is the harmonic frequency of the electric source, t is time.m Eqs. (6) and (9), the total magnetic force Fa applied to the armature can be givenI I0 sinxt 0 6 t 6 t1;I I0 t1 6 t 6 t2;

t=s

>: 9tile cannot be obtained easily for a given rail length. For the third mode, the efciency of the railgun system is rela-high, and a relatively large muzzle velocity of the projectile can be obtained for a given rail length. So, it is a properting current mode.rmally, the third operating current mode consists of multiple pulse currents (see Fig. 3). It can be written asthe railgun after shot exit, so the efciency of the railgun system is relatively low. For the second mode, the breech electricenergy vanishes after shot exit, so the efciency of the railgun system is relatively high, but a large muzzle velocity of thety of the projectile can be obtained easily for a given rail length, but about 50% of the breech electric energy remains in

Fig. 3. Jointed pulse operating current.qi4ph2w2 b h=2 0 h=2 y y02 z z02

xy y02 z z02 x2

q x xiy y02 z z02 x x12

q264

375dy0dz0dydzdx

l04ph2wc

Z b2ww

Z h=2h=2

Z c0

Z h=2h=2

x x0y y02 x x02

zy y02 x x02 z2

q z b 2wy y02 x x02 z b 2w2

q264

375dy0dx0dydz; i 1;2 and 3:

e motion position xi (i = 1, 2 and 3) of the armature can be calculated as below:e differential equation of the armature motion is

md2x

dt2 f dx

dt Fa 0; 11

m is the mass of the armature and projectile, f is the viscous friction coefcient between the rail and the armature, Fa istal magnetic force applied to the armature.

Substituting Eqs. (6) and (9) into (11), yields

md2x1dt2

f dx1dt KaI20 sin2xt;md

2x2dt2

f dx2dt KaI20;md

2x3dt2

f dx3dt KaI20e2ts :

8>>>>>:

12

The initial conditions are x1jt=0 = 0 and dx1dtt0

0, using these initial conditions, the solutions of Eq. (12) can be given asbelow

x1 KamI20

2f 2 KamI202f 2

KamI202f 28x2m2

e

fmt KaI202f t

KamI202f 28x2m2 cos 2xt

fKaI20

4xf 216m2x3 sin 2xt;

x2 mKaI20

2f 2 mKaI20

2f 28x2m2

epf

2mx 2Kam3 I20x2f 44f 2x2m2h i

efmt KaI20f t

pKaI204fx

mKaI20f 2 ;

x3 mKaI20

2f 2 mKaI20

2f 28x2m2

epf

2mx 2Kam3I20x2f 44f 2x2m2h i

efmDt2 2m2KaI20

2mf 2f 3s

n oe

fmt Kas2I204m2fs e

2ts KaI20Dt2f pKaI04fx

KasI202f :

8>>>>>>>>>:

13

4. Internal forces and stresses in the rail

The electromagnetic forces on the rails of the railgun act as an outward, mutually repulsive load. The rails are held byinsulating pieces of asbestos-reinforced resin contained inside a steel tube. The insulating pieces can be considered as anelastic body. Hence, the rails can be considered as an elastic foundation beam (see Fig. 4). The distributed electromagnetic

Th

d v qx

whereTh

Let P denote a concentrated force at the origin of the axis x, using the boundary conditions of the rail, and the displace-ment

bx

1470 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476Fig. 4. Forces on the rail when armature is running.v 2k

e cos bx sin bx: 16

From Eq. (16), the other internal forces and displacements can be obtainedv of the rail can be given

Pbv ebxC1 cos bx C2 sinbx ebxC3 cos bx C4 sinbx: 15v is the displacement of the rail in the direction y, b is a coefcient, b 4EIz; q(x) is the distributed load on the rail.e general solution of Eq. (14) isdx4 4b4v

EIz; 14

k4

qe force balance equation of the rail is

4forces are applied to the rail which is supported by an elastic foundation with stiffness k. E is the modulus of elasticity of therail material, Iz is the sectional modular of the rail.

8For

Fro

(i)

z

(ii)

l w24y2qn>

5. Res

Usthe ru

railgun, the magnetic force is quite small, it grows obviously with increasing coordinate x, it gets to the maximum

(2)

L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476 1471value near half of the running distance of the armature, and then it drops and becomes quite small near the armature.As the running distance of the armature increases, the distribution of the magnetic force on the rail does not changes.It is still quite small near the entrance of the railgun or near the armature. However, the maximum magnetic force onthe rail grows slightly.forces between the two rails along with parameters w and b. Here, l0 = 4p 107 H/m, c = 10 mm, L = 1 m. Figs. 5 and 6show:

(1) The magnetic force between the two rails only occurs after the running armature. It does not exist in front of the arma-ture. On the partial rail with the magnetic force, the magnetic force is not distributed uniformly. At entrance of theults and discussions

ing above equations, the changes of the magnetic force distribution between the two rails along the parameter h andnning position of the armature are investigated (see Fig. 5). Fig. 6 shows changes of the distribution of electromagnetic 0 16Iz ebsn cosbs ndneblsxcosbl s x sin bl s x::sinbs ndneblsxcosbl s x sin bl s x;s R lx0 w24y2qlxn16Iz ebn cos bndn

R x0

w24y2qlxn16Iz

ebn cosbndn

R l0 w24y2qn16Iz ebsncos bs n sinbs ndneblsx sin bl s xR

>>>>>>>>>>>>>

21r R x0 yqlxn4bIz ebncosbn sinbndnR lx0

yqlxn4bIz

ebncos bn sin bndn ybIz

R l0

qn2 e

bsn cos bs ndneblsx sin bl s x R l0 qn4 ebsncos bs nn

8>>>>>>>>>>> R x0 yqlxn4bIz ebncos bn sin bndnR lx0

yqlxn4bIz

ebncosbn sinbndn;s R lx0 w24y2qlxn16Iz ebn cos bndn

R x0

w24y2qlxn16Iz

ebn cosbndn

R l0 w24y2qlxn16Iz ebncos bn sin bnh i

dnebx sin bx

R l0 w24y2qlxn16Iz ebn cos bnh i

dnebxcosbx sinbx:

>>>>>>>>>>>>>>>>>>:

20

For the points after the running armature (0 < x < l)r ybIR l0

qlxn2 e

bn cosbnh i

dnebx sin bx R l0 qlxn4 ebncos bnh

sinbndnebxcosbx sin bxn o8>>>the point after the running armature (0 < x < l), the internal forces in the rail are

v R x0 bqlxn2k ebncosbn sinbndn R lx0 bqlxn2k ebncos bn sin bndn;M R x0 qlxn4b ebncos bn sinbndn R lx0 qlxn4b ebncosbn sinbndn;Q R lx0 qlxn2 ebn cosbndn R x0 qlxn2 ebn cos bndn:

8>>>:

19

m Eqs. (18) and (19), the equations for the bending stress and the shear stress in the rail can be given as below:

For the points in front of the running armature (x > l)h dydx Pb2

k ebx sin bx;

M EI d2ydx2

P4b ebxcosbx sinbx;Q EI d3y

dx3 P2 ebx cosbx;

>>>>>:

17

where h is the rotation angle of the rail section,M is the bending torque on the rail section, Q is the shear on the rail section.As above stated, the distributed electromagnetic forces are applied to the rail. The force distribution changes as the arma-

ture runs. It occurs after the running armature. Based on Eqs. (16) and (17), the internal forces in the rail caused by the run-ning distributed electromagnetic force can be obtained by means of superposition method.

For the point in front of the running armature (x > l), the internal forces in the rail are

v R l0 bqn2k ebxlncos bx l n sin bx l ndn;M R l0 qn4b ebxlncos bx l n sinbx l ndn;Q R l0 qn2 ebxln cos bx l ndn:

8>>>:

18

1472 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476(3) As the running distance of the armature increases further, the acting range of the large magnetic force on the railgrows and the distribution of the large magnetic force is nearly uniform.

(4) As the height of rail increases, the magnetic force drops at the same current. As the width of rail increases, the mag-netic force drops at the same current. As the distance between two rail increases, the magnetic force drops at the samecurrent. For the different height of rail, the effects of the change of the height of rail on the magnetic force are nearlyidentical. As the width of rail increases, the effects of the width change of the rail on the magnetic force become smal-ler and smaller. As the distance between two rail increases, the effects of the distance change on the magnetic forcebecome smaller and smaller as well.

The distribution and changes of electromagnetic forces to the rail applied by armature are investigated as shown in Fig. 7.It shows:

Fig. 5. Distribution of electromagnetic forces between the two rails.

Fig. 6. Changes of the distribution of electromagnetic forces between the two rails along with w and b(l = 1 m).

L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476 1473(1) The magnetic force to the rail applied by armature only occurs after the running armature as well. It does not exist infront of the armature. At entrance of the railgun, the magnetic force is quite small, it grows slowly with increasingcoordinate x. As the position is near the armature, the magnetic force grows quickly.

(2) As the running distance of the armature increases, the distribution of the magnetic force on the rail does not changesas well. The magnetic force grows quickly as the position is near the armature. However, the length of the partial railwith the small magnetic force increases. The amplitude of the electromagnetic force does not change.

(3) As the height h of the rail increases, the magnetic force to the rail applied by armature grows obviously near the arma-ture, and then it drops slightly. However, it is not nearly inuenced at other place of the rail by the height h.

(4) As the distance b between two rails increases, the magnetic force to the rail applied by the armature grows. Theincrease along with distance b is less obvious than that along with the height h near the armature. However, it is moreobvious than that along with the height h at other places of the rail.

(5) As the width w of the rail increases, the magnetic force to the rail applied by the armature grows. As the width c of thearmature rail increases, the magnetic force to the rail applied by the armature drops. The effects of the width c on themagnetic force are more obvious near the armature.

Distribution and changes of the total electromagnetic forces to the rail applied by the armature and another rail are inves-tigated (see Fig. 8). Fig. 8 shows:

Fig. 7. Distribution and changes of electromagnetic forces to the rail applied by armature.

1474 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476Fig. 8. Distribution of the total electromagnetic forces on the rail (l = 1 m).(1) The total magnetic force to the rail applied by the armature and another rail only occurs after the running armature aswell. It does not exist in front of the armature. At entrance of the railgun, the magnetic force is quite small, it growsquickly with increasing coordinate x, and gets to a steady value and remains constant, which is mainly dependent onthe magnetic force applied by another rail. Near the armature, the magnetic force grows quickly, and then it dropsslightly, which is mainly dependent on the magnetic force applied by the armature.

(2) The effects of the height h and the width w of rail on the total magnetic force are obvious. Under the constant current,the total magnetic force grows with decreasing the height h and the width w.

(3) The effects of the distance b between two rails on the total magnetic force are the most obvious. Under the constantcurrent, the total magnetic force grows with decreasing the distanceb. When the distance b is equal to 10 mm, thedistribution of the total magnetic force on the rail changes. Near the armature, the magnetic force does not growquickly, but drops quickly, which is mainly dependent on the magnetic force applied by another rail because of thesmall distance b.

(4) The effects of the width c of the armature on the total magnetic force are quite small at places distant from the arma-ture. The width c has large effects on the magnetic force on the rail near the armature. Under the constant current, thetotal magnetic force grows with decreasing the width c.

The total electromagnetic forces on the rail under jointed pulse operating current are investigated (see Fig. 9). It shows:

(1) At the initial state (t = 0), the electromagnetic forces at any point on the rail are all zero. At stage one (rapid rise stageof the current), the electromagnetic forces on the rail grow quickly with increasing time t. At stage two (constant stageof the current), the electromagnetic forces on the rail remain nearly a constant value with increasing time t. At stagethree (slow set stage of the current), the electromagnetic forces on the rail drop slowly with increasing time t. Aftershot exit (t = 2 ms), the electromagnetic forces at any point on the rail become zero again.

(2) The magnetic forces at the entrance of the railgun are zero at initial state (t = 0), But they are not always zero. As time tincreases, they grow, get to a steady value, and then drop slowly. Of course, they are smaller than those at other pointson the rail at the same time.

(3) The magnetic forces near the armature are the maximum in the rail at the same time. However, they are also zero atinitial state (t = 0) or end state (t = 2 ms).

(4) At the end of the stage two, the electromagnetic force on the rail near the armature is the maximum. It should be takenas the dangerous point for strength design of the rails.

L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476 1475Fig. 9. The total electromagnetic forces on the rail under jointed pulse operating current.

Fig. 10. Distribution of the bending stresses in the rail.

Distribution of the bending stresses in the rail is investigated (see Fig. 10). It shows:

(1) The bending stresses in the rail caused by the magnetic force occur both after the running armature and in front of it.Although, the magnetic force does not exists in front of the armature. At entrance of the railgun, the bending stressesare quite small, it grows quickly as the position is near the armature, and in front of the armature the bending stressesbecome negative and grows quickly in the negative direction. After that, the bending stresses drop quickly and becomezero.

(2) The positions of the positive and negative peaks of the bending stress change along with motion of the armature. Thepositive peak always occurs after the running armature. The negative peak always occurs in front of the running arma-

1476 L. Xu, Y. Geng / Applied Mathematical Modelling 36 (2012) 14651476ture. As the running distance of the armature increases, the positive stress peak grows. It gets to the maximum nearexit of the railgun.

(3) As stiffness k increases, the bending stresses in the rails drop. It is proper that modulus of elasticity for the insulatingpieces contained inside a steel tube should be taken as large as possible.

(4) As the height h and the width w of rail increases, the bending stresses in the rails drop as well. It means that largerheight h and the width w of rail should be taken for decreasing bending stresses in the rails. However, it will causeincrease of the railgun dimension. Hence, height h and the width w of rail should be taken as moderate values forgetting proper size and strength of the railgun.

6. Conclusions

In this paper, the equations for the total magnetic force on the rails are deduced. Besides this, the equations for thechanges of the magnetic force along with time and the running position of the armature under jointed pulse operating cur-rent are presented. Changes of the magnetic force distribution between the two rails along with the running position of thearmature and other parameters are investigated. The total electromagnetic forces on the rail under jointed pulse operatingcurrent are analyzed. The distribution and changes of the bending stresses in the rail are studied. A number of results areobtained. The results show:

(1) The total magnetic force on the rail only occurs after the running armature. At entrance of the railgun, the magneticforce is quite small. Near the armature, the magnetic force grows quickly. The total magnetic force depends on thedistance between two rails, the height and the width of rail.

(2) At the end of the stage two (constant stage of the current), the electromagnetic force near the armature is the max-imum. It should be taken as the dangerous point for strength design of the rails.

(3) The bending stresses in the rail caused by the magnetic force occur both after the running armature and in front of it.The positive peak always occurs after the running armature. As the running distance of the armature increases, thepositive stress peak grows.

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