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Finite element analysis of the magnetomechanical behavior of a toroidal coil with magnetic mirrors
Nuclear Engineering and Design 74 (1982 1982) 339-346 North-Holland Publishing Company
FINITE ELEMENT ANALYSIS OF THE MAGNETOMECHANICAL OF A TOROIDAL COIL WITH MAGNETIC MIRRORS Kenzo MIYA, T0shiyuki TAKAGI
BEHAVIOR
and Tohru TAKAGHI
Nuclear Engineering Research Laboratory, University of Tokyo, Tokai-Mura, lbarakL Japan Received January 1983
A toroidal ~roidal simulator is used to study the magnetomechanical behavior of a superconducting D-shaped coil s~ set at the cente between m sectored ferromagnetic plates. Since ince the magnetic field distribution between the two plates is expected ~ected t~ to be almost th~ is that created by toroidal field coils ils arrayed in a torus, the toroidal simulator may be expected to show almost al the sam~ same as magnetomechanical ~tomechanical behavior as that of the full torus coils. To verify its validity experimentally, buckling and vibration test, arried out in a liquid helium cryostat. ~tat. In addition numerical analyses were performed for the magnetic field fie distributior were carried to reveal :al its similarity and for the experimental "imental results from the toroidal simulator to confirm its validity. Its validity is full3 demonstrated in the present study.
1. Introduction Magnetoelastic coupling phenomena between current carrying elastic bodies and electromagnetic field have become an important problem in engineering fields of magnetic fusion reactor technology, especially in the superconducting coil engineering [1]. The instability of the toroidal field coils among the problems associated with the coil engineering was first investigated by Moon et al. [2], [3] and then by Miya et al. [4], [5]. Studies by M o o n et al. were rather presented with an emphasis on the experimental demonstration although the modal analysis of the data was performed. Those b y Miya et al. aimed to predict the experimental results with application of the numerical tool of finite element method, The prediction was successfully achieved with an introduction of the theory of orthotropy [6] to an evaluation of the elastic stiffness and with a consideration of electric work due to the directional change of a current density vector caused by a rotation of the coil. Akin et al. [7] showed the numerical solution for the magnetoelastic buckling of a current carrying straight beam conductor placed at the center between two parallel beam conductors. In the problem the two side conductors are fixed rigidly and the center conductor is partially free to deflect. This problem can bee solved analytically and the buckling currents calculated tted for rough mesh divisions by Akin et al. converge to the exact 0029-5493/83/0000-0000/$03.00
solution obtained by us [8]. The prediction 9rediction of magneto elastically coupled vibration of the toroidal toroit coils wa~ accomplished by combining the Fast Lapla )lace Transforn with the finite element scheme [9]. In these thes studies, th~ o magnetic stiffness, which is a natural consequence co~ mutual inductance between coils, plays an essential rol~ for understanding of the magnetoelasticall netoelastically interactin~ ~lementation by th~ phenomena and for a numerical implemen finite element formulation. In the present study the validity of the toroida simulator was investigated as to whether it does or doe~, not give the same magnetoelastic behavior avior as that of th~ toroidal field coils arrayed in a torus whil while the experi. mental verification of its validity was dem demonstrated b3 M o o n [10] with buckling and vibration tes tests. Here performed are the numerical analyses for a r magnetic r fielc distribution between the two ferromagneti tgnetic plates anc verify further the for the buckling and vibraton tests to verit economial experiment, validity. F r o m a viewpoint of an economia applicability foJ such a simulator could be of potential apl: char. an investigation of the instability and vibration vit acteristics of symmetrically arrayed coils as seen in there toroidal and solenoidal coil systems. However, He might be a restriction as to a magnitude of the saturated magnetic induction of the ferromagnetic p31ates because the plates cannot serve as a magnetic n symmetric field when the field intensity ris~ the saturated field of the plate.
© 1983 N o r t h - H o l l a n d
K. Miya et aL/ Magnetomechanicalbehaviorof a toroidalcoil
340
SUS Bucking~
2. Theoretical background of magnetoelasticity of toroidai
simulator
A distribution of toroidal magnetic field created by the eight-coil full torus is shown schematically in fig. 1a, where the magnetic field by poloidal coils is not considered. As is evident from the figure, there are eight fold symmetries with eight radial symmetric planes, of which two planes, making an angle of 45 °, are drawn in the figure. Thus there could be a prediction that the magnetic field generated between the two planes is simulated by one coil set at the center between two ferromagnetic plates as shown in fig. lb. The ferromagnetic plates in the toroidal simulator serve as a magnetic mirror of the contour lines of magnetic induction because they enter normally to the plates. Top and elevated views of the toroidal simulator used in the present experiment are shown in fig. 2. The ferromagnetic plates are rectangular and were manufactured from mild steel of a permeability of 7000 in MKS units. Three-dimensional analysis is, in fact, necessary for an exact evaluation of the magnetic field between two ferromagnetic plates but actually too complex to perform because a very wide analysis region must be provided including the air space outside the set. Thus two-dimensional analysis was performed for the first approximation of the field distribution assuming that the magnetic field is similar to one created by two straight coils of infinite length between two ferromag-
~ ~
-~
"~
~
~2L'~
/2Z.5"/
" ~ " -~?-~0 ~ , ~.. ~ -,,~.~ 15 . . . . . . ~ ) ~ - . ---~_, , -w
! 135
Fig. 2. Top view of toroidal simulator and elevated view of D-shape coil.
netic plates of finite width and infinite length as is done by Moses and Young [ 11]. The finite element analysis of the two-dimensional field distribution is very common [12,13] and was carried out in the previous paper [14]. The method developed in [14] can be applicable to the present two-dimensional problem, too.
45*
(b)
Ferromagnetic ~plote
~..
Toroidol coil
Flux lines by toroidol coils
coil
,.~/~~Test
2.1. Finite-element analysis of magnetic field and coupled vibration
(a)
~r~
Toroidal simulator
Fig. 1. Schematic distribution of toroidal field bby an 8 coil full torus with ferromagnetic plates.
K. Miya et al. / Magnetomechanical behavior of a toroidal coil
Maxwell's equations determining the static magnetic field is described simply as (see ref. [14] for the complete derivation) 1 rot(rot .4 ) = 0,
(1)
if the vector potential A is used. Here # is the permeability ( = 4¢r × 10 -7 for the air space and = 7000 for the ferromagnetic plate). The magnetic induction B can be calculated by B = rot A.
(2)
A solution of eq. (1) is obtained with an application of finite element method where the two-dimensional quadrilateral simplex element is adopted for minimization of a functional of eq. (1). The magnetic induction B is computed from eq. (2) after the vector potential A is solved in the region of the space and the ferromagnetic plates, As is well known, the equation of the coupled vibration of the current carrying coils is a natural consequence of the Lagrange's equation which is expressed
The first and second terms of this equatior uatlon are conventional for the finite element implementatiol )lementation [16] and sc are not formulated furthermore here. The third one is dependent on the relative movements betw~ between coils and is calculated below.
2.2. Calculation of perturbed magnetic energy Although the evaluation of the third terr term in eq. (6) i., elegant evaluagiven in the Appendix in ref. [4], a more el tion of the term is shown in the following. IIn general th~ shown in fig.." magnetic energy created by two coils as sh is convieniently evaluated by
t ~ o [ ( j o ) d S ( 1 ) f f j t(~2 ) d S (2), JJ D
where superscripts refer to the coil numbei number 1 and 2 an( dSU) = dx~i)dx(i), jU) is the current densi density of the itt coil and D is a distance between two [~oints of thi respective coils after deformation. It is approximate( with
by,
1
1
d oL - ~ ~w j - ~w = F,
z,
Ro
(3)
where L is the Lagrangian of the present system given by eq. (4), w is a lateral deflection of the coil and F is an external force acting on the coil. The Lagrangian is constructed like, L = T - U + W,
(4)
where
T=½ffo(c~)2hdxdy, u = ½ff(,)T(o)dxdy,
I ('"
.:lr[ /~\ 4-5X,10] I * \ / x(t) 2=
[
(5b)
w )//~/
. (21 X2
12l
R0
pt~,l~
X(21
X'''j , X3
1 0
Owff AJhdxdy=°
°
(6)
")'
where R 0 is a position vector between the 1two points ol the coils before they deflect and • is a difference o displacement vectors of the points. Eq. (~ (8) can be de rived by either Taylor's expansion or Le~gendre poly nomial. Considering a special relation between twt Cartesian coordinates, x~ l) and x[ 2), wh which make ai
In t h e above expressions, T is kinetic energy, U is strain energy and W is magnetic energy; h, #, (c) and (o) are the thickness, the displacement velocity, generalized strain and stress, respectively. A substitution of eqs. (5a), (5b) and (5c) into eq. (4) and then the eq. (4) into eq. (3) yields the following
2
1 (Ro.r)+3
(81
(5a)
ffo ,hdxdY+½ ff{,:(o}dxdy
(71
4¢r J J
W(12) =
Fig. 3. Coordinates systems set to toroidal coils.
K. Miya et al. / Magnetomechanicalbehavior of a toroidal coil
342
angle of O as shown in fig. 3, the following is obtained:
R o • r = - w t l ) x l 2) sin 0
+
W(2)Xl 1) sin O,
(9)
where w ~) is a lateral deflection of the ith coil. A substitution of eq. (9) into eq. (8) yields 1
1 xl 2) x} ~) = R0 + wl ~ - sin 0 - w 2 ~ - sin 0 Ro R0 v(2). ,.(2) 1 ] 3 ~1 ~ l sin20_ w¢~)wt~j 2 R~ 2R~ J
+
3x}l)" x}Z) sin20 ) w(l)w(2 )
COSO
+
R3
R~
3. Experimental and numerical results
3 x~l)'xl 1) +
1 ) sin20 - 2R 3 / wt2)w t2),
2 R~00
(10)
where the following relation is used:
r = ( - w ~2) sin O, 0, w ~2) cos 0 - w~l)).
(11)
The magnetic energy W (~2) is calculated as follows with substitution of eq. (10) into eq. (7), W(12)_ W(12)+ W(I2)+ W2(IZ)+ W(13) + W4(12)+ W5(12) ' where (12) j~2)
W,(12)_ lit0 f f j o ) d S t l ) f f _ _ d S t 2 0
- ~
m(I2) =
J
J J
)
J J R0
'
t~° f f w " ) J C " d S t ' ) f f x l 2 ' J t 2 ) d S ' 2 ) s i n 4~r J J J J R3o
0,
x(l)j( I )
W~,Z)= _ ~ O f f w ( 2 ) j t 2 ) d S , 2 ~ f f ~ d S ~ , ) s i n O 4~r J J 14ztl2)- tz° f fw(1)w(1)j(l)dS 0) 3 - 4~ J J x
Wl~12~ and W~ 12) in eq. (12) yield electroma gnetic forces acting on No. 1 and No. 2 coils, respecti~ ;pectively. For the these toroidal coil system being symmetrically arrayed, a forces cancel out because W,,(k'k+:)= - W~(k.k-t) ~ ( i = 1, 2) for two pairing coils [No. (k + l) and No. N (k - l)] as to No. k coil. Wot~2) is the magnetic energy stored in the system before deflection occurs and does not n, contribute to the electromagnetic force. Although eq. eq (12) is the expression for the magnetic energy created by No. 1 and of No. 2 coils, it can be applied to any two combination co the toroidal coils by changing the angle 0 when summed up for the total magnetic energy.
ff(3xl2~xl 2~_2R5o
sin20 _ 2R31 ) j(2)d S(2,
'
3.1. Experimentalprocedure simula The top view of the toroidal simulator and the ~erconducting coil are elevated view of the small superconduct for the shown in fig. 2. The coil was fabricated newly n present experiment almost in the same waly as the coils used in the previous experiments. The angle made by the two ferromagnetic plates is 45 °, which whick means that the toroidal coil is a simulator of the eight-coil full torus. A size of the square plate is 250 mm × 250 mm and its thickness is 15 mm. The test coil was w~ secured to the wedge at the center of the experimental apparatus. The apparatuses with and without a lateral lateral support are also shown in figs. 4a and 4b, respectively. Three experiments of the buckling test, two vibration tests of the first and second lateral modes, were perfc ,erformed in the liquid helium cryostat. The second lateral mode of late~ vibration was caused by inserting a lateral support at the center of the coil in the opposite side of the wedge. As were used in the previous experiments foil type strain gages for low temperature measurement were measu~ cemented to the coil for a detection of the bending strain through a X Y chart recorder for the buckling test and a synchroscope for the vibration tests.
3.2. Experimental results and numerical analysis
f f w'2'w'2>'r2'ds':'
= It°4~
-~x ~ ×
2R 5
W5~'z)= ~2_ f f
1
j(~dS(1),
s i n 2 0 - 2R~
w~',J~l)dS~') d
×
R~
Rs0
sin20 w~2)J(2)dS(2)"
The experimental result of the buckling test is shown in fig. 5 where the bending strain and the coil current are plotted in the ordinate and the abscissa, respectively. A close examination of the buckling curve clarifies that it is approximately a hyperbola not a parabola. If it were the parabola, the curve might be called a deflection one which could be caused by a si~nifican! amount of initial misalignment. Since the 1: rent is not determined clearly from the curv
K. Miya et aL / Magnetomechanicalbehavior of a toroidal coil
~o 400
=
2oo
~
3oo1-~ 1
o/ o
")~L
Bendingslrolnlcurrenlscpoled,xlO(Amps}/ SouthweUplot for n~gnetoeloslicbuck~,//~
/
Lvv /
F /
0~)
~
10
, 005
, i o,
I
I
/
/
I
I
I
20 :50 40 50 60 70 Toroid(]l field coil current , I , Amps/lum
Fig. 5. Buckling curve of toroida] simulator.
i¸¸¸¸¸
~
76.0 A / T . The buckling current obtain o b t a i n e d from th( experiment of the eight-coil full torus w~ was 78.4 A / q which was reported in [5]. Since b o t h the bucklin l currents can be thought in a fairly good aagreement, wt conclude that the toroidal simulator can rreproduce ac curately the buckling b e h a v i o r of the full torus coils it the pairing mode. The two-dimensional analysis of the rI magnetic fiel( was conducted as the solution of eq. (1) fo for a n u m e r i c a verification of the magnetic field coinci& coincidence betweet the toroidal simulator a n d the full torus co coils. A graphic display of the c o n t o u r lines inside the two tferromagneti~ plates is shown in fig. 6. As m e n t i o n e d b~ before the tw( straight coils carry currents of opposite ddirection witl each other a n d are located at two island islands on the lin, A - A ' in fig. 6. A m a g n i t u d e of the current is 1900 A. A is well k n o w n a n d shown later the magne netic field nea the inside coil is the highest. In fig. 7 iis shown th~ magnetic field distribution on the line A - A ' with th, d o t t e d curve. O n the other h a n d a n accura accurate solution o the magnetic field is given by Moses and Y o u n g [ 11 ] fo N pairs of coil arrayed in a torus by
B(R) Fig. 4. Photographs of the toroidal simulator: (a) for the first mode of vibration, (b) for the second mode of vibration. The small coils shown are copper vibration exciting coils,
well plot [17] of the curve was utilized a nld d is s h o w n in urately in this the same figure. T h e buckling current, accuratel case, can b e d e t e r m i n e d from a tangent of the line as
=
#oNl( i 1 ) + ) N ~ 1+ ( R / R , ) N - 1 ( R 2 / R - 1 '
(13 where R 1 a n d RE are distances from the center of th, torus to the inside a n d outside straight coils, respec tively. 1 is the total current carried by the coil an( /~0 = 4~r × 10 -7. T h e field calculated by eq. (13) is showl ~r~ec~n wlth th, with the solid line in fig. 7 for compari c o m p u t e d field in the toroidal simulator. " field o n the axis between the two coils of t
344
K. Miya et a L / Magnetomechanical behavior of a toroidal coil
A'
,0t
)02
A Fig. 6. Magnetic field distribution in toroidal simulator.
in a rough agreement while those in the air space between the two coils show discrepancy with each other, This discrepancy may be attributed to the finite size of the ferromagnetic plate. However, the rough agreement inside the conductors can be the reason for the fairly good agreement between the buckling currents from the toroidal simulator and the full torus. The validity of the toroidal simulator was also con-
0.125
.~
o, loc
- -
Moses ond Young
.....
Numericol
results
for toroidel sirnulef,
0075 '~ 0.0 5 C "'
0025
=
o.ooo---
firmed from the vibration tests. An example exa of the dynamic strain measured at the point near the wedge i, shown in fig. 8. Scales of the ordinate and the abscissa are 250 # strain and 50 ms per division, respectively The upper straight line, which is reached after a rapid increase of the current, is an output of the currenl flowing in the vibration exciting coil of 300 3( turns. The coil current rises stepwise to 4.0 A / t u r n and a the strain increases rapidly in correspondence with the rapid increase of the current and shows the dampping vibration around the mean value which corresponds to the static deflection due to the magnetic force given by 1 I x Be (B, is the magnetic field generated by the vibration vibr~ exciting coil). The toroidal coil current is 50.0 A / T in the vibration test shown in fig. 8. A compa 9arison of the experimental results with the numerical analysis anal is shown in fig. 9 where the dynamic strain shown showr in fig. 8 is plotted with a reversed sign of the strain. The T numerical result is also plotted with solid points. Both results demonstrate the very good agreement with witt each other. F r o m this agreement we also conclude that the vibration of the full torus of toroidal coils can be simulated sim by the toroidal simulator for the pairing mode of vibration ~erformed for the because the numerical analysis was perfol eight-coil full torus. In the numerical analy.,~sis the damping factor of 15.0 was used. relation In fig. l0 is shown the so called linear lir between squared frequency and squared coil current. The kind of model marked to three lines in the figure is related to the modelling of orthogonality of the coil and is the same as used for the numerical analysis in the
II
' '",. ,"
o ~].
~ Distance from center
-~ --
~1
-0025 ~ by Moses and Fig. 7. Magnetic field profiles of full torus (by Young) and the toroidal simulator (by FEM analysis).
Fig. 8. Dynamic strain of the toroidal simulator and current change of the vibration exciting coil. Current of = 50 A/T, current of the vibration exciting horizontal scale = 50 ms/div., vertical scale = 25
K. Miya et a L / Magnetomechanical behavior of a toroidal coil
f
800
,'~
FEM - FLT
rp = 40 A/turn
~/ ~,
//
l
_c2
f [r = 39 9 A/turn
'¢,
://
Experiment
._
\~
/
, .~,
_
/ ~, 4 0 0 -~
~
'~
2oo
o'.,
o12 Time
0'.3
(sac)
Fig. 9. Comparison of the experimental result with the numerical one for the toroidal simulator (lst mode).
previous experiment [5]. The experimental results show the linear relation between the squared frequency and the squared current but the numerical analyses show the discrepancy with the experiment. The result shown in fig. 9 corresponds to the data marked with "A" in fig. 10 and was obtained by using Model II. It should be noted that the discrepancy is magnified in fig. 10 because squared values were plotted. For example, the buckling currents determined from an extrapolation of the two lines for the experiment and the numerical analysis with Model II are 79.4 ( = 63f6ffO0-) and 76.8 ( = 5 9 ¢ ~ ) A, respectively. The relative error has a very small value of 3.2%. The largest discrepancy occurs for the analysis with Model I, where the relative error is 15.5% ( = 1 - 45v/4500/79.6). These values can be thought
L '~ -~
Experiment Numerical results
-"~"-..
~)
~ ~
x Lx x
o
I00
Model I ~
-
4. Conclusion The validity of the toroidal simulator was w; verified b3 comparing the buckling current from the simulator si witt those from the eight coil full torus. The verification vel wa: furthermore confirmed by the numerical analyses at of th~ magnetic field distribution and the exper ~erimental dat," from the vibration test.
Acknowledgement This work has been done with the financial fina suppor of the Controlled Thermonuclear Reacto lctor Committeq (Chairman, Prof. M. Akiyama), Faculty of Engineering the University of Tokyo. The authors would like t~ express their sincere gratitude to Mr. K. Someya St for hi: assistance to the experiment.
References o
I "', 200L
to agree well with the buckling currents from fror the toroida simulator (76.0 A) and the eight coil full ttco r u s (79.8 A ) tha' And thus it is concluded from the good agreement a~ the toroidal simulator can yield almost the tt same rela. tion between the frequency and the coil current c as tht full torus. The similar results from the experim ;riment and th~ analysis were obtained for the second lateral lat mode o: erroJ vibration and are not shown here. The relative J between the experiment and the analysis is in the rangt of 3% to 13%. The buckling current for the second mod~ times greater that of experiment was 173.2 A, that is 2.2 time,, that of the first mode of experiment, which which is caused b~ the stiffening effect.
~
~ ~
2000
I
" " . ~ ~ ~ x """ """x "~-4000 6000
I~
[ ( A / T ) z]
Fig. 10. Comparison of I 2 - f 2 curves from the ex experiment and the numerical analysis for the toroidal simulator )r (lst Mode).
[1] W.H. Gray, C.J. Long and W.C.T. Stoddart, Stoddar Nucl. Engr8 Des. 58 (1980) 191-205. [2] F.C. Moon, J. Appl. Phys. 47, No. 3 (1976). [3] F.C. Moon, ASME J. Appl. Mech. 46 (1979) 145-150. [4] K. Miya, T. Takagi and M. Uesaka, ASME J. Mech. Des 41 (1980) 91-107. Sot [5] K. Miya, T. Takagi, M. Uesaka and K. Someya, ASMEJ Appl. Mech. 49 (1982) 180-186. [6] M. Uemura and N. Yamada, J. Society oof Material Sci ence of Japan (Zairyo) 24, No. 257 (1965) 156-163. [7] J.E. Akin and W.C.T. Stoddart, ProceedinlLgs of 7th Sym posium on Engineering Problems of Fusion Fu Research IEEE (1977) 722-775. [8] K. Miya and M. Uesaka, Nucl. Engrg. 1 275-775.
346
K. Miya et al. / Magnetomechanical behavior of a toroidal coil
[9] K. Miya, M. Uesaka and F.C. Moon, ASME J. Appl. Mech. 49, No. 3 (1982) 594-600. [10] F.C. Moon, ASME J. Mech. Des. 41 (1980) 77-90. [11] R.W. Moses and W.C. Young, Proceedings of 6-th Symposium on Engineering Problems of Fusion Research, IEEE (1975) 917-921. [12] P. Silvester, IEEE PAS-92 (1973) 1274-1281. [13] M.V.K. Chari, IEEE PAS-93 (1974) 62-69.
[14] K. Miya, T. Takagi and Y. Ando, ASME J. Appl. Mech. 47 (1980) 377-382. [15] W.K.H. Panovsky, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1961). [16] O.C. Zienkiewicz(McGraw-Hill, London, 1971). [17] R.V. Southwell, Proc. Royal Soc. London, Series A, Vol. 135 (1932) 601.
FINITE ELEMENT ANALYSIS OF THE MAGNETOMECHANICAL OF A TOROIDAL COIL WITH MAGNETIC MIRRORS Kenzo MIYA, T0shiyuki TAKAGI
BEHAVIOR
and Tohru TAKAGHI
Nuclear Engineering Research Laboratory, University of Tokyo, Tokai-Mura, lbarakL Japan Received January 1983
A toroidal ~roidal simulator is used to study the magnetomechanical behavior of a superconducting D-shaped coil s~ set at the cente between m sectored ferromagnetic plates. Since ince the magnetic field distribution between the two plates is expected ~ected t~ to be almost th~ is that created by toroidal field coils ils arrayed in a torus, the toroidal simulator may be expected to show almost al the sam~ same as magnetomechanical ~tomechanical behavior as that of the full torus coils. To verify its validity experimentally, buckling and vibration test, arried out in a liquid helium cryostat. ~tat. In addition numerical analyses were performed for the magnetic field fie distributior were carried to reveal :al its similarity and for the experimental "imental results from the toroidal simulator to confirm its validity. Its validity is full3 demonstrated in the present study.
1. Introduction Magnetoelastic coupling phenomena between current carrying elastic bodies and electromagnetic field have become an important problem in engineering fields of magnetic fusion reactor technology, especially in the superconducting coil engineering [1]. The instability of the toroidal field coils among the problems associated with the coil engineering was first investigated by Moon et al. [2], [3] and then by Miya et al. [4], [5]. Studies by M o o n et al. were rather presented with an emphasis on the experimental demonstration although the modal analysis of the data was performed. Those b y Miya et al. aimed to predict the experimental results with application of the numerical tool of finite element method, The prediction was successfully achieved with an introduction of the theory of orthotropy [6] to an evaluation of the elastic stiffness and with a consideration of electric work due to the directional change of a current density vector caused by a rotation of the coil. Akin et al. [7] showed the numerical solution for the magnetoelastic buckling of a current carrying straight beam conductor placed at the center between two parallel beam conductors. In the problem the two side conductors are fixed rigidly and the center conductor is partially free to deflect. This problem can bee solved analytically and the buckling currents calculated tted for rough mesh divisions by Akin et al. converge to the exact 0029-5493/83/0000-0000/$03.00
solution obtained by us [8]. The prediction 9rediction of magneto elastically coupled vibration of the toroidal toroit coils wa~ accomplished by combining the Fast Lapla )lace Transforn with the finite element scheme [9]. In these thes studies, th~ o magnetic stiffness, which is a natural consequence co~ mutual inductance between coils, plays an essential rol~ for understanding of the magnetoelasticall netoelastically interactin~ ~lementation by th~ phenomena and for a numerical implemen finite element formulation. In the present study the validity of the toroida simulator was investigated as to whether it does or doe~, not give the same magnetoelastic behavior avior as that of th~ toroidal field coils arrayed in a torus whil while the experi. mental verification of its validity was dem demonstrated b3 M o o n [10] with buckling and vibration tes tests. Here performed are the numerical analyses for a r magnetic r fielc distribution between the two ferromagneti tgnetic plates anc verify further the for the buckling and vibraton tests to verit economial experiment, validity. F r o m a viewpoint of an economia applicability foJ such a simulator could be of potential apl: char. an investigation of the instability and vibration vit acteristics of symmetrically arrayed coils as seen in there toroidal and solenoidal coil systems. However, He might be a restriction as to a magnitude of the saturated magnetic induction of the ferromagnetic p31ates because the plates cannot serve as a magnetic n symmetric field when the field intensity ris~ the saturated field of the plate.
© 1983 N o r t h - H o l l a n d
K. Miya et aL/ Magnetomechanicalbehaviorof a toroidalcoil
340
SUS Bucking~
2. Theoretical background of magnetoelasticity of toroidai
simulator
A distribution of toroidal magnetic field created by the eight-coil full torus is shown schematically in fig. 1a, where the magnetic field by poloidal coils is not considered. As is evident from the figure, there are eight fold symmetries with eight radial symmetric planes, of which two planes, making an angle of 45 °, are drawn in the figure. Thus there could be a prediction that the magnetic field generated between the two planes is simulated by one coil set at the center between two ferromagnetic plates as shown in fig. lb. The ferromagnetic plates in the toroidal simulator serve as a magnetic mirror of the contour lines of magnetic induction because they enter normally to the plates. Top and elevated views of the toroidal simulator used in the present experiment are shown in fig. 2. The ferromagnetic plates are rectangular and were manufactured from mild steel of a permeability of 7000 in MKS units. Three-dimensional analysis is, in fact, necessary for an exact evaluation of the magnetic field between two ferromagnetic plates but actually too complex to perform because a very wide analysis region must be provided including the air space outside the set. Thus two-dimensional analysis was performed for the first approximation of the field distribution assuming that the magnetic field is similar to one created by two straight coils of infinite length between two ferromag-
~ ~
-~
"~
~
~2L'~
/2Z.5"/
" ~ " -~?-~0 ~ , ~.. ~ -,,~.~ 15 . . . . . . ~ ) ~ - . ---~_, , -w
! 135
Fig. 2. Top view of toroidal simulator and elevated view of D-shape coil.
netic plates of finite width and infinite length as is done by Moses and Young [ 11]. The finite element analysis of the two-dimensional field distribution is very common [12,13] and was carried out in the previous paper [14]. The method developed in [14] can be applicable to the present two-dimensional problem, too.
45*
(b)
Ferromagnetic ~plote
~..
Toroidol coil
Flux lines by toroidol coils
coil
,.~/~~Test
2.1. Finite-element analysis of magnetic field and coupled vibration
(a)
~r~
Toroidal simulator
Fig. 1. Schematic distribution of toroidal field bby an 8 coil full torus with ferromagnetic plates.
K. Miya et al. / Magnetomechanical behavior of a toroidal coil
Maxwell's equations determining the static magnetic field is described simply as (see ref. [14] for the complete derivation) 1 rot(rot .4 ) = 0,
(1)
if the vector potential A is used. Here # is the permeability ( = 4¢r × 10 -7 for the air space and = 7000 for the ferromagnetic plate). The magnetic induction B can be calculated by B = rot A.
(2)
A solution of eq. (1) is obtained with an application of finite element method where the two-dimensional quadrilateral simplex element is adopted for minimization of a functional of eq. (1). The magnetic induction B is computed from eq. (2) after the vector potential A is solved in the region of the space and the ferromagnetic plates, As is well known, the equation of the coupled vibration of the current carrying coils is a natural consequence of the Lagrange's equation which is expressed
The first and second terms of this equatior uatlon are conventional for the finite element implementatiol )lementation [16] and sc are not formulated furthermore here. The third one is dependent on the relative movements betw~ between coils and is calculated below.
2.2. Calculation of perturbed magnetic energy Although the evaluation of the third terr term in eq. (6) i., elegant evaluagiven in the Appendix in ref. [4], a more el tion of the term is shown in the following. IIn general th~ shown in fig.." magnetic energy created by two coils as sh is convieniently evaluated by
t ~ o [ ( j o ) d S ( 1 ) f f j t(~2 ) d S (2), JJ D
where superscripts refer to the coil numbei number 1 and 2 an( dSU) = dx~i)dx(i), jU) is the current densi density of the itt coil and D is a distance between two [~oints of thi respective coils after deformation. It is approximate( with
by,
1
1
d oL - ~ ~w j - ~w = F,
z,
Ro
(3)
where L is the Lagrangian of the present system given by eq. (4), w is a lateral deflection of the coil and F is an external force acting on the coil. The Lagrangian is constructed like, L = T - U + W,
(4)
where
T=½ffo(c~)2hdxdy, u = ½ff(,)T(o)dxdy,
I ('"
.:lr[ /~\ 4-5X,10] I * \ / x(t) 2=
[
(5b)
w )//~/
. (21 X2
12l
R0
pt~,l~
X(21
X'''j , X3
1 0
Owff AJhdxdy=°
°
(6)
")'
where R 0 is a position vector between the 1two points ol the coils before they deflect and • is a difference o displacement vectors of the points. Eq. (~ (8) can be de rived by either Taylor's expansion or Le~gendre poly nomial. Considering a special relation between twt Cartesian coordinates, x~ l) and x[ 2), wh which make ai
In t h e above expressions, T is kinetic energy, U is strain energy and W is magnetic energy; h, #, (c) and (o) are the thickness, the displacement velocity, generalized strain and stress, respectively. A substitution of eqs. (5a), (5b) and (5c) into eq. (4) and then the eq. (4) into eq. (3) yields the following
2
1 (Ro.r)+3
(81
(5a)
ffo ,hdxdY+½ ff{,:(o}dxdy
(71
4¢r J J
W(12) =
Fig. 3. Coordinates systems set to toroidal coils.
K. Miya et al. / Magnetomechanicalbehavior of a toroidal coil
342
angle of O as shown in fig. 3, the following is obtained:
R o • r = - w t l ) x l 2) sin 0
+
W(2)Xl 1) sin O,
(9)
where w ~) is a lateral deflection of the ith coil. A substitution of eq. (9) into eq. (8) yields 1
1 xl 2) x} ~) = R0 + wl ~ - sin 0 - w 2 ~ - sin 0 Ro R0 v(2). ,.(2) 1 ] 3 ~1 ~ l sin20_ w¢~)wt~j 2 R~ 2R~ J
+
3x}l)" x}Z) sin20 ) w(l)w(2 )
COSO
+
R3
R~
3. Experimental and numerical results
3 x~l)'xl 1) +
1 ) sin20 - 2R 3 / wt2)w t2),
2 R~00
(10)
where the following relation is used:
r = ( - w ~2) sin O, 0, w ~2) cos 0 - w~l)).
(11)
The magnetic energy W (~2) is calculated as follows with substitution of eq. (10) into eq. (7), W(12)_ W(12)+ W(I2)+ W2(IZ)+ W(13) + W4(12)+ W5(12) ' where (12) j~2)
W,(12)_ lit0 f f j o ) d S t l ) f f _ _ d S t 2 0
- ~
m(I2) =
J
J J
)
J J R0
'
t~° f f w " ) J C " d S t ' ) f f x l 2 ' J t 2 ) d S ' 2 ) s i n 4~r J J J J R3o
0,
x(l)j( I )
W~,Z)= _ ~ O f f w ( 2 ) j t 2 ) d S , 2 ~ f f ~ d S ~ , ) s i n O 4~r J J 14ztl2)- tz° f fw(1)w(1)j(l)dS 0) 3 - 4~ J J x
Wl~12~ and W~ 12) in eq. (12) yield electroma gnetic forces acting on No. 1 and No. 2 coils, respecti~ ;pectively. For the these toroidal coil system being symmetrically arrayed, a forces cancel out because W,,(k'k+:)= - W~(k.k-t) ~ ( i = 1, 2) for two pairing coils [No. (k + l) and No. N (k - l)] as to No. k coil. Wot~2) is the magnetic energy stored in the system before deflection occurs and does not n, contribute to the electromagnetic force. Although eq. eq (12) is the expression for the magnetic energy created by No. 1 and of No. 2 coils, it can be applied to any two combination co the toroidal coils by changing the angle 0 when summed up for the total magnetic energy.
ff(3xl2~xl 2~_2R5o
sin20 _ 2R31 ) j(2)d S(2,
'
3.1. Experimentalprocedure simula The top view of the toroidal simulator and the ~erconducting coil are elevated view of the small superconduct for the shown in fig. 2. The coil was fabricated newly n present experiment almost in the same waly as the coils used in the previous experiments. The angle made by the two ferromagnetic plates is 45 °, which whick means that the toroidal coil is a simulator of the eight-coil full torus. A size of the square plate is 250 mm × 250 mm and its thickness is 15 mm. The test coil was w~ secured to the wedge at the center of the experimental apparatus. The apparatuses with and without a lateral lateral support are also shown in figs. 4a and 4b, respectively. Three experiments of the buckling test, two vibration tests of the first and second lateral modes, were perfc ,erformed in the liquid helium cryostat. The second lateral mode of late~ vibration was caused by inserting a lateral support at the center of the coil in the opposite side of the wedge. As were used in the previous experiments foil type strain gages for low temperature measurement were measu~ cemented to the coil for a detection of the bending strain through a X Y chart recorder for the buckling test and a synchroscope for the vibration tests.
3.2. Experimental results and numerical analysis
f f w'2'w'2>'r2'ds':'
= It°4~
-~x ~ ×
2R 5
W5~'z)= ~2_ f f
1
j(~dS(1),
s i n 2 0 - 2R~
w~',J~l)dS~') d
×
R~
Rs0
sin20 w~2)J(2)dS(2)"
The experimental result of the buckling test is shown in fig. 5 where the bending strain and the coil current are plotted in the ordinate and the abscissa, respectively. A close examination of the buckling curve clarifies that it is approximately a hyperbola not a parabola. If it were the parabola, the curve might be called a deflection one which could be caused by a si~nifican! amount of initial misalignment. Since the 1: rent is not determined clearly from the curv
K. Miya et aL / Magnetomechanicalbehavior of a toroidal coil
~o 400
=
2oo
~
3oo1-~ 1
o/ o
")~L
Bendingslrolnlcurrenlscpoled,xlO(Amps}/ SouthweUplot for n~gnetoeloslicbuck~,//~
/
Lvv /
F /
0~)
~
10
, 005
, i o,
I
I
/
/
I
I
I
20 :50 40 50 60 70 Toroid(]l field coil current , I , Amps/lum
Fig. 5. Buckling curve of toroida] simulator.
i¸¸¸¸¸
~
76.0 A / T . The buckling current obtain o b t a i n e d from th( experiment of the eight-coil full torus w~ was 78.4 A / q which was reported in [5]. Since b o t h the bucklin l currents can be thought in a fairly good aagreement, wt conclude that the toroidal simulator can rreproduce ac curately the buckling b e h a v i o r of the full torus coils it the pairing mode. The two-dimensional analysis of the rI magnetic fiel( was conducted as the solution of eq. (1) fo for a n u m e r i c a verification of the magnetic field coinci& coincidence betweet the toroidal simulator a n d the full torus co coils. A graphic display of the c o n t o u r lines inside the two tferromagneti~ plates is shown in fig. 6. As m e n t i o n e d b~ before the tw( straight coils carry currents of opposite ddirection witl each other a n d are located at two island islands on the lin, A - A ' in fig. 6. A m a g n i t u d e of the current is 1900 A. A is well k n o w n a n d shown later the magne netic field nea the inside coil is the highest. In fig. 7 iis shown th~ magnetic field distribution on the line A - A ' with th, d o t t e d curve. O n the other h a n d a n accura accurate solution o the magnetic field is given by Moses and Y o u n g [ 11 ] fo N pairs of coil arrayed in a torus by
B(R) Fig. 4. Photographs of the toroidal simulator: (a) for the first mode of vibration, (b) for the second mode of vibration. The small coils shown are copper vibration exciting coils,
well plot [17] of the curve was utilized a nld d is s h o w n in urately in this the same figure. T h e buckling current, accuratel case, can b e d e t e r m i n e d from a tangent of the line as
=
#oNl( i 1 ) + ) N ~ 1+ ( R / R , ) N - 1 ( R 2 / R - 1 '
(13 where R 1 a n d RE are distances from the center of th, torus to the inside a n d outside straight coils, respec tively. 1 is the total current carried by the coil an( /~0 = 4~r × 10 -7. T h e field calculated by eq. (13) is showl ~r~ec~n wlth th, with the solid line in fig. 7 for compari c o m p u t e d field in the toroidal simulator. " field o n the axis between the two coils of t
344
K. Miya et a L / Magnetomechanical behavior of a toroidal coil
A'
,0t
)02
A Fig. 6. Magnetic field distribution in toroidal simulator.
in a rough agreement while those in the air space between the two coils show discrepancy with each other, This discrepancy may be attributed to the finite size of the ferromagnetic plate. However, the rough agreement inside the conductors can be the reason for the fairly good agreement between the buckling currents from the toroidal simulator and the full torus. The validity of the toroidal simulator was also con-
0.125
.~
o, loc
- -
Moses ond Young
.....
Numericol
results
for toroidel sirnulef,
0075 '~ 0.0 5 C "'
0025
=
o.ooo---
firmed from the vibration tests. An example exa of the dynamic strain measured at the point near the wedge i, shown in fig. 8. Scales of the ordinate and the abscissa are 250 # strain and 50 ms per division, respectively The upper straight line, which is reached after a rapid increase of the current, is an output of the currenl flowing in the vibration exciting coil of 300 3( turns. The coil current rises stepwise to 4.0 A / t u r n and a the strain increases rapidly in correspondence with the rapid increase of the current and shows the dampping vibration around the mean value which corresponds to the static deflection due to the magnetic force given by 1 I x Be (B, is the magnetic field generated by the vibration vibr~ exciting coil). The toroidal coil current is 50.0 A / T in the vibration test shown in fig. 8. A compa 9arison of the experimental results with the numerical analysis anal is shown in fig. 9 where the dynamic strain shown showr in fig. 8 is plotted with a reversed sign of the strain. The T numerical result is also plotted with solid points. Both results demonstrate the very good agreement with witt each other. F r o m this agreement we also conclude that the vibration of the full torus of toroidal coils can be simulated sim by the toroidal simulator for the pairing mode of vibration ~erformed for the because the numerical analysis was perfol eight-coil full torus. In the numerical analy.,~sis the damping factor of 15.0 was used. relation In fig. l0 is shown the so called linear lir between squared frequency and squared coil current. The kind of model marked to three lines in the figure is related to the modelling of orthogonality of the coil and is the same as used for the numerical analysis in the
II
' '",. ,"
o ~].
~ Distance from center
-~ --
~1
-0025 ~ by Moses and Fig. 7. Magnetic field profiles of full torus (by Young) and the toroidal simulator (by FEM analysis).
Fig. 8. Dynamic strain of the toroidal simulator and current change of the vibration exciting coil. Current of = 50 A/T, current of the vibration exciting horizontal scale = 50 ms/div., vertical scale = 25
K. Miya et a L / Magnetomechanical behavior of a toroidal coil
f
800
,'~
FEM - FLT
rp = 40 A/turn
~/ ~,
//
l
_c2
f [r = 39 9 A/turn
'¢,
://
Experiment
._
\~
/
, .~,
_
/ ~, 4 0 0 -~
~
'~
2oo
o'.,
o12 Time
0'.3
(sac)
Fig. 9. Comparison of the experimental result with the numerical one for the toroidal simulator (lst mode).
previous experiment [5]. The experimental results show the linear relation between the squared frequency and the squared current but the numerical analyses show the discrepancy with the experiment. The result shown in fig. 9 corresponds to the data marked with "A" in fig. 10 and was obtained by using Model II. It should be noted that the discrepancy is magnified in fig. 10 because squared values were plotted. For example, the buckling currents determined from an extrapolation of the two lines for the experiment and the numerical analysis with Model II are 79.4 ( = 63f6ffO0-) and 76.8 ( = 5 9 ¢ ~ ) A, respectively. The relative error has a very small value of 3.2%. The largest discrepancy occurs for the analysis with Model I, where the relative error is 15.5% ( = 1 - 45v/4500/79.6). These values can be thought
L '~ -~
Experiment Numerical results
-"~"-..
~)
~ ~
x Lx x
o
I00
Model I ~
-
4. Conclusion The validity of the toroidal simulator was w; verified b3 comparing the buckling current from the simulator si witt those from the eight coil full torus. The verification vel wa: furthermore confirmed by the numerical analyses at of th~ magnetic field distribution and the exper ~erimental dat," from the vibration test.
Acknowledgement This work has been done with the financial fina suppor of the Controlled Thermonuclear Reacto lctor Committeq (Chairman, Prof. M. Akiyama), Faculty of Engineering the University of Tokyo. The authors would like t~ express their sincere gratitude to Mr. K. Someya St for hi: assistance to the experiment.
References o
I "', 200L
to agree well with the buckling currents from fror the toroida simulator (76.0 A) and the eight coil full ttco r u s (79.8 A ) tha' And thus it is concluded from the good agreement a~ the toroidal simulator can yield almost the tt same rela. tion between the frequency and the coil current c as tht full torus. The similar results from the experim ;riment and th~ analysis were obtained for the second lateral lat mode o: erroJ vibration and are not shown here. The relative J between the experiment and the analysis is in the rangt of 3% to 13%. The buckling current for the second mod~ times greater that of experiment was 173.2 A, that is 2.2 time,, that of the first mode of experiment, which which is caused b~ the stiffening effect.
~
~ ~
2000
I
" " . ~ ~ ~ x """ """x "~-4000 6000
I~
[ ( A / T ) z]
Fig. 10. Comparison of I 2 - f 2 curves from the ex experiment and the numerical analysis for the toroidal simulator )r (lst Mode).
[1] W.H. Gray, C.J. Long and W.C.T. Stoddart, Stoddar Nucl. Engr8 Des. 58 (1980) 191-205. [2] F.C. Moon, J. Appl. Phys. 47, No. 3 (1976). [3] F.C. Moon, ASME J. Appl. Mech. 46 (1979) 145-150. [4] K. Miya, T. Takagi and M. Uesaka, ASME J. Mech. Des 41 (1980) 91-107. Sot [5] K. Miya, T. Takagi, M. Uesaka and K. Someya, ASMEJ Appl. Mech. 49 (1982) 180-186. [6] M. Uemura and N. Yamada, J. Society oof Material Sci ence of Japan (Zairyo) 24, No. 257 (1965) 156-163. [7] J.E. Akin and W.C.T. Stoddart, ProceedinlLgs of 7th Sym posium on Engineering Problems of Fusion Fu Research IEEE (1977) 722-775. [8] K. Miya and M. Uesaka, Nucl. Engrg. 1 275-775.
346
K. Miya et al. / Magnetomechanical behavior of a toroidal coil
[9] K. Miya, M. Uesaka and F.C. Moon, ASME J. Appl. Mech. 49, No. 3 (1982) 594-600. [10] F.C. Moon, ASME J. Mech. Des. 41 (1980) 77-90. [11] R.W. Moses and W.C. Young, Proceedings of 6-th Symposium on Engineering Problems of Fusion Research, IEEE (1975) 917-921. [12] P. Silvester, IEEE PAS-92 (1973) 1274-1281. [13] M.V.K. Chari, IEEE PAS-93 (1974) 62-69.
[14] K. Miya, T. Takagi and Y. Ando, ASME J. Appl. Mech. 47 (1980) 377-382. [15] W.K.H. Panovsky, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1961). [16] O.C. Zienkiewicz(McGraw-Hill, London, 1971). [17] R.V. Southwell, Proc. Royal Soc. London, Series A, Vol. 135 (1932) 601.