EDDY CURRENTS TORQUE MODEL FOR SPIN STABILIZED EARTH SPACECRAFT

Transcription

1 E URRETS TORQUE MOEL OR SPI STAILIZE EARTH SPAERAT Mr eíl. P. S. Znrd Group of Orbtl ynm nd Plnetology UESP São Pulo Stte Unverty -Gurtnguetá SP- rzl e- ml: Iur Mrtnez Puente Qurell Group of Orbtl ynm nd Plnetology UESP São Pulo Stte Unverty -Gurtnguetá SP- rzl Hélo Kot Kug IPE - rzln Inttute for Spe Reerh Abtrt. An nlytl pproh for the eddy urrent torque tng on the pn-tblzed tellte preented. It umed n nlned dpole model for the Erth mgnet feld nd the method of vergng the eddy urrent torque over eh orbtl perod ppled to obtn the omponent of the torque n the tellte body frme referene ytem. The development re preented n term of the men nomly nd ontn term of eond order n eentrty. It oberved tht the eddy urrent torque ue n exponentl dey of the ngulr veloty mgntude. umerl mplementton performed wth dt of the S nd S rzln tellte how the greement between the developed nlytl oluton nd the tul tellte behvor. Keyword. Atttude of the rtfl tellte, pn-tblzed perft, eddy urrent torque, ngulr veloty mgntude, exponentl dey.. Introduton Th work ddree the rottonl moton dynm of pn tblzed Erth tellte (whh h the pn x long the geometr tellte x), through n nlytl pproh for tttude predton. The emph pled on modelng of the eddy urrent torque oted wth thee tellte. A pherl oordnte ytem fxed n the tellte body ued to loted the pn x of the tellte n relton to the terretrl equtorl ytem. The dreton of the pn x pefed by the rght enon () nd the delnton () whh re repreented n the g.. Kˆ ĵ kˆ = ˆ Rotton pln 0 o - Equtorl pln Ĵ Î î gure. Spn x orentton ( ŝ ): Equtorl Sytem ( Î ), Sytem of the Stellte ( î, ĵ, kˆ ), rght enon () nd delnton () of the pn x.,ĵ, Kˆ The eddy urrent torque pper due to the nterton of uh urrent rultng long the tellte truture h nd the Erth feld. In th pper the torque nly performed through the modelng of the nlned Erth mgnet dpole, whh orentton depend on the mgnet olttude nd on the endng node of the mgnet plne. Eentlly n nlytl vergng method ppled to determne the torque over n orbtl perod. To ompute the verge omponent of eddy urrent torque n the tellte body frme referene ytem (tellte ytem), n verge n tme n term of the men nomly utlzed, whh nvolve rotton mtre dependent on the orbt element, the mgnet olttude, the endng node of the mgnet plne nd the rght enon nd delnton of the tellte x. The nluon of th torque on the rottonl moton dfferentl equton of pn tblzed tellte yeld the ondton to derve n nlytl oluton. The oluton how lerly tht the eddy urrent torque ue n exponentl dey of the ngulr veloty mgntude long the tme. umerl mplementton performed wth dt of the S nd S rzln tellte how the greement between the nlytl oluton for ngulr veloty mgntude nd the tul tellte behvor.

2 . Geomgnet feld An nlned Erth mgnet dpole model umed n th pper. It orentton depend on the mgnet olttude (β) nd on the endng node of the mgnet plne (η). The mgnet referene ytem, whh x z m long the dpole vetor, β nd η re repreented n the g.. z m β Z y m Geomgnet eletl Equtor t Equtor η xm X gure - Mgnet ytem (O x m y m z m ) nd Equtorl l (O XZ) It well known tht the Erth mgnet dpole model (Thom nd pellr, 4; Wertz,78) my be expreed by: =! 4 π µ o r [ kˆ ( î kˆ ) î ] m m where! the mgnet moment of Erth feld mgntude, µ o the permeblty of free pe, r the rdu vetor mgntude of the tellte, kˆ m the unt vetor long the dpole vetor nd î the unt vetor long the rdu vetor of the tellte ( r ). The unt vetor kˆ m nd î n be expreed n the tellte ytem through rotton mtre dependent on the orbt element, rght enon nd delnton of the tellte pn x nd the ngle β nd η.. Eddy urrent torque The torque ndued by eddy urrent re ued by the perft pnnng moton. It known (Wertz, 78) tht the eddy urrent produe torque whh preee the pn x nd ue n exponentl dey of the pn rte. If W the perft ngulr veloty vetor nd p ontnt oeffent whh depend on the perft geometry nd ondutvty, th torque my be modeled by: = p x ( x W) () Here the mgnet torque developed for pn tblzed tellte. In th e, the perft ngulr veloty vetor nd the tellte mgnet moment re long the z-x, nd ndued eddy urrent torque n the expreed by (Kug et l., 87): = p W x ( x kˆ) () where x, y, z re the omponent of the geomgnet feld n the tellte fxed ytem. Thee omponent re obtned n term of the geoentr nertl omponent of the geomgnet feld nd the rght enon nd delnton of the tellte. ()

3 4. Men eddy urrent torque In order to obtn the men eddy urrent torque, t neery to ntegrte the ntntneou torque (), over one orbtl perod ( T ):, gven for = m T t T dt t (4) where: t the tme, t the ntl tme nd T the orbtl perod. hngng the ndependent vrble to the ft vryng true nomly, the men eddy urrent torque n be obtned by (Qurell, 00): = m π υ π r υ h dυ (5) where υ the true nomly t ntnt t nd h the pef ngulr moment. Sne the ntntneou torque gven by () nd ( e ) π ( e ), h r = eo υ = () T where the em-mjor x nd e the eentrty of orbt, the men eddy urrent torque (5) beome: = R m υ π W [ kˆ ( ) ] υ m î kˆ m î [ kˆ ( î kˆ ) î ] kˆ { }( eo υ ) 4 dυ m m (7) wth p! R = π µ o ( / e ) (8) To evlute the ntegrl of (7) we wll ue the ellpt expnon of the true nomly n term of the men nomly M (rouwer & lemene, ), nludng term up to frt order n the eentrty (e). Then the preent development n be pplble for ellptl orbt wthout lo of preon. or mplfton of the ntegrl we wll onder the ntl tme for ntegrton equl to the ntnt tht the tellte pe through the pergee. The omponent of the unt vetor kˆ m n the tellte ytem depend on the mgnet olttude (β) nd endng node of the mgnet plne (η) nd rght enon () nd delnton () of the pn x (Qurell, 00; Thom & pellr, 4). In th pper, we wll onder (Thom nd pellr,4): η = η o bm nd ωe T b = π () where pergee nd η o the ntl poton of the endng node of the geomgnet equtor t the ntnt the tellte t the ωe the ngulr veloty of the Erth. The omponent of the unt vetor î S n the tellte ytem depend on endng node orbt ( ), orbtl nlnton ( ), the true nomly ( ν ) nd rght enon ( ) nd delnton ( ) of the pn x (Qurell, 00). or one orbtl perod the ngle,, ν,, nd β re ontnt. Thu, ung trgonometry properte nd fter exhutng but mple lgebr development, the men eddy urrent torque n be expreed by (Qurell, 00): m = x î y ĵ z kˆ wth x, y nd z term preented n the Appendx. (0)

4 5. Applton The vrton of the ngulr veloty, the delnton nd the enon rght of the pn x re gven by the Euler equton n pherl oordnte (Kug et l, 87): W # = z Iz # = y Iz W # = x () Iz W o where I z the moment of nert long the pn x, x, y nd z re omponent of externl torque n the tellte body frme referene ytem (tellte ytem). Then t poble to oberve tht the eddy urrent torque ffet the ngulr veloty mgntude nd the pn x. In th pper t wll be nlyzed the ngulr veloty mgntude. y ubttutng z gven n (A.), the vrton of the ngulr veloty mgntude n the expreed : d W d t = k d t () wth k z R = () Iz If the prmeter k ondered ontnt for one orbtl perod, the nlytl oluton of eq. () : W = W 0 e k t (4) where W 0 the ntl ngulr veloty. Then when k < 0 the ngulr veloty mgntude dey exponentlly. The reult obtned by omputer mplementton of the developed theory, ung dt of S nd S rzln tellte (gven n tble ) re hown n the g. nd 4. The g. 5 nd preent the devton obtned durng the perod of the tet, ompred wth the IPE' ontrol enter rhve. The behvour of ngulr veloty mgntude for the S nd S tellte how the greement between the propoed nlytl oluton nd the tul tellte behvor. Tble rzln Stellte t (S nd S ). S t 4/07/ S t /04/00 (meter) E ( ) ( ) ω ( ) I z (Kg m ) 4.5 W(rpm) 0.7. o ( ).4 5. o ( ) β ( ).4.4 µ o (Weber/A m) 4 π π. 0-7! (Weber m) M (A m )

5 0. Anlytl oluton Atul dt 8 w(rpm ) y gure Evoluton of ngulr veloty of S Anlytl Soluton Atul dt W (rpm ) y gure 4 Evoluton of ngulr veloty for S Angulr Veloty devton for S evton (rpm)) 0,4 0, 0, 0, 0-0, -0, Tme (dy) gure 5 Angulr veloty devton for S.

6 Angulr veloty devton for (S) evton (rpm) 0, 0, 0, 0-0, -0, Tme (dy) gure Angulr veloty devton for S.. Summry The eddy urrent torque model w dued n th pper, onderng pplton to pn tblzed tellte. The men omponent of th torque n the tellte body frme referene ytem were obtned through the development of n nlytl pproh. Anly of the nlytl oluton how lerly tht the eddy urrent torque ue n exponentl dey of the ngulr veloty mgntude long the tme. The theory developed ount lo for orbt element tme vrton, not retrted to rulr orbt. Suh pproh gve re to ome hundred of trgonometr ntegrl whh were olved nlytlly. The theory w oded n P mro-omputer. Then the progrm w exeuted ung the dt of S nd S rzln tellte. Reult hve hown the greement between the nlytl oluton for the ngulr veloty nd tul tellte behvor. or the perod of the tet, the men dfferene w n the order of 0.0rpm for S nd 0.04rpm for S, well wthn the ury level n whh the tttude determnton ytem of IPE ontrol enter etmte the ngulr veloty mgntude. 7. Referene rouwer, O. ; lemene, G.M.,, Method of eletl mehn, Adem Pre, ew ork. rrr, V.; Guede, U. T. V., 4, Atttude ontrol pet for S nd S, RM - J. of the rz. So. Mehnl Sene, Vol. XVI - Spel Iue, pp Kug, H. K.; errer, L...; Guede, U. T. V., 87, Atttude nd mneuver multon for the pn tblzed rzln tellte, Tehnl Report of IPE, IPE-47, São Joé do mpo, SP. Kug, H. K.; W...; Guede, U. T. V., 87, Atttude dynm for pn tblzed tellte, Tehnl Report of IPE, IPE -440, São Joé do mpo, SP. Qurell, I. M. P., 00, Spn Stblzed Stellte Atttude Propgton, Mter The, UESP, mpu of Gurtnguetá, SP. Thom, L..; ppelr, J. O., 4, Atttude determnton nd predton of pn-tblzed tellte, The ell Sytem Tehnl Journl, July. Wertz, J. R., 78, Sperft Atttude etermnton nd ontrol,. Redel Publhng ompny, ordreht, Hollnd. Appendx ( ) x = R W β Ax x x xd ( )( ( ) ( ) ( ) ( ) xb xe ( ) ( x xf ) ( ) ( ) ( x xf ) x xd ( ) ( ) ( ) ( )

7 {( ( ) ( ) ( ) ) / 4} ( ){ xd xe ( ) } ( x xf ) ( ) ( x xf ) ( ) ( ) ( ) / 4 β ( x xd ) ( ) ( xb xe ) ( ) ( x xf ) ( ) β IV [ IV [W J b O] [ ( d ) 5 b ( ) ]]{( ( )( d 7 ( ) ( ) ( ) ( L b ) b b ( ( ) ) b 8 d ( ) ( b Q ) ( d ) b 4 ( d ) 7 b ( d ) ( ) ( ) ( ) ( ) ( b Q ) b 4 ( d ) 7 b ( d ) ( ) ( O b S ) 5 b ( d ) 7 0 b ( d ) ( ( ) ( ) ) ( ) / 4 ( b T) b 8 ( d ) b ( d ) 4 ( ) ( Q U ) 7 b ( d ) ( b ) ( ) ( Q b U ) ( 7 b ( d ) ( b ) ( ) ( ) ( ) / 4 ( b ) Z7 b ( d ) Z Z b Z ( d ) Z4 ( ) ( b ) b Z0 ( d ) Z Z b Z5 ( d ) Z ( ) ( b ) Z b Z ( d ) Z4 b Z ( d ) Z ( ) ( b ) Z b Z ( d ) Z4 b Z ( d ) Z ( ) (A.) y = R W β A ( ) ( ) x x x xd

8 ( ) ( ) ( ) ( ) xe xb ( ) xf x ( ) ( ) ( ) ( ) xf x ( ) ( ) xd x ( ) ( ) xe xb ( ) xf x Sn Sn o o ( ) ( ) xf x ( ) ( ) 4 / ( ) ( ) β E xd x x ( ) ) ( xe xb ( ) xf x ( ) ( ) xf x β IV IV ( ) d b O J ( ) ( ) 7 d b 5 ( ) ( ) ( ) ( ) b L Sn Sn 8 b b b ( ) d Q b ( ) ( ) d 4 b d b 7 ( ) d ( ) ( ) ( ) ( ) [ ] Q b 4 b ( ) b 7 b d ( ) ( ) d ( ) 7 d b 5 b S O ( ) d 4 b ( ) ( ) ( ) d 4 b ( ) T b ( ) d 8 b ( ) 4 d b b 7 b U Q - ( ) d ( ) ( ) d [ ] U Q b ( ) d b 7 ( ) d ( )

9 / 4 A b ( V V ) { } b Z7 b ( d ) Z Z b Z ( d ) Z4 ( ) Sn Sn [ b b Z0 ( d ) Z Z b Z5 ( d ) Z b Z b Z ( d ) Z4 b Z ( d ) Z [ b ] Z b Z ( d ) z4 b Z ( d ) Z ( ) (A.) z = R W β Ax x ( x xd ) ( ) ( ) ( ) ( ) ( xb xe ) ( x xf ) ( ) ( x xd ) ( ( )) ( ) ( ) / 4 } ( xb xe ) ( x xf ) ( ) ( x xf ) ( ) ( ) / 4 β Ex Sen ( x xd ) ( ) ( xb xe ) ( x xf ) { [ Ae e ( e e ) ] ( Ee e ( Ge He ) ) ( ) ( Ie Je ( Le Me ) ) ( e Oe ( Pe Qe ) ) 0 ( R S ( T U ) ) } β e e e e 4 4 ( J bo ) ( d ) 5 b ( ) ( ) d 7 ( ) ( ) ( ) ( L ) b b b 5 b 8 ( d ) ( b Q)

10 ( d ) b 4 ( d ) b 4 ( d ) 7 b ( ) ( ) ( ) d ( ) ( b Q) W b 4 ( d ) 7 b ( d ) ( ) ( O b S ) 5 b ( d ) 7 0 b ( d ) ( ) ( ) ( ) / 4 ( b T ) b 8 ( d ) b ( d ) 4 ( Q b U ) 7 b ( d ) ( b ) ( ) ( Q b U ) 7 b ( d ) ( b ) ( ) ( ) / 4 ( A b ( V V )) ( b ) Z7 b ( d ) Z Z b Z ( ) Z4 ( ) ( ) b Z0 ( d ) Z Z b Z5 ( d ) Z b Z b Z ( d ) Z4 b Z ( d ) Z ( ( b ) Z b Z ( d ) Z ( d ) Z ( ) 4 b β ( 4b 4b 4b4 4b5) β β (A.) where = Sn nd = o nd ll oeffent re expltly derbed n Qurell ( 00).