Thruster Modulation for Unsymmetric Flexible Spacecraft with Consideration of Torque Arm Perturbation

Transcription

1 hruer Modulaon for Unymmerc Flexble Sacecraf wh onderaon of orue rm Perurbaon a Shgemune anwak Shnchro chkawa a Yohak hkam b a Naonal Sace evelomen gency of Jaan 2-- Sengen ukuba-h barak b eo Unvery 3-- Hyoh a-ku Yokohama-h anagawa brac Mo of earh obervaon aelle have a ngle olar array due o he reuremen moed by he ocal enor vbly. h reuremen make acecraf confguraon hghly unymmerc whch roduce arcular dynamc characerc. When he acuaor uch a ga-je hruer ued for ranlaonal orb conrol ranonal mbalance orue caued even f he um of ranlaonal force vecor a hrough he cener of ma. n h aer h arcular dynamc for unymmerc flexble acecraf formulaed and he mehod o reduce he ranonal mbalance rooed. nroducon Fg. how he confguraon of he dvanced Earh bervng Saelle (ES) whch wa launched n 996 by Naonal Sace evelomen gency of Jaan and ha a ngle olar array exendng abou 23 m and lowe naural freuency le han. Hz. ES ha 4 hruer whch are arallel o he V drecon and hey have ame orue arm lengh hough a he begnnng of ranonally and maxmum rae abou.25 deg/. n h aer hee dynamc formulaed and analyzed. erved euaon how he nomnal orue arm whch correond o he dance from he cener of ma o he hruer nozzle are erurbed by he added erm; whch wll be called orue arm erurbaon. reul mbalance orue are roduced. nd hen a mehod o reduce he effec of orue arm erurbaon rooed. n h mehod he me when he reacon conrol yem Fgure - ES onfguraon (RS) wched from on-modulaon o off-modulaon deermned for each hruer o mnmze he V aude devae magnude of he ranonal mbalance orue. he valdy and uefulne of hee cheme are verfed hrough numercal mulaon reul.

2 Euaon of Moon acecraf model whch con of a man rgd body and a flexble olar array formulaed where ymbol are defned n able and zero marx of dmenon j denoed by j and un marx of dmenon by U. lde oeraor r for an arbrary vecor [ r r r ] defned a follow. r r3 r2 r 2 3 r3 r r2 r hen he euaon of moon gven a follow. M F () where Î a Î a d Ð Ð d à dl à Î Ð 3 3 ( m m ) H dl U Y Q H Î 2 3 M 2 23 Ð 3 32 à ÎU F à Ð Q F Q - U n 3 / H / H m -m 2 2 H à Y Q Î Ð U nn known from E.() ha all of aude dynamc ranlaonal one and rucural flexble one are couled wh each oher becaue nera marx M no dagonal. So durng V aude dynamc acceleraed no only by he drec conrol orue bu alo by he acceleraon erm of dlacemen and flexble deformaon. hee dl à

3 coulng effec are rereened a a orue arm erurbaon n he nex econ. N Newonan reference frame Man body Solar array ll yem formed by and ener of ma of ener of ma of ener of ma of Q Jon beween and R h hruer ( n ) n Number of flexble mode n Number of hruer x y h z m m Roll ax Pch ax Yaw ax Ma of Ma of / nera enor of / nera enor of F Q Roaonal coulng marx beween and Y Q ranlaonal coulng marx beween and ude conrol orue a d f lacemen force h hruer force H..M. beween and Poon vecor elemen from o Q Poon vecor elemen from o Q R Poon vecor elemen from o R w ngular velocy of lacemen velocy of v lacemen of h mode ( n ) Freuency of h mode ( n ) amng rao of h mode ( n ) ude angle of a d lacemen of dl lacemen of flexble mode able Nomenclaure orue rm Perurbaon nh h Subue Ê n R Ê d f a f no E.() and ung alace ranformaon he ranfer roery from he hruer nu f o he aude angle a derved a follow. a nh ( ) G ( ) ( ) G ( ) Ê { ( ) } f ( ) (2)

4 ' & & ) $ ( " where G ( ) ( ) % R 2 2 nh dl dl 2 ( ) # Ê! ( ) ( 2 dl dl 2 ( 2 2 ) ( ) { } f ( ) 2) ) 23 a dynamc a nomnal orue arm or dance from o R and ( ) erurbaon erm whch called orue arm erurbaon. oh G ( ) and ( ) nclude he effec of flexbly. f he acecraf ha relavely ymmerc confguraon he effec of ( ) are no gnfcan o only roere of ( ) G are condered. u f he acecraf ha rong unymmerc confguraon he become gnfcan. effec of ( ) n general hruer are arrayed ymmery wh reec o he cener of ma ha nh mean Ê. f( ) - 3. Fg.2 how wo hruer are algned o have ame nomnal orue arm lengh wh reec o bu known from E(2) ha boh hruer have he erurbaon arm. h make roaonal cener beng cloe o hown n Fg 2. hee effec are more hycally underood a follow; a he momen of hruer frng flexble rucure are end o be lef n he nera ace becaue of he nera force h make he roaonal cener beng cloe o he cener of ma of he man body from he cener of ma of he whole yem. Such roere are dcued n ref () omenaon for orue rm Perurbaon n general all hruer whch are arallel o he V drecon are wched from on-modulaon o off-modulaon a he ame me o all hruer are oened a he ame me a he begnnng of V. h reul n large ranonal durbance orue. n h econ a mehod o reduce P roaon cener :Pc.m. :enre yem c.m. :man bodyc.m. RS hruer uch mbalance rooed. n h mehod he me when he hruer are wched from on-modulaon o off-modulaon deermned for each oher o mnmze he magnude of he ranonal mbalance orue. enoed nvere f / Flgh recon x 3 2 Fgure 2 orue rm Perurbaon f y Man ody

5 J M 5 ;?; V X 4 W Z 9 V > ; < ; alace ranformaon a a a magnude of e nu a f and an mbalance 6 orue a whch correond o he f durbance orue wren a follow. 7 f ( ) Ò Ñ 2 2 a 2( dl dl ) 327 Ó 8 Ò Ñ 2 a 2( dl dl ) 328 á f â (3) Ó ã n general ma of man body much larger han ha of flexble ar n h cae E(3) aroxmaely rewren a follow. ( ) 2 a 32 f 2dag[ co ] 32 f FEG H : dag where w ³ RN ( n) are mode freuence. E.(4) agan aroxmaed a follow f he mo gnfcan mode condered. á â ã (4) ( ) ˆ cow (5) where Q P R R a domnan mode freuency and ˆ coulng coeffcen. herefore oal orue denoed by S ( ) gven a follow where S ( ) he me doman exreon of ( ) n E.(2). n [ h Y u ˆ co f (6) ÊU { } ( ) ( ) ( ) where a oenng me of h hruer. ranonal aude roery can be mroved o mnmze he eak of E.(6). h oluon deend on he hruer locaon o one of oluon gven for he raccal acecraf model n he nex econ. Numercal Smulaon ude dynamc of ES mulaed. ES ha 8 hruer (-8) mouned on he roll- urface and 4 of hem (5-8) are arallel o he V drecon hown n Fg.3.

6 e o h g w v u d h r ` dc b k j h g d f w v u c r fr a normal cae a ame a ES mulaed; ha ( 5 8). E.(6) rewren a follow. ] ( )_ 4^ ˆ co\ (7) Nex a cae where 7 ` ` a a 5 6 / 8 ` 2b / mulaed. E.(6) rewren a follow. mln f ( ) u( ){ 7 ˆ co } f 7 u { ˆ 5 co( )} f 5 yxz r yxz r 2 u { 6 ˆ co( )} f 6 u { 8 ˆco( 2 )} f 8 (8) uu of E.(7) and (8) are hown n Fg.4(a) and (b) reecvely. Maxmum mbalance orue n Fg.4(a) 4 me larger han (b). Fg.5(a) how dynamc mulaon reul when 4 hruer are x Fgure 3 hruer Vecor z y (a) Normal Fgure 4 urbance orue (nalycal Soluon) wched o off-modulaon a he ame me. ranonal large aude devaon een n h fgure whch ame henomenon een n raccal elemery daa.

7 (a) Normal (b) Prooed Fgure 5 Smulaon Reul Fg.5(b) how mulaon reul when 4 hruer are wched o he off-modulaon earaely. reul ranonal aude devaon eak almo dmnhed. oncluon n h aer he dynamc orue arm erurbaon he arcular dynamc of he umymmerc flexble acecraf formulaed a fr. erved euaon how he nomnal orue arm are erurbed by he erurbaon arm. nd hen he mehod o reduce he effec of orue arm erurbaon rooed. n h mehod hruer for V are wched from on-modulaon o off-modulaon wh arorae me nerval and arorae order where h euence are decded analycally wh conderaon of he domnan flexble mode. hrough numercal mulaon reul effecvene of rooed mehod verfed. Reference [] Shnchro chkawa Shgemune anwak and Yohak hkam: ynamc naly of Unymmerc Flexble Sacecraf Proceedng of he nernaonal Symoum on Sace echnology and Scence Moroka Jaan 2.