Dynamic Modeling and Vibration Control for Spacecraft s Solar Array Jian-Ping JIANG 1,a, *, Rui XU 2,b

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1 Intrnatonal Confrnc on Mchancs and Cvl Engnrng (ICMCE 2014) Dynamc Modlng and Vbraton Control for Spaccraft s Solar Array Jan-Png JIANG 1,a, *, Ru XU 2,b 1,2 Collg of Arospac Scnc and Engnrng, Natonal Unvrsty of Dfns chnology, NO.137, Yanwach Strt, Changsha, Hunan , P.R.Chna a janpng202@163.com, b walpc_xuru@139.com *Corrspondng author Kywords: Actv Vbraton Control, Solar Array, Pzolctrc Actuators. Abstract. Dynamc modlng and actv vbraton control for a typcal flxbl solar array of spaccraft by usng pzolctrc actuators s dscussd. It s dffrnt from th smplfd bam or plat modl for th solar array n th ltrat, a coupld dynamc modl of th solar array wth pzolctrc actuators s dvlopd frstly. Espcally, a four-nod rctangular pzolctrc plat lmnt s prsntd to dscrtz th smart solar array. Scondly, a LQR controllr s dsgnd to control th vbraton of th solar array. Numrcal smulaton shows that th vbraton of solar array could b supprssd ffcntly by usng pzolctrc actuators. Introducton Solar array s th vry mportant part of a spaccraft, whch wll supply th lctrc powr. Howvr, th spaccraft s solar array could b vry larg and flxbl, whos modal frquncs and dampng ratos ar rlatvly low. In ordr to mt th hghr prcson rqurmnt of spaccraft, th vbraton control of solar array bcoms mor and mor mportant [1]. In rcnt yars, smart matrals, such as pzolctrc matrals, hav bn usd xtnsvly as dstrbutd snsors and actuators for vbraton control of flxbl structurs. h actv supprsson of vbraton for spaccraft flxbl booms usng pzolctrc actuators was dscussd n Rf.[2]. Hu and Ma[3] prsntd an approach to vbraton rducton of flxbl spaccraft actuatd by on-off thrustrs, by applyng th varabl structur control tchnqu. h spaccraft to b nvstgatd s a hub wth a cantlvr flxbl bam appndag. Garca [4] dvlopd an actv control mthod to supprss vbraton of flxbl rbbd antnna structurs usng pzocramc componnts as both snsor and actuator smultanously. Howvr, most publshd paprs n th ara of actv vbraton control for th flxbl appndags of spaccraft ar basd on dynamc modls of cantlvr bam or plat structurs[2]~[9]. Actually, most of th solar arrays of spaccraft ar mult-panl dployabl structurs. hs dployabl solar array oftn mployd rgd honycomb panls ntrconnctd wth sprng-drvn hngs, bndng mchansm and lctrcal harnssng. hus, a smplfd bam or plat modl dos not prdct th dynamc charactrstc accuratly of th solar array systm. hs papr s am to study th vbraton control of a typcal dployabl solar array usng surfac bondd pzolctrc actuators. Frstly, a coupld dynamc modl of th solar array wth pzolctrc actuators s dvlopd basd on th Hamlton s prncpl. Aftr that, a vbraton controllr for th smart solar array s prsntd. Numrcal rsults dmonstrat th valdty and ffcncy of proposd dsgn schm. Fnally, som concludng rmarks ar drawn. Dynamc Modls of th Smart Solar Array Lt us consdr a typcal mult-panl dployabl solar array llustratd n Fg.1, whch conssts of solar panls, trpod, hngs and pzolctrc actuators. Each actuator conssts of two pzolctrc patchs bondd to oppost facs of th panl. h outr lctrods of th patchs ar lctrcally connctd togthr and th plat, whch s groundd, s usd as th othr lctrod for both patchs of th actuator par. In ths confguraton, whn an xtrnal voltag s appld to th patchs, on patch wll xpand and h authors - Publshd by Atlants Prss 320

2 th othr contract, thus producng a curvatur n th panl. h vbraton of th solar panl wll b supprssd n ths way. Fg. 1 A ypcal Spaccraft S Smart Solar Array In ordr to dvlop th dynamc modl of th mult-body dployabl structur, th fnt lmnt mthod s mployd. hus, th trpod and hngs of th solar array ar modld by usng a two-nod 3D bam lmnt. hs lmnt has sx dgrs of frdom (DOFs) at ach nod, whch stffnss and mass matrcs can b found n many txtbook. h compost panl of solar array wth surfac bondd pzolctrc matral bcoms typcal lamnatd plat. In ordr to formulat th govrnng quatons for ths lamnatd plat, th followng assumptons ar adoptd: (a) th plat s thn and pzolctrc actuators ar ntgratd nto th lamnatd compost substrat as pls; (b) th lamnat s prfctly bondd, lastc and orthotropc n bhavor wth small strans and dsplacmnts; (c) pzolctrc actuators ar mad of homognous and sotropc dlctrc matrals and hgh lctrc flds and cyclc flds ar not nvolvd. Basd on th classcal lamnatd plat thory (CLP), th dsplacmnt fld of th lamnatd plat wll b wrttn as x, y, z u x, y, vx, y, z v x, y z w x, x, y, z w x, y u z w y w (1) whr (u 0, v 0, w 0 ) ar th dsplacmnt of th pont on th rfrnc plan of th lamnat plat. An assumpton of ths thory s that ach layr undrgos th sam dsplacmnt as th md-plan of th lamnatd plat, whch also mans th dsplacmnt fld vctor would b th sam for th ach layr. Accordng to ths, th stran quatons would b wrttn as u0 v0 u0 v0 xx z,, 2 2 yy z 2 xy z w w w x x y y y x xy (2) Basd on ths assumptons a lnar consttutv rlatonshp s adoptd for th pzolctrc compost panl, whch can b xprssd as ζ Qε E (3) D ε pe (4) whr ζ xx yy xy s th lastc strss vctor and ε xx yy xy symmtrc lastc consttutv matrx, th pzolctrc strss coffcnts matrx, th lastc stran vctor, Q th E E x E y Ez th lctrc fld vctor, D D x D y Dz th lctrc dsplacmnt vctor and p th symmtrc dlctrc matrx, n th lmnt local systm (x, y, z) of th lamnat. 321

3 A four-nod rctangular plat lmnt wth fv dgrs of frdom at ach nod s usd. W dnot th nodal dsplacmnt vctor of ach nod by δ u, v, w, x, y and dnot th dsplacmnt vctor of all nods n th lmnt by qp δ 1 δ2 δ3 δ 4. Consquntly, th dsplacmnt vctor of any pont wthn an lmnt can b xprssd n trms of all nodal dsplacmnt by mans of ntrpolaton functons N such as 4 u 0 v 0 w 0 x y p 1 u N δ Nq hrfor, th strans wthn an lmnt can b xprssd n trms of all nodal dsplacmnts n th lmnt as (5) xx yy xy 0 z p ε B B q Bq (6) h lctrcal potntals ar assumd constant for ach pzolctrc layr wthn ach lmnt. Hnc thr s on addtonal potntal dgr of frdom for ach pzolctrc layr to rprsnt th pzolctrc bhavor and th vctor of lctrcal dgrs of frdom s φ 1,, n (7) h lctrc fld vctor can b rprsntd for an arbtrary lmnt E B φ (8) whr B s th lctrc-potntal matrx rlatng th lctrc fld vctor wth th potntals. By usng th gnralzd Hamlton s prncpl, th fnt lmnt quatons of moton for th lamnatd plat wth coupld lctromchancal proprts can b obtand. M q Kq K φ F u u K q K φ Q (9) whr M s th mass matrx, K s th lastc stffnss matrx, K s th dlctrc stffnss matrx and Ku K u ar th couplng matrcs btwn lastc mchancal and lctrcal ffcts. q, φ ar th systm gnralzd dsplacmnts vctor and voltags at actuator, rspctvly. F, Q ar th mchancal loads vctor and th appld lctrcal chargs, rspctvly. Vbraton Control of Smart Solar Array An optmal controllr s dsgnd to mnmz a cost functon, or prformanc ndx, whch s proportonal to th rqurd masur of th systm s rspons and to th control nputs rqurd to attnuat th rspons, such as th lnar quadratc rgulator (LQR). h fdback gan G s chosn to mnmz a quadratc prformanc ndx of th form 1 J dt 2 x Qx φ Rφ 0 (10) whr Q and R ar symmtrc postv sm dfnt wghtng matrcs, and thr lmnts ar slctd to provd sutabl prformanc. Assumng th control voltag φ Gx, thn th dtrmnaton of control gan G can b rducd to solvng a matrx Rccat quaton 322

4 1 A P PA PBR B P Q 0 (11) 1 hus th control gan for stat fdback s thn obtand by G R B P. h wghtng matrx Q for th modal spac quadratc prformanc ndx s usually assumd to b dagonal and taks th form Q dag 2 2 1, r,1,,1. Howvr, thr no gnral gudlns for th choc of R, th wghtng matrx for control forcs. A dagonal R, as assumd n most applcatons, may b wrttn as R r r r. dag 1, 2. m Numrcal Applcatons In ths scton, th accuracy of th coupld dynamc modl s dmonstratd frstly by comparng wth th rsults from th Nastran softwar. hn, th ffcntly of supprssng th vbraton nducd by an ntal dsplacmnt s valuatd. h pzolctrc actuators ar mad of PZ-5H pzolctrc cramcs, whch proprts can b gvn n [10]. h gomtry sz of ach actuator s mm 3. h fnt lmnt modl ncluds 40 3D bam lmnts, 300 plat lmnts and 388 nods. Unform structural dampng wth a typcal valu of 0.5% s assumd. Vrtcal dsplacmnt at pont B (as shown n Fgur 1) s slctd as th prformanc output. abl 1 shows th frst sx natural frquncs of th solar array from th proposd modl and th Nastran softwar. Furthrmor, th frquncs of th smart solar array wth pzolctrc actuators ar also lstd n th abl 1. It tlls us that th prsnt modl provds qut good rsults, and th pzolctrc actuators would chang th natural frquncy. Espcally, th maxmz dffrnc of th frst frquncy s 13.78% bcaus of consdrng th four pzolctrc actuators. hus, th accurat dynamc modl should nclud th nflunc of th pzolctrc patchs. ab. 1 h Frst Sx Natural Frquncs from Nastran Softwar and th Proposd Modl Numbr Nastran Prsnt modl wthout actuators Dffrnc Prsnt modl wth actuators Dffrnc % % % % % % % % % % % % Fg. 2 p Dsplacmnt of th Opn and Closd Loop Systm 323

5 Consdr an ntal dsplacmnt fld appld to th solar array wth a tp dsplacmnt qual to 0.1m. ransnt rsponss of th solar array tp dsplacmnt (at pont B shown n Fg.1) ar shown n Fg.2. Rsults whn th control s off ar also shown for comparson. Corrspondng appld actuaton voltags ar plottd n Fgur 3. As can b sn n Fg. 2, th LQR controllr manags to sgnfcantly attnuat th fr tp dsplacmnt wth an admssbl control voltag. h closd-loop 5% sttlng tm s qual to 52s, whch rvals a grat mprovmnt on th rspons attnuaton whn compard wth th opn-loop on. Not that structur natural sttlng tm corrsponds to th stat whn actv control s turnd off, and th structur vbraton s solly supprssd by structural dampng (0.5% rato n th prsnt cas). Hghr dampng rato wll rsult n shortr sttlng tm. Howvr, th dampng rato of larg flxbl spac structurs s usually low. Fg.3 h Control Voltags of th Pzolctrc Actuatros h maxmum actuator voltags ar 189V for actuator 1, 2 and 127V for actuator 3, 4, rspctvly. Bcaus of th symmtry about x-axs, actuators 1 and 2, 3 and 4, hav dntcal control voltags rspons. In practcal dsgn practc, t s mportant to nsur that th maxmum voltags on actuators ar wthn crtan rangs so that pzolctrc patchs wll not xprnc lctrc brakdown. Not that ths rangs ar prdfnd basd on matral proprts and dvc charactrstcs. If th lnar pzolctrc consttutv quatons ar usd to dsgn th structurs as n prsnt study, t s also mpratv to rstrct actuators wthn th lnar rang. h maxmum actuaton voltag n abov two cass s lss than 200V, whch dos not xcd th lnar rang of most pzolctrc cramcs. Fg 4 prsnts th frquncy rspons (bod plot) at th solar array tp (pont B n Fg.1). A 20 db and 10 db attnuaton s achvd wth th LQR controllr for th frst and scond rsonant frquncs, rspctvly. It s also dmonstratd th ffcntly of th proposd actv vbraton schm. Fg. 4 h Frquncy Rspons of th Opn and Closd Loop Systm 324

6 Conclusons hs papr dscusss th coupld dynamc modl and vbraton control of th solar array wth pzolctrc actuators. owards ths nd, th solar array bondd pzolctrc actuators s modld usng fnt lmnt mthods. h controllr dsgn s carrd out by usng LQR. Numrcal rsults dmonstrat th accuratly and valdty of th prsntd dynamc modl and th control schm. Espcally, t s dffrnt from th smplfd bam or plat modl for th solar array n th ltrat, th coupld dynamc modl s basd on th mult-body dployabl structurs, and th nflunc of th pzolctrc actuators s consdrd also. h vbraton control xprmnts of th smart solar array wll b carrd out to dmonstrat th ffcntly of th proposd modl and control schm. Acknowldgmnt hs rsarch was fnancally supportd by th Spcal Foundaton of Excllnt Doctoral hss. Rfrncs [1]D.C. Hyland, J.L. Junkns, R.W. Longman, Actv control tchnology for larg spac structurs, Journal of Gudanc Control and Dynamcs, 16 (1993) [2]S.D Gnnaro, Actv vbraton supprsson n flxbl spaccraft atttud trackng. Journal of Gudanc, Control, and Dynamcs, 21 (1998) [3]Qngl Hu, Guangfu Ma, Varabl structur control and actv vbraton supprsson of flxbl spaccraft durng atttud manuvr, Arospac Scnc and chnology, 9 (2005) [4]E. Garca, J. Dosch, D.J. Inman, h applcaton of smart structurs to th vbraton supprsson problm, Journal of Intllgnt Matrals Systms and Structurs, 3 (1992) [5]E.F.Crawly, J.d Lus, Us of pzolctrc actuators as lmnts of ntllgnt structurs, AIAA Journal, 25 (1987) [6]E.F.Crawly, K.B.Lazarus, Inducd stran actuaton of sotropc and ansotropc plats, AIAA Journal, 29(1991) [7]H.S. zou, H.Q. Fu, A study of sgmntaton of dstrbutd pzolctrc snsors and actuators. Part I: hortcal analyss, Journal of Sound and Vbraton, 172 (1994) [8]Rddy J.N., On lamnat compost plats wth ntgratd snsors and actuators. Eng Struct, 21 (1997) [9]H.S.zou, Intgratd dstrbutd snsng and actv vbraton supprsson of flxbl manpulators usng dstrbutd pzolctrc J. Robot. Syst., 6 (1989) [10]Y.H.Lm, V.V. Varadan, V.K. Varadan, Closd loop fnt-lmnt modlng of actv/passv dampng n structural vbraton control, Procdng of SPIE Conf. Smart Struct. Matr, 3039 (1997)