two masses in orbit

a DYNAMICS OFTWO SPRING-CONNECTED MASSES INORBIT* é LI-SHENG WANG! andSHYH-FENG CHENG a Associate Professor, InstituteofApplied.|Mechanics, National Taiwan University, Taipei, .Taiwan, R.O.C. (‘e-mail: [email protected]) (Received: 19July 1995; accepted: 18March 1996) Abstract. This paper discusses relative equilibria (orsteady motions) andtheir stability forthe dynamics ofthesystem oftwospring-connected masses inacentral gravitational field. Thesystem canberegarded asasimplified model fortheTethered Satellite System (TSS), where thetether ismodeled bya(linear ornonlinear) spring. Intheprevious studies oftheTSS problem, itwas typically assumed thatthecenter ofmass islocated atthemassive oneofthetwoend-masses, andmovesonagreat-circle orbit.However, forthesimplesystemtreatedinthispaper,itisproved thatnongreat-circle relative equilibria doexist. Some fundamental concepts ofthedynamics ofan arbitrary assembly moving inacentralgravitational fieldarediscussed. Thenotionofsteadymotionsused inengineering literature islinked with thenotion ofrelative equilibria ingeometric mechanics. Numerical computations show some interesting nongreat-circle relative equilibria forthespring- connected system, Radial relative equilibria, which correspond tothestation-keeping mode forTSS, arethen introduced. Within theframework ofsymmetry and reduction, their stability properties are investigated byadopting thereduced energy-momentum method, which takes theadvantage ofthe intrinsic symmetry structure. Itisshown thatforpractical configurations, thesystem atradial relative equilibria isstable ifsome conditions aresatisfied. Keywords:Springsystem,symmetry, relativeequilibrium, relativestability, tethered satellitesystem 1. Introduction This paper discusses therelative equilibria and their stability forthemotions of two spring-connected masses moving inacentral gravitational field. The system, composed oftwo end-point masses connected byanelastic spring, cf.Figure 1, istermed asthespring system. Itcanberegarded asasimplified model forthe Tethered Satellite System (TSS), which contains asatellite (orshuttle orbiter) connected toasubsatellite with along tether. There have been many interesting discussions onthis subject, especially after this idea ofTSS was putforth by Colombo etal.(1974). According to(Bekey, 1987), theearliest report onsuch idea was described byTsiolkovskii in1895, where an‘anchored tower’ from the surface oftheEarth tothealtitude ofthegeostationary orbit was conceived. The problems regarding dynamics andcontrol ofthese large systems inorbit have been investigated bymany researchers, cf.(Christ andEisley, 1970; Bainum etal.,1987; Beletskii and Levin, 1981, 1991; Levin, 1983; Liaw and Abed, 1990; deMatteis and deSocio, 1990; Pasca etal.,1991; Breakwell and Janssens, (1992); Penzo *This work waspartially supported bytheNational Science Council, Republic ofChina, under ‘grant NSC-83-0208-M-002-082. Theauthors would liketothank W.-T. Chou forsome computational assistance. CelestialMechanics andDynamical Astronomy 63:289-312, 1996.©1996Kluwer Academic Publishers. Printed intheNetherlands. ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : 290 LI-SHENG WANGANDSHYH-FENG CHENG é etal.,1989), andthereferences therein. Inthese literature, either distributed model Ei orlumped system were considered. While theanalysis ofthedistributed system H wasobserved tobequitedifficult todealwith,manyofprevious discussions treated thetether asamassless rigid barconnecting twopoint masses. Furthermore, itis mostly assumed that thecenter ofmass ofthesystem islocated atthemassive oneofthetwoend-masses, which moves onagreat-circle orbit,i.e.acircularorbitcentering atthecenterofthe field. However, even atsteady motions, these assumptions may notbevalid. Infact, forthesimple system treated inthispaper, it isproved that thenongreat-circle relative equilibria doexist; namely, thecenter of field andthecircular orbit traced bythecenter ofmass form acone. Accordingly, thedynamical behavior ofthemore exact model without theclassic assumptions becomes interesting. The assumptions made inthispaper areasfollows (J)theattraction center isatrest intheinertial frame; (2)thespring ismassless and undergoes extensive orcompressive deformation along only onedirection; (3)thegravitational attraction between thetwo end bodies isneglected. After constructing thekinetic energy and thepotential energy, itisobserved thatthesystem possesses SO(3)-symmetry. Ingeometic mechanics, cf.(Abraham and Marsden, 1978), such symmetry induces certain reduction ofthedynamics and thenotion ofrelative equilibria can bedefined, which actually corresponds tothenotion ofsteady motions intheliterature. The operations ofTSS mainly consist ofthree phases: deployment, station-keeping, andretrieval. Inthestation- keeping mode, theTSS iscontrolled such that thetether isstraight while pointing toward theEarth. This configuration isinfact attherelative equilibria. With these observations, thetechniques indealing with symmetry, reduction, and stability analysis can beused. Here thePrinciple ofSymmetric Criticality (Palais, 1979) isapplied toobtain conditions forrelative equilibrium, without thederivation of complete dynamical equations. Similar techniques canbeapplied tomore complete models, such astheonetreated in(Wang etal.,1994). Itwas proved intheliterature, e.g. (Wang etal.,1991), that ifthesystem is arigid body, theconfiguration ofTSS atthestation-keeping mode isstable for anapproximate model. This leads toward theso-called gravity-gradient stabiliza- tion technique. However, since thetether isnotrigid, thestability property may bedestroyed. Ithasbeen observed in(Amold, 1987) that numerical simulations showed theinstability ofsystems with very long tethers. Consequently, conditions forthestability oflong tethers with certain flexibility arevery interesting from either theoretical orpractical points ofview, cf.(Bainum and Evans, 1974; Belet- skii and Levin, 1985; Misra and Modi, 1987; Liuand Bainum, 1988), etc. For the stability oftherelative equilibrium, (Wang, etal.,1994) hasapplied thereduced energy-momentum method ontheTSS where thetether ismodeled asanonlinear string. However, duetothecomplexity oftheproblem, explicit conditions were not ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 291 é derived there. Inorder togainsome insights ontheeffects offlexibility andmake Ei thecomputations tractable, thetether ismodeled asamassless nonlinear spring in FS thispaper. , Asdiscussed inthesurvey article ofMisra and Modi (Misra andModi, 1987), even theequations ofmotion governing thedynamics ofalong tether system inthe station-keeping phase arequite complicated, notmention thestability analysis. As described above, thespring system indeed hasSO(3)-symmetry. Accordingly, itis intuitive totake thecombination ofenergy andmomentum asaLyapunov function candidate todetermine thenonlinear stability. This leads totheso-called energy- momentum method. Moreover, theintrinsic structure ofthespring system allows onetoapply thereduced energy-momentum method, which takes theadvantage of thesymmetry structure tosimplify thecomputations. The whole geometric frame- work canbefound in(Simo etal.,1991; Simo etal.1991; Wang andKrishnaprasad, 1992). Here themethod isoutlined andapplied tothespring system. Inthe following discussions, thesystem under consideration isdescribed in Section 2,along with thediscussions ofsome fundamental concepts ofthedynamics ofanarbitrary assembly moving inorbits.Theabstract framework ofsymmetry is discussed inSection 3,inwhich thesymmetry ofthespring system isobserved. The aforementioned Principle ofSymmetric Criticality isthen applied togetthe conditions ofrelative equilibria forthespring system inSection 4.Itisproved that with themodel treated inthispaper, nongreat-circle relative equilibria doexist. It isthecenter ofgravity (instead ofthecenter ofmass) that must trace agreat-circle orbit attherelative equilibria. Inparticular, ifthetwo end-masses arenotequal, there must exist anongreat-circle relative equilibrium. Numerical computations to find some nongreat-circle relative equilibria were performed. Itisobserved that while theorbit ofthecenter ofmass isinfact very close toagreat circle, the system may undergo significant deflections from thevertical (totheorbital plane) atrelative equilibria. The notion ofrelative stability isthen introduced inSection 5,along with an outline ofthereduced energy-momentum method applicable tothespring system. Section 6performs thestability analysis fortheradialrelativeequilibria andderives certain conditions forlinear ornonlinear springs. Inparticular, thecase oflinear spring isfurther explored; anditisfound that forpractical configuration (with the ratio ofthedistances between theouter spacecraft andinner satellite being smaller than 1.74), thesystem isstable ifthespring isstiff enough. Some concluding remarks aregiven inSection 7. 2.System Description The physical system under investigation, i.e.thespring system, isdepicted in Figure 1.Letthetwo end-masses bem,and ms, respectively. Denote thevectors from theattraction center Otom,and m,byaand5,respectively. The configu- ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 292 LL-SHENG WANGANDSHYH-FENG CHENG z4 m, a 0Y Xx Figure 1.Two spring-connected bodies inacentral gravitational field. ration space ofthespring system canbemodeled asQ=R°xIR3.With theonly external force —thegravitational force, thekinetic energy andthepotential energy ofthesystem canbefound tobe, ei Mayena ,Moye 7,6) =FFllal? +PUP, a2 2 and HM, pM V(a,b)=—Tor—T+W((la—ll), 2) lial] ell respectively, where thenotation ||-||denotes theEuclidean norm, andyisthe gravitational constant ofthefield. Inthepotential energy, thefirst two terms represent thegravitational potential energy, andthelastterm Wdenotes theelastic potential energy stored inthe(linear ornonlinear) spring. Inparticular, foralinearly elastic spring, W=}k(||a—b|| —0),withthespring constant kandthereference length éo(the initial length ofthespring without experiencing anyforce). From thekinetic energy andthepotential energy, thesystem Lagrangian canbe constructed along with theequations ofmotion being derived through theEuler— Lagrange’s equation. However, tostudy therelative equilibria andtheir properties, itisnotnecessary toperform thesometimes complicated computations. Infact, it will beshown later that thesystem hasanintrinsic symmetry structure, and thus some modern treatment canbeapplied. Foranassembly, such astheonedescribed above, moving inacentral gravita- tional field, itscenter ofgravity Gisdefined bythevector r,satisfying iA1[r True STeTEKudia) @)Uroll>—MeotarJassembty|\r||* where risthe vector from the center ofthe field toamass element dm, and Moat=Jassembly4 isthetotalmass.Lettheright-hand sideof(3)bedenoted byavector a.Itcan bechecked that a Ter ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 293 é Note thatwhile thecenter ofmass ofarigid body isfixed relative tothebody, its a center ofgravity changes with respect todifferent configurations ofthebody. For FS example, consider asystem consisting oftwoequalpointmasses connected bya 200kmlong massless rigid bar.When thesystem isputinataut configuration with thelowerpointmassattheradius 6,800kmLowEarthOrbit(LEO), thecenter ofgravity ofthesystem isatradius 6,897.83 km, which isaway from thecenter ofmassabout2.17km.Ontheotherhand,ifthesystem ispositioned abouthalfofthesynchronous orbit,e.g.20,000 km,Gisabout20,099.3 km,orabout0.7km from thecenter ofmass. Clearly, theposition ofthecenter ofgravity varies with thelocation oftheassembly. Other examples show that itvaries also with respect todifferent orientations, cf.(Rimrott, 1989). From Newtonian framework, itisnotdifficult toderive Cr.|Ug Cre, Ma _9, 4)de*TrsIP ® where r,denotes theposition vector ofthecenter ofmass oftheassembly. Itisalso noted that thetorque generated bygravitational forces about thecenter ofgravity Gisinfact =pr T—1,)Xaadm=0. Loca ”)Irrl3 Insome literature, cf.(Marion andThornton, 1988), thecenter ofgravity isdefined tobethepoint about which thegravity torque vanishes. Ontheother hand, the torque about thecenter ofmass is =r Ty (r—re)X—dm=pMouireX—25, Docs oeTelP “me” Teall? which typically does notvanish. Inlater discussions, thedifference between center ofmass and center ofgravity provides some physical insight onthenotion of nongreat-circle relative equilibrium. 3.Symmetry and Relative Equilibrium ‘Amechanical system may possess some symmetry sothat thenotion ofrelative equilibrium canbeintroduced. Foraspacecraft moving inacircular orbit,the steady motions refer tocertain configurations with which theassembly isstation- aryrelative toauniformly rotating frame located attheorigin ofthefield. This corresponds tothesymmetry ofthesystem withrespect totherotation groupSO(3) ingeometric mechanics. The special orthogonal group SO(3) consists oftherota- tionmatrices B,with B’B=1,anddet(B) =1,where 1denotes the3x3identity matrix, and1€IR.Lettheconfiguration space ofthemechanical system bedenoted ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 294 LL-SHENG WANGANDSHYH-FENG CHENG é byQ,withtheassociated tangent bundle TQ.Actions ofSO(3) onQandTQmay é bedefined as . $:S0(3)xQ@—Q, 87:so(3)x TQ@—TQ, respectively. Ifthekinetic energy 7’:TQ—Bandthepotential energy V:Q >R areinvariant under theactions, then itissaid that themechanical system has theSO(3)-symmetry. With thissymmetry, thephase space ofthesystem canbe reduced. The dynamics intheoriginal phase space can bealso reduced toyield thereduced dynamical equations. The equilibrium ofthereduced dynamics is termed therelative equilibrium ofthemechanical system. With respect tothe SO(3)-symmetry, therelative equilibrium isnothing more than thesteady motion oftheassembly. Accordingly, themethodology offinding relative equilibria and investigating their behaviors canbeused tostudy thesteady motions. Inparticular, forthesystem under consideration, theconfiguration space isQ=R?x3.The group actions onQandTQcanbeconstructed as 9(B,(a,b))=(Ba,Bb), ©1(B, (a,b,a,b)) =(Ba,Bb,Ba,Bb), respectively, with B€SO(3). Itcanbeeasily checked thatthekinetic energy (1) and thepotential energy (2)areindeed invariant under theactions, i.e. T(#7(B, (a,b,a,b))) =T(a,b, a,b), V(®(B,(a,b))) =V(a,b), forallB€SO(3). The spring system thus hastheSO(3)-symmetry. Tofind the configurations atthe relative equilibria, the Principle ofSymmetric Criticali- ty,cf.(Palais, 1979; Wang and Krishnaprasad, 1989), can beapplied, which is described below. THEOREM 1.Consider asimple mechanical system withsymmetry (Q,T,V,G), of(Abraham andMarsden, 1978), where QisaRiemannian manifold with metric K,Tisthekinetic energy, Vdenotes thepotential, andGisthesymmetry (Lie) group. Wehave therelation K(z)(02,%) =2T(v2), with vz€TQ. Given €€G,theLiealgebra oftheLiegroup G,theposition element ofarelative equilibrium corresponding to€isacritical pointoftheaugmented potential Ve:Q +R, Ve(a) =V(#) —3K(2)(EQ(2); €9(@)), where £qistheinfinitesimal generator associated with €. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 295 é Forthespring system, letting thevector €denote theangular velocity ofthe é uniformly rotating frame, theaugmented potential is . Ve(a,b) =V(a,b) ~T((Exa,€ xb) BM,_pM =7 +W(\\a—bl)— WalTay* Me ms—yExaExa)—(Exb,Exd). 6) Thecritical points ofVegive risetorelative equilibria orsteady motions. The same SO(3)-symmetry canbefound foranarbitrary assembly moving in theorbit, with thecorresponding relative equilibria being classified asfollows. DEFINITION 1.The relative equilibrium iscalled great-circle ifthecenter of massmoves onagreat-circle orbit,i.e.,thecenter ofthefieldresides ontheplane ofmotion, orr,-€=0.Otherwise, itiscalled nongreat-circle. Lettheoperator ‘~*denote theisomorphism from R?toso(3) (thespace of 3x3skew-symmetric matrices), defined by 0 -w3 wr wr w= w; 0-w |,wher w=] w|eR © -w, wr 0 ws Auniformly rotating frame withconstant angular velocity €canbethenrepresented bymatrix exponential e€*,which maps skew-symmetric matrices torotation matri- ces.With these notations, therelative equilibria canbecharacterized byfinding all 0's(constant vectors intheinertial frame) suchthatr(t)=e€'r, forallr(t)’s inthe assembly. Itisnotdifficult toshow thatthere exist r<.andry.(independent oftime) suchthatr,(t) =e€'r,,, andr,(t) =e€¢r,.. Substituting theabove expressions in (4),weobtain 28, Hlgo €brco+TG =02. 12)rool? Taking inner product on(7)with £,wefind To =0, (8) from which thefollowing theorem isproved. THEOREM 2.Atrelative equilibria, thecenter ofgravity r,isperpendicular to theconstant angular velocity oftheassembly, i.e.,rgtraces agreat circle. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 296 LL-SHENG WANGANDSHYH-FENG CHENG é Ontheother hand, taking inner product on(7)withrj,wehave Bi =[IEPUlrooll(raoPeo)=Ell?lIrall(ts-Fe)s (9) which isthemodified Kepler’s third lawattherelative equilibria. When theassem- blycontains only one point mass, thepoints Gand C’coincide with each other; hence theabove formula (9)reduces totheclassical Kepler’s third lawforcircular orbits. While thecenter ofgravity always traces agreat-circle orbit atrelative equilib- rium, thecenter ofmass may not. This fact isshown inthefollowing section. We remark thatthenotion ofnongreat-circle relative equilibrium wastermed ‘oblique regular motions’ in(Aboelnaga andBarkin, 1979;Barkin, 1985), wherearigid body system was considered. Itwas also described in(Wang etal.,1991) and numerically justified in(Wang etal.,1992). 4.Relative Equilibria oftheSpring System This section describes interesting relative equilibria forthespring system. Inpartic ular, theexistence ofthenongreat-circle relative equilibria isproved analytically, along with some numerical examples. Asdiscussed intheprevious section, weseek thecritical points oftheaugmented potential Vg.Thefirstderivative oftheaugmented potential isobtained asfollows, DvV¢(a,b) -(6a,6b) pg4-6a+pm, 66+W"(\la—Bll) (6—66)+ =pm, an —bi). .(6a—lla? lel |la—I) +1(€Ga)-6a+ms(Eb)-6b, where theprime ‘’’denotes thederivative ofWwith respect toitsargument. By thePrinciple ofSymmetric Criticality, theconditions ofrelative equilibria areas follows, HMmade zz, 1Ge~be“ot makbac+W'Ja.be=0, (10) pmsbe 3 1Ge—bepimibe b.—W'227e_=0, WWopFmethe Wb° “ where ae=||ae|] andbe=||bell. The terms containing W’represent theelastic forces onthespring. When the spring becomes more andmore rigid andthelength ||a.—be|| atrelative equilibrium approaches thereference length fo,thespring system becomes arigid barsystem. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System x DYNAMICS OFSPRING-CONNECTED MASSES 297 éz : ry y b m, a [e] x Figure 2.The frame system. Itiseasily checked that (10) and (11) areinvariant under thetransformation R,=Bae,Ry=Bbe,andQ=BE,whereB€SO(3)isthetransformation matrix from theinertial frame toanew frame and Ra, Rs, @arevectors inthenew frame. With this observation, asuitable frame issought tomake theproblem tractable. The frame adopted here issuch thatthe«--axis being parallel tothespring, with the z-axis being perpendicular toboth a,andb,,and they-axis completing thetriad, cf.Figure 2.Asforthecase thata,andb,areparallel, wemay arbitrarily choose az-axis without loss ofgenerality. With respect tothechosen frame, thevectors canbeexpressed asRy= (2ayYoy0)", Re=(2%,Yc,0)7,and =(M,M,M3)". Itisfurther assumed that ©,isgreater than z,.Under this setting, (10) and (11) arerewritten assix equations, t_ WwW (G+Wee+MMaye+WR=a (12) 120—(G+Bue+wes=O, a3) (M120 +Nryc)M3 =0, (14) 2492 zy_W!(M+ Wty+Maye+Hs=™ (is) Maes—(H+Bye+ws=0, (16)6 (Qixy +Mryc)M3 =0, a7) where Ra=||Ral|, andRy=||Rz||. The solutions oftheabove equations corre- spondtotherelativeequilibria ofthe spring system. Let ¢=24—4>0.Define Re=(Le,Yo,0)"with (ma+mp)zc =MaLa +Mya}. Itiseasily derived that ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : 298 LI-SHENG WANGANDSHYH-FENG CHENG é tq=ttmel/(me+ ms),—ty=_—Mal/(mMa +ms). (1g) 5 Givenpy,ma,ms,é,andR-=||Rel],theprocess ofsolving theequations (12)-(17) : canbedivided intothefollowing twocases. 123#0 The equations ofrelative equilibria canbesimplified as, 2=0, Maye =0, 23#0, (19) aw' [-w8+9)+5]m=nT (20) (-23+fi)te=0, (21) -(034.93)44)0,-@(03+03)+Alm= (22) 93+&)ye=0. (23) ( R Subtracting (21) from (23), weget 1 1 ‘The discussions may befurther separated depending onwhether y.=0ornot. First, fory,#0,2must be0,and R,=Ry=R.The solution canbeeasily obtained from (19-23) as Ra=(36,0010), Ro=(—Hlsyey0)", and2=(0,0,95)", with 12amf(mom) ,Bat. 4\mat+m RB This isagreat-circle relative equilibrium with W’=0,which indicates thatthere isnoforce inthespring, cf.Figure 3.Atsuch relative equilibria, thetwo point masses move onthe same circular orbit. Next, weturn tothesituation y,=0.Since 2,<0<aqisnotphysically interesting, itisfurther assumed that x,>«y>0.Thus, Ra=(2450,0)", Ry=(24,0,0)", Re=(2,0,0)". ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 299 : m, {o) m, Figure 3.Relative equilibrium with zero elastic force. This isinfact thecase ofradial relative equilibrium, i.e.thespring deforming along aradial axis. The frame canbethen selected such that 2=0.With these observations, theconditions forrelative equilibria arefurther simplified as %=0, %=0, 40, y%=0, (24) 1 a ™a w (-98+5)m=. (26) = ms Asaconsequence, theconfiguration ofaradialrelativeequilibrium isderived, with Ry=(20,0,0)", Rs=(24,0,0)", —@=(0,0,95)", a Ma,mM 93=—_“§__(™e,™3°(ma+me)Re(3*“*), (3-23) 7) yt—HimatMs(a ~Tp. "=Ug-+m)eaaph *® This isagreat-circle relative equilibrium with tensile elastic forces, cf.Figure 4. I.23=0 Forsuch case, theequations ofrelative equilibria become _¢2 Za__W!Ve+UDI +HE=~ (28) Mate —Nye+HHS=O, (29) ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 300 LI-SHENG WANGANDSHYH-FENG CHENG a m, Figure 4.Radial relative equilibrium _@2 ze_WwW!Yep+UMye +Has=m 30) M224 —Nye+WHS=0. (G1)b For ye=0,thesolution leads totheradial relative equilibrium asdiscussed in case I,byinterchanging 22and23.Therefore werestrict ourattention tothecase ye#0.Assume first that R,=Ry=R,which implies 2=—zp. The above equations arerewritten as 2 ZW!—2eq +MUAY +HRS=a? (32) 1200 —Nive+HAE=O (33) 2 ay_W!Gay +UM. +wRS=To (34) Maze~Oye+WHE=0. (5) From (33)and(35), itcanbeproved that.=97R3, which implies further that Q=0.Thus, from (32) and (34), wemust have m, =my. Consequently, the solution forrelative equilibrium isobtained as Ra=(36,450), Re=(—34,40,0)", —%=(MN,0,0)", with = = 2_ _malMa=™, Yo=Rey=Fe» W'=—Spr<0. ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 301 Bi m, Figure5.Relative equilibrium withcompressive elasticforceandma=mb. This relative equilibrium isalso great-circle, atwhich thespring with compressive elastic force isperpendicular totheorbital plane andmgandmyareequidistant to theattraction center, cf.Figure 5. Ontheother hand, forR,#Ry,from (29) and (31), weobtain 11 36 (La—£»)%1.Q2 =—HYeBR #0, (36) which implies 2;#0andQ)#0.Byadding mgx(29) andmgx(31), itis found that (mg+m4)(Mee—My)% =n(TE+™)y,£0, er)RB” R which implies 22 —Q1ye #0.From (28) and(30), wehave (17q+ms)(Qaite~Mrye)%=w|MT*+MEP)40, (38) Ry R, Define Mala |Myth Ma,Mp fp== += #0; id =(s+a) ue#0. fomTit+pt#0,andfy(+m) ux From (37) and (38), itisobserved that %__fe a ty Since fave~fyte=(FT) (a,ayy#0 2cyt=ma+m RBRTa—Zh)Ye ” ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 302 LL-SHENG WANGANDSHYH-FENG CHENG é weknow that : 2.01+yor#0, 9) . which indicates that there isnogreat-circle relative equilibrium forthis case. Moreover, itcanbeeasily verified thatthevectors r,andr,defined inSection 2 arenotparallel toeach other. Eliminating 2;andQ2from (36), (37), and(38), oneobtains f=fefit fyh =0, (40) where fa 11 =3- +. d f=|—-s)u. A3RPandfo(z#)ue Now ifma=ms=m,then 11 1 1 1 11R24R-@)(S[+s5G54+5)+5>|-xa+Re-e)ritwentTe)*Bake Fornatural configurations, wehaveR2+R?>R2>@,andthus(40)implies R,=Ry.Consequently, thefollowing theorem isconcluded. ‘THEOREM 3.Attherelative equilibrium ofanatural configuration, wheretheconstraint forceiscompressive, Ra=Rsifandonlyifma=ms. Inparticular, thecase ofm,#msleads toward thenongreat-circle relative equilibria, inwhich Ry=(2a,¥er0)", Re=(2854e,0)", —@=(M4,%, 0)", which areallonthery-plane. For such case, equation (40) needs tobesatisfied, Letc,=R.cos@ andy,=R.sin9. Given 1,Re,¢,M,, andms,theformula (40) canbewritten asanequation of8.Consequently, nongreat-circle relative equilibria canbeobtained bysolvingf()=0.Notethat@isindependent ofj:or2. Since f(0) <0,f(x) >0,andthatf(6) isacontinuous functionforR.>& (which istruefornatural configuration), there exists asolution forf(@) =0.On theother hand, thefirst derivative offcanbefound tobe df lye 1{3 1 o(3 1Fg) =—e 3o,4h 3ha0”)=in+ms)™(Bw+eg)+™(Ree,+Rs}t 11 1,1) 22y2 +mam[xa+2(gei)-(s+)l}: ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 303 6 oO g Dm Uc a Zz é y x = sLA\ : a Figure 6,Nongreat-circle relative equilibrium, which canbeproved tobepositive. Asaconsequence, theequation f(9) =0has oneandonly onesolution inthedomain 0€(0,7]forma4mpandRy>é. Withthevalueof0,thevariables 2;and22canbecomputed fromtheformula = fyB, %=—frh, where p-,| He (ltCfefy \Ri RR)” The elastic force onthespring canbethen obtained from either (28) or(30). This leads immediately tothefollowing theorem. THEOREM 4.For natural configurations (R, >@),thedynamics ofthespring system hasexactly onenongreat-circle relative equilibrium ifand only ifma#mp. The nongreat-circle relative equilibriaaredepictedinFigure6,wherethespring also undergoes compressive forces. Note that when thespring becomes more and more rigid, thespring system approaches arigid barsystem, which consists oftwopoint masses connected bya massless rigid link. The analyses also show that there arenongreat-circle relative equilibria fortherigid barsystem. While thenongreat-circle relative equilibrium forarigid body was numerically obtained in(Wang etal.,1992), itisanalytically proved inthis paper. Tojustify theexistence ofnongreat-circle relative equilibria, some numerical computations were performed tosolve f(8) =0fordifferent R/£ andmq/(1ma +ms) (itmay beassumed that ma<m»without loss ofgenerality). The results arepresented in Table I. Let¢denote theangle between therotating axis andthez-axis oftheframe system, c.f.Figure 6.The angle between 9and R,isthen 9—¢.The configura- tions forsome nongreat-circle relative equilibria arenext computed, and shown in ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 304 LI-SHENG WANGANDSHYH-FENG CHENG é Table I.Solutions off(@) =0 8 @(deg) Re/t=2 Rft=20 Re/f=200 —-Re/é =2000 —™*_ 0.01 10036489817 91,05265856 —90.10528084 —_90.01052809 mame —™_=01 98.41353280 90.85926471 90.08594349 —_90,00859436 mame —™_ =02 9627869712 90.64441681 90.06445759 90.00644577 mem —™_ 503 94.17070995 90.42959610 90.04297171 _90.00429718 met me —_ =04 92.08080378 90.21479351 —90.02148585 —_90.00214859 Ma +m Table II.Nongreat-circle relative equilibria ma(kg) mo(kg) 9(deg) ¢(deg) 6(deg) 1 9999 91.073934 1.073934 -0.00000026839 10 9990 91.071999 1.072002 -0.0000026767 100 9900 91.052659 1.052684 —0.0000260481000 9000 90.858264 0.859458 —0.000193352000 8000 9.644416 0.644674 —-0.00025784 Table II.Here 6=@—¢—90°, anditisassumed thatR,=7,000km, £=350km, andjt=4x10!4(m?/s?), thegravitational constant oftheEarth. Note that 6#0implies thatthecorresponding relative equilibrium isnongreat- circle. Although thisangle israther small, thedeflection ofthespring system from thevertical (oftheorbital plane), represented by¢,issignificant. Forlonger tether atLEO, this attitude drift may reach tens ofdegrees. This effects oforbit-attitude coupling canbesignificant forsystems with long tethers such astheorbit tower or theskyhook. 5.Relative Stability and theReduced Energy-momentum Method Relative equilibria aretheequilibria ofthereduced dynamical equations after performing thereduction process with respect tothesymmetry. They arecalled relatively stable ifthecorresponding equilibria arestable forthereduced dynam- icsinthesense ofLyapunov. Recall that inclassical mechanics, cf.(Goldstein, 1980), theRouthian reduction process canbeperformed forsystems having cyclic coordinates. Ingeometric language, thisreduction corresponds toasymmetry with ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES. 305 é Euclidean group R”.Thestability ofthesteady motion ignoring thecyclic coordi- 5 nates isinfactequivalent totherelative stability ofrelative equilibrium. g Forgeneral reductions, suchastheonetreated here,thesymmetry groupmaynot beR®.The classical Routhian approach isthus notdirectly applicable. Ontheother hand, recent development ingeometric mechanics enables onetoassess thestability ofsteady motions viatheso-called reduced energy-momentum method, cf.(Simo etal.,1991; Wang etal.,1992). There thesum ofenergy map and momentum map isused toobtain conditions forrelative stability. With theterminologies in geometric mechanics, cf.(Abraham andMarsden, 1978), themethod applying to thesystem under consideration canbebriefly outlined asfollows. @®Pick €€R3. (ii)Find a.,besuch thatDVe(ae, be)=0,cf.(5). (iii)Compute thepremomentum map J:QxB3>B3, F(a,b,&)=Tioex(@,b)E, where I?isthelocked inertia tensor, defined through theformula, (Coiocx(asb)n) =K(a,)((¢ xa,¢xb), (nxa,nxb)), for¢,7€R3. Thenlet1,=I(ae,be,€). (iv)Find thekernel ofDJat(a,,be, €). (v)Compute thesubgroup Gy. ={B€S0(3):Bu. =He} andtheassociated tangent space ofthegroup orbit, T(a,,5.,¢)(Gue (aesbe, €)),wheretheactionofG,,on(a,b,€)isdefinedtobe B,(a,b, €)+>(Ba,Bb,BE). (vi) Find thesubspace Ssuch that kerDJ(a.,be,€) =5©Tansbe,e)(Ge *(deybes))- (vii) Define thebilinear form T(de, be,€)+(a1, 661,1)-(642, 662, 02) =(M15Toca@esbe)M2) +D?Ve(@es be)+(Sar,61)«(Sa2,5b2). Then therelative equilibrium isrelatively stable iftheassociated quadratic formonSispositive definite. Here thebilinear form isinfact thesecond variation oftheenergy-momentum map induced onanappropriate space. For more complicated systems, thegen- eral form ofthereduced energy-momentum method canbefound intheabove- ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : 306 LI-SHENG WANGANDSHYH-FENG CHENG é mentioned references. Themethod listed above shall befollowed stepbystepto d determine thestability ofradial relative equilibrium inthenextsection. 6.Relative Stability ofRadial Relative Equilibria The radial relative equilibria discussed inSection 4correspond tothestation- keeping modeforTSS.Thestability ofsuchmodeisimportant duringtheoperation ofTSS. However, theclassical energy method isnotadequate forthesystem under consideration. Accordingly, thereduced energy-momentum method isemployed inthis section toprove thestability ofradial relative equilibria. Forsimplicity, werewrite theconditions forradial relative equilibrium (27) as Ra=(a,0,0)", Rs=(b,0,0)", a>b>0, and2=(0,0,w)?, with maat+msb 4 =n5 41 He[ae +my”? a and 2 2 2+ab+b?)mamyw’ wi=tab +mam? (4_4)50, 42) mabe mar (78) > @ Without loss ofgenerality, wechoose €=(0,0,w). Then Step (ii)described in Section 5isfulfilled bytheradial relative equilibria. InStep (iii), themetric ICcan befound tobe K(a,b)((v1,¥2), (Wi,W2))=ma(vi,W1) +ms(v2,2). Thusthelocked inertia tensor I?,.,(a,b) canbederived as 1),(a,b) =—mga@a —mybb. Thepre-momentum mapJ:QxB3>R3isthen T(a,b, €)=Tex(a,b)€ =—madia€ —mybbE. Accordingly, thederivative ofJis Di{a,b,€)-(6a,5b,n) =m,(2a€ —G@)Sa +ms(26E —€b)6b —(maa +mybb)n. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 307 j Atradial relative equilibrium, a,=ae1,be=be;and€=wes, where e,’s é denote theunit vectors ofthecoordinate axes oftheuniformly rotating frame. g Thus, 000 Tcy(desbe) =e] 010), withI,=mga? +mpb?. oo Since thelocked inertia tensor issingular atsuch radial relative equilibrium, the block diagonalization result in(Simo etal.,1991; Wang andKrishnaprasad, 1992) cannot bedirectly used; however, thereduced energy-momentum method still applies. Atradial relative equilibrium, wehave Ide,be,€)=[ewes=pe. Next itisrequired tofindthekernel space ofDJattheradial relative equilibria. Let6a=(641,542,603)", 5b=(6b,6b2,663)”andm=(m,1,73)".Itcanbe checked thatthekernel ofDJ(a-,be,€) is ker(D(ae,b., €))={(¢a,66,0): maasas+myb6b3=0, m=0,m=-%(myasay+inyo6)}.(43) InStep(v),weneedtofindTa,»,,¢)(Gu° (de,Be,€)), thetangent space onthe group orbit, where Gy, ={e:q//e3}- From theabove observation, thetangent space isimmediately obtained, Taebese)(Gue *(desbe,€))={(ader, aber, 0):a€ R}. (44) Obviously, Tya,,.,€)(Gue *(debe,€))isasubspace ofker(DJ(ae,be,€)).From (43)and(44),theSpace5inStep(vi)canbeexpressed as Ss{(a,5b,):607=6by=m=0,mgabas+mybbbs=0, dwm=~F(meabay +mmybbbs)} (45) Step (vii) isnext performed. The quadratic form corresponding toTonScan beobtained as ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : 308 LL-SHENG WANGANDSHYH-FENG CHENG : T(a.,be,€) «(6a,6b,)-(6a,6b,n) 4u? 2, =F(mmgabay +mdb)+mg[-(%+.)fa+S603]+ 2,+m,[-(#+*)562+fats]+ +W"-(6a;—5b1)?+W’-(6a3—5b3)?/(a —). (46) Since W’>0anda >b>0,itiseasily checked thattheterms containing a3and 663intheabove expression arealways positive. Hence todetermine the positive-definiteness, itisonly necessary toconsider thefollowing terms 2Bes|mde? —(240?)my+Ww"bah+ Te a 2+|Simpor-(#+*)my+W"|865+ Te 63 ‘Aw?+2(“mmc -w)501564. (a7) Defining 4m? ap 4myb?=me Ey pLBST ae PT RR Fin(47) issimplified as F=(mqw?u +W")baz +(myw?v +W")66}+ +2(4u?margab/I. —W")6a16b). The necessary andsufficient condition forF’being positive-definite isthat maw*u+W" >0, (48) and (mu+mvtSeenst) ow" 252+(w-‘onan amy!>0. 9)Re The results aresummarized inthefollowing theorem. ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System ‘ DYNAMICS OFSPRING-CONNECTED MASSES 309 é THEOREM 5.Foraspring system whose spring characteristics aregoverned by d theelastic potential energy function W(possibly nonlinear), theradial relativeg equilibrium isstableif(48)and(49)aresatisfied. Checking theconditions onstability thus requires theknowledge oftheelastic potential energy W.Itisnoted that intheclassical approach, such astheone in(Beletskii and Levin, 1991), thereduced system isobtained byderiving the equations ofmotion relative toaconstantly rotating frame. The stability analysis isthen performed with anappropriate Hamiltonian. Ithasbeen checked that such method leads toasimilar quadratic form asin(46), however, without thefirst positive term. Asaconsequence, theconditions obtained bythereduced energy- momentum method areweaker than those byclassical energy method. Next weconsider alinearly elastic spring whose elastic potential energy is W(a-) =}k(a—b- b), wherekisthespringconstant andfpisthereference length.From(42),wehave w'epmamp(a? +ab+b?)x@62(m,a +myb) > where x=(a—b)/(a— b—fo)>1.Asufficient condition forstability canbe then obtained as 2 Araup ttt so, (50) mtr and .wrtrtl 16mr? B=CT xX+waa > (51) where Ct=mutv4Ser =m, pat Smee OFre my? = Itcan bechecked that At={(r3)m? +[(r5474 +0)+(39>—20?=3r)]m+ +P +?+r)x- (PF+2) /Ph, where h=(m+1/r)(m+4 1?)>0. ©Kluwer Academic Publishers +Provided bytheNASA Astrophysics Data System : 310 LI-SHENG WANGANDSHYH-FENG CHENG 10 q nr Ke 14 10 sie ead “2 yr 10 ey /10° : 2 10 m 10° Figure 7.Xwia With respecttomandr. Since r>1andx>1,thecoefficients ofm?,m!andm°intheabove expression ofA*areallpositive. Thus A*isalways positive. Ontheother hand, wehave Ct=[rm +(—2r6 +375—13+6r?—3r)m? + +(-3r5 +6r4—13+3r—2)m+r3]/1h, which canbechecked numerically tobepositive if1<r(= a/b) <1.74, which isaphysically accepted range. Within such range, B*ispositive ifandonly if (m+1?)[16mr?/(mr? +1)?—uv]_ x?Gr+rtl) =Xmin(m1)- Some values ofXmin(m, 7)with respecttomandrareshowninFigure7,whereitis found thatpractically isgreater than Xmin. Consider, forexample, aradial relative equilibrium ofthe TSS with m,=10,000kg, mp=100kg, anda=7,350,000m, b=7,000,000m. Then r=1.05 and m=100, Since 1<r <1.74, the radial relative equilibrium isstable ifx>Xmin(m, 1),where Xmin(m, 7)=1.05138. One equivalent condition isthatthereference length 1 > (1-——) (a= 6)=17103.7m. Xin ©Kluwer Academic Publishers *Provided bytheNASA Astrophysics Data System : DYNAMICS OFSPRING-CONNECTED MASSES 311 é If£9=340,000 m,wehave x=35,andthus therelative equilibrium isstable. : With respect totheEarth’s gravitational field, thecorresponding spring constant H must bek=0.011 (Nt/m), which implies thattheproduct ofYoung’s modulus E ofthetether material andthecross section area Aofthetether isequal toEA = key=3,730.45 (Nt). Accordingly, forthematerial Alumina with E=69GPa, the cross-section area shouldbeA=5.4x10-8m?. Fortherigid barsystem, which isthelimiting case ofthespring system with xX—00,thecondition isalways satisfied andthus theradial relative equilibrium is stable. 7.Conclusions Inthispaper, thedynamical behaviors ofthesystem oftwospring-connected point masses moving inacentral gravitational field were discussed. Itwas observed that thespring system inorbit possesses theSO(3)-symmetry. Attherelative equilibrium corresponding tothissymmetry, itisthecenter ofgravity (instead of thecenter ofmass) that must trace agreat-circle orbit. This leads tothenotion of nongreat-circle relative equilibria, inwhich thecenter ofmass traces anongreat circle. For thespring systems under consideration, such nongreat-circle relative equilibria exist ifandonly ifthetwoend-masses areunequal. Tostudy thestability oftheradial relative equilibria, thereduced energy-momentum method was applied tosuccessfully obtain some stability conditions, For general (linear ornonlinear) springs, theconditions areintermsofthesecondderivatives ofthe elastic potential energyfunctions. Inparticular, foralinearspring,explicitconditions wereobtained such that, forcertain configuration, theradial relative equilibrium ofTSS become stable ifthetether ischosen tobestiff enough with suitable material constants. Although theradial relative equilibria arestable formost physically acceptable configurations, theanalysis inthispapermaybecomeessential formorechallenging designs such astheorbital tower (Pearson, 1975) ortheskyhook (Moravec, 1977). References Aboelnaga M.Z.andBarkinY.V.:1979,Stationary motionofarigidbodyinthefieldofattractionofasphere. Astron. J.,Acad. Sci, USSR. 56,881-887. Abraham, R.andMarsden,J.E.:1978,Foundations ofMechanics. Benjamin/Cummings, Reading, 2nd edition, 1978. 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