Static analysis of synchronism deployable antenna

Guan et al.1J Zhejiang Univ SCIENCEA 2006 7(8):1365-13711365Jourmal of Zhejiang University SCIENCE AISSN 1009-3095 (Print); ISSN 1862-1775 (Online)www.zju.edu.cn/jzus; www.springerlink.comJzuSE-mail: jzus@zju.edu.cnStatic analysis of synchronism deployable antennaGUAN Fu-ling"', SHOU Jian-jun', HOU Guo-yong', ZHANG Jing-jie('Department of Civil Engering, Zhejiang Universiy, Hangzhou 310027, China)(Chongqing Research Institute of Building Science, Chongqing 400015, China)'E-mail: ciegfl@zju.edu.cnReceived Mar.17, 2006; revision accepted Apr. 23, 2006Abstract: A 3D synchronism deployable antenna was designed, analyzed, and manufactured by our research group. This an-tenna consists of tetrahedral elements from central element. Because there are springs at the ends of some of the rods, spider jointsare applied. For analysis purpose, the structure is simplified and modelled by using 2D beam elements that have no bendingstifness. Displacement vectors are defined to include two translational displacements and one torsional displacement. The stiff-ness matrix derived by this method is relatively simple and well defined. The analysis results generated by using software de-veloped by our research group agreed very well with available test data.Key words: Space deployable structures, Analysis design, Reflector antennadoi: 10.1631jzus.2006.A1365CLC number: V414.2INTRODUCTION .values using both finite difference and finite elementtechniques. Armero and Ehrlich (2006) proposedLarge antennas for space applications have some some new finite elements for thin Euler-Bernoullidifferences as compared to their counterparts de- beams that incorporate the softening hinges to de-signed and manufactured for ground usage. For a scribe the beam's deformation. Yau (2006) suggestedlarger space antenna, the applied loads in space canbe a finite element model for buckling analysis of ta-several orders of magnitude less than the gravity pered I-beams subjected to torsional moments. Maloading it is subjected to when being fabricated and (2006) analyzed the ideal displacement amplificationassembled on the ground. Since the effect of gravity ratio of a bridge-type flexure hinge based on the ki-cannot be removed from any experimental verifica- nematic theory.tion performed on the ground, we must rely on the useTwo types of special beam elements developedof efficient and highly accurate analysis methods for specifically for modelling and analyzing largedesigning and verifying large space antennas. Tibert 3D-synchronism deployable space antennas will be(2002) provided the state-of-the-art of deployable studied in this paper. The first type is a one-beamantenna. And many researches have been trying to element with torsion springs formed by two elementsanalyze the beam with torsion moments.of the first type that are linked by a pin joint. As someNa and Kim (2006) analyzed the deployment of loads are always zero, the stiffness matrices can bea multi-link flexible structure with the Timoshenko simplified and used to perform static and dynamicbeam theory and determined the equations of motion analyses of the antenna.using Hamilton's principle. Serna et al.(2006) pre-sented a set of equivalent uniform moment factorREDUCTION FORMULATIONProject (No. 863-2-4) supported by the National Basic ResearchFig.1 show中国煤化工。e analyzed.Program (863) of ChinaYHCNM HG.1366Guan et al. IJ Zhejiang Univ SCIENCE A 2006 7(8):1365-1371[In+ptp{}n+) = {}(+p)，(3)If the number of zero load is p, i.e., l,=0, thesystem of equilibrium Eq.(1) can be split into thefollowing two coupled systems of m and p linearequations, respectivelyA.σ。+A.σp=lm,(4.1)A.σ+A.σ,=0.(4.2)a)If App is not singular, system Eq.(4.2) can be solvedfor σp, in terms of σnσ,=-ApAmσ。(5)Subtituting Eq.(5) into Eq.(4.1) yields .(Am-A.AbpA.m)σ,=Iw(6)b)Fig.1 Antenna structure model. (a) Stowed state;That is(b) Deployable stateA°o*=l',(7)This is a beam-and-truss structure (Yue et al., 2001; whereZhao and Guan, 2005). The equilibrium equation of(8)this structure can be derived in terms of generalizedstresses in a Cartesian coordinate system. Formula-So that we can obtain (Zhang, 2001)tions are rewritten as follows (Pellegrino and Cal-ladine, 1986; Zhang, 2001):(Am -AMATA)d. =8,-AAPTe。(9)[I0+pxp}m+t) = {)m+p,(1)This is a new system of compatibility equations, withwhere A is flexibility rectangular matrix, σ is gener- an nxm coefficient matrix. The internal work isalized stress, l is generalized loads. The equilibriumequation can also be expressed in terms of generalizedσ ε=σ%e, +σjep.(10)displacements[AT ]m+p){d}mop) = E}a+p)，)(2)The reduced flexibility matrix iswhere AT is gomeric compatibilit (m+p)(n+p) F"=Fm - rmAGAm -AAFm+AHATpACHAmmatrix, d is displacement of node (m+p), ε is gener-(11)alized strain. This equation is valid only for smalldeformation.The reduced sifness matrix isFor a linear-elastic material, the generalizedstresses σ are related to the generalized strains ε byK'=(F*)-'=Fml-AApFmpthe matrix of member F, or just flexibility matrix by中国煤化工(12)the relationshipYHCNMHG.Guan et al.1J Zhejiang Univ SCIENCEA 2006 7(8):1365-13711367MATRIX OF FIRST TYPE OF MACRO-ELEMENTwhere M;, M} are torsion moments of the spring, mi,Fig.2 shows a one-beam element that has torsionmi2, mjl, m;2 are generalized stresses loaded at the endsspring. joints at each end. The torsion spring joints canof torsion spring, T is axile force, L is length of thebe connected to other elements or spider frictionlessbeam, superscript 1 represents the number of macro-pins. A torsion spring joint transmits the force com-element. At the nodes in Fig.3c, force equilibrium andponent and bending moment. When the joint revolvesdisplacement compatibility must be maintained. Theequilibrium equation is(turns), it also transmits shear force.The beam is a 3D structure of a plane beam.0]nn0fix0 0 -1/L 1/LrlfFig.2 First type of macro-element-1Mm,2 + myEquilibrium matrixM,fxf2:Macro-element 1 can be separated into a beamM}and two torsion springs as shown in Fig.3.00m]; + m2,y000mj(14)(9) +.(2)As mn+my and mj1+m2y are always zero, the equilib-Torsion spring im(a)Torsion springj rium equation reduced to[ fix1mi m2+m,m;+m, mz0 -1/L 1/Lfi2m2i(15)f2x0 1/L -1/L|L M2"2(c)l m;」Fig.3 Reduced macro-element 1. (a) Separated forceThe coefficient matrix of Eq.(15)is A.model; (b) Force combination; (C) Force simplificationIn the state of Fig.3a, the equilibrium equation isGeometric harmonious matrixLet matrix A" turn, get (A )' , which is a reducedgeometric harmonious matrix.mnThe reduced displacement vector ism2| fid*=[dx d: η d2x d2 n]，(16)|[M;1/.T|m, (13)where 17;, η are generalized displacements of turnangle about torsion springs.The reduced generalized strain vector is .-1/L0|f2m2,yε=[} eq内](17)_ m2.The gener中国煤化工be dividedYHCNM HG.1368Guan et al. IJ Zhejiang Univ SCIENCE A 2006 7(8):1365-1371into Em, &Ep.EAI00_ E4e'P4Q20Pzφ+φ'(18)([)2(L)2[虫+φ) ]Q24PA21R24(L)21whereEAlEALε,=|φ+φ(19)PA_ Q24PA Q0([)2 [(正 L中+中), IQ4_Q0izROFlexibility matrix(23)For linear-elastic material behavior, flexibilitymatrix is given bywhere A is area of the beam, L is length of the beam,=1/0+f+2(+52)*+3(i)}]，P=fr+h+6i', Q=f+3i',000Q2=f+3i', R=f+2i', R2=f+2i'. If f=f=f, the stiff-ness matrix can be further simplified.0L'/EA'000e|φ|(20)i' 2il 0 |M2|MATRIX OF SECOND TYPE OF MACRO-[o);ELEMENTAs shown in Fig.4, macro-element 2 consists ofwhere f,f2 are flexibility of springs, il=L'/(6EL), A'two macro-element 1 jointed in the middle by a fric-is section area. Flexibility equation can be written astionless pin.「LIEA' 0「Tf+2i'M+o==子f2+2i' I M2[4+好」(a)(21)+o=子(b)「L/EA' 0=cd子(22)(C)i'f2+2il_Fig.4 Macro-element 2. (a) Separated force model; (b)Force combination; (c) Force simplificationStiffness matrixFromAσ=1',(A)'d'=ε,F σ=ε，the reducedIt is the key part of a deployable structure. Thestiffness matrix can be expressed as .middle pin joint that links the two macro-elements 1can turm by pin joint transfer axial force. The middleK*=two torsion中国煤化工noment t"TYHCNMH G.Guan et al.1J Zhejiang Univ SCIENCEA 2006 7(8):1365-13711369macro-element 1. Fig.4a depicts the bending moment)0loaded on macro-element 1, Fig.4b shows all the0 -1/L01/LCT'forces on the macro and Fig.4c is the simplification.So that it has characteristics of a plane beam. Simi-Mlarly，we can get the equilibrium equation of12macro-element 2, that isT2M;00-1/L1/L000 1/L 1/L20 1/L' -1/Lrl~01-10 -1)00M|=[fxf: m,fxf2 m:,rmefme m"]'. (26)M,|0 1/L -1/L 0 -1/L2 1/1Since fumx=fm=m";=0, A'=AmAppAm,AM40 0 1/E2 1/I_M2_is the reduced equilibrium matrix,「-1)fx7=[ff m;fmwfme m" 2xf2 m]', (24)正Tf2 ||M(27)where m" is torque of torsion spring in the middlefz.'two nodes, L,A4 are length, area of the second beamfz:|respectively.22L:2The flexibility equation can be written aswhereL'2=L'+L2.「L)(00The reduced geometrical harmonious matrixB*=(A )' corresponding to reduced load vector is0I0 i' f+2i'd*=[d、d; η d2x d2 η] (28)00(EAThe reduced generalized strain vector becomesf2 +2i_x[T' M M3 T2 M、M2]'=[e' 中+申内+ψe内+书内+丁",(25) :=q+曲LP +φ;[内+子」[φ +φwhere i' =L/(6Er),i2= L2 /(6EI}).|0正-正Load components of middle node m are alwaysequal to zero, so that the degrees of freedom can be=φ+φ +(Z1L2)( +φ;+φ.+φ?)|. (29)reduced and Eq(26) can be derived from Eq.(24) byprimary transform[4+中; +(L/L*)(中+φ);+中+?)」中国煤化工MHCNMH G.1370Guan et al.1.J Zhejiang Univ SCIENCE A 2006 7(8):1365-1371The stiffiness matrix of macro-element 2 was diameter flD=0.4， area of strut cross sectionassembled from macro-element 1. The inverse matrix A=0.13345 cm', Young's modulus E=210 GPa, den-of reduced flexibility matrix is shown assity of mass p=7.8 g/cm'.「EA/L000.560.52(F*)'=a -b(30)-baI0.440.26、0.26^0.44The reduced stiffness matrix of macro-element 2is( 0.25EA00.26 I 0.52L2S S0.430.5K*E2S_ S2S_ S|(b)SbFig.5 A revolving parabolic deployable antenna.(a) Orthographic drawing; (b) Plan drawing(31)whereS=a+b,CONCLUSIONa=(/72+4)/(2f*+10fi+12i7),(32)b=(/2+2i)/(2f2+10fi+12i7).(33)We applied the reduction formula of equilibrium,compatibility and flexibility matrices in the forceObviously, the reduced siffness matrix is of method Pellegrino et al, 1992) to derive the siffnesspositive determinate symmetry and the linear reducedmatrix of complicated new elements. In this paperequilibrium equation can be solved.two types of special beam elements are discussed.The first type consists of one beam with torsion springinstalled at its end. The second type consists of twoSTATIC ANALYSISfirst type elements linked by pin joint. Since someloads are always zero, the stiffness matrix can beThis antenna consists of two types of elements,simplified. We analyze the static deformation of de-one is macro-element 2, consisting of upper and lowerployable structure by new elements. The count resultchord struts, the other is truss element, composed of is corrected compared with the experiment data.abdominal struts. Stiffness matrix is assembled in theglobal coordinate system. Finally, the linear equationsare solved.ACKNOWLEDGEMENTFig.5 shows a revolving parabolic deployableantenna, whose design parameters are: aperture di-In this paper we got a lot of help from Dr. Mi-ameter D= 2.2 m, ratio of focal distance to aperture chael C. Lu, whom we thapk verv much.中国煤化工YHCNMH G.Guan et al. /J Zhejiang Univ SCIENCEA 2006 7(8):1365-13711371ReferencesSerna, M.A, Lopez, A, Puente, I, Yong, D.J, 2006. Equiva-Armero, F., Ehrlich, D. 2006. 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Journal of the International Association for Shell1620350605]and Spatial Sructures, 46:195-204.JzusEditors-in-Chief: Pan Yun-heISSN 1009-3095 (Print); ISSN 1862-1775 (Online), monthlyJournal of Zhejiang UniversitySCIENCE Awww.zju.edu.cn/jzus; www.springerlink.comjzus@zju.edu.cnJZUS-A focuses on“Applied Physics & Engineering”➢Welcome Your Contributions to JZUS-AJournal of Zhejiang University SCIENCE A warmly and sincerely welcomes scientists all overthe world to contribute Reviews, Articles and Science Letters focused on Applied Physics & Engi-neering. Especially, Science Letters (3- -4 pages) would be published as soon as about 30 days (Note:detailed research articles can still be published in the professional journals in the future after ScienceLetters is published by JZUS-A).中国煤化工MHCNMH G.