2514118285 This paper presents a study of a square membrane creased according to the Miuraori folding pattern When the membrane is allowed to expand from its packaged con guration it initially expands elastically under zero corner forces Starting fro ID: 72167
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SystematicallyCreasedThin-FilmMembraneStructures AlessandroPapa andSergioPellegrino UniversityofCambridge, Cambridge,EnglandCB21PZ,UnitedKingdom DOI:10.2514/1.18285 Thispaperpresentsastudyofasquaremembrane,creasedaccordingtotheMiura-orifoldingpattern.Whenthe membraneisallowedtoexpandfromitspackagedcon guration,itinitiallyexpandselasticallyunderzerocorner forces.Startingfromthisnaturallyexpandedcon guration,thepaperinvestigatesthestressdistributionandthe load-displacementrelationshipwhenin-plane,diagonalloadsareappliedatthecorners.Itisfoundthatout-of-plane bendingisthemainload-carryingmodeand,forstressmagnitudestypicalofcurrentsolar-saildesigns,thebehavior ofthemembraneremainslinearelastic.Asimpleanalyticalmodel,originallyproposedforrandomlycreased membranes,isshowntopredictwithgoodaccuracytheload-displacementrelationshipofthecorners.Ituses I.Introduction T HEREiscurrentlymuchinterestintheuseofthin- lm membranestructuresforavarietyoffuturegossamerspacecraft missions.Insomeoftheproposedstructuralarchitectures,itis envisagedthatthin lmswillbepackagedbymeansofaregular arrangementofcreases,orfoldlines,andhencethequestionarisesof howsuchprecreasedfoilsbehavewhentheyaredeployedinorbit. OnepackagingschemeisknownasMiura-ori[1],Fig.1.Anideal Miura-orisurfaceisamechanicallinkageofthin, atplates connectedbyfrictionlesshinges;thislinkagehasasingledegreeof kinematicfreedom.Amembranefoldedsuchthatthepatternofthe resultingcreasesmimicsthearrangementofthehingesinanideal Miura-orisurfacecanbepackagedanddeployedef ciently[2,3]. Becauseoftheseproperties,theMiura-oricreasepatternwasselected forthepresentresearch. anddetailed niteelementsimulations)ofasquaremembranewith speci cdimensionsandcreasesarrangedaccordingtotheMiura-ori foldingpattern.Randomlycreasedthin lmshavebeenextensively investigatedbyMurphey[4],andthepresentstudyisanextensionof Murphey sworkto lmswherethecreasesarearranged systematically,accordingtoarepeatingpattern.Theaimofour studyistodeterminetheshapeofthemembraneandtheload- displacementrelationshipforin-plane,diagonalloadingofthe corners,startingfromacon gurationthatmightbeconsideredasthe unstresseddeployedshapeofthemembrane,i.e.,thecon guration thatthemembranewillexpandto,intheabsenceofanyexternal forces.Themagnitudesofthecornerforcesarechosensuchasto generatestresslevelsontheorderof0.02MPaatthecenterofa m thickmembrane.Weshowthatasimpleanalyticalmodel consistingoftwocreasedbeams,derivedfromMurphey smodelfor randomlycreasedmembranesandcharacterizedbyonlyasmall numberofdirectlymeasurableparameters,predictsthedisplace- mentsofthecornersquiteaccurately. Thepaperislaidoutasfollows.SectionIIdescribesthelayoutof thecreases.SectionIIIpresentsanexperimentalstudyofa 0 : 5 0 : 5m membranecreasedaccordingtotheMiura-oripattern.The experimentaltechniqueisexplainedandmeasurementsoftheinitial creaseangleareobtained.SectionIVpresentsadetailed nite elementsimulationofthismembrane.SectionVcompares experimentalmeasurementswith niteelementresults.Finally, Sec.VIpresentsasimpleanalyticalmodelforthemembrane sload- displacementbehavior,whosepredictionsarethencomparedwith the niteelementresults.SectionVIIconcludesthepaper. II.CreaseGeometry Therearetwosetsofcreases,asshowninFig.1.Thoseinthe rst set,called primarycreases ,formthemainhillandvalleyfoldsofthe packagingscheme;eachofthesecreasesisfoldedinthesamesense throughoutitslength,anditneverbecomesstraight.Thecreasesin thesecondset,called secondarycreases ,formalternatehilland valleyfoldsandbecomealignedwhenthemembraneis at.Adetail ofthecreasegeometryona atmembraneisshowninFig.2a. Thedegreeoffreedomusedtodescribethedegreeofdeployment ofaMiura-orisurfaceisthedeploymentangle ,de nedastheangle x - y ofthefullydeployedsurface (seeFig.2b). Thegeometricfeaturethatdeterminesthedeploymentpath,and thereforethedegreeofcouplingbetweentheexpansionofthe membranealongthetwodiagonaldirections,isthedefectangle , de nedinFig.2a.TheexpansionratiosofaMiura-orisurface,inthe x and y directions,arerelatedto by[1] OA OA max cos sin 1 sin cos (1) OB OB max cos cos sin 1 sin cos (2) III.ExperimentalStudy Theexperimentalstudyhadtwoprincipalaims.First,toobtaina representativevaluefortheinitialdeploymentangle 0 tobeusedin the niteelementsimulations.Second,tomeasuretheload- displacementrelationshipforthecornersofthemembraneandsotest thevalidityoftheelasticbehaviorassumption. PresentedasPaper1975atthe46thAIAA/ASME/ASCE/AHS/ASC Structures,StructuralDynamicsandMaterialsConference,Austin,Texas, 18 21April2005;received18June2005;revisionreceived11September AlessandroPapaandSergioPellegrino.PublishedbytheAmericanInstitute ofAeronauticsandAstronautics,Inc.,withpermission.Copiesofthispaper maybemadeforpersonalorinternaluse,onconditionthatthecopierpaythe $10.00per-copyfeetotheCopyrightClearanceCenter,Inc.,222Rosewood Drive,Danvers,MA01923;includethecode0022-4650/08$10.00in correspondencewiththeCCC. MechanicalEngineeringStudent,DepartmentofEngineering, TrumpingtonStreet;Alessandro.Papa@cantab.net. ProfessorofStructuralEngineering,DepartmentofEngineering, TrumpingtonStreet;currentlyProfessorofAeronauticsandCivil Engineering,CaliforniaInstituteofTechnology,1200EastCalifornia Boulevard,MailCode301-46,Pasadena,CA91125;sergiop@caltech.edu. AssociateFellowAIAA. J OURNALOF S PACECRAFTAND R OCKETS Vol.45,No.1,January February2008 10 A.Membrane A 25 m thickaluminizedKaptonmembranewasused.The requiredcreasepatternwassetupasfollows.A 5 5 panelMiura-ori creasepatternwasmarkedwitha nepermanentmarkerononeside ofa 500 500mm squaremembrane.Thesecondaryconvex creases,alongthe x direction,wereformedbyrunningaballpoint pen rmlyalongthemarkedcreaselines.Themembranewasthen turnedandthesecondaryconcavecreasesformedinasimilar fashion.Byplacingthemembraneoveracompliantsurface consistingofadozensheetsofA4paperandcarefullycontrollingthe appliedforce,ahomogenoussetofsecondarycreaseswasformed. ForanidealMiura-orisurfaceatsmallanglesofdeployment,the amplitudeofthecreasesinthe x directionisconsiderablylargerthan theamplitudeofthecreasesinthe y direction.Toallowforthis difference,theprimarycreaseswereformedusingadifferent techniquetothatusedtoformthesecondarycreases.Athinplateof Kevlarwascutintoarectangularshape,withthelengthofitsminor edgematchingthelengthoftheprimarycreases.Foreachprimary crease,theminoredgeoftheKevlarplatewasplacedtocoincidewith itsdesiredpositionandthemembranecarefullyfolded180degover theKevlarplate,avoidinganydamagetothemembrane.Arollerwas thenrunalongthelengthoftheKevlaredge,throughthefolded membrane,therebyformingatightcreaseinthemembrane.This processwasrepeateduntilthecreasepatternwascomplete. B.Apparatus Adetaileddescriptionoftherigusedfortheseexperimentscanbe foundin[5].ThecreasedmembranecanbeseeninFig.3,attachedto ahorizontalsteelframe.Notethatthecornershavebeencutatan angleandreinforcedwithedgetabs,madeofKaptontapelooped overasteelrod.Thesecornerreinforcementshavethepurposeof reducingthemaximumstressinthemembraneandsoavoidplastic deformationinthecornerregions. Theforcesappliedtothecornersofthemembrane,whichranged from0to0.2N,weremeasuredwithstraingaugedcantileverbeams tiedwithalightKevlarcordtoasmallholeinthemembranecorner tab.Giventhatthemembrane stotalweightisapproximately0.11N, theexperimenthadtobecarriedoutwiththemembranesupportedon a atsurface.Asteelplatewasplacedonwoodblocksupportsinthe interiorofasquaresteelframe,suchthatitstopsurfacewasatthe sameheightasthestraingaugedcantileversattachedtothecornersof thesteelframe,asshowninFig.3. AKeyenceLK-081charge-coupleddevice(CCD)laserwas mountedtothesteelrigsothatitcouldscanthemembraneand measureitssurfacepro le.Theresolutionofthislaseris 3 m overa rangeof 15mm . Photogrammetrywasusedtotracktherelativedisplacementsof twooppositecornersofthemembrane.Thirtyblackmarkerswere af xedtothetwocornersofthemembrane(whichhadpreviously beencoveredwithathincoatofwhitespray)andninewhitemarkers wereaf xedtoeachcorrespondingcorneroftherig;allofthese targetswerepaperdiskswithadiameterof9mm.Photographsofthe completemembraneweretakenwitha4.0megapixeldigitalcamera, andtheimageswereprocessedwiththesoftwarePhotomodelerPro 4.0.Theaveragestandarddeviationofthemeasurementswas 0.11mm,or6.0%ofthemaximumdisplacementmeasured. C.SurfacePro le Theaveragecreaseangleofthemembraneintheunloadedstate wasmeasuredtobe 0 152 : 4deg .Hencethecorrespondingvalue of 0 ,seeFig.4,is 0 180 0 2 13 : 8deg (3) Theaveragecreaseangleofthemembraneinthefullyloadedstate wasmeasuredas 1 156 : 0deg ;thisisa2.4%increasefrom 0 . Fig.1Miura-oripackagingscheme[1]. Hill folds Valley folds a) Flat membrane b) Perspective view OA B x y Secondary creases Primary creases B A O C C z Fig.2Detailsofcreasegeometry,defectangle ,anddeploymentangle . Fig.3Testrig,includingCCDlaser,cantileverstraingauges,and targetmarkers. 175180185190195 -10 -8 -6 -4 -2 0 2 4 6 8 10 Distance Perpendicular to Crease (mm) Scan Amplitude (mm) 0 0 Fig.4Measuredpro leofaprimarycrease. PAPAANDPELLEGRINO 11