DYNAMICSANDCONTROLOFSYSTEMSINVOLVINGTHEELASTICA By SherylSuetYingChau ADISSERTATION Submittedto MichiganStateUniversity inpartialfullmentoftherequirements forthedegreeof MechanicalEngineering{DoctorofPhilosophy 2020 ABSTRACT DYNAMICSANDCONTROLOFSYSTEMSINVOLVINGTHEELASTICA By SherylSuetYingChau Theelasticacanundergoalargedeformationwhilestayingwithinthee lasticlimit.This hasmotivateddiverseapplicationsfromcomputergraphicstoys hingandsoftrobotics. Inthiswork,weinvestigatethedynamicsandcontrolofanumbero fderentsystemsthat entailtheelastica.Werstinvestigatethemechanicsofpolevaultin g,whereanathlete usesalongexiblepoletotransformkineticenergytopotentialene rgytocrossahigh barrier.Thepole,whichisanelastica,isusedtotemporarilystoreth ekineticenergyofthe athleteandchangethedirectionofmotionoftheathletefromtheh orizontaldirectionto theverticaldirection.Inthisstudy,theathleteisreplacedbyapo intmassforthesakeof simplicity,butthequalitativebehaviorofthecombinedsystemisfoun dtobeverysimilar tothatobservedinthesport.Polevaultersareknowntoachievea greaterheightbydoing additionalworkinbendingthetipofthepole;similarresultsareseenw hennon-conservative workisdoneontheelasticaduringthevaultingmaneuver.Similartopo levaulters,runners withlower-limbprostheses,commonlyuseC-shapedelasticalegstot emporarilystoreenergy duringthecontactphaseandretrieveitpriortotheightphase.T heforce-displacement characteristicsofcircular-shapedelasticasarethereforeofgr eatinterestinapplicationswhere elasticaelementsareusedassprings.ThedeformationofC-shape dandO-shapedelastica springs,subjectedtounilateralconstraintsduetogroundcont act,isstudiedundervertical loading;thehardeningandsofteningbehaviorsofthesespringsar eobservedintheabsence orpresenceoffriction.Symmetricandasymmetricloadingsarealso investigatedwiththe objectiveofbetterunderstandingthedynamicsofmass-elastica systems;theseinvestigations revealthepossibilityofdesigningrobotscapableoflocomotionthro ughhopping.Weexplore avariable-structuremass-elasticahopper;thestnessofthee lasticaischangedbychanging itsboundaryconditions.Theforce-displacementcharacteristics ofthevariable-structure elasticaareusedtomodulatetheenergyofthesystemandcontro lthehoppingheight. Finally,rollinglocomotionisconsideredusingaexiblewheelthatisaclos edelastica.In additiontoitsbendingstness,theelasticaisassumedtohavemass andtheequationsof motionarederivedusingextendedHamilton'sprinciple.Thearc-lengt hconstraintofthe elasticaintroducesnonlinearityintheequationsofmotionandthegr oundcontactleads tounilateralconstraints.Theweakformoftheequationsofmotio nareobtainedtosolve thedynamicsusingtheniteelementmethods;thepenaltymethodis usedtoenforcethe groundconstraint.Fortheparticularcaseofconstantvelocitym otion,theeectsofthenon- dimensionalweightandvelocityparametersarestudied.Itisobser vedthatthefrequencyof therollingelasticareducesasthevelocityincreases;thisresembles thedivergenceproblem inaxiallymovingmaterials. Copyrightby SHERYLSUETYINGCHAU 2020 Idedicatethisworktomygrandmaandmyparents,whoseuncondit ionalloveandsupport continuetogivemestrengthtoovercomeobstaclesandbecometh epersonthatIamtoday. v ACKNOWLEDGEMENTS Iamdeeplyindebtedtomyadvisor,Dr.RanjanMukherjee,forhisc ontinuousguidanceand supportthroughoutmyPhDcareer.Ireallyappreciatethathealw ayskepthisocedoor openfordiscussionandadvice. IwouldliketoextendmysincerethankstomyPhDcommittee,including Dr.Andre Benard,Dr.C.Y.Wang,andDr.GuomingZhu,whoprovidedmewithag reatamountof guidanceandconstructivefeedback. Manythankstothemembersofmylab,includingMahmoud,Nilay,Sand ers,Connor, Amer,Kristina,fortheircontinuoussupportonacademicandpers onalmatters. IgratefullyacknowledgethesupportprovidedbytheNationalScie nceFoundationGrad- uateResearchFellowshipProgram. IwouldliketoextendmydeepestgratitudetomymomShukTungSiu,m ydadLeonard Chau,andmysisterSpringChau.Iwouldn'thavecomethisfarwithou tthesupportand nurtureofmyfamily.ThankyouformakingmewhoIamtodayandalwa ysbeingbymy side. IcannotbegintoexpressmythankstomygrandmaCaiyunYe,whoh asbeenshowering mewithherloveandspoilingmesinceIwasababy. Iamalsoverygratefultomyfriendsfortheiremotionalsupporta ndtheadventureswe shared.TheyhavemadethePhDjourneymucheasiertoendure. vi TABLEOFCONTENTS LISTOFTABLES ..................................ix LISTOFFIGURES ..................................x LISTOFALGORITHMS ..............................xv Chapter1Introduction ..............................1 1.1Background....................................1 1.2ThePole-VaultProblem.............................2 1.3Circular-ShapedElasticas............................4 1.4ElasticHoppingRobots.............................5 1.5ARollingElastica.................................7 Chapter2KinetictoPotentialEnergyTransformationUsing aSpring asanIntermediary ...................................8 2.1Overview......................................8 2.2PointMassandLinearSpringSystem...................... 9 2.2.1ProblemStatement............................9 2.2.2EquationsofMotion...........................10 2.2.3KinetictoPotentialEnergyTransformation.............. 11 2.3PointMassandElasticaSystem.........................1 6 2.3.1ProblemStatement............................16 2.3.2ElasticaDeformation...........................17 2.3.3DynamicsofPointMass.........................19 2.3.4KinetictoPotentialEnergyTransformation.............. 20 2.4Mass-ElasticaSystemwithExternalTorque............... ...24 2.4.1ModcationoftheMassElasticaSystem................24 2.4.2MaximizingPotentialEnergyoftheMass...............25 2.4.3ADimensionalCaseStudy........................29 Chapter3Force-DisplacementCharacteristicsofCircular -ShapedMass- lessElastica .......................................31 3.1Overview......................................31 3.2MathematicalModelofElastica......................... 32 3.2.1GeneralProblemFormulation......................32 3.2.2Circular-ShapedElasticaProblems...................34 3.2.3SolutiontotheBoundary-ValueProblem................35 3.3C-ShapedElasticaunderVerticalLoading................. ..36 3.3.1InitialCongurationoftheC-ShapedElastica............. 36 3.3.2ElasticaRollswithoutSlippingDuetoFriction............37 3.3.3ElasticaRollsWhileSlippingDuetoNoFriction...........41 vii 3.4O-ShapedElasticaunderVerticalLoading................. ..44 3.4.1SymmetricLoadingwithPointandLineContact...........44 3.4.2DynamicsofSymmetricMass-ElasticaSystem.............4 7 3.4.3AsymmetricLoadingwithPointandLineContact...........49 3.4.4DynamicsofAsymmetricMass-ElasticaSystem............5 5 Chapter4AVariable-StructureMass-ElasticaHopper ...........59 4.1Overview......................................59 4.2HoppingUsingaVariable-StructureSpring -TheConcept...................................59 4.3Fixed-StructureMass-ElasticaHopper.................. ...61 4.3.1SystemDescription............................61 4.3.2ForceandElasticaDeformationRelationship..............6 1 4.3.3HoppingDynamics:FlightandContactPhases............64 4.3.4Simulation.................................65 4.4Variable-StructureMass-ElasticaHopper................ ....66 4.4.1SystemDescription............................66 4.4.2EnergyRegulationforHoppingHeightControl............. 67 4.4.3Simulation.................................70 Chapter5DynamicsandSimulationsofaRollingElastica ........72 5.1Overview......................................72 5.2SystemDynamics.................................72 5.2.1KinematicsoftheRollingElastica....................72 5.2.2EquationsofMotionUsingHamilton'sPrinciple............74 5.2.3ImposingConstraintonLineContact..................82 5.3FormulationoftheFiniteElementModel.................... 83 5.3.1DiscretizationoftheEquationsofMotion...............83 5.3.2ApplyingBoundaryConditions.....................87 5.3.3DirectTimeIntegration.........................88 5.4SteadyStateSolutionInvestigation..................... ..89 5.5NumericalSimulations..............................90 5.5.1SystemParameters............................90 5.5.2Eectsof wonSystemDynamics....................91 5.5.3Eectsof_ sonSystemDynamics....................93 5.5.4EectsofInitialCongurations.....................99 Chapter6Conclusion ...............................101 APPENDIX........................................105 BIBLIOGRAPHY.....................................12 2 viii LISTOFTABLES Table2.1:Eectofinitialangle 0 onthesolutiontotheTPBVPforthemass-spring systeminFig.2.1with " =0 :1.........................15 Table2.2:Eectofinitialangle ˚ 0 onthesolutiontotheproblemofcompleteenergy transformationforthemasswithelasticafor w =6.............23 Table2.3:ComparisonofsolutiontrajectoriesoftheelasticainFig.2.1 3withand withoutexternaltorqueforderentinitialangles ˚ 0 andfor w =6....27 Table2.4:PercentagegainsinmaximumheightreachedbythemassinF ig.2.13for derentinitialangles ˚ 0 andthreederentcombinationsof( tmax ;w )...28 Table3.1:Simulationresultsofthesymmetricmass-elasticasystemin Fig.3.12....49 Table3.2:Internalforcesandmomentsatthepointofapplicationo ftheloadand lengthofcontactofelasticawiththegroundforasymmetricloading with q 2f 4 ;8 ;12 ;15 :5 g at =30 .........................54 ix LISTOFFIGURES Figure2.1:Systemcomprisedofapointmassandahingedspringinthe vertical plane,isshowninitsinitialconguration..................9 Figure2.2:TrajectoriesofthemassinFig.2.1with " =0 :1forvederentvalues ofinitialvelocity v 0 ,startingfrom 0 = ˇ= 6................13 Figure2.3:TrajectoriesofthemassinFig.2.1forvederentvalues of " ,starting from 0 = ˇ= 6withinitialvelocity v 0 =0 :607...............13 Figure2.4:TrajectoriesofthemassinFig.2.1with " =0 :1:SolutionstotheTPBVP forderentvaluesof 0 ...........................14 Figure2.5:Variationofmaximumheight h andinitialenergy e withinitialangle 0 forderentvaluesof " forthemass-springsysteminFig.2.1.......15 Figure2.6:Ahingedelasticaintheverticalplanewithatipmass M isshowninits (a)Initialconguration,and(b)anarbitraryconguration.... ....16 Figure2.7:Freebodydiagramsofthetipmass M andaninnitesimalsegmentof theelasticaoflength dS ...........................17 Figure2.8:Methodofsolvingeachtimestepoftemporaldynamicsof masstogether withspatialdeformationofelastica.....................20 Figure2.9:TrajectoriesofthetipoftheelasticainFig.2.6isshownfor veder- entinitialvelocities v 0 :theelasticaisinitsundeformedshapeatthe beginningandendofeachtrajectory....................21 Figure2.10:CongurationsoftheelasticainFig.2.6atderentinstan tsoftimethat resultsincompletetransformationofkineticenergyofthemassint o potentialenergy................................22 Figure2.11:TrajectoriesofthemassinFig.2.6with w =6:SolutionstotheTPBVP forderentvaluesof ˚ 0 ...........................23 Figure2.12:Variationofmaximumheight h andinitialenergy e withinitialangle ˚ 0 forderentvaluesof w forthemass-elasticasysteminFig.2.6......24 Figure2.13:Ahingedelasticaintheverticalplanewithtipmass M andexternaltip torque Tisshowninits(a)Initialconguration,and(b)anarbitrary conguration.................................24 x Figure2.14:CongurationsoftheelasticainFig.2.13atderentinsta ntsoftime thatresultincompletetransformationofthetotalenergyintopo tential energyofthemassfor(a) tmax =0 :0and(b) tmax =0 :5.........26 Figure2.15:Plotofderentcomponentsofthenon-dimensionalen ergy,eachex- pressedasafractionoftheinitialnon-dimensionalenergyofthem ass, asafunctionoftimeforthetwocaseswheretheexternaltorque is(a) absentand(b)present............................29 Figure3.1:Deformedcongurationofaninitiallystraightelastica... .......32 Figure3.2:AC-shapedelasticaunderloading:itiscomprisedofanFES anda VC-LES....................................34 Figure3.3:AnO-shapedelasticaisshowninitsundeformedanddefor medcong- urations:thedeformedcongurationissymmetricwithrespectto the verticalandiscomprisedofaVC-LESandaCC-LES...........35 Figure3.4:InitialcongurationoftheC-shapedelasticafor q =0..........37 Figure3.5:Force-displacementcurvesforaC-shapedelasticarollin gwithoutslipping withxedtipslope (1)=( ˇ );thecurvescorrespondtoderent valuesof incrementedby5 overtheinterval[0 ;45 ]..........39 Figure3.6:Force-displacementcurvesforaC-shapedelasticarollin gwithoutslipping withfreetipslope;thecurvescorrespondtoderentvaluesofth einitial angle incrementedby5 overtheinterval[0 ;45 ]............39 Figure3.7:DeformedshapesoftheC-shapedelasticawith(a)xed tip-slopeand (b)freetip-slopeunderderentvaluesofverticalloading q ,q 2 [0 ;15] forinitialangle =20 ...........................40 Figure3.8:Force-displacementcurvesforaC-shapedelasticawith xedtipslope (1)=( ˇ )onafrictionlesssurface;thecurvescorrespondtoderent valuesof incrementedby5 overtheinterval[0 ;45 ]..........42 Figure3.9:(a)DeformedshapesoftheC-shapedelasticawithxed tip-slope (1)= ( ˇ ), =30 ,onafrictionlesssurfaceforfourderentvaluesof verticalloading,(b)plotsof " ,˙ andtheirsumasafunctionofvertical loading q ,q 2 [0 ;15]forxedangle =30 ................43 Figure3.10:(a)UndeformedC-shapedelasticaonafrictionlesssur facewithinitial angle =30 (b)deformedshapesoftheelasticawithfreetipslope underderentvaluesofverticalloading q ,q 2f 0 :001 ;3 ;6 ;9 ;12 ;15 g startingfromtheinitialcongurationin(a);notethatallcongur ations havetheslopeof ˇ ,i .e., =0 ,althoughthetipslopeisfree......44 xi Figure3.11:Force-displacementrelationship,lengthoflinecontact ,andfourdif- ferentcongurationsofO-shapedelasticaundersymmetricloadin g...46 Figure3.12:Symmetricmass-elasticasystem................. ......47 Figure3.13:Methodofsolvingthedynamicsofthesymmetricmass-e lasticasystem inFig.3.12...................................48 Figure3.14:O-shapedelasticasubjectedtoasymmetricloading:(a )undeformedcon- gurationand(b)deformedconguration.................5 0 Figure3.15:(a)ThedeformedelasticainFig.3.14(b)iscomprisedofth erightand leftelasticas(b)theleftelasticaisshowninitstransformedcongu ration togetherwiththerightelastica.......................51 Figure3.16:FlowchartforsolvingthedeformedshapeoftheO-sha pedelasticaunder asymmetricloading.............................53 Figure3.17:Deformedshapesoftheelasticaforasymmetricloading withfourder- entloadsat =30 :q = f 4 ;8 ;12 ;15 :5 g ..................54 Figure3.18:Asymmetricmass-elasticasystem................ ......56 Figure3.19:Methodofsolvingthedynamicsoftheasymmetricmass- elasticasystem inFig.3.18...................................57 Figure3.20:Derentcongurationsoftheasymmetricmass-elast icasystemandtra- jectoryofthemassinCartesiancoordinates................ 58 Figure4.1:Amass-springhopperwith(a)aspringofconstantstn ess K ,and(b) aspringofvariablestness,switchingbetween K and K , K>K .In (a),thevelocitiesofthemassattouchdownandtakeoarethesa me, equalto V .In(b),thevelocityofthemassattakeoisgreaterthanthe velocityattouchdown, V>V .Aftertakeo,thestnessisswitched backto K ...................................60 Figure4.2:Axed-structuremass-elasticasystemwithshouldera ngle hoppingon theground..................................61 Figure4.3:Free-bodydiagramsofthemassandelastica......... .......62 Figure4.4:Force-displacementcurvesfortheelasticahalfinFig.4.3f orshoulder angles =15 ;30 ;45degwith a =1 = 2 ˇ ...................63 Figure4.5:Congurationsofthexed-structuremass-elasticas ystemat(a)release time,(b)timeoftouchdown t cs ,and(c)timeatwhichtheelasticahas themaximumdeformation..........................65 xii Figure4.6:Avariable-structuremass-elasticasystemisshowninits equilibriumcon- gurationwithshoulderangles(a) ,and(c) ,(b)non-equilibriumcon- gurationofthesystemimmediatelyaftertheshoulderanglehasbe en changedfrom to .............................66 Figure4.7:Derentintermediatecongurationsofavariable-stru cturemass-elastica systemduringthe j -thhop.Theshoulderangleswitchesbetween and .......................................67 Figure4.8:Avariable-structuremass-elasticasystemhoppingont heground;the shoulderangleswitchesbetween and , > ,togainheight.....68 Figure4.9:Thevariable-structuremass-elasticasystemhoppingt oaheightof H = 0 :60mfromaninitialheightof H =0 :24m................70 Figure5.1:Deformedcongurationofanelasticarollingonthegroun d........72 Figure5.2:Velocitydiagramforelasticawithpointcontact........ .....73 Figure5.3:Velocitydiagramforelasticawithlinecontact.......... ....83 Figure5.4:Anelementiscomprisedoftwonodes;eachnodehasved egrees-of- freedom.Thesubscriptsindicatethenodenumber............ 84 Figure5.5: ypositionvs. ˝ for w= f 45 :10 ;94 :41 ;375 :85 g .Thedashedlinesshow the ypositionof pwhentheelasticaisinstaticequilibrium.......92 Figure5.6:Theinitial,staticequilibriumandmaximumdeformedcongur ationsof theelasticawith w=45 :10.........................93 Figure5.7:Theinitial,staticequilibrium,andmaximumdeformedcongu rationsof theelasticawith w=94 :41.........................93 Figure5.8:Theinitial,staticequilibrium,andmaximumdeformedcongu rationsof theelasticawith w=375 :85.........................94 Figure5.9: ypositionvs. ˝ for w=53 :45and_ s=4 :95.Blacksolidline:FEM result;bluedottedline:COMSOLresult.................94 Figure5.10: yposition(bluesolidline)and xposition(redsolidline)of pvs. ˝ for w=94 :41and_ s=5 :07.Thebluedashedlineshowsthemean ypositionofthenode.............................95 Figure5.11: yposition(bluesolidline)and xpostion(redsolidline)of pvs. ˝ for w=94 :41and_ s=12 :67.Thebluedashedlineshowsthemean ypositionofthenode.............................96 xiii Figure5.12: yposition(bluesolidline)and xpostion(redsolidline)of pvs. ˝ for w=94 :41and_ s=20 :27.Thebluedashedlineshowsthemean ypositionofthenode.............................96 Figure5.13:Deformedcongurationsoftheelasticaat ˝ =0 :358for w=94 :41and threederentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g ........97 Figure5.14:Deformedcongurationsoftheelasticaat ˝ =0 :667for w=94 :41and threederentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g ........97 Figure5.15:Deformedcongurationsoftheelasticaat ˝ =0 :726for w=94 :41and threederentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g ........98 Figure5.16:Non-dimensionalmaximumtensileforce^ ˙ max oftheelasticawith w= 94 :41vs. ˝ for_ s= f 5 :07 ;12 :67 ;20 :27 g withasamplingfrequencyof380.98 Figure5.17: yposition(bluesolidline)and xposition(redsolidline)of pvs. ˝ for w=53 :45and_ s=4 :95.Thebluedashedlineshowsthemean ypositionofthenode.............................99 Figure5.18:Initialcongurationofelasticaat ˝ =0anddeformedcongurationsof theelasticaat ˝ = f 0 :715 ;1 :083 g for w=53 :45and_ s=4 :95. aisthe semi-majoraxisand bisthesemi-minoraxis...............100 Figure5.19:Non-dimensionalmaximumtensileforce^ ˙ max oftheelasticawith w= 53 :45vs. ˝ for_ s=4 :95withasamplingfrequencyof103.........100 xiv LISTOFALGORITHMS Algorithm1Directtimeintegrationusingthecentralderencemeth od.......88 xv Chapter1 Introduction 1.1Background TheelasticaproblemwasrstposedbyJamesBernoulliin1691;howe ver,thecomplete solutionwaslaterobtainedbyEulerusingvariationalapproachwher ethebendingenergyof theelasticcurveisminimized[1].Aninterestingpropertyofelasticaist hatitcanundergo largeoveralldeformationwhilestayingwithintheelasticlimit;forthisr eason,theelasticahas continuouslyattractedtheinterestofresearchersfromvariou seldsthroughouttheyears.In computerscience,forexample,researchersusetheminimizationo ffunctionalsrelatedtothe elasticaenergytosolvevariousproblemsincomputervisionandimage processing[2].Some physicistshavebeeninterestedinstudyingthecrackofawhipcaus edbysupersonicmotionof thetipofthewhipinair;highspeedcamerashavebeenusedtocaptu rethemotionofthetip togainbetterunderstandingofthemechanics[3].Inrecreational sports,researchersstudied themechanicsofyshingfordesignoptimizationoftheylineandy rodsystem[4]. Asadvancementinmaterialscienceandtechnologycontinuetoope nuppossibilitiesfor precisecontrolandconstructionofsoftstructures,theeldo fsoftroboticsisgainingalot ofattentionfromresearchers[5].Safehuman-robotinteraction andversatileoperationin derentterrainsaresomeoftheincentivesfordevelopingsoftro bots,whichmakesthe elasticaagoodcandidateasitallowslarge-scaleelasticdeections[6], derentfromsome oftheotherwell-knownbeammodelssuchastheEuler-Bernoullibea mandtheTimoshenko beam.Euler-Bernoullibeamisthemostcommonlyusedbeammodel;itis simplebut itisonlyapplicableforsmalldeformations.Inmanyengineeringapplica tions,theEuler- Bernoullibeamissucienttocapturetheessentialcharacteristic softhedeformingstructure. Forlargedeformation,Timoshenkobeamincludessheardeformatio nandrotationalbending eects;however,thismodelisonlysuitableforthickbeams.Inthis work,wepresent 1 severalproblemsinvolvingthedynamicsofsystemsentailingtheelas ticathatundergoes largedeformation.Inpolevaulting,anathleteusesalongexiblepole (anelastica)to transformkineticenergytopotentialenergytocrossahighbarr ier.Forcircular-shaped elasticas,theforce-displacementcharacteristicsleadtoapplicat ionssuchasthelower-limb prostheses.Intheaboveproblems,theelasticacanbeassumedt obemasslessastheweight oftheelasticaisinsigncantintheanalysisofthesystems.Intheca sewherethedynamics oftheelasticaisprominentsuchasaexiblewheelsystem,theprob lembecomesmore dculttosolve.Aexiblewheelisoftenmodeledtohavesmalldeform ations;however, arollingexiblestructurethatcanundergolargedeformationisofin terestasitcanbe versatiletobeoperatedinderentterrains. 1.2ThePole-VaultProblem PolevaultingisanOlympicsportwhereanathletetriestoreachthema ximumheightby transformingkineticenergyintopotentialenergywiththehelpofa longexiblepole.The heightreachedbythevaulterismeasuredbyacrossbarplacedhor izontally,andasequenceof complicatedbodymaneuvershastobeperformedbythevaulterto crossoverthebar.While performingthesemaneuvers,vaultersapplybendingmomentstha tdopositiveworkonthe pole;thisworkislaterreturnedtothevaulterintheformofaddition alpotentialenergy whenthepoleregainsitsundeformedshape.Thepropertiesofthe poleplayanimportant roleinvaultingperformance.Withthedevelopmentofstronger,ligh ter,andmoreexible poles,vaultershavebeenabletoreachgreaterheights.Thepoles usedbyathletesnowadays aregenerallymadeofcompositematerials,whichareverylightweight andaccountforonly ˇ 3%oftheweightofavaulter.Thepoleessentiallybehaveslikeamassle ssspringthat temporarilystoresenergyandreturnsittothevaulter.Themech anicsofpole-vaultingand areviewoftheassociatedliteraturecanbefoundinthe2010paper [7]. Thepole-vaultproblemhasbeenwidelystudiedintheeldofbiomechan ics.Hubbard[8] studiedthedynamicsofarealvaulterandpolesystembymodelingth epoleasanelastica 2 andthehumanasseveralhingedrigidbodiescoupledbytorquegene ratorsthatsimulate muscleaction.Griner[9]alsomodeledtheexiblepoleasanelasticaand explainedtherole ofpositiveandnegativebendingmomentsduringthedeectionands traighteningphasesof thepoleonvaultperformance.EkevadandLundberg[10,11]used niteelementmodels toinvestigatetheoptimumstnesstoavoidbucklingandtheoptimum polelengththat maximizestheratioofthenalpotentialenergytotheinitialkinetic energyofthevaulter, inthepresenceoflimitsontheappliedbendingmoments.Inanother niteelementmodel- basedstudy[12],itwasshownthathigherbendingstnessofthepo le,uptoacertainlimit, resultsinhighervaultingheights.AsimplemodelwasusedbyLinthorn e[13]toillustrate theadvantagesofaexiblepoleoverarigidpole;aexiblepolelowerse nergylossesand improvesperformancebyenablinglowertake-oangles.Usingadyn amicoptimalcontrol model,Liuetal.[14]showedthatforagivensetofanthropomorphic dataandrunning speed,thereexistsanoptimalsetofparameters(take-oangle ,polestness,andgripping height)thatmaximizesvaultingheight. Severalresearchershaveinvestigatedtheperformanceofpole -vaultersthroughvideo recordings.Earlyworkinthisarea[15]investigatedthetransform ationofkineticenergyinto potentialenergyofcollegepole-vaultersusingaberglasspole.Ab iomechanicalanalysisof thetopperformancesinanOlympiccompetitionwascarriedouttore latevaultingperfor- mancetopoleparameters,grip-height,andspeedandangularmom entumofthevaulters[16]. Recentstudies[17,18]haveusedthekinematicdataofexperience dvaulterstogetherwith ground-poleforcedatatoconductdetailedanalysisoftheenergy exchangebetweenvaulters andthepole,includingthemuscularworkdonebythevaulters.Aninv ersedynamicanalysis ofanine-segmentplanarrigidbodymodelwasusedtoanalyzetheda taofelitevaulters[19] forunderstandinghowmuscleactivitiescontributetoreorientatio nofthebodyandvaulting performance.Recentworkbasedonathree-dimensionalrigidbod ymodel[20]highlighted theroleofbendingmomentsonvaultingperformance. Thepole-vaultingproblemhasalsobeenstudiedintheeldofrobotics ,wherethevaulter 3 andpolesystemhasbeenmodeledasanunderactuatedsystem.Fu kushimaetal.[21] developedapole-vaultrobotandmodeleditusingaexiblebeamandap oint-mass-actuator atthetipofthebeam;itwasshownthatactivebendingmomentsapp liedbytheactuator improvedvaultingperformance.Amoreadvancedversionofthero botwaslaterdeveloped byNishikawaetal.[22];inthiswork,thepolewasmodeledusingfourrigid linksconnected byspringsandthevaulterwasmodeledasasinglependulum.Thepole- vaultproblemis alsodiscussedin[6]asanapplicationofrobotswithsoftarms. 1.3Circular-ShapedElasticas Oneofthebasicelementsofmechanicalsystemsisthespring,comm onlydescribedbythe physicalembodimentofahelicalcoil.Typicallymodeledasmasslesswith constantstness, thespringstoreselasticstrainenergywithoutdissipation.Othere mbodimentsofthespring withuniquepropertiesandcharacteristicshavebeenusedforawid evarietyofapplications. Examplesareleafspringsinvehiclesuspension,C-shapedatspring sinswitchesandclamps, andcircularring-springsinbuersofrailwaycarriages[23].Therunn ing-speccprostheses arecommonlydesignedtobeaC-orJ-shapedspringtoecientlysto reelasticenergyand thenreturnintheformofmechanicalenergybacktotheuserdur ingthecontactphaseofthe stancelegwhilerunning[24].Theseandotherembodimentsofthespr ingcanbemodeled aselasticas,andearlyworkonsuchmodelingwasdonebyFrisch-Fay [25]. FollowingtheworkbyFrisch-Fay[25],thedeformationandenergyst orageproperties ofexibleelementshavebeenextensivelystudiedusingthetheoryo felastica.Motivated bytheapplicationofpistonringsininternalcombustionengines,Shin oharaandHara[26] studiedthedeformationofafullcircleC-shapedelastica;theeec toftangentialloadsat theopenendswasinvestigated.Usinghomotopymethods,Wangan dWatson[27]extended thestudytoproblemswheretheundeformedshapeoftheelastica isnotrestrictedtoa fullcircle.SrpcicandSajeinvestigatedlargedeformationsoft hincurvedplanebeamsof constantinitialcurvature,includingaC-shapedspring,usingnonlin earstrain-displacement 4 relations[28].Consideringplasticbehaviorresultingfromlargelocals train,Wang[29] studiedthedeformationcharacteristicsofathinringcrushedirre versiblybytwopointloads. CannarozziandMolari[30]performednonlinearanalysisofplanare lasticcurvedbeamsusing avariationalprincipleintermsofstresscomponents. Severalresearchershavestudiedthedeformationofelasticasin contactwithrigidsurfaces andotherdeformablestructures.WuandPlunkett[31]studiedth eproblemofcontactbe- tweenuniformcircularringsorcylindersusingasetofend-loadedela sticas;thedeformation oftwounequalringspressedbetweenrigidanvilswasobtainedinclos ed-formusingelliptic integrals.PlautandKlusman[32]determinedtheequilibriumshapesof stackeddissimilar geosynthetictubesondeformablefoundations.Plautetal.[33]de terminedtheequilibrium shapeofastraightelasticathatisbentandpusheddownonaatsu rface.Itwasshown thatforvariousdistancesofseparationbetweentheclampedend s,theelasticacanassume symmetricandasymmetricshapes,makepointorlinecontact,andb uckleinthecentral region.LuandChen[34]conductedexperimentswithanelasticaplac edinsideacircular channelwithclearanceandloadedatbothends;theoreticalinves tigationsweresubsequently performedbydividingtheelasticaintosub-domainsbasedoncontac tconditionsobserved experimentally. Besidesproblemsofstaticequilibrium,ChenandRo[35]studiedthevib rationand stabilitycharacteristicsofabuckledelastica,clampedatoneendan dfreetoslideattheother end,constrainedbyano-axispoint.Plaut,etal.investigatedthe vibrationcharacteristics ofverticalexibleloopsandcompressednestedrings[36,37]. 1.4ElasticHoppingRobots SomeoftheearlyworkonhoppingrobotsiscreditedtoRaibert[38],w hostudiedthedynam- icsandcontrolofaone-leggedrobotandpresentedpioneeringex perimentalwork.Following thiswork,manyresearchershaveinvestigatedhoppinggaitsandt heircontrol.Blickhan[39] consideredasimplemass-springsystemtostudyrunningandhoppin ggaitsandhowthese 5 gaitsareaectedbythelandingvelocity.Dummeretal.[40]andMicha lskaetal.[41]used themass-springmodeltodesigncontrollerstostabilizethehopping gaitandLapshin[42] investigatedtheproblemofheightcontrolofaone-leggedrobot. Brown[43]designeda robotwithabow-stringmechanismthatiswell-suitedforleggedlocom otion.Saranliet al.[44]introducedthewell-knownSLIP(springloadedinvertedpend ulum)modelofthe two-dimensionalhopper.CherouvimandPapadopoulos[45]andMat hisandMukherjee[46] designedcontrollersforregulatingtheapex-heightofhoppingrob ots. Toextendthecapabilitiesofhoppingrobots,recentworkshaveinv estigateddesignsfor hoppingonroughterrain.Saranlietal.[47]developedthehexapod RHexwithsixcompliant legsthatareC-shapedelasticatofacilitatewalkingandjumping.Gallo wayetal.[48] proposedvariablestnesslegsfortheRHexforrobustlocomotion .Wheeledmobilerobots capableofjumpingusecable-loadedsprings[49],solenoids[50],andpn eumaticcylinders[51]. AminiaturejumpingrobotwasdesignedbyZhaoetal.[52].Cable-loade delasticelements wereusedtostoreenergy,whichissuddenlyreleasedforjumping. Softmaterialshavealsobeenusedtomakedeformablestructures capableofrollingand hopping.Kovacetal.[53]developedaminiaturesphericalrobottha tcanreorientitselfafter landing.Armouretal.[54]designedarobotwithmetalhoopspringst hatformasphereand thatcanrollandjump.Yamadaetal.[55]developedajumpingrobot basedonanelastica thatutilizessnap-throughbuckling.SugiyamaandHirai[56]propos edadeformablerobot thatcancrawlandjumpbychangingitsshape.ReisandIida[57]des ignedahoppingrobot basedonfreevibrationofanelasticacurvedbeam.Tolley,etal.[58] designedasoftjumping robotthatusesacombinationofpneumaticandchemicalactuator stoexecutedirectional jumpingmaneuvers.Althoughdeformablerobotshavemanyadvan tages,precisecontrolof jumpingcanbeachallengingproblem. 6 1.5ARollingElastica DasandMukherjeestudiedthedynamicsofarollingdiskwiththreeun balancemasses;a self-propellingmechanismwasdevelopedtocontrolthemovemento ftherigiddisk[59].From theliteraturereviewofanelastichoppingrobot,wehaveseensome oftheexiblerolling robotsthatarecapableoflocomotiononvariousterrains.Thedyn amicsofaexiblerolling systemsuchastireshasbeenwidelystudied.KimandSavkoorperfo rmedananalysisof contactproblemoffree-rollingofpneumatictiresmodeledaselastic aringonaviscoelastic foundation[60].Lopezetal.modeledthevibrationsofadeformedro llingtireusingnite elementmethod[61].HoganandForbesmodeledandstudiedthevibra tionsofarolling exiblecircularringusingtheLagrangianformulationandtheRayleigh -Ritzmethod[62]. HuangandSoedelinvestigatedthevibrationofrotatingringsonela sticfoundation[63]. Manyoftheseinvestigationsarebasedonsmalldeformationassum ptions;foramoregeneral approachtostudytherollingelasticaproblem,axiallymovingmaterials areconsidered. Someoftheearlyworksonaxiallymovingmaterialsstemmedfromthes tudyofband sawvibrationsbyMote[64].Inextensionofpreviouswork,Wickerta ndMotestudied thevibrationofaxiallymovingcontinuasuchasmagnetictapesandth eCorioliseectsof thesystem[65].HwangandPerkinsderivedatheoreticalmodeltha tdescribesthenonlin- earresponseofacontinuousbandrotatingabouttworotatingwh eels[66].Furthermore, Banichuketalstudiedavarietyofproblemsinvolvinganalysesofsta bilityofaxiallymoving materials[67]. 7 Chapter2 KinetictoPotentialEnergyTransformationUsinga SpringasanIntermediary 2.1Overview Motivatedbythemechanicsofpole-vaulting,weinvestigateaseque nceofthreeproblems ofincreasingcomplexitywheretheunderlyingobjectiveistocomplet elyconvertthekinetic energyofamassmovinginthehorizontaldirectionintopotentialen ergyusingaspringasan intermediary.Foragivensetofnon-dimensionalsystemparamete rs,numericalsimulations areperformedtodetermineoptimalcombinationsofinitialcondition sthatresultincomplete transformationofthekineticenergyandworkdonebyexternalf orcesintopotentialenergy. Insection2.2westudythesimpleproblemofapointmassandlinearspr ing;themassis assumedtoattachitselftothespringwithaninitialvelocityinthehor izontaldirection anddetachitselfwhenthespringhasnostoredenergyandmassha snohorizontalvelocity. Theensuingtwo-pointboundaryvalueproblem(TPBVP)issolvedinan on-dimensional framework 1 forderentsetsofsystemparametersandinitialconditions.The samenon- dimensionalproblemisstudiedinsection2.3byreplacingthelinearsprin gwithanelastica. Acomparisonoftheresultswiththoseobtainedinsection2.2indicate sseveralsimilarities anddissimilarities.Theprobleminsection2.3ismodedinsection2.4byint roducing externaltorquesthatcandopositiveworkontheelasticabutare subjectedtoconstraints. Resultsobtainedinthissectionarecomparedwiththoseobtainedins ection2.3toshowthe gaininpotentialenergyofthemassduetoadditionalworkbytheex ternaltorques. 1 AdimensionalversionofthisproblemwasstudiedbyHoepner[68]. 8 2.2PointMassandLinearSpringSystem 2.2.1ProblemStatement Considerthesystemcomprisedofthepointmass M andthespringofstness K andlength R intheverticalplane,showninFig.2.1.Thespringcanrotatefreelyab outthe X axis (outoftheplaneofthepaper)and denotesitsanglewiththe Y axis,measuredcounter- clockwise.Theundeformedlengthofthespringis R 0 anditsinitialangleis 0 .Themass isassumedtohaveaninitialvelocity V 0 alongthenegative Y axiswhenitcomesincontact with,andattachesitselfto,thefreeendofthespring.Themassc andetachitselffromthe springatanytime.Theobjectivehereistocompletelytransformth ekineticenergyofthe massintoitspotentialenergy.Assumingnolossofenergy,themas swillthenreachthe maximumheightgivenby H = R 0 sin 0 + V 2 0 = 2 g (2.1) where g istheaccelerationduetogravity. Y ZK M g = 0 V0 R= R0 Figure2.1:Systemcomprisedofapointmassandahingedspringinthe verticalplane,is showninitsinitialconguration. 9 2.2.2EquationsofMotion TheequationsofmotionofthepointmassandspringsysteminFig.2.1a rederivedusing Lagrange'smethod[69].Forchoiceofgeneralizedcoordinates R and ,theequationsof motionare M R MR _ 2 K ( R 0 R )+ Mg sin =0(2.2a) R +2 _R _ + g cos =0(2.2b) andtheinitialconditionsare: R (0)= R 0 ; (0)= 0 _R (0)= V 0 cos 0 ;_ (0)=( V 0 =R 0 )sin 0 Wenon-dimensionalizeEq.(2.2)byintroducingthefollowingnon-dimens ionaltime,po- sition,andvelocityvariables: ˝ = t p K=M;r = R=R 0 ;v 0 =( V 0 =R 0 ) p M=K (2.3) Thenon-dimensionalequationsofmotionare r 00 r 0 2 (1 r )+ " sin =0(2.4a) r 00 +2 r 0 0 + " cos =0(2.4b) where( :) 0 denotesthederivativeof( :)withrespecttonon-dimensionaltime ˝ .Thenon- dimensionalstaticdeectionofthespringduetotheweightofthem assis " ,where " = =R 0 ; , Mg=K (2.5) 10 Thenon-dimensionalinitialconditionsare: r (0)=1 ; (0)= 0 r 0 (0)= v 0 cos 0 ; 0 (0)= v 0 sin 0 (2.6) whichcanalsobewrittenas r (0)=1 ; (0)= 0 ;r 0 (0)sin 0 = 0 (0)cos 0 (2.7) 2.2.3KinetictoPotentialEnergyTransformation Forcompletetransformationofthekineticenergyofthemassinto itspotentialenergy,the followingtwoconditionsmustbesatised: r ( ˝ f )=1 ;r 0 ( ˝ f )cos ( ˝ f )= r ( ˝ f ) 0 ( ˝ f )sin ( ˝ f )(2.8) where ˝ = ˝ f isthetimewhenthemassshoulddetachitselffromthespring.Indee d,the abovetwoconditionsimplythatthevelocityofthemassispurelyinthe vertical(upward) direction 2 andthespringhasnostoredenergy.Ifthemassdetachesitselff romthespring atthistime,itsdynamicswillbegivenbytherelation Z = g ) z 00 = " where " ,denedasthenon-dimensionalstaticdeectionofthespringdue totheweightof themassinEq.(2.5),isalsothenon-dimensionalaccelerationduetog ravity.UsingEqs.(2.1), (2.3)and(2.5),itcanbeshownthatthemaximumnon-dimensionalhe ightreachedbythe masswillbe h = H=R 0 =sin 0 +( v 2 0 = 2 " )(2.9) 2 Toeliminatethepossibilityofthemasshavingadownwardvelocity,wer estricttherangeof to(0 ;ˇ ). 11 Thedynamicsofthemass-springsystemisdescribedbyfourstate variables: r ,r 0 , and 0 .Sincethenaltime ˝ f isalsounknown,wehavetosolveaTPBVPwhereEq.(2.8) providestwoconditionsthathavetobesatisedatthenaltimean dEq.(2.7)providesthree conditionsthathavetobesatisedattheinitialtime. WerstsolvetheTPBVPbyassuming " and 0 tobeconstants.Asolutionwasobtained throughtrialanderrorforthespecccasewhere " =0 :1and 0 = ˇ= 6rad:Eqs.(2.4a)and (2.4b)wereintegratedforwardintimeusingtheinitialconditionsinEq .(2.6)forderent valuesofvelocity v 0 .TheresultsareshowninFig.2.2usingthenon-dimensionalCartesian coordinatesystem yz ,where y = r cos ,z = r sin .Ofthevederentcasesshownin Fig.2.2,only v 0 =0 :432satisestheveboundaryconditionsat ˝ f =3 :337,whichprovides thesolutiontotheTPBVP.At ˝ f ,thepolarcoordinatesofthemassarefoundtobe r ( ˝ f )=1 :0 ; ( ˝ f )=1 :315rad Ifthemassdetachesitselffromthespringatthistime,whichismark edby inFig.2.2, itsinitialkineticenergywillbecompletelyconvertedintopotentialen ergyatsomefuture time ˝ ˝ f =3 :337.Themaximumheightreachedbythemasswillbe h =1 :433,whichis computedusingEq.(2.9).TheresultsinFig.2.2indicatethatforaxed valueof " ,thereis auniquevalueofinitialvelocity v 0 thatsatisestheTPBVP;thissolutionresultsinunique valuesof ˝ f and ( ˝ f ).Thisimpliesthat ( ˝ f )cannotbepre-specedforaxedvalueof " .Inthenextsimulation,weimposetheadditionalcondition: ( ˝ f )= ˇ= 2;thisimpliesfrom Eq.(2.8)thatthefollowingconditionsneedtobesatisedattheterm inaltime: r ( ˝ f )=1 ; ( ˝ f )= ˇ= 2 ; 0 ( ˝ f )=0(2.10) Sinceweimposeanadditionalconditionat ˝ = ˝ f ,wenowvary " andtreat 0 asconstant. For " =0 :36,wend v 0 =0 :607satisesEq.(2.10)throughtrialanderror.Thetrajectories forvederentvaluesof " areshowninFig.2.3fortheinitialangle 0 = ˇ= 6radand 12 0.00.40.81.20.01.0 -0.6y z 0 = ˇ= 6 =1 : 315 v 0 =0 :49 v 0 =0 :40 v 0 =0 :432 v 0 =0 :37 v 0 =0 :46 r =1 Figure2.2:TrajectoriesofthemassinFig.2.1with " =0 :1forvederentvaluesofinitial velocity v 0 ,startingfrom 0 = ˇ= 6. 0.01.0 -0.60.00.40.81.2y z 0 = ˇ= 6 " =0 :36 " =0 :33 " =0 :38 " =0 :37 " =0 :35 r =1 Figure2.3:TrajectoriesofthemassinFig.2.1forvederentvalues of " ,startingfrom 0 = ˇ= 6withinitialvelocity v 0 =0 :607. v 0 =0 :607.For " =0 :36,thesixboundaryconditionsgivenbyEqs.(2.7)and(2.10)are satisedat ˝ f =4 :95.Ifthemassdetachesitselffromthespringatthistime,whichis markedby inFig.2.3,itsinitialkineticenergywillbecompletelyconvertedintopote ntial energyatsomefuturetime ˝ ˝ f =4 :95.Themaximumheightreachedbythemasswill be h =1 :011,whichiscomputedusingEq.(2.9). ForthesimulationsshowninFig.2.3,thevalueof " wasvariedtoaccountfortheaddi- 13 tionalconstraint ( ˝ f )= ˇ= 2.Insteadofxingthevalueof ( ˝ f ),itispossibletosolvethe problembyspecifying v 0 or y ( ˝ f )foraconstantvalueof 0 .Alternately,theproblemcanbe solvedforaxedvalueoftheinitialenergyofthemass.Thenon-dim ensionalinitialenergy ofthemasscanbeexpressedas e = 1 KR 2 0 MgR 0 sin 0 + 1 2 MV 2 0 = " sin 0 + 1 2 v 2 0 (2.11) ItisclearthatEq.(2.11)imposesaconstraintonthevaluesof " ,v 0 and 0 ;anyoneofthese threevariablescanbespecedandtheothertwocanbevariedto ndtherightcombination ofvaluesthatsolvestheTPBVP.Wewillnotexaminethesecasesany further;instead,we willxthevalueof " andexploresolutionstotheTPBVPforderentinitialangles 0 .For " =0 :1,thesolutiontrajectoriesareshowninFig.2.4andselectresultsar eprovided inTable2.1.Theseresultsindicatethatas 0 increases,theinitialvelocity v 0 decreases, ˝ f increases,and h and e decrease 3 .Itisinterestingtonotethat ( ˝ f )initiallydecreasesbut laterincreasesas 0 increases;itbecomesdculttondasolutionfor 0 > 58deg. 3 Atthemaximumheight h ,themasshaszerovelocity;thus e = "h .0.01.00.01.00.50.5y z 0 =10 0 =15 0 =20 0 =25 0 =30 0 =35 0 =58 0 =54 0 =48 r =1 Figure2.4:TrajectoriesofthemassinFig.2.1with " =0 :1:SolutionstotheTPBVPfor derentvaluesof 0 .14 Table2.1:Eectofinitialangle 0 onthesolutiontotheTPBVPforthemass-springsystem inFig.2.1with " =0 :1. 0 (deg) ˝ f ( ˝ f )(rad) v0 h e 10 3 : 033 1 : 496 0 : 801 3 : 382 0 : 338 20 3 : 130 1 : 383 0 : 596 2 : 118 0 : 212 30 3 : 337 1 : 315 0 : 432 1 : 433 0 : 143 48 4 : 283 1 : 325 0 : 234 1 : 017 0 : 102 58 5 : 429 1 : 399 0 : 166 0 : 986 0 : 099 Althoughboth h and e decreasewhen 0 isincreasedforaxedvalueof " (aswehave seenfromTable2.1),achangeinthevalueof " hasoppositeeectsonthe h - 0 and e - 0 curves.Anincreaseinthevalueof " movesthe h - 0 curvedownwardbutmovesthe e - 0 curveupward-seeFig.2.5.Thisimpliesthatforthesameinitialangle 0 ,alargeramount ofenergycanbetransformedusingalargervalueof " butthemaximumheightreachedby themasswillbelower. 1.00.10.53.51.00.10.050.50 0 (rad) 0 (rad) h e " =0 :25 " =0 :20 " =0 :10 " =0 :25 " =0 :20 " =0 :10 Figure2.5:Variationofmaximumheight h andinitialenergy e withinitialangle 0 for derentvaluesof " forthemass-springsysteminFig.2.1. 15 2.3PointMassandElasticaSystem 2.3.1ProblemStatement WeconsiderthepointmassandspringsysteminFig.2.6wherethesprin gisanelasticaof length L andexuralrigidity EI ;E istheYoung'smodulusofelasticityand I isthearea momentofinertiainbending.Theelasticacanrotatefreelyaboutth e X axis,whichpoints outoftheplaneofthepaper.Initsinitialconguration,theelastic aisundeformedandsub- tendsangle ˚ 0 withthe Y axis,measuredcounter-clockwise.Inthedeformedconguratio n, theslopeoftheelasticaatapointlocatedatadistance S ,S L ,alongitslengthfromthe hingedendisdenotedby ˚ ( S ).Themassisassumedtohaveaninitialvelocity V 0 alongthe negative Y axiswhenitcomesincontactwith,andattachesitselfto,thefreee ndofthe elastica.Themasscandetachitselffromtheelasticaatanytime.He re,theobjectiveisto completelytransformthekineticenergyofthemassintoitspotent ialenergy.Assumingno energyloss,themasswillthenreachthemaximumheight: H = L sin ˚ 0 + V 2 0 = 2 g (2.12) (a)(b) YY ZZ EI M M g ˚ ( S ) V0 L S ˚ 0 Figure2.6:Ahingedelasticaintheverticalplanewithatipmass M isshowninits(a) Initialconguration,and(b)anarbitraryconguration. 16 where g istheaccelerationduetogravity. 2.3.2ElasticaDeformation Theelasticaisanonlinearspringandgeneratesforces P and Q (alongthe Y and Z axes) thatactonthemass-seeFig.2.7.Themagnitudeoftheforces P and Q dependonthe changeinthecoordinatesofthelocationofthemassfromtheinitial time.Toinvestigate thedeformationoftheelastica,weconsideraninnitesimalsegmen toflength dS .Fromthe free-bodydiagramofthiselementshowninFig.2.7,wehave[25,70] ( d M =dS )= EI ( d 2 ˚=dS 2 )= Q cos ˚ P sin ˚ (2.13a) ( dY=dS )=cos ˚ (2.13b) ( dZ=dS )=sin ˚ (2.13c) where Y and Z arespatialvariablesand M denotesthebendingmoment.Sixboundary conditionsareneededtosolveEq.(2.13)asitiscomprisedofoneseco nd-orderandtworst- orderderentialequationsandinvolvestwounknowns P and Q .Ifthelocationofthemass Y ZM M ˚ ( S ) ˚ ( S ) P P P Q Q Q S M ( S ) M ( S + dS ) Mg dY dZ dS Figure2.7:Freebodydiagramsofthetipmass M andaninnitesimalsegmentofthe elasticaoflength dS .17 is( Y;Z )=( A;B ),thenwehavethefollowingboundaryconditions: at S =0:( d˚=dS )=0 ;Y =0 ;Z =0 at S = L :( d˚=dS )=0 ;Y = A;Z = B Intheaboveboundaryconditions,wehave( d˚=dS )=0atbothendssincenomomentis appliedontheelastica. Tonon-dimensionalizeEq.(2.13),thefollowingnon-dimensionalpositio nandforcevari- ablesareintroduced: y = Y=L;z = Z=L;s = S=L p = P ( L 2 =EI ) ;q = Q ( L 2 =EI ) Thisresultsinthefollowingnon-dimensionalequations: ( d 2 ˚=ds 2 )= q cos ˚ p sin ˚ (2.14a) ( dy=ds )=cos ˚ (2.14b) ( dz=ds )=sin ˚ (2.14c) andnon-dimensionalboundaryconditions: at s =0:( d˚=ds )=0 ;y =0 ;z =0 at s =1:( d˚=ds )=0 ;y = a , A=L;z = b , B=L (2.15) Together,Eqs.(2.14)and(2.15)deneaTPBVPwhichcanbeusedto nd p and q when a and b aregiven. 18 2.3.3DynamicsofPointMass Forthepointmass(seeFig.2.7),theequationsofmotioncanbewritt enas M ( d 2 e Y=dt 2 )= P (2.16a) M ( d 2 e Z=dt 2 )= Q Mg (2.16b) where e Y and e Z aretemporalvariables.Theinitialconditionsat t =0are e Y = L cos ˚ 0 ;( d e Y=dt )= V 0 e Z = L sin ˚ 0 ;( d e Z=dt )=0 Tonon-dimensionalizeEq.(2.16),thefollowingnon-dimensionalpositio n,time,velocity,and forcevariablesareintroduced: e y = e Y=L; e z = e Z=L;w = Mg ( L 2 =EI ) ˝ = t p EI=ML 3 ;v 0 = V 0 p ML=EI Thisresultsinthefollowingnon-dimensionalequations: ( d 2 e y=d˝ 2 )= p (2.17a) ( d 2 e z=d˝ 2 )= q w (2.17b) andnon-dimensionalinitialconditionsat ˝ =0: e y =cos ˚ 0 ;( d e y=d˝ )= v 0 e z =sin ˚ 0 ;( d e z=d˝ )=0 (2.18) ThedynamicsofthemasscanbesolvedbyintegratingEq.(2.17)nume ricallystartingfrom 19 Spatial DomainElastica DeformationTime Domain Mass Dynamics( a;b )=( e y; e z ) ( p;q ) Eqs.(2.14),(2.15) Eqs.(2.17),(2.18) Figure2.8:Methodofsolvingeachtimestepoftemporaldynamicsof masstogetherwith spatialdeformationofelastica. theinitialconditionsinEq.(2.18).Thisrequiresthevaluesoftheforc es p and q tobe updatedateachtimestep.Thisupdaterequiressolvingthespatial deformationofthe elasticainEq.(2.14)fortheboundaryconditionsinEq.(2.15) 4 ,where a and b aretheupdated coordinatesofthemassateachtimestep.Theprocedureformar chingthrougheachtime stepofintegrationisillustratedwiththehelpofFig.2.8. 2.3.4KinetictoPotentialEnergyTransformation Tocompletelytransformthekineticenergyofthemassat ˝ =0intoitspotentialenergyat somefuturetime,thethreefollowingconditionsneedtobesatised at ˝ = ˝ f :d e y=d˝ =0 ;d e z=d˝ 0 ;˚ ( s )=constant(2.19) where ˝ f isthetimewhenthemassshoulddetachitselffromtheelastica.Inde ed,theabove threeconditionsimplythatthevelocityofthemassispurelyintheupw arddirectionand theelasticahasnostoredenergy.Ifthemassdetachesitselffro mtheelasticaatthistime, itsdynamicswillbegivenby ( d 2 Z=dt 2 )= g ) ( d 2 z=d˝ 2 )= w SimilartoEq.(2.9),atsomefuturetime ˝ ˝ f ,themaximumheightreachedbythemass 4 TheMatlabfunctionbvp4c[71]isusedtosolvetheelasticadeformat ionTPBVP 20 willbe h = H=L =sin ˚ 0 +( v 2 0 = 2 w )(2.20) where H isdenedinEq.(2.12).Sincethenaltime ˝ f and v 0 areunknown,wehavetosolve aTPBVPintimewhereEq.(2.18)providesfourconditionsthathaveto besatisedatthe initialtimeandEq.(2.19)providestwoconditionsthathavetobesatis edatthenaltime. Throughtrialanderror,wesearchforasolutionfortheTPBVPfo rthespecccasewhere w =6and ˚ 0 = ˇ= 6rad.FollowingtheproceduredescribedinFig.2.8,wesearchforan initialvelocity v 0 forwhichtheconditionsinEq.(2.19)aresatised.Theresultsaresh own inFig.2.9usingthenon-dimensionalCartesiancoordinates e y and e z .Ofthevederent casesshowninFig.2.9,only v 0 =3 :23satisestheboundaryconditionsat ˝ f =0 :607, wheretheelasticaregainsitsundeformedshapeandsubtendsthe angle ˚ ( s )=1 :415rad =81 :073deg.Ifthemassdetachesitselffromtheelasticaat ˝ = ˝ f ,markedby inFig.2.9, itskineticenergywillbecompletelyconvertedintopotentialenergy atsomefuturetime ˝ ˝ f =0 :607.Themaximumheightreachedbythemasswillbe h =1 :369,whichis computedusingEq.(2.20).Thedeformedshapesoftheelasticaate ightderentinstants 0.01.0 -0.40.41.0 0.7e y e z ˚ 0 = ˇ= 6 v 0 =3 :23 v 0 =3 :75 v 0 =2 :75 v 0 =3 :00 v 0 =3 :50 ( e y 2 + e z 2 )=1 Figure2.9:TrajectoriesofthetipoftheelasticainFig.2.6isshownfor vederentinitial velocities v 0 :theelasticaisinitsundeformedshapeatthebeginningandendofea ch trajectory. 21 1.0 0.50.00.01.0 -0.2trajectory of the tip of the elastica e y e z ˚ 0 = ˇ= 6 ˚ ( s )=1 : 415 Figure2.10:CongurationsoftheelasticainFig.2.6atderentinstan tsoftimethatresults incompletetransformationofkineticenergyofthemassintopoten tialenergy. oftimeduringtheinterval[0 ;0 :607]areshowninFig.2.10alongwiththeinitialandnal undeformedshapes.Thenon-dimensionalinitialenergyofthemas scanbeexpressedas e = L EI MgL sin ˚ 0 + 1 2 MV 2 0 = w sin ˚ 0 + 1 2 v 2 0 (2.21) andwecannowinvestigatetheeectoftheinitialangle ˚ 0 ontheinitialvelocityrequired, themaximumheightreached,andtheamountofenergyconverted .For w =6,thetrajec- toriesareshowninFig.2.11andselectresultsareprovidedinTable2.2. Acomparisonof theseresultswiththoseobtainedforthemass-springsystem,sh owninFig.2.4andprovided inTable2.1,indicateseveralsimilaritiesandsomederences.Inboth cases,thesolution trajectoriesaresimilarforabroadrangeofinitialangles;however ,forlargeinitialangles, theelasticaundergoesmuchlessdeformationandrotationcompar edtothespring.Forboth cases,anincreaseintheinitialanglerequiresalowermagnitudeofth einitialvelocity v 0 and resultsinlowervaluesofmaximumheight h andenergytransformed e .Also,itispossible tondsolutiontrajectoriesforapproximatelythesamerangeofin itialangles:[10 ;58]deg forthespringand[15 ;60]fortheelastica.Themainderencebetweenthetwocasesisth e timerequiredbythespringandtheelasticatoregainitsundeformed conguration.This 22 0.01.00.01.00.50.5e y e z ˚ 0 =15 ˚ 0 =20 ˚ 0 =25 ˚ 0 =30 ˚ 0 =35 ˚ 0 =45 ˚ 0 =60 ˚ 0 =55 ˚ 0 =50 ˚ 0 =40 ( e y 2 + e z 2 )=1 Figure2.11:TrajectoriesofthemassinFig.2.6with w =6:SolutionstotheTPBVPfor derentvaluesof ˚ 0 .Table2.2:Eectofinitialangle ˚ 0 onthesolutiontotheproblemofcompleteenergy transformationforthemasswithelasticafor w =6. ˚ 0 (deg) ˝ f ˚ ( s )at ˝ f (rad) v0 h e 15 0 : 710 1 : 535 4 : 373 1 : 852 11 : 11 20 0 : 681 1 : 498 3 : 959 1 : 648 9 : 889 30 0 : 607 1 : 415 3 : 230 1 : 369 8 : 216 40 0 : 507 1 : 319 2 : 532 1 : 177 7 : 062 50 0 : 374 1 : 216 1 : 780 1 : 030 6 : 180 60 0 : 169 1 : 113 0 : 750 0 : 913 5 : 477 time,denotedby ˝ f ,decreasesfortheelasticabutincreasesforthespringastheinit ialangle increases. Weplotthe h - ˚ 0 and e - ˚ 0 curvesinFig.2.12forderentvaluesof w .Theoveralltrends aresimilartothoseobservedforthemass-springsysteminFig.2.5wit htheexceptionthat the h - ˚ 0 curvesofthemass-elasticaintersecteachotherwhereasthe h - 0 curvesofthe mass-springdonot. 23 1.00.20.82.21.00.2412 ˚ 0 (rad) ˚ 0 (rad) h e w =7 w =6 w =5 w =7 w =6 w =5 Figure2.12:Variationofmaximumheight h andinitialenergy e withinitialangle ˚ 0 for derentvaluesof w forthemass-elasticasysteminFig.2.6. 2.4Mass-ElasticaSystemwithExternalTorque 2.4.1ModicationoftheMassElasticaSystem WereconsiderthepointmassandelasticasysteminFig.2.6andassume thatanexternal torque Tisappliedtothetipoftheelastica-seeFig.2.13.Thistorquewillbedesig nedto (a)(b) YY ZZ EI M M g ˚ ( S ) TV0 L S ˚ 0 Figure2.13:Ahingedelasticaintheverticalplanewithtipmass M andexternaltiptorque Tisshowninits(a)Initialconguration,and(b)anarbitrarycongu ration. 24 addenergytothemass-elasticasystem(dopositivework)subjec ttotheconstraint Tmax T Tmax (2.22) where Tmax > 0isthemaximumtorquethatcanbeapplied.Theobjectivehereisto completelytransformthetotalenergycomprisedofthekineticen ergyofthemassandthe workdonebytheexternaltorqueintopotentialenergyofthema ss.Thebehaviorofthe modedsystemcanbeinvestigatedbyfollowingtheprocedureoutlin edinsection2.3.3. Asdescribedtherein,Fig.2.8describestheproceduretomarchthr ougheachtimestepof integration:Eqs.(2.14),(2.17)and(2.18)remainunchangedbutEq .(2.15)hastobemoded totheform: at s =0:( d˚=ds )=0 ;y =0 ;z =0 at s =1:( d˚=ds )= t;y = a;z = b t, TL=EI;a , A=L;b , B=L (2.23) where tisthenon-dimensionalexternaltorque,and a and b arethenon-dimensionalcoor- dinatesofthetipoftheelastica. 2.4.2MaximizingPotentialEnergyoftheMass Tocompletelytransformthetotalenergycomprisedofthekinetic energyofthemassand theworkdonebytheexternaltorqueintopotentialenergyofth emass,theconditionsin Eq.(2.19)needtobesatisedatsometime ˝ = ˝ f ,where ˝ f isthetimewhenthemass shoulddetachitselffromtheelastica.Furthermore,tomaximizeth epotentialenergyofthe mass,theworkdonebytheexternaltorqueshouldbemaximized.T hiscanbeaccomplished bychoosing tasfollows t=sgn f d d˝ [˚ (1)] g tmax ;tmax , ( L=EI ) Tmax (2.24) 25 wherethesgn( :)functionreturns+1or 1dependingonwhether( :)ispositiveornegative. Throughtrialanderror,wesearchforasolutionforthespeccc asewhere w =6, ˚ 0 =20 deg,and tmax =0 :5.TheconditionsinEq.(2.19)aresatisedfor v 0 =3 :894and ˝ f =0 :691, wheretheelasticaregainsitsundeformedshapeandsubtendsthe angle ˚ ( s )=1 :489rador 85 :313deg.Ifthemassdetachesitselffromtheelasticaatthistime,t hemaximumheight reachedbythemasswithexternaltorquewillbe h =1 :869.Forthesamevaluesof w and ˚ 0 ,Table2.2indicatesthat tmax =0 :0resultsin h =1 :648.Thedatafor tmax =0 :0and tmax =0 :5arepresentedtogetherinTable2.3;theyindicateagaininthemaxim umheight ofthemassby13 :4%.Figure2.14showstheinitialandnalcongurations(undeform ed) oftheelasticaalongwithsevenintermediatedeformedconguratio nsfor tmax =0 :0and tmax =0 :5.Itcanbeseenfromtheseguresthatthetipoftheelasticainitia llyrotatesin theclockwisedirectionandapplicationofthetipmoment t= tmax causesthetiptobend furtherdownward;thisresultsinmorestrainenergybeingstored intheelastica.Asthe elasticastraightensout,thetipoftheelasticarotatesinthecoun ter-clockwisedirectionand applicationofthetipmoment t=+ tmax addsmoreenergytothesystem.Sincetheadded 0.01.0 -0.4trajectory of the tip of the elastica 1.0 0.5 0.01.0 0.5 0.00.01.0 -0.4trajectory of the tip of the elastica e y e y e z e z (a) (b) Figure2.14:CongurationsoftheelasticainFig.2.13atderentinsta ntsoftimethat resultincompletetransformationofthetotalenergyintopotent ialenergyofthemassfor (a) tmax =0 :0and(b) tmax =0 :5. 26 Table2.3:ComparisonofsolutiontrajectoriesoftheelasticainFig.2.1 3withandwithout externaltorqueforderentinitialangles ˚ 0 andfor w =6. ˚ 0 (deg) tmax v0 ˝ f h ( h=h ) 20 0 : 0 3 : 959 0 : 681 1 : 648 0 : 5 3 : 894 0 : 691 1 : 869 +13 : 4% 25 0 : 0 3 : 585 0 : 647 1 : 493 0 : 5 3 : 514 0 : 658 1 : 683 +12 : 7% 30 0 : 0 3 : 230 0 : 607 1 : 369 0 : 5 3 : 164 0 : 620 1 : 528 +11 : 6% 35 0 : 0 2 : 881 0 : 560 1 : 265 0 : 5 2 : 830 0 : 580 1 : 412 +11 : 6% 40 0 : 0 2 : 532 0 : 507 1 : 177 0 : 5 2 : 503 0 : 536 1 : 310 +11 : 3% 45 0 : 0 2 : 170 0 : 445 1 : 099 0 : 5 2 : 183 0 : 492 1 : 228 +11 : 7% energyiscompletelygivenbacktothemass,themaximumheightreac hedbythemassis higherforthecasewith tmax =0 :5. Figure2.14presentsresultsforthespecccasewhere ˚ 0 =20deg.Similarresults wereobtainedforotherinitialanglesandareprovidedinTable2.3.Th isdataindicates thatforallinitialangles,thepercentagegaininmaximumheightfort hemassismorethan 11%;also,consistentwithearlierobservations,themaximumheight islargerforsmallinitial angles ˚ 0 .Itisinterestingtonotethattheinitialvelocitiesofthemassandth etimetaken fortheelasticatoregainitsundeformedcongurationarequitesim ilarfor tmax =0 :0and tmax =0 :5. TheresultsinTable2.3showthepercentagegainsinmaximumheightfo rthespecc casewhere w =6and tmax =0 :5.Numericalsimulationsindicatethatlargevaluesofboth w and tmax donotyieldfeasiblesolutionsbutsmallervaluesof w andlargervaluesof tmax resultinhigherpercentageheightgains.Theresultsofthreespec ccasesarepresentedin Table2.4forafewinitialangles. TheresultsinTables2.3and2.4comparethemaximumheightreachedb ythemassin thepresenceandabsenceoftheexternaltorque.Thesolutions totheTPBVPswithand 27 Table2.4:PercentagegainsinmaximumheightreachedbythemassinF ig.2.13forderent initialangles ˚ 0 andthreederentcombinationsof( tmax ;w ). ( tmax ;w ) ˚ 0 (deg) 20 25 30 (0 : 60 ; 5) 16.7 15.4 15.2 (0 : 50 ; 6) 13.4 12.7 11.6 (0 : 43 ; 7) 11.5 10.5 9.7 withoutexternaltorquedonotresultinthesameinitialvelocityoft hemassandtherefore thepercentagegaininheightcannotbesimplyattributedtothewor kdonebytheexternal torque. Toillustratetheprocessinwhichsomeportionoftheinitialenergyof themassistem- porarilystoredasstrainenergyoftheelastica,andthencomplete lyconvertedintopotential energyofthemass,weplotthederentcomponentsofthenon-d imensionalenergyasa functionofnon-dimensionaltime.Fortheparticularcasepresent edinFig.2.14,weplotthe followingquantitiesinFig.2.15: 1.non-dimensionalkineticenergyofthemassassociatedwithhoriz ontalmotion, 2.non-dimensionalkineticenergyofthemassassociatedwithvert icalmotion, 3.non-dimensionalpotentialenergyofthemass, 4.non-dimensionalstrainenergyenergyoftheelastica, 5.non-dimensionalworkdonebytheexternaltorque, eachexpressedasafractionoftheinitialnon-dimensionalenergy ofthemass.Theseenergy fractionsaredenotedby 1 ,2 ,3 ,4 ,and 5 ,respectively,inFig.2.15.Forbothcases, withandwithoutexternaltorque,itcanbeseenthatfor ˝ 2 [0 ;˝ f ]:thekineticenergy associatedwithhorizontalmotion 1 goestozero;thekineticenergyassociatedwithvertical motion 2 increasesfromzeroandreachesitsmaximum;andthestrainenerg yoftheelastica 4 increasesfromzero,reachesitsmaximum,andthengoesbacktoz ero.At ˝ = ˝ f ,the 28 0.41.2 0.00.8 0.41.2 0.00.8 1.0 0.01.2 0.0˝ ˝ 4 4 1 1 2 2 3 3 5 ˝ f =0 : 681 ˝ f =0 : 691 ˝ =1 : 146 ˝ =1 : 230 (a) (b) Figure2.15:Plotofderentcomponentsofthenon-dimensionalen ergy,eachexpressedas afractionoftheinitialnon-dimensionalenergyofthemass,asafu nctionoftimeforthetwo caseswheretheexternaltorqueis(a)absentand(b)present. massdetachesitselffromtheelastica;hereafter,thekineticene rgyassociatedwithvertical motion 2 decreasestozeroandthepotentialenergyofthemass 3 reachesitsmaximum value.Theexternaltorqueaddsenergy 5 tothesystemduringtheperiod ˝ 2 [0 ;˝ f ];therefore,thepotentialenergy,whichisexpressedasafractio noftheinitialenergy,reaches amaximumvaluegreaterthanunity.Intheabsenceofexternalto rque,thepotentialenergy ofthemass 3 reachesamaximumvalueofunity. 2.4.3ADimensionalCaseStudy Wepresentresultsfromadimensionalcasestudywherethesyste mparametersaresimilar tothosedocumentedinadultmalepole-vaulting[8].Assumingthemass ofthepole-vaulter tobeequaltothetipmass,thesystemparameterswerechosena sfollows: M =80kg ;L =4 :5m ;EI =2000Nm 2 ;Tmax =100Nm Althoughthemaximumtorqueof Tmax =100Nmiswellwithinthephysicalcapabilitiesof apole-vaulter[19],thereareseveralderencesbetweentheass umptionsmadeinthiswork 29 andtheactualpole-vaultmaneuver.Here,itisassumedthatthee xternaltorquecanbe appliedfortheentiredurationoftimewhenthepoleisdeformed;inco ntrast,pole-vaulters havetoreorienttheirbodiesandarethereforeunabletoapplythe maximumtorqueatall times.Consistentwiththetakeoangleofadultmalepole-vaulters[1 6],theinitialangleis chosenas ˚ 0 =20deg.Thesimulationresultsindicatethatcompletetransformat ionofthe kineticenergyofthemassandtheworkdonebytheexternaltorq ueintopotentialenergy ofthemassisachievedwith V 0 =9 :75m/s ;t f =1 :55s ;H =6 :70m Thedatafortheeightnalistsinthe1992olympiceventareprovided in[16].Itcanbe veredfromthesedatathattheinitialvelocity V 0 =9 :75misquitesimilartotherunning speedofthecontestants,whichliesintherangeof8 :95 9 :74m/s;thetimerequiredfor themaneuver, t f =1 :55s,isalsosimilartothetimeof ˇ 1 :50srequiredbytheathletes. Themaximumheightreached, H =6 :70m,ishigherthanthecurrentworldrecordof6 :16 m;butthiscanbeattributedtotheseveralsimplifyingassumptions madeinthiswork.In additiontotheassumptionsofpointmassandapplicationofthemaxim umtorqueforthe entiredurationofthemaneuver,itwasassumedthatthepoleismas sless,noenergyislost, themasshasnohorizontalvelocityatthenaltime,andthekinetic energyofthemassand workdonebytheexternaltorqueiscompletelytransformedintop otentialenergy. 30 Chapter3 Force-DisplacementCharacteristicsofCircular-Shaped MasslessElastica 3.1Overview Inthischapter,weusenumericalmethodstoinvestigatetheforc e-displacementcharacteris- ticsofcircular-shapedelasticas.Inparticular,westudythedefo rmationofC-andO-shaped elasticasrestingonthegroundunderverticalloading;eachproble misstudiedbydividing theelasticaintosegmentsthatcanbeclassedasfree,loadedwith constantcurvature,and loadedwithvariablecurvature.FortheC-shapedelastica,wecons iderthetwocaseswhere theverticalloadingresultsintheelasticarollingwithoutslippingdueto thepresenceof highfriction,androllingwhileslippingduetotheabsenceoffriction.Fo rtheO-Shaped elastica,bothsymmetricandasymmetricloadingareinvestigatedalo ngwiththetransition frompointtolinecontact.Someoftheresultsareobservedinthed ynamicbehaviorof mass-elasticasystems.Usinganon-dimensionalframework,thep roblemsstudiedarepre- sentedintheorderofincreasingcomplexityintermsofsolvingthetw o-pointboundaryvalue problemswithunknownparametersandboundaryconditions.Finite elementmethodshave alsobeenusedtosolveelasticaproblems[72{78].However,slidingand rollingcontactdueto frictionandlinecontactduetounilateralconstraintsrequirespec ialmethodsthatincrease numericalcomplexityandcomputationalcost.Thischapterisorga nizedasfollows.Model- ingofcircular-shapedelasticasispresentedinsection3.2.TheC-sh apedelasticaisstudied insection3.3andtheO-shapedelasticaisstudiedinsection3.4. 31 3.2MathematicalModelofElastica 3.2.1GeneralProblemFormulation Consideraninitiallystraightmasslesselasticaoflength L andexuralrigidity EI ,shownin itsdeformedcongurationinFig.3.1.Oneendoftheelasticaisxedat theoriginofthe XY Cartesiancoordinateframe;theotherendissubjectedtoforce s P and Q alongthenegative X andnegative Y directions,respectively,andacounter-clockwisebendingmoment M .An arbitrarypointontheelasticaislocatedatanarc-lengthof S ,S 2 [0 ;L ],fromthexed end;theslopeoftheelasticaatthispointisdenotedby ( S ).Fortheinnitesimalsegment oflength dS ,themomentbalanceequationis: ( d =dS )= EI ( d 2 =dS 2 )= Q cos P sin (3.1) where( S )isthebendingmomentintheelastica,asshown.Theboundarycond itionsare splitbetweenthetwoends: at S =0: = 0 ;at S = L :( d=dS )= M=EI X Y ( S ) ( S ) P P P Q Q Q S 0 ( S ) ( S + dS ) dX dY dS M L Figure3.1:Deformedcongurationofaninitiallystraightelastica. 32 TheshapeoftheelasticaintheCartesiancoordinateframecanbeo btainedfromthekine- maticequations: ( dX=dS )=cos ; ( dY=dS )=sin (3.2) andtheboundaryconditions X (0)= Y (0)=0.Byintroducingthefollowingnon-dimensional variables: x = X=L;y = Y=L;s = S=L p = P ( L 2 =EI ) ;q = Q ( L 2 =EI ) ;m = M ( L=EI ) thenon-dimensionalequationsofEqs.(3.1)and(3.2)areobtained: ( d 2 =ds 2 )= q cos p sin (3.3a) ( dx=ds )=cos (3.3b) ( dy=ds )=sin (3.3c) Thenon-dimensionalboundaryconditionsare: at s =0: = 0 ;x =0 ;y =0 at s =1:( d=ds )= m (3.4) Equation(3.3)iscomprisedofonesecond-orderandtworst-ord erderentialequations. TheycanbesolvedusingthefourboundaryconditionsinEq.(3.4).In thenextsection,we willstudycircular-shapedelastica, i.e. ,elasticathathasaninitialcurvatureintheabsence ofloading. 33 3.2.2Circular-ShapedElasticaProblems Anarbitrarycongurationofanon-dimensionalC-shapedelastica ,thatmakespointcontact withtheground,isshowninFig.3.2.Forces p and q areappliedexternallyatthetip( s =1) oftheelasticaanditisassumedthatthegroundcangeneratesuc ientlylargehorizontal forcestopreventslipping.Theportionoftheelasticabetweenthe pointincontactwiththe groundandthetip, s 2 [; 1],canbereferredtoastheLoadedElasticaSegment(LES); theremainingportion, s 2 [0 ; ],isnotsubjectedtoanyloadandcanbereferredtoas theFreeElasticaSegment(FES).Typically,theLEShasaVariableCu rvature;therefore, itisappropriatetorefertoitastheVC-LES.Thefree-bodydiagra msofboththeFES andVC-LESarealsoshowninFig.3.2;theunknownforce p ,theunknownmoment m ,and thevalueof canbeobtainedfromthesolutionoftheboundary-valueproblem,w hichis discussedinthenextsection. Anon-dimensionalO-shapedelastica,thatinitiallymakespointconta ctwiththeground, isshowninFig.3.3.Notethathalfthecircumferenceoftheelasticaisit scharacteristiclength. Figure3.3alsoshowsthedeformedcongurationthatmakeslinecon tactwiththeground whentheexternalforce2 q isappliedtothetopoftheelastica.Thedeformedconguration issymmetricwithrespecttotheverticalaxisanditissucienttoinve stigateonesymmetric p p p q q q s =0 s =0 s = s = s = s =1 s =1 FES m m VC-LES Figure3.2:AC-shapedelasticaunderloading:itiscomprisedofanFES andaVC-LES. 34 p p 2 q xx y y (0 ; 0) s =0 s = s = s =1 s =1 m VC-LES q q y m CC-LES Figure3.3:AnO-shapedelasticaisshowninitsundeformedanddefor medcongurations: thedeformedcongurationissymmetricwithrespecttothevertic alandiscomprisedofa VC-LESandaCC-LES. halfofthedeformedshape.Figure3.3showsthattherighthalfisco mprisedoftwosegments: thesegmentthatmakeslinecontact 1 withtheground, s 2 [0 ; ],andtheremainingsegment, s 2 [; 1].ThesegmentthatmakeslinecontactwiththegroundhasaConst antCurvature 2 andcanbereferredtoasCC-LESwhereastheremainingsegmenth asavariablecurvature andisthereforeaVC-LES.Thefree-bodydiagramoftheVC-LESis showninFig.3.3;the unknownforce p ,theunknownmoments m , m ,andthevalueof canbeobtainedfromthe solutionoftheboundary-valueproblem,whichisdiscussedinthenex tsection. 3.2.3SolutiontotheBoundary-ValueProblem Underloading,aC-oranO-shapedelasticacanbecomprisedofacom binationofsegments thatarecharacterizedasanFES,aCC-LES,andaVC-LES.ForaV C-LES,itisnecessary tosolveaboundary-valueproblem(BVP)asthecurvatureofthee lasticasegmentvaries alongitslength.Sincethecircular-shapedelasticahasaninitialradiu sofcurvature,the 1 Theloadingonthelinesegmentmaynotbeuniform;itisimplicitlyassumed thatthelinesegment doesnotbuckle. 2 ItcanbeseenfromEq.(3.3a)thataconstantvalueof implies( d=ds )=0,whichinturnimpliesthat thecurvatureisconstantandequaltozero.Itwillbeseenlatert hatthecurvatureisconstantbutnotequal tozeroforaninitiallycurvedelastica. 35 derentialequationsfromEq.(3.3)aremodedto: ( d 2 ˚=ds 2 )= q cos( ˚ + s=ˆ ) p sin( ˚ + s=ˆ ) ;˚ ( s ) , ( s ) s=ˆ (3.5a) ( dx=ds )=cos( ˚ + s=ˆ ) (3.5b) ( dy=ds )=sin( ˚ + s=ˆ ) (3.5c) where ˚ ( s )istheslopeoftheelasticarelativetoitsnominalcircularshapeand ˆ isthecon- stantnon-dimensionalradiusofcurvatureoftheundeformedcir cular-shapedelastica.The boundaryconditionswillbederentforderentproblems;anumbe rofderentproblems willbestudiedinsections3.3and3.4.Inalloftheseproblems,thebou ndaryconditions aresplitbetweenthetwoboundariesandthisrequiresthesolutiono fatwo-pointBVP (TPBVP).AllTPBVPsinthispaperweresolvedusingtheMATLABfunc tionbvp4c[71]. ForaCC-LES,itisnotnecessarytosolvetheBVPsinceauniformlyco nstantslope, suchas ( s )=0inFig.3.3,implies( d˚=ds )= 1 =ˆ and( d 2 ˚=ds 2 )=0.ForanFES,it isnotnecessarytosolvetheBVPeithersincetherearenoappliedfo rcesandmoments actingontheelasticasegment.Indeed, p = q =0implies( d 2 ˚=ds 2 )=0andthisimplies ( d˚=ds )=constantand( d=ds )=constant. 3.3C-ShapedElasticaunderVerticalLoading 3.3.1InitialCongurationoftheC-ShapedElastica TheinitialcongurationoftheC-shapedsemi-circularelasticainthe absenceofloading ( q =0)isshowninFig.3.4-theslopeoftheelasticatipisequalto (1)=( ˇ ).Since thetotallengthoftheelasticaisunity,theradiusofcurvatureise qualto ˆ =1 =ˇ .Itcan beshownthatthepointofthecontactoftheelastica( s = )withthegroundhasthe coordinates( x a ;0),where = ˇ ;x a = 1 ˇ sin (3.6) 36 s =0 s = s =1 ˆ( x a ; 0) (0 ; 0) x y q y Figure3.4:InitialcongurationoftheC-shapedelasticafor q =0. Underloading,boththepointofcontactoftheelasticawiththegro und(valueof s )andthe locationontheground(valueof x )canchange.Weinvestigatethecaseswheretheelastica rollswithoutslippingduetothepresenceoffrictionandrollswhileslippin gduetolackof friction.Thecaseofrollingandslippinginthepresenceoffrictionisdis cussedwiththehelp ofaremark.Inallthreecases,theelasticamaintainsapointconta ctwiththeground. 3.3.2ElasticaRollswithoutSlippingDuetoFriction Inthiscase,weassumethatthefrictionforceislargeenoughandp reventsthepointof contactoftheelasticawiththegroundfromslipping.Theexternal force q isappliedto thetipoftheelastica( s =1)-seeFig.3.4,andthetipisconstrainedtomovealongthe y directionasthemagnitudeof q isincreased.Weconsidertwocaseswheretheslopeofthe tipoftheelasticais(a)xedand(b)free,respectively.Forbothc ases,thegroundapplies nomomentontheelasticaandtheboundaryconditionsontheVC-LE Sare: at s = + " :˚ =0 ;( d˚=ds )=0 ;x = x a + ";y =0 (3.7) 37 where " isthedistancebywhichtheelasticarollsinthepositive x directionfromtheinitial congurationwheretheelasticahasnoloading, i .e. q =0.Toensure ˚ ( + " )=0,the relationbetween ˚ ( s )and ( s )inEq.(3.5a)ismodedto: ˚ ( s )= ( s ) [s ( + " )] =ˆ = ( s ) ˇ ( s " ) Whentheslopeofthetipoftheelasticaisxed =( ˇ ),theotherboundaryconditions ontheVC-LESare: at s =1: x =0 ;˚ ( s )= ˇ" (3.8) Whentheslopeofthetipoftheelasticaisfree,theotherboundary conditionsontheVC-LES are: at s =1: x =0 ;( d˚=ds )=0(3.9) TheVC-LESisdescribedbytworst-orderandonesecond-order ordinaryderentialequa- tions-seeEq.(3.5).This,alongwiththefactthatthehorizontalfo rceatthetipofthe elastica( p inFig.3.2)andthedistancerolledbytheelastica( " )duetoendloadingare unknown,impliesthatsixboundaryconditionsarerequiredforsolvin gthederentialequa- tions.FouroftheseboundaryconditionsareprovidedbyEq.(3.7)a ndtheremainingtwo areprovidedbyEq.(3.8)orEq.(3.9). TheTVBVPdescribedbyEq.(3.7)andEq.(3.8)orEq.(3.9)cannotbeso lveddirectly usingtheMATLABfunctionbvp4c[71]sincetheleftboundarylocatio nisunknown.Starting fromaninitialguess ,thetruevalueof " attheleftboundaryisfoundbyusingastandard gradientsearchalgorithm[79]wherethevalueof isiterativelyupdatedby + d ,where d = 1 s ˆ d˚ ds s = + d˚ ds s = + + s ˙ 1 d˚ ds s = + (3.10) and s isasmallnumberchosenappropriatelyforaccuratelycomputingth enumerical derivativeof( d˚=ds ).Theiterativeprocedureisterminatedwhen( d˚=ds ) ˇ 0atthepoint 38 0.00.40.20510 15 y q =45 =0 Figure3.5:Force-displacementcurvesforaC-shapedelasticarollin gwithoutslippingwith xedtipslope (1)=( ˇ );thecurvescorrespondtoderentvaluesof incrementedby 5 overtheinterval[0 ;45 ].0.00.40.20510 15 y q =45 =0 Figure3.6:Force-displacementcurvesforaC-shapedelasticarollin gwithoutslippingwith freetipslope;thecurvescorrespondtoderentvaluesoftheinit ialangle incrementedby 5 overtheinterval[0 ;45 ].ofcontact. Simulationresultsarerstpresentedforthecasewheretheslope ofthetipoftheelastica 39 0.00.6 0.4 0.20.00.2 -0.20.00.2 -0.20.00.60.4 0.2(a)(b) xx y y q =0 q =0 q =5 q =5 q =10 q =10 q =15 q =15 Figure3.7:DeformedshapesoftheC-shapedelasticawith(a)xed tip-slopeand(b)free tip-slopeunderderentvaluesofverticalloading q ,q 2 [0 ;15]forinitialangle =20 .isxed, i .e., (1)=( ˇ ).Figure3.5showstheforce-displacementcurvesoftheelastica forderentvaluesof intheinterval[0 ;45 ],inregularincrementsof5 .Itcanbeseen thatthecurveissteeperwhen islarger;thisimpliesthattheC-shapedelasticabecomes sterwhen islarger.Forthecasewheretheslopeofthetipisfree,Fig.3.6shows the force-displacementcurvesforderentvaluesoftheinitialangle , 2 [0 ;45 ],alsoin incrementsof5 .AcomparisonofFigs.3.5and3.6indicatesthat,similartothecasewhe re thetipslopeisxed,theelasticabecomessterforlargervaluesof .However,forthesame valueofverticalloading,theelasticawithfreetip-slopehasalarger displacementcompared totheelasticawithxedtip-slopeforthesamevalueofinitialtipslope .Thisimpliesthat theelasticawithxedtip-slopeissterthantheelasticawithfreetip -slope.Thiscanalso beveredfromFig.3.7,whichshowsthedeformationoftheelasticaf orderentlevelsof loading,forthespecccasewheretheinitialangle =20 .Anotherobservationthatcan bemadefromFigs.3.5and3.6isthatthexedtip-slopeelasticabehave slikeahardening spring;incontrast,thefreetip-slopeelasticabehaveslikeasofte ningspringforsmallvalues ofinitialangle andlikeahardeningspringforlargervaluesof .40 3.3.3ElasticaRollsWhileSlippingDuetoNoFriction Inthiscase,weassumethatthereisnofrictionbetweentheelastic aandtheground;con- sequently,theelasticacanrollandslipastheverticalloadingchang es.Similartoour investigationinsection3.3.1,theexternalforce q isappliedtothetipoftheelastica( s =1) -seeFig.3.4,andthetipisconstrainedtomovealongthe y directionasthemagnitudeof q isincreased.Onceagain,weconsidertwocaseswheretheslopeoft hetipoftheelastica is(a)xedand(b)free,respectively.Forbothcases,thegroun dappliesnomomentonthe elasticaandtheboundaryconditionsontheVC-LESare: at s = +( " + ˙ ): ˚ =0 ;( d˚=ds )=0 ;x = x a + ";y =0 (3.11) where " isthedistancebywhichtheelasticahasrolledinthepositive x directionfromthe initialcongurationwheretheelasticahasnoloading, i .e. q =0;and ˙ istheadditional decreaseinthelengthoftheVC-LESduetoslipping.Tosatisfytheb oundarycondition ˚ ( + " + ˙ )=0,therelationbetween ˚ ( s )and ( s )inEq.(3.5a)ismodedtotheform ˚ ( s )= ( s ) [s ( + " + ˙ )] =ˆ = ( s ) ˇ ( s " ˙ ) Whentheslopeofthetipoftheelasticaisxed =( ˇ ),theotherboundaryconditions ontheVC-LESare: at s =1: x =0 ;˚ ( s )= ˇ ( " + ˙ )(3.12) Whentheslopeofthetipoftheelasticaisfree,theotherboundary conditionsontheVC- LESareidenticaltothoseinEq.(3.9).TheVC-LESisdescribedbytwo rst-orderandone second-orderordinaryderentialequations-seeEq.(3.5).This, alongwiththefactthatthe distancerolledbytheelastica( " )andthereductioninthelengthofVC-LESduetoslip( ˙ ) areunknown,impliesthatsixboundaryconditionsarerequiredfors olvingthederential 41 equations 3 .FouroftheseboundaryconditionsareprovidedbyEq.(3.11)andt heremaining twoareprovidedbyEq.(3.12)orEq.(3.9). TheTVBVPdescribedbyEq.(3.11)andEq.(3.12)orEq.(3.9)cannotbe solveddirectly usingtheMATLABfunctionbvp4c[71]sincetheleftboundarylocatio nisunknown.Starting fromaninitialguess( 1 ; 2 )for( ";˙ ),thecorrectvaluesof " and ˙ attheleftboundarycanbe founditerativelybyusingtheMATLABfunctionfsolveoranygradien tsearchalgorithm[79]. Theiterativeprocedureisterminatedwhenboth( d˚=ds ) ˇ 0and p ˇ 0atthepointof contact. Simulationresultsarepresentedforthecasewheretheslopeofth etipoftheelasticais xed, i .e., (1)=( ˇ ).Figure3.8showstheforce-displacementcurvesoftheelasticaf or derentvaluesof intheinterval[0 ;45 ],inregularincrementsof5 .Itcanbeseenthat thecurvesbecomesteeperastheloadisincreased,implyingthatall congurationsofthe elasticabehaveasahardeningspring.Thecurvesalsobecomestee perasthevalueof is 3 Notethatthehorizontalforce p (seeFig.3.2),iszeroduetotheabsenceoffrictionbetweentheelas tica andtheground. 0.00.40.20510 15 y q =45 =0 Figure3.8:Force-displacementcurvesforaC-shapedelasticawith xedtipslope (1)= ( ˇ )onafrictionlesssurface;thecurvescorrespondtoderentva luesof incremented by5 overtheinterval[0 ;45 ].42 increased;however,thederencebetweenthecurvesismargina lcomparedtothatbetween thecurvespresentedintheprevioussection.Thesimilarityofthec urvesinFig.3.8indicates thattheforce-displacementcharacteristicsoftheelasticaaren earlyindependentofthexed tip-slope,whichisnotquiteintuitive.Forthespecccaseof =30 ,thedeformedshapes oftheelasticaareshowninFig.3.9(a)forfourderentvaluesofver ticalloading q .Asthe loadingisincreased,thecontactpointonthegroundchangesalong withachangeinthe lengthoftheVC-LES.Thechangeinthelocationofthecontactpoin t,whichisequalto distancerolledbytheelastica " ,andthechangeinthelengthoftheVC-LESduetoslipping, whichisequalto ˙ ,areplottedasafunctionoftheverticalloading q inFig.3.9(b).Itis clearthatthemajorityofthechangeinlengthoftheVC-LESisduet oslipping.Forthe casewheretheslopeofthetipisfree,theelasticaadjustsitscon gurationsuchthatthe tipslopebecomesequalto ˇ ( =0 )immediatelyaftersomeverticalloadingisapplied, independentoftheinitialorientation-seeFig.3.10.Thisdiscretecha ngeintheconguration istheresultofzeromoment,zerohorizontalforce,andequalan doppositeverticalforceat thetwoboundaryconditions. 0.00.3 0.2 0.115.0-0.20.0 -0.40.00.6 0.4 0.2(a)(b) 0.0x y q =0 q =5 q =10 q =15 q " + ˙ ˙ " Figure3.9:(a)DeformedshapesoftheC-shapedelasticawithxed tip-slope (1)=( ˇ ), =30 ,onafrictionlesssurfaceforfourderentvaluesofverticalload ing,(b)plotsof " ,˙ andtheirsumasafunctionofverticalloading q ,q 2 [0 ;15]forxedangle =30 .43 0.00.6 0.20.00.2 -0.2-0.20.0 -0.40.00.60.4 0.2x x y y q =0 q ˇ 0 q =15 q increasing Figure3.10:(a)UndeformedC-shapedelasticaonafrictionlesssur facewithinitialangle =30 (b)deformedshapesoftheelasticawithfreetipslopeunderdere ntvaluesof verticalloading q ,q 2f 0 :001 ;3 ;6 ;9 ;12 ;15 g startingfromtheinitialcongurationin(a); notethatallcongurationshavetheslopeof ˇ ,i .e., =0 ,althoughthetipslopeisfree. Remark1. Tosolvetheproblemwheretheelasticacanrollwhileslippi nginthepresence offriction,wecomputethemaximumnon-dimensionalfricti onforceas f s = q .Fora givenvalueof q ,wecomputethevalueof p assumingthatthereisnoslipping, i.e. ,usingthe procedureinsection3.3.1.If p f s ,thereisnoneedtochangethesolution.For p>f s , weallowtheelasticatorollandslipusingtheprocedureins ection3.3.2tillthecondition p = f s issatised. 3.4O-ShapedElasticaunderVerticalLoading 3.4.1SymmetricLoadingwithPointandLineContact TheO-shapedelasticaisshowninFig.3.3.Itisassumedthatthevertic alload2 q isapplied atthehighestpointontheelasticaintheabsenceofloading.Underlo ading,theelastica deformssymmetricallyabouttheverticalanditsucestoinvestiga teone-halfofitsdeformed shape.Inthegeneralcase,thedeformedelasticawillbecomprise dofaCC-LESoflength andaVC-LESoflength(1 )asshowninFig.3.3.ThelengthoftheCC-LESisequalto 44 zerowhentheexternalload2 q isequaltozero,becomesnonzerowhentheexternalloadis greaterthansomecriticalvalue q cr ,andincreasesastheexternalloadincreasesbeyondthe criticalvalue, i.e. ,q>q cr .Ifthevalueof isknown,theboundaryconditionsfortheVC-LESare: at s = :˚ =0 ;x = ;y =0(3.13) Toensure ˚ ( )=0,therelationbetween ˚ ( s )and ( s )inEq.(3.5a)hastobemodedto theform ˚ ( s )= ( s ) [s ]=ˆ = ( s ) ˇ ( s ) Sincetheslopeoftheelasticaat s =1is = ˇ ,theotherboundaryconditionsofthe VC-LESare: at s =1: ˚ = ˇ;x =0(3.14) TheVC-LESisdescribedbytworst-orderandonesecond-order ordinaryderentialequa- tions-seeEq.(3.5).Intheseequations,thehorizontalforce p isunknownasitisaninternal force-seeFig.3.3.TheveboundaryconditionsinEqs.(3.13)and(3.1 4)aresucientfor solvingthederentialequationsinEq.(3.5). Ifthevalueof isunknown,anadditionalboundaryconditionisneeded: at s = :( d˚=ds )= 1 =ˆ (3.15) Asimilarprocedurewasusedin[37]toobtainthedeformedshapesof circularelasticas.It shouldbementionedthat p =0whentheelasticahaspointcontactand p< 0whenthe elasticahaslinecontact. Theforce-displacementrelationshipoftheO-shapedelasticaunde rsymmetricloading, alongwiththevariationof ,isshowninFig.3.11;similarresultscanbefoundin[37,Figs.5 45 and6].Theelasticamaintainspointcontactwiththegroundfor q q cr =13 :75 4 and haslinecontactfor q>q cr .Thevalueof increaseswithincreaseinthevalueof q and reachesamaximumvalueof ˇ 0 :2when q =40.Theforcedisplacementrelationship isfoundtobealmostlinear;thisiscounter-intuitivesincetheelastica undergoesalarge deformation;fourdeformedcongurationsoftheelasticaaresh ownforfourderentvalues of q = f 10 ;20 ;30 ;40 g .Itshouldbenotedthatthenondimensionaldisplacementofthe elasticausesthelengthofthesemi-circleasthecharacteristicleng th-seeFig.3.3.Forthe samereason,thenondimensionalforceplottedinFig.3.11is q althoughthenetforceacting ontheO-shapedelasticais2 q .4 Itshouldbenotedthatthevalueof q cr isquitesensitivetothetoleranceusedtodistinguishaline contactfromapointcontact. 020 40 0.00.60.40.20.00.20.4 10 30 y q q cr =13 : 75 Figure3.11:Force-displacementrelationship,lengthoflinecontact ,andfourderent congurationsofO-shapedelasticaundersymmetricloading. 46 3.4.2DynamicsofSymmetricMass-ElasticaSystem Thealmost-linearforce-displacementrelationshipoftheO-shaped elasticaindicatesthatthe stnessoftheelasticaisalmostconstant.Thiscanbeveredthro ughsimulationsofthe dynamicsofamassattachedtotheelastica,withtheelasticaacting asaspring.Consider theverticalmotionofamass M undertheeectofgravity;themassisattachedtothetop oftheO-shapedelasticaandisreleasedfromrestattheinitialtime- seeFig.3.12.The non-dimensionalweightofthemassandthenon-dimensionaltimeva riablearerstdened asfollows: w =( M gL 2 =EI ) ;˝ = t p EI= M L 3 where g istheaccelerationduetogravity.Theequationofmotionofthemas sanditsinitial conditionsare: d 2 ~ y d˝ 2 =2 q w; ~ y (0)=2 =ˇ; d ~ y d˝ (0)=0(3.16) where~ y isthenon-dimensionaltemporalvariabledescribingthepositionoft hemass,and q istheverticalforceappliedbytheelasticatiponthemass.Thevalue of q isinitially M M g x y (0 ; 0) Figure3.12:Symmetricmass-elasticasystem. 47 Spatial DomainElastica DeformationTime Domain Mass Dynamics~ y q Eqs.(3.5),(3.13),(3.14),(3.15) Eqs.(3.16) Figure3.13:Methodofsolvingthedynamicsofthesymmetricmass-e lasticasystemin Fig.3.12. zerowhentheelasticaisundeformedandperfectlycircular.Byinte gratingEq.(3.16)with respecttotime,thepositionofthemass~ y ( ˝ )isobtained.TheTPBVPinEq.(3.5)isthen solvedbyusingtheboundaryconditionsinEq.(3.13),additionallyEq.(3 .15)ifthereisline contact,andthefollowingboundaryconditions 5 :at s =1: ˚ = ˇ;x =0 ;y =~ y ( ˝ )(3.17) ThesolutiontotheTPBVPprovidesthevalue q whichisusedtoupdateEq.(3.16).The procedureformarchingthrougheachtimestepofintegrationisillus tratedinFig.3.13.Sim- ulationswerecarriedoutforeightnon-dimensionalweightsinthera nge w 2 [5 ;40];for eachcase,themotionofthemasswasfoundtobesinusoidalinnatu re.Table3.1provides thevaluesofthefollowingvariablesthatdescribethemotionofthem ass:non-dimensional amplitude( A ),maximumvalueof ( max ),non-dimensionaltimeperiodofoscillation( T ), non-dimensionalfrequency( ! =2 ˇ=T ),andnon-dimensionaleectivestnessoftheelas- tica( k = ! 2 ).ThedatainTable3.1indicatesthatthedecreaseinthestnessof theelastica isrelativelysmallcomparedtothechangeintheweightofthemasswh en max =0;for max > 0,thestnessremainsapproximatelyconstant. ItshouldbeclearfromtheinitialconditionsinEq.(3.16)thatthesimula tionresults inTable3.1wereobtainedbyreleasingthemassfromtheheightwhere theelasticahas nodeformation.Therefore,thevaluesof T ,! and k reportedare\average"values,where 5 Notethatthevalueof isdeterminedusingtheprocedurediscussedinsection3.4.1. 48 Table3.1:Simulationresultsofthesymmetricmass-elasticasystemin Fig.3.12. w A max T ! k 5 0.0259 0 0.4607 13.638 186.00 10 0.0558 0 0.4865 12.915 166.80 15 0.0897 0 0.5111 12.293 151.13 20 0.1265 0.070 0.5261 11.943 142.63 25 0.1630 0.125 0.5273 11.916 141.99 30 0.1979 0.165 0.5243 11.984 143.62 35 0.2322 0.190 0.5219 12.039 144.94 40 0.2666 0.205 0.5225 12.025 144.61 averagingoccursoverthenetverticaldisplacementofthemass. Theslopeoftheforce- displacementcurveinFig.3.11canbeusedtocompute\local"valuesof thestnessofthe elastica.Thesevalues,whichcorrespondtoacertaininitialdeform ationoftheelasticaora certainvalueof q or w ,willbederentfromtheaveragevaluesinTable3.1.Forthesake ofcompleteness,itshouldbementionedthatthelocalstnessfor q =0or w =0canbe computedanalytically[80]. 3.4.3AsymmetricLoadingwithPointandLineContact Forasymmetricloading,itisassumedthattheelasticaisxedattheo riginandtheexternal loadisappliedatthepointontheundeformedelasticathatsubtends angle withthe verticalintheclockwisedirection-seeFig.3.14(a).Thedeformedela sticaiscomprisedof twoparts,namely,therightelasticaandtheleftelastica.Theright elasticaisthelength oftheelasticainwhichthepointofapplicationoftheloadcanbereach edfromtheorigin inacounter-clockwisedirection;inthegeneralcase,itwillbecompr isedofaCC-LESand aVC-LES,asshowninFig.3.14(b).Theleftelasticaisthelengthofthe elasticainwhich thepointofapplicationoftheloadcanbereachedfromtheorigininac lockwisedirection; thisentirelengthwillbeaVC-LESduetotheboundaryconditionsatt heorigin.Following theconventionusedintheprevioussectionsthatthenon-dimensio nallengthofthesemi- circularelasticaisequaltounity,thecoordinatesofthepointofap plicationoftheloadon 49 (a)(b) VC-LES VC-LES CC-LES x x y y q 6=0 (0 ; 0) q =0 Figure3.14:O-shapedelasticasubjectedtoasymmetricloading:(a )undeformedcongura- tionand(b)deformedconguration. theundeformedelasticaare( x;y )=[(1 =ˇ )sin ; (1 =ˇ )(1+cos )].Accordingly,thelengths oftherightandleftelasticaare(1 =ˇ )and(1+ =ˇ ),respectively.Todeterminethe equilibriumcoordinatesofthepointofapplicationoftheload q ,itisnecessarytosolvetwo TPBVPssimultaneouslyfortherightandleftelasticas;wewillusesub scripts r and ` to distinguishbetweenthesetwoelasticas.Therightelasticaiscompris edofaCC-LESwith s r 2 [0 ; ]andaVC-LESwith s r 2 [; 1 =ˇ ],where willhavetobedetermined-see Fig.3.15(a).Tostudytheleftelasticausingthesameconventionast hatusedfortheright elastica,theleftelasticaisrotatedaboutthe y axisby180degrees.Therotatedconguration oftheleftelastica,showninFig.3.15(b),iscomprisedofaVC-LESwith s ` 2 [0 ;1+ =ˇ ].Ifthevalueof isknown,theboundaryconditionsforVC-LESoftherightelasticaa re: at s r = :˚ r =0 ;x r = ;y r =0(3.18) Toensure ˚ r ( )=0,therelationbetween ˚ ( s )and ( s )inEq.(3.5a)hastobemodedto theform ˚ r ( s r )= r ( s r ) [s r ]=ˆ = r ( s r ) ˇ ( s r ) 50 (a)(b) s ` =0 s r =0 x x y y s r =(1 =ˇ ) s ` =(1+ =ˇ ) s r = x x y m ` q ` p ` m r q r p r Figure3.15:(a)ThedeformedelasticainFig.3.14(b)iscomprisedofth erightandleft elasticas(b)theleftelasticaisshowninitstransformedcongurat iontogetherwiththe rightelastica TheotherboundaryconditionsfortheVC-LESoftherightelastica are: at s r =(1 =ˇ ): x r = x ;y r = y ˚ r = ˇ [1 ( =ˇ ) ]= + ˇ (1 ) ( d˚ r =ds r )= m r (3.19) where x ,y , and m r areunknowns.FortheVC-LESofthetransformedleftelastica,t he boundaryconditionsattheoriginare: at s ` =0: ˚ ` =0 ;x ` =0 ;y ` =0(3.20) 51 TheotherboundaryconditionsfortheVC-LESofthetransforme dleftelasticaare: at s ` =(1+ =ˇ ): x ` = x ;y ` = y ˚ ` = ˇ [1+ =ˇ ]= ( + + ˇ ) ( d˚ ` =ds ` )= m ` (3.21) where m ` isanadditionalunknown. The14boundaryconditionsgiveninEqs.(3.18)-(3.21)arenotallinde pendent.Since x r (1 =ˇ )= x ` (1+ =ˇ )= x ,y r (1 =ˇ )= y ` (1+ =ˇ )= y ,˚ r (1 =ˇ )and ˚ ` (1+ =ˇ )dependonthesingleparameter ,and m r = m ` 6 ,Eqs.(3.18)-(3.21)represent 10independentboundaryconditions.BoththerightVC-LESandth etransformedleftVC- LESaredescribedbytworst-orderandonesecond-orderder entialequationinEq.(3.5) involvingtwounknowntipforces;however,since p r = p ` 6 and q r + q ` = q (where q isthe externalload),the10independentboundaryconditionsaresuc ientforsolvingthetwo TPBVPs.Thesolutionwillprovidethedeformedshapeoftheelastica ,thepositionand slope( x ,y , )andtheinternalforcesandmoments( p r ,p ` ,q r ,q ` ,m r ,m ` )atthepointof loading. TheTVBVPsdescribedbyEqs.(3.18)-(3.21)arecoupledandcannot besolveddirectly usingtheMATLABfunctionbvp4c[71].Startingfromaninitialguess( 1 , 2 , 3 )forthe positionandslope( x ,y , )ofthepointofloading,thecorrectvaluescanbefound iterativelybyusingtheMATLABfunctionfsolveoranygradientsear chalgorithm[79].The iterativeprocedureisterminatedwhen p r p ` ˇ 0 ;q r + q ` ˇ q;m r m ` ˇ 0(3.22) 6 Sincethebendingmoments(horizontalforces)atthetipoftherig htandleftVC-LESshouldbeequal andopposite,thebendingmoments(horizontalforces)atthetip oftherightandtransformedleftVC-LES areequal. 52 No, solve forYes Yes No Left VC-LESRight VC-LES Solve TPBVP in Solve TPBVP in gradient search Include =0 q =0 Eqs.(3.5),(3.18),(3.19) Eqs.(3.5),(3.20),(3.21) q = q + q Eq.(3.23) y r ( s ) > 0 8 s q = q des Terminate Figure3.16:FlowchartforsolvingthedeformedshapeoftheO-sha pedelasticaunderasym- metricloading Ifthevalueof isunknown,anadditionalboundaryconditionisneeded: at s r = :( d˚=ds )= 1 =ˆ (3.23) Foraspecedvalueof andagivenvalueof q = q des ,thedeformedshapeoftheelastica canbecomputedusingthealgorithmshowninFig.3.16.Intheowchar tofthealgorithm, thecondition y r ( s ) > 0 8 s isusedtocheckthatnoportionoftherightelasticaviolates theunilateralconstraintimposedbytheground.Thevalueof ,whichisthelengthofthe CC-LESfortherightelastica,isdeterminedbyincludingEq.(3.23)inth eTPBVPofthe rightelastica. Weprovidesimulationresultsforasymmetricloadingwithfourderen tnon-dimensional loadsat =30 :q = f 4 ;8 ;12 ;15 :5 g ;itisnotpossibletonumericallysolvethecoupled 53 0.0-0.20.20.4 0.00.20.40.6undeformed O-shaped elastica 0.00.20.10.005 0.000VC-LES CC-LES0.1050.14x y =30 q =4 q =8 q =12 q =15 : 5 Figure3.17:Deformedshapesoftheelasticaforasymmetricloading withfourderentloads at =30 :q = f 4 ;8 ;12 ;15 :5 g Table3.2:Internalforcesandmomentsatthepointofapplicationo ftheloadandlength ofcontactofelasticawiththegroundforasymmetricloadingwith q 2f 4 ;8 ;12 ;15 :5 g at =30 q p r = p l q r q l m r = m l 4 0 :214 3 :303 0 :697 0 :271 0 8 0 :650 6 :814 1 :185 0 :508 0 12 1 :837 10 :794 1 :206 0 :617 0 15 :5 3 :705 14 :548 0 :952 0 :527 0 :105 TPBVPsfor q> 15 :5,whichwillbeexplainedlater.Theundeformedelasticaandits deformedshapesundertheseloadsareshowninFig.3.17;theequilibr iumpositionsofthe pointofloadingareshownwiththehelpofthe\ "symbol.Forthefourderentloads, theinternalforcesandmomentsandthelengthofcontactofthe elasticawiththeground areshowninTable3.2.Severalobservationscanbemadefromthed ata:themajorityof theverticalloadissupportedbytherightelastica,thehorizontal internalforceincreasesas theloadincreases,andthelengthoftheelasticaincontactwiththe groundismuchsmaller 54 thanthatdepictedbythedeformedshapesinFig.3.17.Ofthefourlo adcasesconsidered, theelasticamakeslinecontactwiththegroundonlyfor q =15 :5.Acloserlookatthe neighborhoodofthepointwheretherightelasticatransitionsfrom aCC-LEStoaVC-LES, showninFig.3.17,indicatesthataportionoftheVC-LESisverycloset otheground: y 7 :58 10 5 for s r ˇ x 2 (0 :105 ;0 :14) andhasalmostzerocurvature.ThelengthofthisVC-LESsegment (withalmostzero curvature)increasesastheloadincreasesandthismakesitdcult tonumericallysolvethe problembeyondacertainvalueoftheload.Thenumericaldcultyco uldstemfromthe simplifyingassumptionsmadeinthepaper.Forexample,itwasassume dthatthelengthof theCC-LESoftherightelasticawillincreaseastheloadincreases;t hismaynotbeagood assumptionastherightelasticamaybuckle.Toaccuratelysimulatet hedeformationofthe elasticaunderhigherloads,itmaybenecessarytoobservethebeh avioroftheelasticain experimentsandmakesuitablechangestotheassumptions. 3.4.4DynamicsofAsymmetricMass-ElasticaSystem Thenumericaldcultyinsolvingthestaticequilibriumcongurationof theasymmetrically loadedelasticaforlargevaluesofloadcanbeattributedtothedcu ltyinmatchingthe threeforceandmomentboundaryconditionsoftherightandlefte lasticas-seeEq.(3.22). Thisproblemdoesnotexistwhenwesimulatethedynamicsofthemass -elasticasystem undertheeectofgravity-seeFig.3.18,sincetheunbalancedforc esandmomentsnow resultinaccelerationsofthemass.ThesysteminFig.3.18isassumedt ohavemass M ,radiusofgyration R ,andangleofasymmetry .Thenon-dimensionalequationsofmotion 55 M M g x y Figure3.18:Asymmetricmass-elasticasystem. ofthemassare d 2 ~ x d˝ 2 = p r p ` ; d 2 ~ y d˝ 2 = q r + q ` w " d 2 ~ d˝ 2 # = 1 r 2 ( m r m ` ) ;r , R L (3.24) where~ x ,~ y and ~ arethenon-dimensionaltemporalvariablesdescribingtheposition and orientationofthemass,and r isthenon-dimensionalradiusofgyration.Ifweassumethe masstohavethesameorientationasthatoftheelasticaatthepoin twhereitisattached anditisreleasedfromrest,theinitialconditionsare ~ x (0)=(1 =ˇ )sin ; d ~ x d˝ (0)=0 ~ y (0)=(1 =ˇ )(1+cos ) ; d ~ y d˝ (0)=0 ~ (0)= ˇ ; " d ~ d˝ # (0)=0 (3.25) ByintegratingEq.(3.25)withrespecttotime,thepositionandorient ationofthemass ~ x ( ˝ ),~ y ( ˝ ), ~ ( ˝ )areobtained.TheTPBVPinEq.(3.5)isthensolvedseparatelyforth eright 56 Spatial DomainElastica DeformationTime Domain Mass Dynamics~ x; ~ y; ~ p r ;p ` ;q r ;q ` ;m r ;m ` Eqs.(3.5),(3.18),(3.20),(3.23),(3.26),(3.27) Eqs.(3.24),(3.25) Figure3.19:Methodofsolvingthedynamicsoftheasymmetricmass- elasticasystemin Fig.3.18. andleftelasticasbyusingtheboundaryconditionsinEqs.(3.18)and( 3.20),additionally Eq.(3.23)ifthereislinecontact,andthefollowingconditions 7 :at s r =(1 =ˇ ): x r =~ x ( ˝ ) ;y r =~ y ( ˝ ) ;˚ r = ~ ( ˝ )+ ˇ (1 )(3.26) at s ` =(1+ =ˇ ): x ` = ~ x ( ˝ ) ;y ` =~ y ( ˝ ) ;˚ ` = ( ~ ( ˝ )+ + ˇ )(3.27) ThesolutiontothetwoTPBVPsprovidethevalues p r ,p ` ,q r ,q ` ,m r ,m ` whichareusedto updateEq.(3.24).Figure3.19illustratestheprocedureformarchin gthrougheachtimestep ofintegration. SimulationresultsarepresentedinFig.3.20forthefollowingsetofpar ameters: =46 ;w =16 ;r =0 :03 Thissetofparameterswaschosenbytrialanderrorsuchthatth eCartesiantrajectoryof themassis\repeatable"andiseasytovisualize.Here,repeatableim pliesthatthemass movesbackandforthbetweenpoints A and B alongthetrajectoryjoiningthesepoints showninFig.3.20.Themass-elasticasystemhasthreedegrees-of- freedomandrepeatable 7 Notethatthevalueof hastobedeterminedateachtimestepusingtheprocedurediscuss edinsection 3.4.3. 57 0.00.30.6 -0.30.00.30.6x y A B Figure3.20:Derentcongurationsoftheasymmetricmass-elast icasystemandtrajectory ofthemassinCartesiancoordinates. Cartesiantrajectoriesaregeneratedwhenthenaturalfreque nciesofthesystemassociated withtheCartesiandegrees-of-freedomareintegermultiples 8 ofoneanother.Figure3.20 showsvederentcongurationsofthemass-elasticasystem(o rientationofthemassis notshown)atvederentpointsonthetrajectory.Clearly,the elasticaundergoesamuch largerdeformationthanthatseeninFig.3.17forthestaticloadof q =15 :5.Themaximum linecontactinFig.3.17is0 :105whereasthemaximumlinecontactinFig.3.20exceeds0 :4.It shouldbementionedthatnumericaldcultyinsolvingthestaticspro blemwasencountered for q> 15 :5with =30 ;for =46 ,numericaldcultyisencounteredforalowervalue of q ,q> 14 :5.However,thedynamicsproblemwassolvedwithoutdcultyfor w =16with =46 .8 Whenthenaturalfrequenciesarenotintegermultiplesofoneanot her,thetrajectoryofthemasstraces apaththatdoesnotrepeatitselfandisdenseinaboundedregionof theCartesianspace. 58 Chapter4 AVariable-StructureMass-ElasticaHopper 4.1Overview Inthischapter,westudyahoppingrobotthatisavariable-struct uremass-elasticasystem. Therobotchangesthestnessoftheelasticabychangingitsboun daryconditions.The elasticabehaveslikeaspringandhencethemass-elasticasystemst udiedhereisquitesimilar totheSLIPmodel[44].Therationaleforstudyingthisproblemistwof old.Itisnon-trivial tochangethestnessofatraditionalhelicalcoiledspringbutchan gingthestnessofthe elasticabychangingitsboundaryconditionseemsfeasible.Second, themassoftheelastica canbequitesmallandenergylossduetoimpactcanbenegligible;ther efore,theSLIPmodel isagoodapproximationofthemass-elasticasystem.Thischapteris organizedasfollows. Section4.2introducestheconceptofavariable-structurespring formodulatingtheenergyof amass-springsystem.Section4.3describesthedynamicsofaxed -structuremass-elastica hopper.Section4.4describesthedynamicsofavariable-structur emass-elasticahopperwith energyregulationforhoppingheightcontrol. 4.2HoppingUsingaVariable-StructureSpring -TheConcept Avariable-structurespringcanswitchitsstnessfromonevaluet oanother.Alsoreferred toasvariablestnesssprings,theyhavebeenusedinvibrationcon trol[81{83]androbot designs[84{86].Variablestructurespringscanbeusedtochanget heenergyofasystemand isusedheretodesignahoppingmass-springsystem-seeFig.4.1.The mass M isconnected tothetopofthespringandtheotherendofthespringisconnecte dtoamasslessbase.We 59 (a) (b) H H K K K K K K H V V V V g g M M M M M M MM M M M Figure4.1:Amass-springhopperwith(a)aspringofconstantstn ess K ,and(b)aspring ofvariablestness,switchingbetween K and K , K>K .In(a),thevelocitiesofthemass attouchdownandtakeoarethesame,equalto V .In(b),thevelocityofthemassattakeo isgreaterthanthevelocityattouchdown, V>V .Aftertakeo,thestnessisswitched backto K .considertwocases:(a)thestnessofthespringisconstant,eq ualto K -seeFig.4.1(a), and(b)thestnessisswitchedbetweenthevalues K and K , K>K -seeFig.4.1(b).In bothcases,themassisreleasedfromrestfromaheight H abovetheground.Assumingno lossofenergyduetoimpact,themassreturnstothesameheighta ftereachhopintherst case.Inthesecondcase,thestnessofthespringisswitchedto itshighervalue K when thespringisinitsstateofmaximumcompression;thisaddsenergyto thesystemandasa resultthemassreachesaheight H , H>H .Althoughtheideaofchangingtheenergyofthesystembychanging thestnessissimple, practicalimplementationisnottrivial.Inthenexttwosectionsweinv estigatethedynamics ofamass-springsystemwherethespringisanelastica;thestnes softheelasticaischanged bychangingitsboundaryconditions. 60 H V V g M M M M M Figure4.2:Axed-structuremass-elasticasystemwithshouldera ngle hoppingonthe ground. 4.3Fixed-StructureMass-ElasticaHopper 4.3.1SystemDescription Considerthemass-elasticasysteminFig.4.2,wheretheelasticaisuse dinplaceofthespring inFig.4.1(a).Theelasticaisathinexiblestructuralelement,bothen dsofwhichare connectedtothemassandsubtendangle withthehorizontal.Thisanglewillbereferred toasthe\shoulderangle".LikethespringinFig.4.1(a),theelasticais assumedtobe massless.Thehoppingmotionofthemass-elasticasystemiscompris edoftwophases:the ightphaseandthecontactphase.Intheightphase,theelastic aisinitsnominalshape andprovidesnoforceonthemassintheverticaldirection.Inthec ontactphase,theelastica isdeformedandappliesaforceonthemassintheverticaldirection. 4.3.2ForceandElasticaDeformationRelationship Assumingthattheelasticaremainssymmetricaboutthevertical,we constructfreebody diagramsofthemassandone-halfoftheelasticaasshowninFig.4.3.T hetwoendsof theelasticaapplyequalandoppositeforces P inthehorizontaldirection.Theyalsoapply verticalforces Q thatactinthesamedirection.Theforce-displacementrelationship ofthe elasticacanbestudiedbysolvingtheboundaryvalueproblemforone -halfoftheelasticaas 61 PSfrag M M PP P P P P Q Q Q Q Q Mg 2 A X Y Figure4.3:Free-bodydiagramsofthemassandelastica. showninFig.4.3.Theboundaryproblemcanbeposedasfollows[25,70]: EI ( d=dS )= M + QX PY (4.1a) ( dX=dS )=cos (4.1b) ( dY=dS )=sin (4.1c) where EI istheexuralrigidityoftheelastica, X and Y areCartesiancoordinatesofan arbitrarypointontheelasticahalf, istheslopeatthatpoint, S isthearclengthofthe segmentbetweenthepointandthexedend,and M isthebendingmomentatthexed end.SixboundaryconditionsareneededtosolveEq.(4.1)astherea rethreeunknowns P ,Q ,and M .Theboundaryconditionsare: at S =0: =0 ;X =0 ;Y =0 at S = L : = ;X = A;Y = B (4.2) where L isthelengthoftheelasticahalf, A ishalfofthewidthofthemassand B isthe Y coordinateofthetipoftheelasticahalfwhen Q =0.Itshouldbenotedthat B dependson thevalueof .Tonon-dimensionalizeEq.(4.1),thefollowingnon-dimensionalpositio nand 62 forcevariablesareintroduced: y = Y=L;x = X=L;s = S=L p = P ( L 2 =EI ) ;q = Q ( L 2 =EI ) ;m= M ( L=EI ) Thisresultsinthefollowingnon-dimensionalequations: ( d=ds )= m+ qx py (4.3a) ( dx=ds )=cos (4.3b) ( dy=ds )=sin (4.3c) andnon-dimensionalboundaryconditions: at s =0: =0 ;x =0 ;y =0 at s =1: = ;x = a;y = b (4.4) where a = A=L ,b = B=L and = =L .Together,Eqs.(4.3)-(4.4)deneatwo-point boundaryvalueproblem(TPBVP)whichcanbeusedtond p ,q ,and mwhenthetipvertical displacement isgiven.TheTPBVPissolvedforderentvaluesof usingtheMATLAB function bvp 4 c ;theensuingforce-displacement( q - )relationshipisshowninFig.4.4. 0.000.180.0014.0 q =45deg =15deg =30deg Figure4.4:Force-displacementcurvesfortheelasticahalfinFig.4.3f orshoulderangles =15 ;30 ;45degwith a =1 = 2 ˇ .63 4.3.3HoppingDynamics:FlightandContactPhases Wedescribethetemporallocationofthemassusingthevariable ~ Y .Intheightphase,the masshasthefollowingdynamicsduringfreefall: ~ Y = g; ~ Y (0)= H; _~ Y (0)=0(4.5) where H istheheightfromwherethemassisreleased.Usingthenon-dimensio nalvariables ~ y = ~ Y=L;˝ = t p EI=ML 3 ;w = Mg ( L 2 =EI )(4.6) thenon-dimensionalfreefalldynamicsisdescribedbytheequation andinitialconditions: d 2 ~ y=d˝ 2 = w; ~ y (0)= h;d ~ y=d˝ (0)=0 (4.7) where h = H=L .Atsometime ˝ = ˝ cs ,theelasticamakescontactwiththeground.Inthe contactphase,thedynamicsofthemassisdescribedbythefollowin gequationandtemporal conditions: d 2 ~ y=d˝ 2 =2 q w; ~ y ( ˝ cs )= bd ~ y=d˝ ( ˝ cs )= v (4.8) where v isthenon-dimensionalvelocityofthemassatthetimewhentheelast icarstmakes contactwiththeground;itisgivenbytherelation v = p 2 w ( h b )(4.9) Atsomenon-dimensionaltime ˝ = ˝ cf ,˝ cf >˝ cs ,theelasticawillbreakcontactwiththe groundandthemasswillbebackinitsightphase.Sinceitisassumedt hatnoenergy islost,thedynamicsofthemasswillbedescribedbythefollowingequa tionandtemporal 64 conditions d 2 ~ y=d˝ 2 = w; ~ y ( ˝ cf )= bd ~ y=d˝ ( ˝ cf )=+ v (4.10) Thehoppingmotionofthexed-structuremass-elasticasystemc anbedescribedusing Eqs.(4.8)and(4.10);eachhopwillgenerateapairoftouchdownand liftotimes ˝ cs and ˝ cf .4.3.4Simulation Weconsideradimensionalexampleofthexed-structuremass-ela sticasystem.Theelastica materialwasassumedtobespringsteel,whichhasYoung'sModuluso f E =207 10 9 Pa.Thecross-sectionoftheelasticawasassumedtoberectangu larwithathicknessof 0 :70mmandwidthof4 :0cm;thisresultsintheareamomentofinertia I =1 :14 10 12 m 4 .Thetotallengthoftheelasticawasassumedtobe2 L =50cm.Thegeometricand dynamicparametersofthemasswereassumedtobe A = L= 2 ˇ =3 :98cmand m =0 :7 kg.Theshoulderangleisassumedtobe =15deg.Themassisreleasedfromtheheight H =0 :6mat t =0 :0s-seeFig.4.5(a).Thedynamicsofthexed-structuremass-elas tica isnumericallyintegratedforwardintime.Theelasticamakescontact withthegroundwhen 0.0(a)(b)(c) 0.20.40.6M M M t =0 :000s t =0 :324s t =0 :297s Figure4.5:Congurationsofthexed-structuremass-elasticas ystemat(a)releasetime, (b)timeoftouchdown t cs ,and(c)timeatwhichtheelasticahasthemaximumdeformation. 65 themassisataheightof H =0 :168mat t cs =0 :297s-seeFig.4.5(b).Aftercontact, themasscontinuestomoveverticallydownwardanddeformtheelas tica.Atthepointof maximumdeformation,whichoccursat t =0 :324s,showninFig.4.5(c),themasscomes torestmomentarilyat H =0 :117mandsubsequentlystartsmovingupward.Theelastica breakscontactwiththegroundat t cf =0 :353s,notshowninFig.4.5.Thevariables t cs and t cf arerelatedtothenon-dimensionaltimevariables ˝ cs and ˝ cf bytherelationinEq.(4.6). 4.4Variable-StructureMass-ElasticaHopper 4.4.1SystemDescription Weconsideravariable-structuremass-elasticasystemwherethe shoulderanglecanbe changedinstantaneously.Twocongurationsofthesystemwiths houlderanglesangles and areshowninFigs.4.6(a)and(c).Inbothcongurations,themassis inequilib- rium;itsweightissupportedbytheverticalforcesgeneratedbyt heelastica.Iftheangle ischangedtoangle pseudo-statically,theequilibriumheightofthemassisraisedand theworkdoneisequaltogaininthepotentialenergyofthemass.I fthechangeofan- gleoccursinstantaneously,themass-elasticasystemimmediatelya ssumestheconguration showninFig.4.6(b),wherethenetforceonthemassactsintheupwa rddirection.Thiswill resultinthemassmovingupwardsandhaveanupwardvelocitywhenit passesthroughthe equilibriumcongurationinFig.4.6(c).Thisimpliesthataninstantaneou schangeinthe (a)(b) (c) M M M g Figure4.6:Avariable-structuremass-elasticasystemisshowninits equilibriumcongura- tionwithshoulderangles(a) ,and(c) ,(b)non-equilibriumcongurationofthesystem immediatelyaftertheshoulderanglehasbeenchangedfrom to .66 shoulderanglecanaddmoreenergytothesystemthanwhenitischa ngedpseudo-statically. Bychoosing < ,theenergyofthesystemcanbereducedandthereforemodulat ingthe shoulderanglecanbeusedtochangetheenergytosomedesiredva lue.Thisideaisusedto designahoppingsysteminthenextsection. 4.4.2EnergyRegulationforHoppingHeightControl Weinvestigatethevariationofkinetic,potential,andstrainenergie softhemass-elastica systeminitsightandcontactphaseswiththeobjectiveofcontro llingthehoppingheight H bychangingtheshoulderangle .Werstdenethenon-dimensionalkinetic,potential, strain,andtotalenergies t,v,sand eintermsofthedimensionalkinetic,potential,strain andtotalenergies T,V,Sand Easfollows: t= T( L=EI ) ;v= V( L=EI ) ;s= S( L=EI ) ;e= E( L=EI )(4.11) Wedescribeeachhoptobecomprisedoffourphasesandtopassth roughvederent congurations.Thefourphasesandvecongurationsarediscu ssednextandshownin Figs.4.7and4.8. 1.Free-Fall-Inthisphase,themassfallsfreelyundergravity. Atthebeginningofthefree-fallphase,themassisinconguration 1 andhasthe Free FallDeformationRestitutionLaunch H 1 1 1 2 3 4 5 H 1 V V g M M M M M M t ˇ 0 Figure4.7:Derentintermediatecongurationsofavariable-stru cturemass-elasticasystem duringthe j -thhop.Theshoulderangleswitchesbetween and .67 1 1 2 3 4 5 q q h 1 h 2 h 3 , h 4 h 5 h 1 q y Figure4.8:Avariable-structuremass-elasticasystemhoppingont heground;theshoulder angleswitchesbetween and , > ,togainheight. height H 1 m,ornon-dimensionalheight h 1 .Thevelocityofthemassiszero.Theset ofnon-dimensionalenergiesis: [t1 ;v1 ;s1 ;e1 ]=[0 ;wh 1 ;0 ;wh 1 ](4.12) 2.Deformation-Inthisphase,theelasticaisincontactwiththegro undandundergoes compressionasthemassmovesverticallydownward. Atthebeginningofthedeformationphase,themassisincongurat ion 2 andhasthe height H 2 m,ornon-dimensionalheight h 2 ,andadownwardvelocity V 2 m/sornon- dimensionalvelocity v 2 .Thenon-dimensionalvelocity v isrelatedtothedimensional velocitybytherelation v = V p ML=EI 68 Thesetofnon-dimensionalenergiesis: [t2 ;v2 ;s2 ;e2 ]=[ v 2 2 = 2 ;wh 2 ;0 ;wh 1 ](4.13) Attheendofthedeformationphase,themassisinconguration 3 andhasthe height H 3 m,ornon-dimensionalheight h 3 ,andzerovelocity.Theshoulderangleof themass-elasticais andthenon-dimensionalenergiesare: [t3 ;v3 ;s3 ;e3 ]=[0 ;wh 3 ;2 Z h 2 h 3 q ( y ) dy;wh 1 ](4.14) where q istheforceappliedbythemassontheelastica(positivedownward)a sa functionofthe y positionofthemasswithshoulderangle .Thefunction q is plottedinFig.4.8. 3.Restitution-Inthisphase,theelasticaisincontactwiththegrou ndandgradually recoversitsundeformedshapeasthemassmovesverticallyupwar d. Atthebeginningoftherestitutionphase,themassisinconguratio n 4 andhasthe height H 4 = H 3 m,ornon-dimensionalheight h 4 = h 3 ,andzerovelocity.Theshoulder angleofthemass-elasticais ;thechangeofanglefrom to isassumedtooccur instantaneously.Inconguration 4 ,thesetofnon-dimensionalenergiesis: [t4 ;v4 ;s4 ;e4 ]=[0 ;wh 4 ;2 Z h 5 h 4 q ( y ) dy;wh 1 + e ](4.15) where h 5 isthenon-dimensionalheightoftheundeformedelasticawithshould erangle ,H 5 mistheassociateddimensionalheight, q isthecurveplottedinFig.4.8,and e istheareaoftheshadedregioninFig.4.8. 4.Launch-Inthisphase,theelasticaisnotincontactwiththegrou ndandthemass gainsheight. 69 Themassisinconguration 5 atthestartofthelaunchphaseandinconguration 1 attheendofthephase.Inconguration 5 ,themasshastheheight h 5 andan upwardvelocity V 5 m/sornon-dimensionalvelocity v 5 .Theshoulderangleofthe mass-elasticaisis andthenon-dimensionalenergiesare: [t5 ;v5 ;s5 ;e5 ]=[ v 2 5 = 2 ;wh 5 ;0 ;wh 1 + e ](4.16) Fromconguration 5 ,themassstartsmovingupwards.Whenitreachesthemaximum potentialenergyconguration,itisbackinconguration 1 forthenexthop.Theassociated heightis h 1 >h 1 ,orequivalently H 1 >H 1 ,if > ,andviceversa.Note,thattheshoulder anglehastoberesetto beforethemassreachesconguration 1 ;thisswitchingdoesnot changetheenergyofthesystem. 4.4.3Simulation 0.00.20.40.60.00.40.81.21.62.0 y (m) t (s) =15 H 1 =0 :24m =25 H 2 =0 :15m =15 H 1 =0 :33m =25 H 2 =0 :14m =15 H 1 =0 :45m =25 H 2 =0 :13m =15 H 1 =0 :59m =16 H 2 =0 :12m =15 H 1 =0 :60m Figure4.9:Thevariable-structuremass-elasticasystemhoppingt oaheightof H =0 :60m fromaninitialheightof H =0 :24m. 70 Weconsideradimensionalexampleusingtheparametersusedinsect ion4.3.4.The shoulderangleisassumedtobe =15deg.Themassisreleasedfromtheheight H =0 :24 mat t =0 :0sandthetargetheightis H =0 :60m.Tondthevalueof thatprovides theadditionalenergy e toreach H ,asetof q curvesarerstgenerated.The q curves areapproximatedusingpolynomialfunctionsandintegratedtoobt ain e .For H =0 :60m, isfoundtobe45deg.Forarealisticsimulation,wechoosetorestrict j j 10deg sinceabruptlychanging by30degisnotpractical.Thedynamicsofthevariable-structure mass-elasticaisnumericallyintegratedforwardintime.Theresultsin Fig.4.9showthatthe hopperreaches H =0 :6monthefourthhop.Theshoulderangleandtheheightofthe hopperareshownincongurations 1 and 4 foreachhop. 71 Chapter5 DynamicsandSimulationsofaRollingElastica 5.1Overview Inthischapter,wemodelandsimulatethedynamicbehaviorofarollin gelastica.The equationsofmotionoftheelasticasystemarederivedusingHamilton 'sprinciple.Thearc- lengthconstraintoftheelasticaintroducesnonlinearityintheequa tionsofmotionandthe unilateralgroundconstraintleadstoanontrivialcontactproblem ;theseconstraintsmake thesystemnontrivialtosolve.Weobtaintheweakformoftheequa tionsofmotionandthe associatedconstraintsbyusingtheweightedresidualmethodand solvethesystemdynamics usingtheniteelementmethod.Thesteadystatesolutionandthee ectsofthesystem parametersonthedynamicbehavioroftherollingelasticaareinvest igated.Thischapteris organizedasfollows:Theequationsofmotionarederivedinsection5 .2andthediscretization oftheequationsofmotionarepresentedinsection5.3.Thesteady statesolutionoftherolling elasticaisinvestigatedinsection5.4.Resultsofsimulationsareprese ntedinsection5.5. 5.2SystemDynamics 5.2.1KinematicsoftheRollingElastica X s ~r y x g ( x ( s;t ) ;y ( s;t )) O cFigure5.1:Deformedcongurationofanelasticarollingonthegroun d. 72 ConsideranelasticashowninFig.5.1,assumedtohaveamassperunitle ngthof ˆ and alengthof L .Theelasticaisassumedtobeinitiallyrollingandmaintainingpointcontac t withafrictionlessground.Itisalsoassumedthatthereisnoslipatth epointofcontact. X isthehorizontaldistancethatthepointcontact coftheelasticahastraveledfromthe origin O ;and x;y arethelocalCartesiancoordinatesontheelastica.Anarbitraryp ointon theelasticaislocatedatanarc-lengthof s ,s 2 [0 ;L ],fromthecontactpointwhere s =0. Thevector ~r extendingfromtheorigintoanarbitrarypointontheelasticaisexpr essedas: ~r = X ( t ) ^ i + x ( s;t ) ^ i + y ( s;t ) ^ j (5.1) anditsrsttotalderivative 1 canbewrittenas: _~r = dX dt ^ i + Dx Dt ^ i + Dy Dt ^ j (5.2) TofurtherinvestigatethevelocityvectordenedinEq.(5.2),weco nsidertheelasticatohave bothtranslationalandrotationalmotionasshowninFig.5.2.Eachpo intontheelastica experiencesatranslationalvelocityin X andamaterialvelocitythatistangenttothelocal curvature;bothwithamagnitudeof_ s ,where_ s =( ds=dt )isthechangeofarc-lengthwith 1 Thetotal(material)derivative D ( ) =Dt isintheLagrangianperspectiveandthederivative d ( ) =dt is intheEulerianperspective[67]. _ s _ s _ s _ s _ s _ s _ s _ s Figure5.2:Velocitydiagramforelasticawithpointcontact 73 time.Thetranslationalvelocityinthehorizontaldirectioncanbede scribedintheEulerian frameofreferencewhilethematerialvelocitycanbedescribedinth eLagrangianframeof reference.Toensureno-slip,thetranslationalvelocityvectorh astobeexactlyoppositeof thematerialvelocityvectoratthepointofcontact, s =0,asshowninFig.5.2.Nowwetake therstandsecondderivativesofthepositionvectorofanarbitr arypoint ~r withrespectto time, _~r = _s ^ i + @ @t +_ s @ @s x ^ i + y ^ j _~r = _s ^ i + @x @s _s ^ i + @x @t ^ i + @y @s _s ^ j + @y @t ^ j (5.3) ~r = s ^ i + @ @t +_ s @ @s @x @s _s ^ i + @x @t ^ i + @y @s _s ^ j + @y @t ^ j = s ^ i + @ 2 x @s@t _s + @x @s s + @ 2 x @s 2 _s 2 ^ i + @ 2 x @t 2 +_ s @ 2 x @t@s ^ i + @ 2 y @s@t _s + @y @s s + @ 2 y @s 2 _s 2 ^ j + @ 2 y @t 2 +_ s @ 2 y @t@s ^ j ~r = s + @x @s s + @ 2 x @s 2 _s 2 +2 @ 2 x @s@t _s + @ 2 x @t 2 ^ i + @y @s s + @ 2 y @s 2 _s 2 +2 @ 2 y @s@t _s + @ 2 y @t 2 ^ j (5.4) ThevelocityandaccelerationvectorsfromEqs.(5.3)and(5.4)willbe utilizedinthederiva- tionsoftheequationsofmotioninthefollowingsections. 5.2.2EquationsofMotionUsingHamilton'sPrinciple Werstconsiderthekineticenergy,denotedby T ,oftherollingelastica: T = 1 2 ˆ Z L 0 ( _~r _~r ) ds = 1 2 ˆ Z L 0 2 6 6 6 6 4 1 @x @s 2 _s 2 | {z } 1 +2 @x @s @x @t _s | {z } 2 2 @x @t _s | {z } 3 + @x @t 2 | {z } 4 + @y @s 2 _s 2 | {z } 5 +2 @y @s @y @t _s | {z } 6 + @y @t 2 | {z } 7 3 7 7 7 7 5 ds (5.5) 74 Eachofthetermsaredenotedbyanumbertofacilitateintegration later.Ontheother hand,thepotentialenergy V is: V = V 1 + V 2 where V 1 isthegravitationalenergyand V 2 isthebendingstrainenergy.Thegravitational energyis: V 1 = ˆg Z L 0 y |{z} 8 ds (5.6) where g isthegravitationalaccelerationconstant.Thegeneralbendings trainenergy V 2 can bewrittenas: V 2 = 1 2 E Z L 0 I ( s ) " @ 2 x @s 2 2 + @ 2 y @s 2 2 # ds (5.7) where I ( s )istheareamomentofinertia.Ifweassume I ( s )isconstant,thebendingstrain energybecomes: V 2 = 1 2 EI Z L 0 " @ 2 x @s 2 2 + @ 2 y @s 2 2 # | {z } 9 ds (5.8) where EI istheexuralrigidityoftheelastica.Sincetheelasticaisinextensible ,weimpose anarc-lengthconstraintasshownbelow: @x @s 2 + @y @s 2 =1 Toimposethisconstraint,weuseamultiplier ˙ ( s )anddenethezeroworkdonebythe constraintforceas C 1 asshowninthefollowing[87]: C 1 = 1 2 Z L 0 ˙ " @x @s 2 + @y @s 2 1 # | {z } 10 ds (5.9) ˙ actsasaconstraintforcetoensurethearc-lengthconstraintis notviolated;therefore, ˙ itselfhasnodependencyontime.Tondtheequationsofmotion,we applyextended 75 Hamilton'sprincipleandtakethevariationoftheLagrangian L [69]: L = Z t 2 t 1 ( T V 1 V 2 C 1 ) dt =0(5.10) Wenowperformintegrationbypartsfor 1 7 inEq.(5.5).For 1 ,wehave: n Z t 2 t 1 Z L 0 _s 2 1 @x @s 2 dsdt o = Z t 2 t 1 Z L 0 2_ s 2 1 @x @s @x @s dsdt u = 2_ s 2 1 @x @s ;dv = @ @s ( x ) ds du =2_ s 2 @ 2 x @s 2 ds;v = x Then 1 becomes: Z t 2 t 1 2_ s 2 1 @x @s x L 0 Z L 0 2_ s 2 @ 2 x @s 2 xds dt where x (0)= x ( L )=0.For 2 ,wehave: n Z t 2 t 1 Z L 0 2 @x @s @x @t _sdsdt o = Z t 2 t 1 Z L 0 h 2 @x @s @x @t _s | {z } 2-1 +2 @x @s @x @t _s | {z } 2-2 i dsdt where 2-1 isrstbeingintegrated: u =2_ s @x @t ;dv = @ @s ( x ) ds du =2_ s @ 2 x @s@t ds;v = x andweget: Z t 2 t 1 2_ s @x @t x L 0 Z L 0 2_ s @ 2 x @s@t xds dt 76 where x (0)= x ( L )=0.Then,weintegrate 2-2 u =2_ s @x @s ;dv = @ @t ( x ) dt du = 2_ s @ 2 x @s@t +2 s @x @s dt;v = x whichyields: Z L 0 2_ s @x @x x t 2 t 1 Z t 2 t 1 2_ s @ 2 x @s@t +2 s @x @s xdt ds where x ( t 1 )= x ( t 2 )=0.For 3 ,wehave: n Z t 2 t 1 Z L 0 2 @x @t _s dsdt o = Z t 2 t 1 Z L 0 2_ s @x @t dsdt u = 2_ s;dv = @ @t ( x ) dt du = 2 sdt;v = x Then, 3 becomes: Z L 0 2_ sx t 2 t 1 Z t 2 t 1 ( 2 s ) xdt ds where x ( t 1 )= x ( t 2 )=0.For 4 ,weget: n Z t 2 t 1 Z L 0 @x @t 2 dsdt o = Z t 2 t 1 Z L 0 2 @x @t @x @t dsdt u =2 @x @t ;dv = @ @t ( x ) dt du =2 @ 2 x @t 2 dt;v = x 77 Then 4 becomes: Z L 0 2 @x @t x t 2 t 1 Z t 2 t 1 2 @ 2 x @t 2 xdt ds where x ( t 1 )= x ( t 2 )=0.For 5 ,wehave: n Z t 2 t 1 Z L 0 _s 2 @y @s 2 dsdt o = Z t 2 t 1 Z L 0 2_ s 2 @y @s @x @s dsdt u =2_ s 2 @y @s ;dv = @ @s ( y ) ds du =2_ s 2 @ 2 y @s 2 ds;v = y Afterintegratingbyparts, 5 becomes: Z t 2 t 1 2_ s 2 @y @s y L 0 Z L 0 2_ s 2 @ 2 y @s 2 yds dt where y (0)= y ( L )=0.For 6 , n Z t 2 t 1 Z L 0 2 @y @s @y @t _sdsdt o = Z t 2 t 1 Z L 0 h 2 @y @s @y @t _s | {z } 6-1 +2 @y @s @y @t _s | {z } 6-2 i dsdt where 6-1 isrstbeingintegrated: u =2_ s @y @t ;dv = @ @s ( y ) ds du =2_ s @ 2 y @s@t ds;v = x andweget: Z t 2 t 1 2_ s @y @t y L 0 Z L 0 2_ s @ 2 y @s@t yds dt 78 where y (0)= y ( L )=0.Thenweintegrate 6-2 :u =2_ s @y @s ;dv = @ @t ( y ) dt du = 2_ s @ 2 y @s@t +2 s @y @s dt;v = y whichyields: Z L 0 2_ s @y @s y t 2 t 1 Z t 2 t 1 2_ s @ 2 y @s@t +2 s @y @s ydt ds where y ( t 1 )= y ( t 2 )=0.For 7 , n Z t 2 t 1 Z L 0 @y @t 2 dsdt o = Z t 2 t 1 Z L 0 2 @y @t @y @t dsdt u =2 @y @t ;dv = @ @t ( y ) dt du =2 @ 2 y @t 2 dt;v = y Then 7 becomes Z L 0 2 @y @t y t 2 t 1 Z t 2 t 1 2 @ 2 y @t 2 ydt ds where y ( t 1 )= y ( t 2 )=0.Nowweintegrate 8 inEq.(5.6) n ˆg Z t 2 t 1 Z L 0 ydsdt o and 8 becomes: ˆg Z t 2 t 1 Z L 0 ydsdt 79 Nextweintegrate 9 inEq.(5.8): n 1 2 EI Z t 2 t 1 Z L 0 @ 2 x @s 2 2 + @ 2 y @s 2 2 dsdt o = 1 2 EI Z t 2 t 1 Z L 0 2 @ 2 x @s 2 @ 2 x @s 2 | {z } 9-1 +2 @ 2 y @s 2 @ 2 y @s 2 | {z } 9-2 dsdt where 9-1 isrstbeingintegrated: u =2 @ 2 x @s 2 ;dv = @ 2 @s 2 ( x ) ds du =2 @ 3 x @s 3 ds;v = @x @s andthenweintegratebypartsagain: Z t 2 t 1 2 @ 2 x @s 2 @x @s L 0 Z L 0 2 @ 3 x @s 3 @x @s ds dt u =2 @ 3 x @s 3 ;dv = @ @s ( x ) ds du =2 @ 4 x @s 4 ds;v = x Then 9-1 becomes: Z t 2 t 1 2 @ 2 x @s 2 @x @s L 0 2 @ 3 x @s 3 x L 0 + Z L 0 2 @ 4 x @s 4 xds dt where x (0)= x ( L )=0and ( @x=@s )(0)= ( @x=@s )( L )=0.Similarto 9-1 ,9-2 becomes: Z t 2 t 1 2 @ 2 y @s 2 @y @s L 0 2 @ 3 y @s 3 y L 0 + Z L 0 2 @ 4 y @s 4 yds dt 80 where y (0)= y ( L )=0and ( @y=@s )(0)= ( @y=@s )( L )=0.Finally,weintegrate 10 inEq.(5.9): n 1 2 Z t 2 t 1 Z L 0 ˙ " @x @s 2 + @y @s 2 1 # dsdt o = Z t 2 t 1 Z L 0 ˙ @x @s @x @s | {z } 10-1 + ˙ @y @s @y @s | {z } 10-2 dsdt wherewerstintegrate 10-1 :u = ˙ @x @s ;dv = @ @s ( x ) ds du = @ @s ˙ @x @s ds;v = x andweget: Z t 2 t 1 ˙ @x @s x L 0 Z L 0 @ @s ˙ @x @s xds dt where x (0)= x ( L )=0.Similarly, 10-2 becomes: Z t 2 t 1 ˙ @y @s y L 0 Z L 0 @ @s ˙ @y @s yds dt where y (0)= y ( L )=0.AftercollectingallthetermsinEq.(5.10),weget Z t 2 t 1 Z L 0 n 1 2 ˆ 2_ s 2 @ 2 x @s 2 2_ s @ 2 x @s@t 2_ s @ 2 x @s@t 2 s @x @s +2 s 2 @ 2 x @t 2 1 2 EI 2 @ 4 x @s 4 + @ @s ˙ @x @s o xdsdt + Z t 2 t 1 Z L 0 n 1 2 ˆ 2_ s 2 @ 2 y @s 2 2_ s @ 2 y @s@t 2_ s @ 2 y @s@t 2 s @y @s 2 @ 2 y @t 2 1 2 EI 2 @ 4 x @s 4 + @ @s ˙ @y @s ˆg o ydsdt =0 81 Since x and y areindependent,weequatethetermsassociatedwith x tobezeroandthe termsassociatedwith y tobezeroseparately.Thisprovidestheequationsofmotion,which arenonlinear: ˆ s + s @x @s +_ s 2 @ 2 x @s 2 +2_ s @ 2 x @s@t + @ 2 x @t 2 + EI @ 4 x @s 4 @˙ @s @x @s ˙ @ 2 x @s 2 =0(5.11a) ˆ s @y @s +_ s 2 @ 2 y @s 2 +2_ s @ 2 y @s@t + @ 2 y @t 2 + EI @ 4 y @s 4 @˙ @s @y @s ˙ @ 2 y @s 2 + ˆg =0(5.11b) andthearc-lengthconstraintassociatedwithEq.(5.11)is: @x @s 2 + @y @s 2 =1(5.12) where x ( s;t ), y ( s;t ), ˙ ( s ). 5.2.3ImposingConstraintonLineContact Itwasinitiallyassumedthattheelasticamaintainspointcontactwitht heground.However, whenthedeformationislarge,theelasticacanreachashapeforwh ichpartsoftheelastica haveanegative y coordinate.Toensurethegroundconstraintisnotviolated,weimp ose thefollowingconstraint: y 0(5.13) Thisislikelytoresultinlinecontactwiththegroundasopposedtopoint contact.Equation (5.13)cannotbedirectlyincorporatedintothederivationsofthee quationsofmotion;this groundconstraintwillbeimplementedbyapplyingthepenaltymethod intheniteelement model,whichwillbediscussedlater.Inthecaseoflinecontact,thev elocitydiagramis illustratedinFig.5.3.Itcanbeseenthattheentireelasticasegmentw ithlinecontact willhavetranslationalvelocityvectorandmaterialvelocityvector oppositeofeachother, resultinginanetzerovelocity 2 .Fortherestofthechapterweinvestigatethespecialcase 2 Itisassumedthatthereisnobucklinginthelinecontactsegment. 82 where_ s isconstant,whichmeans s iszero. _ s _ s _ s _ s _ s _ s _ s _ s _ s _ s Figure5.3:Velocitydiagramforelasticawithlinecontact 5.3FormulationoftheFiniteElementModel 5.3.1DiscretizationoftheEquationsofMotion Weconsiderthespecialcasewhere_ s isconstant;then,Eq.(5.11)becomes: ˆ _s 2 @ 2 x @s 2 +2_ s @ 2 x @s@t + @ 2 x @t 2 + EI @ 4 x @s 4 @˙ @s @x @s ˙ @ 2 x @s 2 =0(5.14a) ˆ _s 2 @ 2 y @s 2 +2_ s @ 2 y @s@t + @ 2 y @t 2 + EI @ 4 y @s 4 @˙ @s @y @s ˙ @ 2 y @s 2 + ˆg =0(5.14b) Theassociatedarc-lengthconstraintinEq.(5.12)isrequiredforth ediscretization.The groundconstraintinEq.(5.13)willbeimplementedusingthepenaltyme thod,whichwill bediscussedlater.SinceEq.(5.14)containsfourthorderterms,a tleast C 1 continuityis requiredattheelementboundaries.However,thereisnorequirem entforthemultiplier ˙ .Therefore,forthissystem,wechoosetousetheHermiteinterp olationpolynomialsfor x ( s;t )and y ( s;t )andtheLagrangianinterpolationpolynomialsfor ˙ ( s ).Figure5.4shows anelementoftheelasticawithtwonodes;eachnodehasvedegree soffreedom,including theCartesiancoordinates x ,y ,theslopes( dx=ds ),( dy=ds ),andtensileforce ˙ .Weassume rstorderHermitepolynomialsfor x and y :83 s 1 s 2 x 1 ; @x 1 @s y 1 ; @y 1 @s ˙ 1 x 2 ; @x 2 @s y 2 ; @y 2 @s ˙ 2 Node1 Node2 s Figure5.4:Anelementiscomprisedoftwonodes;eachnodehasved egrees-of-freedom. Thesubscriptsindicatethenodenumber. x ( s;t )= H 01 ( s ) x 1 ( t )+ H 11 ( s ) @x 1 @s ( t )+ H 02 ( s ) x 2 ( t )+ H 12 ( s ) @x 2 @s ( t )(5.15) y ( s;t )= H 01 ( s ) y 1 ( t )+ H 11 ( s ) @y 1 @s ( t )+ H 02 ( s ) y 2 ( t )+ H 12 ( s ) @y 2 @s ( t )(5.16) where H 01 ( s )=1 3 s s 1 s 2 s 1 2 +2 s s 1 s 2 s 1 3 H 11 ( s )=( s 2 s 1 ) s s 1 s 2 s 1 s s 1 s 2 s 1 1 2 H 02 ( s )= s s 1 s 2 s 1 2 3 2 s s 1 s 2 s 1 H 12 ( s )=( s 2 s 1 ) s s 1 s 2 s 1 2 s s 1 s 2 s 1 1 TheHermiteinterpolationpolynomialshave C 1 continuity.Nowweassumerstorder Lagrangepolynomialfor ˙ :˙ ( s )= N 1 ( s ) ˙ 1 + N 2 ( s ) ˙ 2 (5.17) 84 where N 1 ( s )=1 s s 1 s 2 s 1 N 2 ( s )= s s 1 s 2 s 1 TheLagrangianinterpolationpolynomials N 1 ;N 2 have C 0 continuity.Afterdeningthe element,weusetheweightedresidualmethod[88]toobtainthewea kformoftheequations ofmotioninEq.(5.14)andthearc-lengthconstraintinEq.(5.12)tofo rmulatethenite elementmodel.Firstweneedtoapplyaspecialcaseforthefourtho rdertermin x (same proceduresaredonefor y ): EI Z L 0 @ 4 x @s 4 wds = EI @ 3 x @s 3 w L 0 EI Z L 0 @ 3 x @s 3 @w @s ds Integratingbypartsagain,weget * 0 EI @ 3 x @s 3 w L 0 * 0 EI @ 2 x @s 2 @w @s L 0 + EI Z L 0 @ 2 x @s 2 @ 2 w @s 2 ds wherethetwonon-integraltermsarezeroduetoperiodicity.The elementalequationof motionin x isthen: Z s 2 s 1 ˆ _s 2 @ 2 x @s 2 wds + Z s 2 s 1 2 ˆ _s @ 2 x @s@t wds + Z s 2 s 1 ˆ @ 2 x @t 2 wds Z s 2 s 1 @˙ @s @x @s wds Z s 2 s 1 ˙ @ 2 x @s 2 wds + Z s 2 s 1 EI @ 2 x @s 2 @ 2 w @s 2 ds =0(5.18) Similarly,theelementalequationofmotionin y is: Z s 2 s 1 ˆ _s 2 @ 2 y @s 2 wds + Z s 2 s 1 2 ˆ _s @ 2 y @s@t wds + Z s 2 s 1 ˆ @ 2 y @t 2 wds Z s 2 s 1 @˙ @s @y @s wds Z s 2 s 1 ˙ @ 2 y @s 2 wds + Z s 2 s 1 EI @ 2 y @s 2 @ 2 w @s 2 ds = Z s 2 s 1 ˆgds (5.19) 85 andtheweakformoftheelementalarc-lengthconstraintis: Z s 2 s 1 @x @s 2 wds + Z s 2 s 1 @y @s 2 wds = Z s 2 s 1 wds (5.20) InEqs.(5.18)-(5.20),wesubstitutetheassumedsolutionsandinte grateeachterm;andthis leadsustothefollowingdiscretizedmatrixequationofmotionasshow ninEq.(5.21).The detailedproceduresandderivationoftheniteelementmodelcanb efoundintheAppendix. M ( ˆ ) U + C ( ˆ; _s ) _U + K ( ˆ; _s;U ) U = F ( ˆ )(5.21) where U istheglobalDOF5 n 1vector, M isa5 n 5 n massmatrix, C isa5 n 5 n Coriolis matrix, K isa5 n 5 n stnessmatrixand F isa5 n 1forcevector. n isthetotalnumberof nodesintheniteelementmodel.Notethat _( ) ; ( )aretherstandsecondderivativeswith respectto t .Here K isanon-constantstnessmatrixduetothenonlinearitiesintroduc ed bythearc-lengthconstraint.Equation(5.21)canalsobewrittena s: M ( ˆ ) U Z + C ( ˆ; _s ) _U Z + K ( ˆ; _s ) U Z +( U Z ) U ˙ = F ( ˆ )(5.22a) ( U Z ) T U Z = p (5.22b) andthematrixformis 2 6 6 6 6 6 6 6 4 M 0 0 0 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 U Z 0 3 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 4 C 0 0 0 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 _U Z 0 3 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 4 K T 0 3 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 4 U Z U ˙ 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 F p 3 7 7 7 7 7 7 7 5 where M isa4 n 4 n matrix, C isa4 n 4 n matrix, K isa4 n 4 n matrix,isa4 n n matrix, F isa4 n 1vector, p isa n 1vector, U Z isa4 n 1vector,and U ˙ isa n 1vector. InEq.(5.22), K ispartitionedsothat K isaconstantsquarematrixassociatedwiththe x ,86 ( dx=ds ), y ,( dy=ds )DOFsandisanon-constantmatrixassociatedwiththe ˙ DOFs. ToimposethegroundconstraintinEq.(5.13),weaddapenaltyterm y 2 i ,where i isthe nodenumberand isthepenaltyparameter,foreverynodethatviolatestheconstr aint. Then,Eq.(5.22a)becomes: M ( ˆ ) U Z + C ( ˆ; _s ) _U Z + K ( ˆ; _s ) U Z +( U Z ) U ˙ = F ( ˆ )if y 0 M ( ˆ ) U Z + C ( ˆ; _s ) _U Z + K ( ˆ; _s )+ Y c Y T c U Z +( U Z ) U ˙ = F ( ˆ )if y< 0 (5.23) where Y c isamatrixwithallentriesequaltozeroinarow,exceptforthe y DOFassociated withthenodethatviolatestheconstraint.Thedimensionsof Y c is4 n r ,where r isthe numberofnodesthatviolatetheconstraint.Theimplementationwill bediscussedlaterin thesection. 5.3.2ApplyingBoundaryConditions Theglobalmatricescanbereducedbyapplyingboundaryconditions .Inthisproblem,it isassumedthattheelasticastaysincontactwiththegroundat( x;y )=(0 ;0)at s =0; therefore,theboundaryconditionsare x (0 ;t )=0 ;y (0 ;t )=0 Inthediscretizedequationsthismeans: x 1 ( t )=0 ;y 1 ( t )=0 Therefore,therowsandcolumnsassociatedwiththedegreesoff reedom x 1 and y 1 canbe removedfromtheglobalmatricesandvectors.Nowthedimensions ofalltheglobalmatrices andvectorsarereducedbytwo. 87 5.3.3DirectTimeIntegration Wechoosetousethecentralderencemethodtoperformdirect timeintegrationofthe time-dependentFEM[88].Thecentralderencemethodisanexplicit timeintegration methodwhichdoesnotrequirethefactorizationofthe(eective) stnessmatrix.Theerror ofthemethodusedhereisoforder t 2 ,where t isthechosentimestep.Thealgorithm ofthetimeintegrationisshownbelowinAlgorithm1. Algorithm1 Directtimeintegrationusingthecentralderencemethod Initialization: Form M , C , K , F ,andinitialize 0 U Z ,0 _U Z ,0 U Z ,0 U ˙ for t =0;select timestep t andcalculatetheintegrationconstants: a 0 = 1 t 2 ,a 1 = 1 w t ,a 2 =2 a 0 ,a 3 = 1 a 2 Initialcalculations: t U Z = 0 U Z t 0 _U Z + a 3 0 U Z ;eectivemassmatrix ^ M = a 0 M + a 1 C for t = t + t do Calculateeectiveloads: t ^ F = F ( K a 2 M ) 0 U Z ( a 0 M a 1 C ) t t U Z Solvethenonlinearequationfordisplacementsat t + t using FindMinimum bypro- vidinganinitialguess t + t 0 U Z and t + t 0 U ˙ :if all y DOFsin t U Z 0 then ^ M t + t U Z t ^ F +( t + t U ˙ )=0(5.24a) ( t + t U Z ) T t + t U Z p =0(5.24b) elseif any y DOF(s)in t U Z < 0 then h ^ M + Y c Y T c i t + t U Z t ^ F +( t + t U ˙ )=0 ( t + t U Z ) Tt + t U Z p =0 endif endfor TheMathematicafunction FindMinimum [89]isusedtosolvethenonlinearmatrixequation inEq.(5.24);thismethodndssolutionbyminimizingthestrainenergyo fthesystem. Thechoiceofthepenaltyparameter dependsonthenumberofelementsandnumerical stability.Itshouldbenotedthatthetimestepandthenumberofele mentsneedtobe carefullychosentoensureconvergenceandstabilityforthisnonlin earniteelementmodel; 88 however,discussiononconvergenceandstabilitywillnotbepresen tedhere. 5.4SteadyStateSolutionInvestigation Fromwhatwelearnedfrompreviouschapters,wecanalsoexpress theelasticaequationsas thefollowing[36]: @ @s = MEI ;@ M@s = Q cos + P sin @x @s =cos ; @y @s =sin @P @s = ˆ r x @Q @s = ˆg ˆ r y where M( s;t )isthebendingmoment.Byassuming_ s isconstantandsubstituting r x and r y fromEq.(5.4),wecanwrite: @ @s = MEI ;@ M@s = Q cos + P sin @x @s =cos ; @y @s =sin @P @s = ˆ h @ 2 x @s 2 _s 2 +2 @ 2 x @s@t _s + @ 2 x @t 2 i @Q @s = ˆg ˆ h @ 2 y @s 2 _s 2 +2 @ 2 y @s@t _s + @ 2 y @t 2 i or @ @s = MEI ; @ M@s = Q cos + P sin @x @s =cos ; @y @s =sin @P @s = ˆh sin @ @s _ s 2 +2 @ @t (cos )_ s + @ 2 x @t 2 i @Q @s = ˆg ˆh cos @ @s _ s 2 +2 @ @t (sin )_ s + @ 2 y @t 2 i 89 Ifweassumeasteadystatesolution,allthetime-dependentterm swilldropoutandwecan directlyintegratetoobtaintheexpressionsfor P and Q .Then( @ M=@s )becomes @ M@s =( ˆgs ˆ _s 2 sin )cos + ˆ _s 2 cos sin = ˆgs cos Theaboveequationshowsthatthesteadystatesolutionforany_ s isthestaticequilibrium solution.Since denestheshapeoftheelastica,thismeansthattheshapedoesn 'tchange with_ s .Weshouldnotethat( @ M=@s )isdenedtobetheshearforce and denesthe shapeoftheelastica;thismeansthat andtheelasticashapedonotchangewith_ s .This steadystatesolutionwillbefurtherstudiedinthefollowingsection 3 .5.5NumericalSimulations 5.5.1SystemParameters Tohighlighttheparametersinthesystem,weusethefollowingnon-d imensionalvariables: x= x=L; y= y=L; s= s=L; _s=_ s p ˆL 2 =EI;˝ = t p EI=ˆL 4 w= ˆgL 3 =EI; ^ ˙ = ˙L 2 =EI (5.25) ˆ isthemassperunitlength, E istheYoung'smodulus,and I istheareamomentof inertia;inrealmaterials,thereisadirectrelationshipbetweenthed ensity(relatedto ˆ )and theYoung'smodulus E .Non-dimensionalizingtheequationsofmotioninEq.(5.14)using 3 Itisveredfromsimulationsthatthecongurationoftheelasticar emainsunchangedintimeifthe initialcongurationisinitsstaticequilibriumconguration. 90 Eq.(5.25)yieldsthefollowingnon-dimensionalequationsofmotion: _s2 @ 2 x@ s2 +2_ s@ 2 x@ s@˝ + @ 2 x@˝ 2 + @ 4 x@ s4 @ ^ ˙ @ s@ x@ s ^ ˙ @ 2 x@ s2 =0(5.26a) _s2 @ 2 y@ s2 +2_ s@ 2 y@ s@˝ + @ 2 y@˝ 2 + @ 4 y@ s4 @ ^ ˙ @ s@ y@ s ^ ˙ @ 2 y@ s2 + w=0(5.26b) alongwiththeconstraints @ x@ s 2 + @ y@ s 2 =1(5.27) y 0(5.28) ItcanbeobservedfromEq.(5.26)thatthethegoverningequation sonlydependontwonon- dimensionalparameters:thenon-dimensionalvelocity_ sandthenon-dimensionalweight w.Therefore,wefocusourdiscussionontheeectsofthetwonon- dimensionalparameterson thedynamicsoftherollingelastica.Fortheniteelementanalysisper formedinthenext sections,wechoosetouseatimestepof t =0 :00001s 4 and44elementstoensurenumerical stabilityandconvergenceaftercarefulconsideration 5 .5.5.2Eectsof wonSystemDynamics Toinvestigatetheeectsof wonthedynamicsofthesystemrepresentedbyEqs.(5.26)- (5.28),thevalueof wisvariedwhile_ sissettobezero.Morespeccally,threederent cases, w= f 45 :10 ;94 :41 ;375 :85 g ,areconsidered 6 ;eachcasehasaderentvalueof wand sameinitialconditionswheretheelasticaisinitsundeformedcongur ation(acirclewith non-dimensionalradiusof r=1 = 2 ˇ )andhaszeroinitialvelocity.Figure5.5showsthe ypositionofthetoppointoftheundeformedelastica,denotedby pinFig.5.6,foraperiod ofnon-dimensionaltime ˝ andFigs.5.6-5.8showtheundeformedinitialconguration,the 4 Notethatthisisdimensionaltimewithaunitofsecond. 5 Thenumberofelementsischosentoappropriatelyaddressthecon tactproblem.Thechoiceofnumberof elementsaectstherequiredtimestepsizeforconvergence,whic hisrelatedtotheCourant-Friedrichs-Lewy (CFL)condition[90] 6 Thethreecasesarechosentoclearlydisplaythecharacteristicso ftheelasticaoverarangeofweights 91 0.00.20.40.60.80.000.050.100.150.200.250.30˝ yw=45 : 10 w=94 : 41 w=375 : 85 Figure5.5: ypositionvs. ˝ for w= f 45 :10 ;94 :41 ;375 :85 g .Thedashedlinesshowthe ypositionof pwhentheelasticaisinstaticequilibrium. staticequilibriumconguration,andthemaximumdeformedcongur ationoftheelasticafor w= f 45 :10 ;94 :41 ;375 :85 g .Thetotalsimulationtimeforallthreecasesis t =2s;however,it isinterestingtonotehowthistimeisrepresentedinthenon-dimensio naltimeframeasshown inFig.5.5.Thenon-dimensionalfrequenciesoftheoscillationfor w= f 45 :10 ;94 :41 ;375 :85 g are9 :434,8 :842,and11 :300.Itisintuitivethatthefrequencydecreasesastheweight increases;andthisiswhatisobservedfor w= f 45 :10 ;94 :41 g .However,inthecaseof w=375 :85,wenoticeanincreaseinfrequency;thisisthecasewheretheela sticaexperiences signcantlinecontact.Thiscanbeduetothefactthattheamount ofelasticaundergoing deformationdecreasesasthelinecontactportionincreases,whic hresultsinincreaseof stnessoftheelastica.AsitcanbeenseeninFigs.5.6and5.7,thedef ormedcongurations remainroundorsomewhatelliptical.However,whentheelasticawith w=375 :85undergoes maximumdeformation,itsshapebecomespeanut-likewithmorethan almosthalfofthetotal lengthbeingincontactwiththeground,asshowninFig.5.8.Figure5.5a lsoshowsthatthe elasticawiththelargest whasthelargestamplitudewhiletheelasticawiththesmallest whasthesmallestamplitude;thisresultisintuitive.Itisalsoobservedt hatthereisdecrease inamplitudeinthecaseof w=375 :85;thiscanbeexplainedbythefactthatnon-physical 92 -0.2-0.10.10.20.050.100.150.200.250.300.35equilibriuminitialmaximumyxpFigure5.6:Theinitial,staticequilibriumandmaximumdeformedcongur ationsofthe elasticawith w=45 :10. -0.2-0.10.10.20.050.100.150.200.250.300.35equilibriuminitialmaximumxyFigure5.7:Theinitial,staticequilibrium,andmaximumdeformedcongu rationsofthe elasticawith w=94 :41. energyislostduetothepenaltymethodusedtoenforcethegroun dconstraint[91]. 5.5.3Eectsof _ sonSystemDynamics Considerthecasewhere w=53 :45and_ s=4 :95wherethetimeresponseissimulated usingthehand-codedFEMonMathematicaandthecommercialsoft wareCOMSOL[92]; 93 -0.2-0.10.10.20.050.100.150.200.250.300.35equilibriuminitialmaximumxyFigure5.8:Theinitial,staticequilibrium,andmaximumdeformedcongu rationsofthe elasticawith w=375 :85. theinitialcongurationoftheelasticaisanundeformedcirclewithar adiusof r.Figure 5.9showsthemaximum ypositionforaperiodofnon-dimensionaltime ˝ ,˝ =[0 ;0 :61]. Itshouldbenotedthattheelasticaisalsotranslatinginthehorizont aldirectionwitha speedof_ s ;here,theresultsarepresentedintheLagrangianframeofrefe renceasdiscussed before.Theaveragenon-dimensionalfrequenciesobtainedfrom FEMandfromCOMSOL are8 :696and8 :929,respectively,whicharesimilar.Thederenceinresultscanbed ueto variousparametertuningandthenumericaldampingappliedinCOMSO L.Insection5.4, 0.10.20.30.40.50.60.280.290.300.310.32F COMSOL˝ yFigure5.9: ypositionvs. ˝ for w=53 :45and_ s=4 :95.Blacksolidline:FEMresult;blue dottedline:COMSOLresult 94 itisobservedthatthedeformationoftheelasticainsteadystates olutionisidenticaltothe deformationoftheelasticainstaticequilibriumsolution,regardlesso fthevalueof_ s.It canbeseenfromFig.5.9thattheelasticaisoscillatingaroundthestea dystatedeformation. Thisresultwillbefurtherexaminedwhenderentcasesofanelastic awith w=94 :41and derentvaluesof_ sarestudied. Inthefollowingsimulations,theverticalandhorizontalpositionsof Node22,whichis locatedatthetopoftheelasticaintheundeformedcongurationa nddenotedbypoint pinFig.5.6,aretracked.Forthecaseof_ s=5 :07,the xand ypositionsof pareplottedfrom ˝ =0to ˝ =1 :32inFig.5.10;theaveragenon-dimensionalfrequencyofthe ypositionofthe nodeis8 :464.Forthecaseof_ s=12 :67,the xand ypositionsof pareplottedfrom ˝ =0 to ˝ =1 :32inFig.5.11;theaveragenon-dimensionalfrequencyofthe ypositionofthenode is6 :534.Forthecaseof_ s=20 :27,the xand ypositionsof pareplottedfrom ˝ =0to ˝ =1 :32inFig.5.12;theaveragenon-dimensionalfrequencyofthe ypositionofthenodeis 4 :888.Itisobservedinallthreecasesthattheamplitudeofthe ypositionisdecreasedwith time;thisresultmayseemcounterintuitiveasthesystemdoesnoth avedissipativeterms. However,weobserveanincreaseinamplitudesofthe xpositionwithtime;thismaybe attributedtothedecreaseinamplitudesin y.Inallthreecases,point pisoscillatingaround 0.00.2 0.81.01.20.25 0.270.280.290.300.310.320.000.020 0 0.08˝ yx˝ =0 :358 ˝ =0 :667 ˝ =0 :726 Figure5.10: yposition(bluesolidline)and xposition(redsolidline)of pvs. ˝ for w=94 :41 and_ s=5 :07.Thebluedashedlineshowsthemean ypositionofthenode. 95 0.00.2 0.81.01.20.25 0.270.280.290.300.310.320.000.02 0.08˝ yx˝ =0 :358 ˝ =0 :667 ˝ =0 :726 Figure5.11: yposition(bluesolidline)and xpostion(redsolidline)of pvs. ˝ for w=94 :41 and_ s=12 :67.Thebluedashedlineshowsthemean ypositionofthenode. 0.00.2 0.81.01.20.250.270.280.290.300.310.32-0.020.000.02 0.080.10˝ yx˝ =0 :358 ˝ =0 :667 ˝ =0 :726 Figure5.12: yposition(bluesolidline)and xpostion(redsolidline)of pvs. ˝ for w=94 :41 and_ s=20 :27.Thebluedashedlineshowsthemean ypositionofthenode. ameanpositionof yˇ 0 :28,whichisthepositionofthenodewhentheelasticaisinstatic equilibriumconguration,showninFig.5.7.Itisinterestingtoseethat thenon-dimensional frequencydecreaseswhen_ sisincreased.Thisobservationresemblesthedivergenceproblem inaxiallymovingmaterials[67],suchasthebandsawproblem[64].Tofurt herillustratethe deformationsoftheelasticawithderentspeeds,thedeformedc ongurationsoftheelastica with_ s= f 5 :07 ;12 :67 ;20 :27 g areplottedinFig.5.13for ˝ =0 :358,Fig.5.14for ˝ =0 :667, andFig.5.15for ˝ =0 :726.Anotherinterestingvariableinthesystemisthemultiplier^ ˙ ,96 -0.2-0.10.10.20.050.100.150.200.250.300.35xy_s=5 :07 _s=12 :67 _s=20 :27 Figure5.13:Deformedcongurationsoftheelasticaat ˝ =0 :358for w=94 :41andthree derentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g .whichisalsodenedtobethenon-dimensionaltensileforceasdiscus sedintheprevious section.Figure5.16showsthemaximum^ ˙ oftheelasticafor_ s= f 5 :07 ;12 :67 ;20 :27 g from ˝ =0to ˝ =1 :32;themaximumtensileforce,denotedby^ ˙ max ,isdenedasthemaximum ofthe^ ˙ valuesofallthenodesinanyarbitrarycongurationoftheelastica .Since^ ˙ max is ahigherorderterm,wechoseasamplingfrequencyof380toremov esomenoiseinthedata -0.2-0.10.10.20.050.100.150.200.250.300.35xy_s=5 :07 _s=12 :67 _s=20 :27 Figure5.14:Deformedcongurationsoftheelasticaat ˝ =0 :667for w=94 :41andthree derentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g .97 -0.2-0.10.10.20.050.100.150.200.250.300.35xy_s=5 :07 _s=12 :67 _s=20 :27 Figure5.15:Deformedcongurationsoftheelasticaat ˝ =0 :726for w=94 :41andthree derentvaluesof_ s,namely_ s= f 5 :07 ;12 :67 ;20 :27 g .inordertobettervisualizetheresults.ItcanbeseenfromFig.5.16t hatthevaluesof^ ˙ max increaseas_ sincreases.Thismeansthatthemaximumtensileforceoftheelastica increases asthespeedofrollingincreases. 0.00.2 0.81.01.20100200300 ^ ˙ max ˝ _s=5 :07 _s=12 :67 _s=20 :27 Figure5.16:Non-dimensionalmaximumtensileforce^ ˙ max oftheelasticawith w=94 :41vs. ˝ for_ s= f 5 :07 ;12 :67 ;20 :27 g withasamplingfrequencyof380. 98 5.5.4EectsofInitialCongurations Intheprevioussections,weusethesameinitialcongurationwher etheelasticaisunde- formedandcircularwitharadiusof r.Inthissection,weinvestigatethedynamicresponse oftherollingelasticawithaderentinitialconguration,namelyanellip se 7 .Theellipsehas aperimeterofunity;thesemi-majoraxisischosentobe: a=0 :175;andthecorresponding semi-minoraxisis: b=0 :143.ThisellipticalinitialcongurationisshowninFig.5.18.For thecaseof w=53 :45and_ s=4 :95,the xand ypositionsof pareplottedfrom ˝ =0to ˝ =1 :32inFig.5.17;thenon-dimensionalfrequencyofthe ypositionofthenodeis8 :861. Point pisoscillatingaroundameanpositionof yˇ 0 :30,whichisthepositionofthenode whentheelasticaisinstaticequilibriumconguration.Alongwiththeinit ialcongura- tionat ˝ =0,Fig.5.18showsthedeformedcongurationsoftheelasticaat ˝ =0 :715and ˝ =1 :083.Figure5.19showsthemaximumtensileforce^ ˙ max oftheelasticafrom ˝ =0 to ˝ =1 :32.Sincetheelasticaisnotinitsundeformedshape(acircle)atinitial time,we expecttoseeahigh^ ˙ max valueinitially,whichturnsouttobe^ ˙ max > 3000 8 asobservedin 7 Wechoose t =8 10 6 sandthenumberofelementstobe44 8 Theactual^ ˙ max isapproximately5 10 4 whichisnotshowninFig.5.19. 0.00.2 0.81.01.20.2850.2900.2950.3000.3050.3100.3150.000.010.020.03 0.05˝ yx˝ =1 :083 ˝ =0 :715 ˝ =0 Figure5.17: yposition(bluesolidline)and xposition(redsolidline)of pvs. ˝ for w=53 :45 and_ s=4 :95.Thebluedashedlineshowsthemean ypositionofthenode. 99 -0.2-0.10.10.20.050.100.150.200.250.300.35xyab˝ =1 :083 ˝ =0 :715 ˝ =0 Figure5.18:Initialcongurationofelasticaat ˝ =0anddeformedcongurationsofthe elasticaat ˝ = f 0 :715 ;1 :083 g for w=53 :45and_ s=4 :95. aisthesemi-majoraxisand bis thesemi-minoraxis. 0.00.2˘ˇˆ ˘ˇ˙ 0.81.01.2050010001500200025003000^ ˙ max ˝ Figure5.19:Non-dimensionalmaximumtensileforce^ ˙ max oftheelasticawith w=53 :45vs. ˝ for_ s=4 :95withasamplingfrequencyof103. Fig.5.19;andthevalueof^ ˙ max isexponentiallydecreasingastheelasticaissettlingaround steadystatesolution. 100 Chapter6 Conclusion Inthisthesis,weconsiderthedynamicsandcontrolproblemsofd erentsystemsinvolving theelastica.Chapter2studiesasequenceofthreeproblemsofinc reasingcomplexitythat graduallycapturetheessentialdynamicsofpole-vaulting.Anon-d imensionalframeworkis usedtostudytheproblemstobetterunderstandtheparameter sandinitialconditionsthat contributetoimprovingpole-vaultingperformance.Thecompletet ransformationofthe kineticenergyofapointmasstravelinginthehorizontaldirectionint opotentialenergyis rstconsideredusingalinearspringasanintermediary.Inthenon- dimensionalframework, theonlysystemparameteristhenon-dimensionaldeectionofthe springduetotheweight ofthemass.Foragiveninitialangleofthespringwithrespecttothe groundwithinacertain range,completeenergytransformationrequiresthemasstohav easpeccinitialvelocity. Thisinitialvelocity,whichisobtainedbysolvingatwo-pointboundaryv alueproblem,is foundtobelargerforsmallerinitialanglesofthespring.Thisimpliesth atalargeamountof kineticenergycanbetransformedintopotentialenergyofthema ssiftheinitialangleofthe springissmall.Itisalsoobservedthatforaspeccinitialangleofthe spring,largervalues ofthenon-dimensionalparameterresultinlargeramountsofener gyconversionbutalower maximumheightreachedbythemass.Inthesecondproblem,theline arspringisreplaced bytheelasticaandtheinvestigationsareagaincarriedoutusingasin glenon-dimensional systemparameter.Itisfoundthatwhentheinitialangleoftheelas ticaisintherange oftakeoanglesusedbypole-vaulters,themainresultsaresimilart othoseobtainedwith thelinearspring.Thisisaninterestingresult,butthepointmassand linearspringsystem cannotbeusedtomodelthebendingmomentsappliedbypole-vaulte rs.Thethirdproblem isthereforestudiedasanextensionofthesecondproblembyinclud ingexternaltorques.A comparisonoftheresultsofthesecondandthirdproblemsindicate thattheinitialvelocities aresimilarforthesametakeoangle;and,therefore,themaximum heightreachedbythe 101 massissigncantlyhigherinthepresenceofexternaltorques.Mo reinterestingly,thisheight gainishigherforasystemwheretheexternaltorquesarelargera ndthemassissmaller.This indicatestherelativeimportanceofmuscleworkovermassinpole-va ulting.Tobetterrelate totheproblemofpole-vaulting,adimensionalexampleisconsideredu singdatapresented intheliterature.Simulationresultsprovideaninitialvelocityofthema ssthatissimilarto therunningspeedoftheathletes;themaximumheightreachedbyt hemassishigherthan theworldrecord,butthiscanbeattributedtotheseveralsimplify ingassumptionsmadein theanalysis.Despitetheseassumptions,thepointmassandelastic asystemwithexternal torqueisfoundtocapturetheessentialdynamicsofpole-vaulting .InChapter3,theforce-displacementcharacteristicsofcircular -shapedelasticassubjected tounilateralconstraintswereinvestigatedusingnumericalmetho ds;aseriesoftwo-point boundaryvalueproblemsweresolvedintheorderofincreasingcomp lexity.Thesolutionsto theseproblemsindicatethattheforce-displacementcharacteris ticsofelasticassigncantly dependontheboundaryconditions.AC-shapedelasticawasrsts tudiedassumingrolling withoutslippingduetothepresenceoffriction;theeectofvertic alloadingwasinvestigated forarbitraryinitialorientationforthetwocaseswherethetipslop ewasxedandfree, respectively.Forxedtip-slope,theelasticabehaveslikeaharden ingspring;forfreetip- slope,theelasticabehaveslikeasofteningspringforsmallinitialorie ntationandlikea hardeningspringforlargeinitialorientation.Intheabsenceoffric tion,loadingresultsin simultaneousrollingandslippingoftheC-shapedelasticawhenthetips lopeisxed.In thecaseoffreetip-slope,theelasticaundergoespureslippingand acquiresatipslopethat isidenticalforallloading,independentoftheinitialorientation.The O-shapedelastica wasrststudiedundersymmetricloading.Withincreaseinloading,th eelasticamakesline contactwiththeground;itbehavesasasofteningspringpriortom akinglinecontactbutlike alinearspringafteritmakeslinecontact.Underasymmetricvertica lloading,theO-shaped elasticadisplacesinbothverticalandhorizontaldirectionsandmak eslinecontactwhen theloadincreases.Fortheassumptionsmade,asolutiontothepro blemcanbeobtained 102 foralimitedrangeoftheverticalload.Thetwo-pointboundaryvalu eproblemissolved simultaneouslyfortwoseparatesegmentsoftheelasticawithmatc hingboundaryconditions anditbecomesincreasinglydculttondasolutionastheloadincreas es.Tosimulatethe behaviorunderlargerloads,someoftheassumptionsmayhavetob emodedguidedby experimentalinvestigations;suchinvestigations,ofcourse,lieint hescopeoffutureresearch. InChapter4,westudyahoppingrobotthatisavariable-structur emass-elasticasystem. Amass-springsystemcancontinuetohoptothesameheightunder theassumptionthat noenergyislostduetoimpactofthespringwiththeground.Achang einthestnessof thespringatthetimeofmaximumdeformationcanchangetheenerg yofthesystemand canthereforebeusedtocontrolthemaximumheightreachedbyt hemass.Thisideawas exploredfurtherbyinvestigatingavariable-structuremass-spr ingsystemwherethespring isanelasticaandthestnessoftheelasticaischangedbyinstantan eouslychangingits boundaryconditions.Thedynamicsofthemass-elasticasystemiss tudiedbysolvingatwo- pointboundaryvalueprobleminanon-dimensionalframework.Thee lasticaisassumed tobemasslessanditsforce-displacementcharacteristicsareder ivedapriori.Thehopping heightiscontrolledbychangingtheshoulderangles,whicharethean glessubtendedby theelasticawiththehorizontalatthepointsofcontactwiththema ss.Simulationresults arepresentedtoshowthatthehoppingheightcanbeincreasedgr aduallybyswitchingthe shoulderanglebetweenderentvaluesatderentphasesoftheh oppingcycle.Thiswork isbasedontheassumptionthattheelasticamakespointcontactwit htheground. InChapter5,weconsiderthedynamicsofarollingelastica.Theequa tionsofmotion oftherollingelasticaarederivedusingextendedHamilton'sprinciple.T hesystemhasa nonlineargeometricconstraintduetoinextensibilityoftheelasticaa ndthegroundimposes aunilateralconstraintonthesystem.Theensuingnonlinearpartia lderentialequations aresolvedbyusingtheniteelementmodel.Tosimulatethedynamicre sponseoftheFEM, weapplythecentralderencemethodwithpenaltymethodfordire cttimeintegration.The steadystatesolutionisrstinvestigated.Afterdiscardingthetim e-dependenttermsfrom 103 theequationsofmotion,itisobservedthatthesteadystatecon gurationoftheelastica doesnotchangewithspeedandremainsidenticaltothecongurat ionoftheelasticain itsstaticequilibrium.Iftheelasticaisinitiallyinthisconguration,then thevariablesin thesystemdonotchangeintime.Tostudytheeectsoftheparam etersonthesystem dynamics,theequationsofmotionarenon-dimensionalized.Thenon -dimensionalequations ofmotionarefoundtobedependentontwonon-dimensionalparam eters,namely,thenon- dimensionalweightandthenon-dimensionalvelocity.Fromnumerica lsimulations,itis foundthatthenon-dimensionalfrequencydecreasesasthenon -dimensionalweightincreases whiletheelasticamaintainspointcontactwiththeground;however, thefrequencyincreases forverylargeweightasalargeportionoftheelasticaexperienceslin econtactwiththe ground.Thisresultcanbeattributedtothefactthattheportion oftheelasticaundergoing deformationreducesasthelinecontactincreases,whichresultsin anincreaseinstness oftheelastica.Asthenon-dimensionalvelocityincreases,thenon -dimensionalfrequency decreasesandthetensileforceoftheelasticaincreases.Thisres ultresemblesthedivergence probleminaxiallymovingmaterials.Inallcases,itisobservedthatthe rollingelastica oscillatesaroundthesteadystateconguration.Theamplitudeso foscillationinthevertical directionforthecaseswithnon-zerovelocityareobservedtobed ecreasing;thisresultmay beattributedtotheincreaseinamplitudesinthehorizontaldirectio n.Someresultsare comparedwiththesimulationresultsobtainedfromCOMSOL;theyar efoundtobesimilar. However,itshouldbenotedthatwhilethehand-codedFEMisabletoh andlederent initialcongurationsoftheelastica,itischallengingtodeneinitialco nditionsspeccally forthissystemusingcommercialsoftware,suchasCOMSOL.Furt hermore,thedcultyin ne-tuningtheparametersfornumericalstabilityinCOMSOLincrea sesastheamountof linecontactoftheelasticawiththegroundincreases. 104 APPENDIX 105 FiniteElementModeloftheRollingElastica Werststartwiththeelementalequationofmotionin x :Z s 2 s 1 ˆ _s 2 @ 2 x @s 2 wds | {z } E 1 + Z s 2 s 1 2 ˆ _s @ 2 x @s@t wds | {z } E 2 + Z s 2 s 1 ˆ @ 2 x @t 2 wds | {z } E 3 Z s 2 s 1 @˙ @s @x @s wds | {z } E 4 Z s 2 s 1 ˙ @ 2 x @s 2 wds | {z } E 5 + Z s 2 s 1 EI @ 2 x @s 2 @ 2 w @s 2 ds | {z } E 6 =0 Nowwesubstitutetheassumedsolutionsandintegrateeachterm. Notethat( ) 0 ;( ) 00 are rstandsecondderivativeswithrespectto s and _( ) ; ( )arerstandsecondderivativeswith respectto t .E1: Z s 2 s 1 ˆ _s 2 @ 2 x @s 2 wds = ˆ _s 2 Z s 2 s 1 H 00 01 ( s ) x 1 ( t )+ H 00 11 ( s ) @x 1 ( t ) @s + H 00 02 ( s ) x 2 ( t )+ H 00 12 ( s ) @x 2 ( t ) @s wds Let w = H 01 :ˆ _s 2 Z s 2 s 1 H 00 01 H 01 ds x 1 + Z s 2 s 1 H 00 11 H 01 ds @x 1 @s + Z s 2 s 1 H 00 02 H 01 ds x 2 + Z s 2 s 1 H 00 12 H 01 ds @x 2 @s Let w = H 11 :ˆ _s 2 Z s 2 s 1 H 00 01 H 11 ds x 1 + Z s 2 s 1 H 00 11 H 11 ds @x 1 @s + Z s 2 s 1 H 00 02 H 11 ds x 2 + Z s 2 s 1 H 00 12 H 11 ds @x 2 @s Let w = H 02 :ˆ _s 2 Z s 2 s 1 H 00 01 H 02 ds x 1 + Z s 2 s 1 H 00 11 H 02 ds @x 1 @s + Z s 2 s 1 H 00 02 H 02 ds x 2 + Z s 2 s 1 H 00 12 H 02 ds @x 2 @s 106 Let w = H 12 :ˆ _s 2 Z s 2 s 1 H 00 01 H 12 ds x 1 + Z s 2 s 1 H 00 11 H 12 ds @x 1 @s + Z s 2 s 1 H 00 02 H 12 ds x 2 + Z s 2 s 1 H 00 12 H 12 ds @x 2 @s E2: Z s 2 s 1 2 ˆ _s @ 2 x @s@t wds =2 ˆ _s Z s 2 s 1 H 0 01 ( s )_ x 1 ( t )+ H 0 11 ( s ) @ _x 1 ( t ) @s + H 0 02 ( s )_ x 2 ( t )+ H 0 12 ( s ) @ _x 2 ( t ) @s wds Let w = H 01 :2 ˆ _s Z s 2 s 1 H 0 01 H 01 ds _x 1 + Z s 2 s 1 H 0 11 H 01 ds @ _x 1 @s + Z s 2 s 1 H 0 02 H 01 ds _x 2 + Z s 2 s 1 H 0 12 H 01 ds @ _x 2 @s Let w = H 11 :2 ˆ _s Z s 2 s 1 H 0 01 H 11 ds _x 1 + Z s 2 s 1 H 0 11 H 11 ds @ _x 1 @s + Z s 2 s 1 H 0 02 H 11 ds _x 2 + Z s 2 s 1 H 0 12 H 11 ds @ _x 2 @s Let w = H 02 :2 ˆ _s Z s 2 s 1 H 0 01 H 02 ds _x 1 + Z s 2 s 1 H 0 11 H 02 ds @ _x 1 @s + Z s 2 s 1 H 0 02 H 02 ds _x 2 + Z s 2 s 1 H 0 12 H 02 ds @ _x 2 @s Let w = H 12 :2 ˆ _s Z s 2 s 1 H 0 01 H 12 ds _x 1 + Z s 2 s 1 H 0 11 H 12 ds @ _x 1 @s + Z s 2 s 1 H 0 02 H 12 ds _x 2 + Z s 2 s 1 H 0 12 H 12 ds @ _x 2 @s E3: Z s 2 s 1 ˆ @ 2 x @t 2 wds = ˆ Z s 2 s 1 H 01 ( s ) x 1 ( t )+ H 11 ( s ) @ x 1 ( t ) @s + H 02 ( s ) x 2 ( t )+ H 12 ( s ) @ x 2 ( t ) @s wds 107 Let w = H 01 :ˆ Z s 2 s 1 H 01 H 01 ds x 1 + Z s 2 s 1 H 11 H 01 ds @ x 1 @s + Z s 2 s 1 H 02 H 01 ds x 2 + Z s 2 s 1 H 12 H 01 ds @ x 2 @s Let w = H 11 :ˆ Z s 2 s 1 H 01 H 11 ds _x 1 + Z s 2 s 1 H 11 H 11 ds @ x 1 @s + Z s 2 s 1 H 02 H 11 ds x 2 + Z s 2 s 1 H 12 H 11 ds @ x 2 @s Let w = H 02 :ˆ Z s 2 s 1 H 01 H 02 ds x 1 + Z s 2 s 1 H 11 H 02 ds @ x 1 @s + Z s 2 s 1 H 02 H 02 ds x 2 + Z s 2 s 1 H 12 H 02 ds @ x 2 @s Let w = H 12 :ˆ Z s 2 s 1 H 01 H 12 ds x 1 + Z s 2 s 1 H 11 H 12 ds @ x 1 @s + Z s 2 s 1 H 02 H 12 ds x 2 + Z s 2 s 1 H 12 H 12 ds @ x 2 @s E4: Z s 2 s 1 @˙ @s @x @s wds = Z s 2 s 1 N 0 1 ( s ) ˙ 1 + N 0 2 ( s ) ˙ 2 H 0 01 ( s ) x 1 ( t )+ H 0 11 ( s ) @x 1 ( t ) @s + H 0 02 ( s ) x 2 ( t )+ H 0 12 ( s ) @x 2 ( t ) @s wds Let w = H 01 : Z s 2 s 1 N 0 1 H 0 01 H 01 ˙ 1 ds x 1 + Z s 2 s 1 N 0 1 H 0 11 H 01 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 0 1 H 0 02 H 01 ˙ 1 ds x 2 + Z s 2 s 1 N 0 1 H 0 12 H 01 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 0 2 H 0 01 H 01 ˙ 2 ds x 1 + Z s 2 s 1 N 0 2 H 0 11 H 01 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 0 2 H 0 02 H 01 ˙ 2 ds x 2 + Z s 2 s 1 N 0 2 H 0 12 H 01 ˙ 2 ds @x 2 @s 108 Let w = H 11 : Z s 2 s 1 N 0 1 H 0 01 H 11 ˙ 1 ds x 1 + Z s 2 s 1 N 0 1 H 0 11 H 11 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 0 1 H 0 02 H 11 ˙ 1 ds x 2 + Z s 2 s 1 N 0 1 H 0 12 H 11 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 0 2 H 0 01 H 11 ˙ 2 ds x 1 + Z s 2 s 1 N 0 2 H 0 11 H 11 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 0 2 H 0 02 H 11 ˙ 2 ds x 2 + Z s 2 s 1 N 0 2 H 0 12 H 11 ˙ 2 ds @x 2 @s Let w = H 02 : Z s 2 s 1 N 0 1 H 0 01 H 02 ˙ 1 ds x 1 + Z s 2 s 1 N 0 1 H 0 11 H 02 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 0 1 H 0 02 H 02 ˙ 1 ds x 2 + Z s 2 s 1 N 0 1 H 0 12 H 02 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 0 2 H 0 01 H 02 ˙ 2 ds x 1 + Z s 2 s 1 N 0 2 H 0 11 H 02 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 0 2 H 0 02 H 02 ˙ 2 ds x 2 + Z s 2 s 1 N 0 2 H 0 12 H 02 ˙ 2 ds @x 2 @s Let w = H 12 : Z s 2 s 1 N 0 1 H 0 01 H 12 ˙ 1 ds x 1 + Z s 2 s 1 N 0 1 H 0 11 H 12 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 0 1 H 0 02 H 12 ˙ 1 ds x 2 + Z s 2 s 1 N 0 1 H 0 12 H 12 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 0 2 H 0 01 H 12 ˙ 2 ds x 1 + Z s 2 s 1 N 0 2 H 0 11 H 12 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 0 2 H 0 02 H 12 ˙ 2 ds x 2 + Z s 2 s 1 N 0 2 H 0 12 H 12 ˙ 2 ds @x 2 @s 109 E5: Z s 2 s 1 ˙ @ 2 x @s 2 wds = Z s 2 s 1 ( N 1 ( s ) ˙ 1 + N 2 ( s ) ˙ 2 ) H 00 01 ( s ) x 1 ( t )+ H 00 11 ( s ) @x 1 ( t ) @s + H 00 02 ( s ) x 2 ( t )+ H 00 12 ( s ) @x 2 ( t ) @s wds Let w = H 01 : Z s 2 s 1 N 1 H 00 01 H 01 ˙ 1 ds x 1 + Z s 2 s 1 N 1 H 00 11 H 01 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 1 H 00 02 H 01 ˙ 1 ds x 2 + Z s 2 s 1 N 1 H 00 12 H 01 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 2 H 00 01 H 01 ˙ 2 ds x 1 + Z s 2 s 1 N 2 H 00 11 H 01 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 2 H 00 02 H 01 ˙ 2 ds x 2 + Z s 2 s 1 N 2 H 00 12 H 01 ˙ 2 ds @x 2 @s Let w = H 11 : Z s 2 s 1 N 1 H 00 01 H 11 ˙ 1 ds x 1 + Z s 2 s 1 N 1 H 00 11 H 11 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 1 H 00 02 H 11 ˙ 1 ds x 2 + Z s 2 s 1 N 1 H 00 12 H 11 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 2 H 00 01 H 11 ˙ 2 ds x 1 + Z s 2 s 1 N 2 H 00 11 H 11 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 2 H 00 02 H 11 ˙ 2 ds x 2 + Z s 2 s 1 N 2 H 00 12 H 11 ˙ 2 ds @x 2 @s 110 Let w = H 02 : Z s 2 s 1 N 1 H 00 01 H 02 ˙ 1 ds x 1 + Z s 2 s 1 N 1 H 00 11 H 02 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 1 H 00 02 H 02 ˙ 1 ds x 2 + Z s 2 s 1 N 1 H 00 12 H 02 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 2 H 00 01 H 02 ˙ 2 ds x 1 + Z s 2 s 1 N 2 H 00 11 H 02 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 2 H 00 02 H 02 ˙ 2 ds x 2 + Z s 2 s 1 N 2 H 00 12 H 02 ˙ 2 ds @x 2 @s Let w = H 12 : Z s 2 s 1 N 1 H 00 01 H 12 ˙ 1 ds x 1 + Z s 2 s 1 N 1 H 00 11 H 12 ˙ 1 ds @x 1 @s + Z s 2 s 1 N 1 H 00 02 H 12 ˙ 1 ds x 2 + Z s 2 s 1 N 1 H 00 12 H 12 ˙ 1 ds @x 2 @s + Z s 2 s 1 N 2 H 00 01 H 12 ˙ 2 ds x 1 + Z s 2 s 1 N 2 H 00 11 H 12 ˙ 2 ds @x 1 @s + Z s 2 s 1 N 2 H 00 02 H 12 ˙ 2 ds x 2 + Z s 2 s 1 N 2 H 00 12 H 12 ˙ 2 ds @x 2 @s E6: Z s 2 s 1 EI @ 2 x @s 2 @ 2 w @s 2 ds = EI Z s 2 s 1 H 00 01 ( s ) x 1 ( t )+ H 00 11 ( s ) @x 1 ( t ) @s + H 00 02 ( s ) x 2 ( t )+ H 00 12 ( s ) @x 2 ( t ) @s w 00 ds Let w 00 = H 01 : Z s 2 s 1 H 00 01 H 00 01 ds x 1 + Z s 2 s 1 H 00 11 H 00 01 ds @x 1 @s + Z s 2 s 1 H 00 02 H 00 01 ds x 2 + Z s 2 s 1 H 00 12 H 00 01 ds @x 2 @s 111 Let w 00 = H 11 : Z s 2 s 1 H 00 01 H 00 11 ds x 1 + Z s 2 s 1 H 00 11 H 00 11 ds @x 1 @s + Z s 2 s 1 H 00 02 H 00 11 ds x 2 + Z s 2 s 1 H 00 12 H 00 11 ds @x 2 @s Let w 00 = H 02 : Z s 2 s 1 H 00 01 H 00 02 ds x 1 + Z s 2 s 1 H 00 11 H 00 02 ds @x 1 @s + Z s 2 s 1 H 00 02 H 00 02 ds x 2 + Z s 2 s 1 H 00 12 H 00 02 ds @x 2 @s Let w 00 = H 12 : Z s 2 s 1 H 00 01 H 00 12 ds x 1 + Z s 2 s 1 H 00 11 H 00 12 ds @x 1 @s + Z s 2 s 1 H 00 02 H 00 12 ds x 2 + Z s 2 s 1 H 00 12 H 00 12 ds @x 2 @s Theelementalequationof y is: Z s 2 s 1 ˆ _s 2 @ 2 x @s 2 wds | {z } E 1 + Z s 2 s 1 2 ˆ _s @ 2 x @s@t wds | {z } E 2 + Z s 2 s 1 ˆ @ 2 x @t 2 wds | {z } E 3 Z s 2 s 1 @˙ @s @x @s wds | {z } E 4 Z s 2 s 1 ˙ @ 2 x @s 2 wds | {z } E 5 + Z s 2 s 1 EI @ 2 x @s 2 @ 2 w @s 2 ds | {z } E 6 = Z s 2 s 1 ˆgds | {z } E 7 TheproceduresaresimilartoaboveasitcontainsE1-E6termswitha nadditionalgravity termE7. E7: Z s 2 s 1 ˆgwds Let w = H 01 :Z s 2 s 1 ˆgH 01 ds Let w = H 11 :Z s 2 s 1 ˆgH 11 ds 112 Let w = H 02 :Z s 2 s 1 ˆgH 02 ds Let w = H 12 :Z s 2 s 1 ˆgH 12 ds Nowthearc-lengthconstraintwillbediscretized: X e 0 B B @ Z s 2 s 1 @x @s 2 wds | {z } E 8 + Z s 2 s 1 @y @s 2 wds 1 C C A = X e Z s 2 s 1 wds | {z } E 9 E8: Z s 2 s 1 @x @s 2 wds = Z s 2 s 1 H 0 01 ( s ) x 1 ( t )+ H 0 11 ( s ) @x 1 ( t ) @s + H 0 02 ( s ) x 2 ( t )+ H 0 12 ( s ) @x 2 ( t ) @s H 0 01 ( s ) x 1 ( t )+ H 0 11 ( s ) @x 1 ( t ) @s + H 0 02 ( s ) x 2 ( t )+ H 0 12 ( s ) @x 2 ( t ) @s wds Let w = N 1 : Z s 2 s 1 N 1 H 0 01 H 0 01 ˙ 1 x 1 ds x 1 + Z s 2 s 1 N 1 H 0 11 H 0 01 ˙ 1 x 1 ds @x 1 @s + Z s 2 s 1 N 1 H 0 02 H 0 01 ˙ 1 x 1 ds x 2 + Z s 2 s 1 N 1 H 0 12 H 0 01 ˙ 1 x 1 ds @x 2 @s + Z s 2 s 1 N 1 H 0 01 H 0 11 ˙ 1 @x 1 @s ds x 1 + Z s 2 s 1 N 1 H 0 11 H 0 11 ˙ 1 @x 1 @s ds @x 1 @s + Z s 2 s 1 N 1 H 0 02 H 0 11 ˙ 1 @x 1 @s ds x 2 + Z s 2 s 1 N 1 H 0 12 H 0 11 ˙ 1 @x 1 @s ds @x 2 @s + Z s 2 s 1 N 1 H 0 01 H 0 02 ˙ 1 x 2 ds x 1 + Z s 2 s 1 N 1 H 0 11 H 0 02 ˙ 1 x 2 ds @x 1 @s + Z s 2 s 1 N 1 H 0 02 H 0 02 ˙ 1 x 2 ds x 2 + Z s 2 s 1 N 1 H 0 12 H 0 02 ˙ 1 x 2 ds @x 2 @s + Z s 2 s 1 N 1 H 0 01 H 0 12 ˙ 1 @x 2 @s ds x 1 + Z s 2 s 1 N 1 H 0 11 H 0 12 ˙ 1 @x 2 @s ds @x 1 @s + Z s 2 s 1 N 1 H 0 02 H 0 12 ˙ 1 @x 2 @s ds x 2 + Z s 2 s 1 N 1 H 0 12 H 0 12 ˙ 1 @x 2 @s ds @x 2 @s 113 Let w = N 2 : Z s 2 s 1 N 2 H 0 01 H 0 01 ˙ 2 x 1 ds x 1 + Z s 2 s 1 N 2 H 0 11 H 0 01 ˙ 2 x 1 ds @x 1 @s + Z s 2 s 1 N 2 H 0 02 H 0 01 ˙ 2 x 1 ds x 2 + Z s 2 s 1 N 2 H 0 12 H 0 01 ˙ 2 x 1 ds @x 2 @s + Z s 2 s 1 N 2 H 0 01 H 0 11 ˙ 2 @x 1 @s ds x 1 + Z s 2 s 1 N 2 H 0 11 H 0 11 ˙ 2 @x 1 @s ds @x 1 @s + Z s 2 s 1 N 2 H 0 02 H 0 11 ˙ 2 @x 1 @s ds x 2 + Z s 2 s 1 N 2 H 0 12 H 0 11 ˙ 2 @x 1 @s ds @x 2 @s + Z s 2 s 1 N 2 H 0 01 H 0 02 ˙ 2 x 2 ds x 1 + Z s 2 s 1 N 2 H 0 11 H 0 02 ˙ 2 x 2 ds @x 1 @s + Z s 2 s 1 N 2 H 0 02 H 0 02 ˙ 2 x 2 ds x 2 + Z s 2 s 1 N 2 H 0 12 H 0 02 ˙ 2 x 2 ds @x 2 @s + Z s 2 s 1 N 2 H 0 01 H 0 12 ˙ 2 @x 2 @s ds x 1 + Z s 2 s 1 N 2 H 0 11 H 0 12 ˙ 2 @x 2 @s ds @x 1 @s + Z s 2 s 1 N 2 H 0 02 H 0 12 ˙ 2 @x 2 @s ds x 2 + Z s 2 s 1 N 2 H 0 12 H 0 12 ˙ 2 @x 2 @s ds @x 2 @s WecanapplythesameproceduresdoneforE8tothetermfor y .AndE9is: E9: Z s 2 s 1 wds Let w = N 1 :Z s 2 s 1 N 1 ds Let w = N 2 :Z s 2 s 1 N 2 ds 114 Elementmatrixassembly Allthetermsobtainedfromintegrationarecollectedtoformthema ss,Coriolis,stness matricesandtheelementforcevector.Theelementmassmatrix M e is: M e = 2 6 6 6 6 4 A 0 0 0 A 0 0 0 0 3 7 7 7 7 5 ~a (1) where ~a = x 1 @ x 1 @s x 2 @ x 2 @s y 1 @ y 1 @s y 2 @ y 2 @s 00 T and A = 2 6 6 6 6 6 6 6 4 R s 2 s 1 H 01 H 01 ds R s 2 s 1 H 11 H 01 ds R s 2 s 1 H 02 H 01 ds R s 2 s 1 H 12 H 01 ds R s 2 s 1 H 01 H 11 ds R s 2 s 1 H 11 H 11 ds R s 2 s 1 H 02 H 11 ds R s 2 s 1 H 12 H 11 ds R s 2 s 1 H 01 H 02 ds R s 2 s 1 H 11 H 02 ds R s 2 s 1 H 02 H 02 ds R s 2 s 1 H 12 H 02 ds R s 2 s 1 H 01 H 12 ds R s 2 s 1 H 11 H 12 ds R s 2 s 1 H 02 H 12 ds R s 2 s 1 H 12 H 12 ds 3 7 7 7 7 7 7 7 5 andtheelementCoriolismatrix C e is: C e =2 ˆ _s 2 6 6 6 6 4 B 0 0 0 B 0 0 0 0 3 7 7 7 7 5 ~ b (2) where ~ b = _x 1 @ _ x 1 @s _x 2 @ _ x 2 @s _y 1 @ _ y 1 @s _y 2 @ _ y 2 @s 00 T and B = 2 6 6 6 6 6 6 6 4 R s 2 s 1 H 0 01 H 01 ds R s 2 s 1 H 0 11 H 01 ds R s 2 s 1 H 0 02 H 01 ds R s 2 s 1 H 0 12 H 01 ds R s 2 s 1 H 0 01 H 11 ds R s 2 s 1 H 0 11 H 11 ds R s 2 s 1 H 0 02 H 11 ds R s 2 s 1 H 0 12 H 11 ds R s 2 s 1 H 0 01 H 02 ds R s 2 s 1 H 0 11 H 02 ds R s 2 s 1 H 0 02 H 02 ds R s 2 s 1 H 0 12 H 02 ds R s 2 s 1 H 0 01 H 12 ds R s 2 s 1 H 0 11 H 12 ds R s 2 s 1 H 0 02 H 12 ds R s 2 s 1 H 0 12 H 12 ds 3 7 7 7 7 7 7 7 5 115 andtheelementstnessmatrix K e is: K e = 2 6 6 6 6 4 C 0 D11 D12 0 C D21 D22 G 1 G 2 0 3 7 7 7 7 5 ~c (3) where ~c = x 1 @x 1 @s x 2 @x 2 @s y 1 @y 1 @s y 2 @y 2 @s ˙ 1 ˙ 2 T and C = c 1 c 2 c 1 = 2 6 6 6 6 6 6 6 6 6 4 ˆ _s 2 R s 2 s 1 H 00 01 H 01 ds + EI R s 2 s 1 H 00 01 H 00 01 dsˆ _s 2 R s 2 s 1 H 00 11 H 01 ds + EI R s 2 s 1 H 00 11 H 00 01 ds ˆ _s 2 R s 2 s 1 H 00 01 H 11 ds + EI R s 2 s 1 H 00 01 H 00 11 dsˆ _s 2 R s 2 s 1 H 00 11 H 11 ds + EI R s 2 s 1 H 00 11 H 00 11 ds ˆ _s 2 R s 2 s 1 H 00 01 H 02 ds + EI R s 2 s 1 H 00 01 H 00 02 dsˆ _s 2 R s 2 s 1 H 00 11 H 02 ds + EI R s 2 s 1 H 00 11 H 00 02 ds ˆ _s 2 R s 2 s 1 H 00 01 H 12 ds + EI R s 2 s 1 H 00 01 H 00 12 dsˆ _s 2 R s 2 s 1 H 00 11 H 12 ds + EI R s 2 s 1 H 00 11 H 00 12 ds 3 7 7 7 7 7 7 7 7 7 5 c 2 = 2 6 6 6 6 6 6 6 6 6 4 ˆ _s 2 R s 2 s 1 H 00 02 H 01 ds + EI R s 2 s 1 H 00 02 H 00 01 dsˆ _s 2 R s 2 s 1 H 00 12 H 01 ds + EI R s 2 s 1 H 00 12 H 00 01 ds ˆ _s 2 R s 2 s 1 H 00 02 H 11 ds + EI R s 2 s 1 H 00 02 H 00 11 dsˆ _s 2 R s 2 s 1 H 00 12 H 11 ds + EI R s 2 s 1 H 00 12 H 00 11 ds ˆ _s 2 R s 2 s 1 H 00 02 H 02 ds + EI R s 2 s 1 H 00 02 H 00 02 dsˆ _s 2 R s 2 s 1 H 00 12 H 02 ds + EI R s 2 s 1 H 00 12 H 00 02 ds ˆ _s 2 R s 2 s 1 H 00 02 H 12 ds + EI R s 2 s 1 H 00 02 H 00 12 dsˆ _s 2 R s 2 s 1 H 00 12 H 12 ds + EI R s 2 s 1 H 00 12 H 00 12 ds 3 7 7 7 7 7 7 7 7 7 5 and D11 = d 11 d 12 d 11 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 0 1 H 0 01 H 01 x 1 ds + R s 2 s 1 N 0 1 H 0 11 H 01 @x 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 01 x 2 ds + R s 2 s 1 N 0 1 H 0 12 H 01 @x 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 11 x 1 ds + R s 2 s 1 N 0 1 H 0 11 H 11 @x 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 11 x 2 ds + R s 2 s 1 N 0 1 H 0 12 H 11 @x 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 02 x 1 ds + R s 2 s 1 N 0 1 H 0 11 H 02 @x 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 02 x 2 ds + R s 2 s 1 N 0 1 H 0 12 H 02 @x 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 12 x 1 ds + R s 2 s 1 N 0 1 H 0 11 H 12 @x 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 12 x 2 ds + R s 2 s 1 N 0 1 H 0 12 H 12 @x 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 116 d 12 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 0 2 H 0 01 H 01 x 1 ds + R s 2 s 1 N 0 2 H 0 11 H 01 @x 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 01 x 2 ds + R s 2 s 1 N 0 2 H 0 12 H 01 @x 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 11 x 1 ds + R s 2 s 1 N 0 2 H 0 11 H 11 @x 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 11 x 2 ds + R s 2 s 1 N 0 2 H 0 12 H 11 @x 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 02 x 1 ds + R s 2 s 1 N 0 2 H 0 11 H 02 @x 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 02 x 2 ds + R s 2 s 1 N 0 2 H 0 12 H 02 @x 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 12 x 1 ds + R s 2 s 1 N 0 2 H 0 11 H 12 @x 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 12 x 2 ds + R s 2 s 1 N 0 2 H 0 12 H 12 @x 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 and D12 = d 21 d 22 d 21 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 1 H 00 01 H 01 x 1 ds + R s 2 s 1 N 1 H 00 11 H 01 @x 1 @s ds + R s 2 s 1 N 1 H 00 02 H 01 x 2 ds + R s 2 s 1 N 1 H 00 12 H 01 @x 2 @s ds R s 2 s 1 N 1 H 00 01 H 11 x 1 ds + R s 2 s 1 N 1 H 00 11 H 11 @x 1 @s ds + R s 2 s 1 N 1 H 00 02 H 11 x 2 ds + R s 2 s 1 N 1 H 00 12 H 11 @x 2 @s ds R s 2 s 1 N 1 H 00 01 H 02 x 1 ds + R s 2 s 1 N 1 H 00 11 H 02 @x 1 @s ds + R s 2 s 1 N 1 H 00 02 H 02 x 2 ds + R s 2 s 1 N 1 H 00 12 H 02 @x 2 @s ds R s 2 s 1 N 1 H 00 01 H 12 x 1 ds + R s 2 s 1 N 1 H 00 11 H 12 @x 1 @s ds + R s 2 s 1 N 1 H 00 02 H 12 x 2 ds + R s 2 s 1 N 1 H 00 12 H 12 @x 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 d 22 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 2 H 00 01 H 01 x 1 ds + R s 2 s 1 N 2 H 00 11 H 01 @x 1 @s ds + R s 2 s 1 N 2 H 00 02 H 01 x 2 ds + R s 2 s 1 N 2 H 00 12 H 01 @x 2 @s ds R s 2 s 1 N 2 H 00 01 H 11 x 1 ds + R s 2 s 1 N 2 H 00 11 H 11 @x 1 @s ds + R s 2 s 1 N 2 H 00 02 H 11 x 2 ds + R s 2 s 1 N 2 H 00 12 H 11 @x 2 @s ds R s 2 s 1 N 2 H 00 01 H 02 x 1 ds + R s 2 s 1 N 2 H 00 11 H 02 @x 1 @s ds + R s 2 s 1 N 2 H 00 02 H 02 x 2 ds + R s 2 s 1 N 2 H 00 12 H 02 @x 2 @s ds R s 2 s 1 N 2 H 00 01 H 12 x 1 ds + R s 2 s 1 N 2 H 00 11 H 12 @x 1 @s ds + R s 2 s 1 N 2 H 00 02 H 12 x 2 ds + R s 2 s 1 N 2 H 00 12 H 12 @x 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 and D21 = d 31 d 32 d 31 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 0 1 H 0 01 H 01 y 1 ds + R s 2 s 1 N 0 1 H 0 11 H 01 @y 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 01 y 2 ds + R s 2 s 1 N 0 1 H 0 12 H 01 @y 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 11 y 1 ds + R s 2 s 1 N 0 1 H 0 11 H 11 @y 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 11 y 2 ds + R s 2 s 1 N 0 1 H 0 12 H 11 @y 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 02 y 1 ds + R s 2 s 1 N 0 1 H 0 11 H 02 @y 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 02 y 2 ds + R s 2 s 1 N 0 1 H 0 12 H 02 @y 2 @s ds R s 2 s 1 N 0 1 H 0 01 H 12 y 1 ds + R s 2 s 1 N 0 1 H 0 11 H 12 @y 1 @s ds + R s 2 s 1 N 0 1 H 0 02 H 12 y 2 ds + R s 2 s 1 N 0 1 H 0 12 H 12 @y 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 117 d 32 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 0 2 H 0 01 H 01 y 1 ds + R s 2 s 1 N 0 2 H 0 11 H 01 @y 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 01 y 2 ds + R s 2 s 1 N 0 2 H 0 12 H 01 @y 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 11 y 1 ds + R s 2 s 1 N 0 2 H 0 11 H 11 @y 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 11 y 2 ds + R s 2 s 1 N 0 2 H 0 12 H 11 @y 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 02 y 1 ds + R s 2 s 1 N 0 2 H 0 11 H 02 @y 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 02 y 2 ds + R s 2 s 1 N 0 2 H 0 12 H 02 @y 2 @s ds R s 2 s 1 N 0 2 H 0 01 H 12 y 1 ds + R s 2 s 1 N 0 2 H 0 11 H 12 @y 1 @s ds + R s 2 s 1 N 0 2 H 0 02 H 12 y 2 ds + R s 2 s 1 N 0 2 H 0 12 H 12 @y 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 and D22 = d 41 d 42 d 41 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 1 H 00 01 H 01 y 1 ds + R s 2 s 1 N 1 H 00 11 H 01 @y 1 @s ds + R s 2 s 1 N 1 H 00 02 H 01 y 2 ds + R s 2 s 1 N 1 H 00 12 H 01 @y 2 @s ds R s 2 s 1 N 1 H 00 01 H 11 y 1 ds + R s 2 s 1 N 1 H 00 11 H 11 @y 1 @s ds + R s 2 s 1 N 1 H 00 02 H 11 y 2 ds + R s 2 s 1 N 1 H 00 12 H 11 @y 2 @s ds R s 2 s 1 N 1 H 00 01 H 02 y 1 ds + R s 2 s 1 N 1 H 00 11 H 02 @y 1 @s ds + R s 2 s 1 N 1 H 00 02 H 02 y 2 ds + R s 2 s 1 N 1 H 00 12 H 02 @y 2 @s ds R s 2 s 1 N 1 H 00 01 H 12 y 1 ds + R s 2 s 1 N 1 H 00 11 H 12 @y 1 @s ds + R s 2 s 1 N 1 H 00 02 H 12 y 2 ds + R s 2 s 1 N 1 H 00 12 H 12 @y 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 d 42 = 2 6 6 6 6 6 6 6 6 6 4 R s 2 s 1 N 2 H 00 01 H 01 y 1 ds + R s 2 s 1 N 2 H 00 11 H 01 @y 1 @s ds + R s 2 s 1 N 2 H 00 02 H 01 y 2 ds + R s 2 s 1 N 2 H 00 12 H 01 @y 2 @s ds R s 2 s 1 N 2 H 00 01 H 11 y 1 ds + R s 2 s 1 N 2 H 00 11 H 11 @y 1 @s ds + R s 2 s 1 N 2 H 00 02 H 11 y 2 ds + R s 2 s 1 N 2 H 00 12 H 11 @y 2 @s ds R s 2 s 1 N 2 H 00 01 H 02 y 1 ds + R s 2 s 1 N 2 H 00 11 H 02 @y 1 @s ds + R s 2 s 1 N 2 H 00 02 H 02 y 2 ds + R s 2 s 1 N 2 H 00 12 H 02 @y 2 @s ds R s 2 s 1 N 2 H 00 01 H 12 y 1 ds + R s 2 s 1 N 2 H 00 11 H 12 @y 1 @s ds + R s 2 s 1 N 2 H 00 02 H 12 y 2 ds + R s 2 s 1 N 2 H 00 12 H 12 @y 2 @s ds 3 7 7 7 7 7 7 7 7 7 5 118 and G 1 = 2 6 4 g 11 g 12 g 13 g 14 g 21 g 22 g 23 g 24 3 7 5 with g 11 = Z s 2 s 1 N 1 H 0 01 H 0 01 x 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 01 @x 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 01 x 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 01 @x 2 @s ds g 12 = Z s 2 s 1 N 1 H 0 01 H 0 11 x 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 11 @x 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 11 x 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 11 @x 2 @s ds g 13 = Z s 2 s 1 N 1 H 0 01 H 0 02 x 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 02 @x 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 02 x 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 02 @x 2 @s ds g 14 = Z s 2 s 1 N 1 H 0 01 H 0 12 x 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 12 @x 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 12 x 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 12 @x 2 @s ds g 21 = Z s 2 s 1 N 2 H 0 01 H 0 01 x 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 01 @x 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 01 x 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 01 @x 2 @s ds g 22 = Z s 2 s 1 N 2 H 0 01 H 0 11 x 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 11 @x 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 11 x 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 11 @x 2 @s ds g 23 = Z s 2 s 1 N 2 H 0 01 H 0 02 x 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 02 @x 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 02 x 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 02 @x 2 @s ds g 24 = Z s 2 s 1 N 2 H 0 01 H 0 12 x 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 12 @x 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 12 x 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 12 @x 2 @s ds and G 2 = 2 6 4 g 11 g 12 g 13 g 14 g 21 g 22 g 23 g 24 3 7 5 with 119 g 11 = Z s 2 s 1 N 1 H 0 01 H 0 01 y 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 01 @y 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 01 y 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 01 @y 2 @s ds g 12 = Z s 2 s 1 N 1 H 0 01 H 0 11 y 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 11 @y 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 11 y 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 11 @y 2 @s ds g 13 = Z s 2 s 1 N 1 H 0 01 H 0 02 y 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 02 @y 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 02 y 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 02 @y 2 @s ds g 14 = Z s 2 s 1 N 1 H 0 01 H 0 12 y 1 ds + Z s 2 s 1 N 1 H 0 11 H 0 12 @y 1 @s ds + Z s 2 s 1 N 1 H 0 02 H 0 12 y 2 ds + Z s 2 s 1 N 1 H 0 12 H 0 12 @y 2 @s ds g 21 = Z s 2 s 1 N 2 H 0 01 H 0 01 y 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 01 @y 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 01 y 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 01 @y 2 @s ds g 22 = Z s 2 s 1 N 2 H 0 01 H 0 11 y 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 11 @y 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 11 y 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 11 @y 2 @s ds g 23 = Z s 2 s 1 N 2 H 0 01 H 0 02 y 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 02 @y 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 02 y 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 02 @y 2 @s ds g 24 = Z s 2 s 1 N 2 H 0 01 H 0 12 y 1 ds + Z s 2 s 1 N 2 H 0 11 H 0 12 @y 1 @s ds + Z s 2 s 1 N 2 H 0 02 H 0 12 y 2 ds + Z s 2 s 1 N 2 H 0 12 H 0 12 @y 2 @s ds andtheelementforcevector F e is: F e = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 0 0 0 0 R s 2 s 1 ˆgH 01 ds R s 2 s 1 ˆgH 11 ds R s 2 s 1 ˆgH 02 ds R s 2 s 1 ˆgH 12 ds R s 2 s 1 N 1 ds R s 2 s 1 N 2 ds 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 120 Globalmatrixassembly Toassembletheglobalmatricesandforcevector,weconstructa connectivitymatrix( CN ) forthedegreesoffreedomofalltheelements.Notethat n -thelementisdenotedas e n .Eachelementhastwonodesandatotaloftendegreesoffreedo m(DOFs)inwhich thecorrespondingDOFcanbereferredtotheDOFvectorofthe rstelementwhere x 1 ! DOF1 ;@x 1 @s ! DOF2 ;y 1 ! DOF3 ;@y 1 @s ! DOF4 ;˙ 1 ! DOF5 ;x 2 ! DOF6 ;@x 2 @s ! DOF7 ;y 2 ! DOF8 ;@y 2 @s ! DOF9 ;˙ 2 ! DOF10.DOF1 DOF5correspondtotherstnode andDOF6 DOF10correspondtothesecondnode. CN = 0 B B B B B B B B B B B B B B @ DOF1DOF2DOF3DOF4DOF6DOF7DOF8DOF9DOF5DOF10 e 1 x 1 @x 1 @s y 1 @y 1 @s x 2 @x 2 @s y 2 @y 2 @s ˙ 1 ˙ 2 e 2 x 2 @x 2 @s y 2 @y 2 @s x 3 @x 3 @s y 3 @y 3 @s ˙ 2 ˙ 3 e 3 x 3 @x 3 @s y 3 @y 3 @s x 4 @x 4 @s y 4 @y 4 @s ˙ 3 ˙ 4 e 4 x 4 @x 4 @s y 4 @y 4 @s x 5 @x 5 @s y 5 @y 5 @s ˙ 4 ˙ 5 . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .e n x n @x n @s y n @y n @s x 1 @x 1 @s y 1 @y 1 @s ˙ n ˙ 1 1 C C C C C C C C C C C C C C A Using CN ,theglobalmass,Coriolis,andstnessmatrices( M;C;K )andtheglobalforce vector F canbeconvenientlyassembled.WedenotetheglobalDOFvectoras U whose dimensionis5 n 1.Thenaldiscretizedequationsofmotion: M ( ˆ ) U + C ( ˆ; _s ) _U + K ( ˆ; _s;U ) U = F ( ˆ ) where M isa5 n 5 n matrix, C isa5 n 5 n matrix, K isa5 n 5 n matrixand F isa5 n 1 vector. 121 BIBLIOGRAPHY 122 BIBLIOGRAPHY [1]RaphLevien.Theelastica:amathematicalhistory. ElectricalEngineeringandCom- puterSciencesUniversityofCaliforniaatBerkeley ,2008. 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