VIBRATIONSUPPRESSIONINSIMPLETENSION-ALIGNEDSTRUCTURESByTingliCaiADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofMechanicalEngineeringŒDoctorofPhilosophy2016ABSTRACTVIBRATIONSUPPRESSIONINSIMPLETENSION-ALIGNEDSTRUCTURESByTingliCaiTension-alignedstructureshavebeenproposedforspace-basedantennaapplicationsthatrequirehighdegreeofaccuracy.Thistypeofstructuresusecompressionmemberstoimparttensionontheantenna,thushelpingtomaintaintheshapeandfacilitatedisturbancerejection.Thesestruc-turescanbeverylargeandthereforesensitivetolow-frequencyexcitations.Inthisstudy,twocontrolstrategiesareproposedforthepurposeofvibrationsuppression.First,asemi-activecon-trolstrategyfortension-alignedstructuresisproposed,basedontheconceptofstiffnessvariationbysequentialapplicationandremovalofconstraints.Theprocessfunnelsvibrationenergyfromlow-frequencytohigh-frequencymodesofthestructure,whereitisdissipatednaturallyduetointernaldamping.Inthisstrategy,twomethodsofstiffnessvariationwereinvestigated,including:1)variablestiffnesshingesinthepanelsand2)variablestiffnesselasticbarsconnectingthepanelstothesupportstructure.Two-dimensionalandthree-dimensionalmodelswerebuilttodemon-stratetheeffectivenessofthecontrolstrategy.Thesecondcontrolstrategyproposedisanactiveschemewhichusessensorfeedbacktodonegativeworkonthesystemandtosuppressvibration.Inparticular,itemploysaslidingmechanismwheretheconstraintforceismeasuredinrealtimeandthisinformationisusedasfeedbacktoprescribethemotionofthesliderinsuchawaythatthevibrationenergyisreducedfromthestructurecontinuouslyanddirectly.Theinvestigationoftheslidingmechanismwasperformednumericallyusingthemodelofanonlinearbeam.Practicalissuesofthiscontrolschemehavebeenconsideredandmeasuressuchasaddingalow-passwastakentoeaserequirementsonthecontrolhardware.Ithasbeenshowninsimulationsthatthesetwocontrolstrategiesareeffectivemechanismstoremoveenergyfromavibratingsystem.Tovalidatethecontrolstrategies,anexperimentalsetupwasbuilt.A3.66meterlongaluminumbeamwasplacedonarigidbenchwithatensiondeviceappliedatoneend.Abelt-drivenactuatorcarriedaslider,whichmovedaxiallyalongthesurfaceofthebeam.Ontheslidinginterface,thesliderimposedaconstraintonthebeamtomaintainzerotransversedisplacement.Rotationattheslidingcontactpoint,controlledbyanelectromagneticbrake,couldbeedinstantaneously,orallowedtovaryfreely.Thesliderwasequippedwithstraingaugesandanencodertomeasuretheconstraintforcefromthebeam.Thesensordatawasfedbackandprocessedinreal-timebyacontrolalgorithmimplementedonaDSPboard.Differentcontrolstrategycombinationshavebeenexperimentedonthesystem.Resultsshowedthat,withlightmaterialdampingpresentinthestructure,thetwocontrolstrategieseffectivelyredistributedthevibrationenergyintothehigh-frequencymodes,whereitwasdissipatednaturallyandquickly.CopyrightbyTINGLICAI2016ACKNOWLEDGEMENTSIwouldliketoexpressmydeepestappreciationtothetwodearadvisorsofmine,Dr.RanjanMukherjeeandDr.AlejandroR.Diaz.vTABLEOFCONTENTSLISTOFTABLES.......................................ixLISTOFFIGURES.......................................xCHAPTER1INTRODUCTIONTOTENSION-ALIGNEDSTRUCTURES........1CHAPTER2VIBRATIONSUPPRESSIONTHROUGHSTIFFNESSVARIATION...52.1Atwo-DOFillustrativeexample...........................52.2Stiffnessvariationinmulti-DOFsystems.......................92.3Modelofatwo-dimensionalsimpletension-alignedstructure............132.3.1Nonlineardynamicmodelofthesupportstructure..............132.3.2Staticequilibriumofthesupportstructure...........152.3.3Lineardynamicmodelofthesupportstructure................162.3.4Hingedpanelarraymodel..........................192.3.5Methodsofstiffnessvariation........................202.3.6Numericalsimulation.............................212.4Modelofathree-dimensionaltension-alignedstructure...............26CHAPTER3VIBRATIONSUPPRESSIONUSINGASLIDINGMECHANISM.....293.1Slidingmechanismdescription............................293.2Equationsofmotionforanonlinearbeamwithaedslider............303.3Numericsimulationoftheslidingmechanismofanonlinearbeam.........343.3.1Finiteelementdiscretizationusingadaptivemesh..............343.3.2Numericissueofcomputingtheconstraintforce:Lagrangemultipliervs.penaltymethod..............................403.4Feedbackcontroldesignoftheslidermotion.....................41vi3.4.1Preliminarydesignoftheslidermotioncontrol...............413.4.2controldesign............................443.5Combiningslidermotionwithstiffnessvariation...................473.5.1Simulationofthesystemundergoingfreevibration.............483.5.2Simulationofthesystememployingslidingcontrolonly..........533.5.3Simulationofthesystememployingslidingcontrolcombinedwithstiff-nessvariation.................................56CHAPTER4EXPERIMENTALSTUDYOFVIBRATIONSUPPRESSIONTHROUGHSTIFFNESSVARIATIONANDSLIDINGMECHANISM..........634.1Experimentdesign...................................634.1.1Tension-alignedstructure...........................634.1.2Stiffnessvariationmechanism........................644.1.3Slidingmechanism..............................654.2Experimentimplementation..............................654.2.1Anonlinearbeam...............................654.2.2Addedmass..................................664.2.3Tensiondevice................................664.2.4Slidingmotioncontrol............................674.2.5Rotationalon/offmechanism.........................684.2.6Slidercontactinterface............................684.2.7Measurementforfeedbackinformationandthesystemstatus........694.2.8Dataacquisition................................704.2.9Real-timecontrolanddataprocessingalgorithm...............714.2.10Initialdisplacementholder..........................714.2.11Completesystem...............................724.3Experimentresults...................................72vii4.3.1Freevibration.................................724.3.2Performanceofcontrolemployingslidingmechanismonly.........794.3.3Performanceofcontrolemployingstiffnessvariationmechanismonly...824.3.4Performanceofcontrolcombiningstiffnessvariationandslidingmechanism854.3.5Discussion...................................89CHAPTER5SUMMARYANDFUTUREWORK.....................91BIBLIOGRAPHY........................................94viiiLISTOFTABLESTable2.1Parametersusedinthe2-DOFsimulations.....................6Table2.2PropertiesofSimulatedTension-AlignedStructure................22Table2.3Firstsixnaturalfrequenciesoftheunconstrainedandtheconstrainedtension-alignedstructureinrad/s...............................24Table3.1Propertiesandgeometryofthesimulatedbeamwithaslidingconstraint......38Table3.2Properties,geometryandotherparametersofthebeamsystemusedinthevalidationsimulation.................................48Table3.3Comparisonofperformanceofdifferentcontrolstrategies.Allrelativedis-placementsandaccelerationsarecomputedusingthemass1initialstatusasreference.......................................62Table4.1Propertiesandgeometryofthebeamusedinexperiment..............66Table4.2Comparisonofperformanceofdifferentcontrolstrategies.Allrelativevalueswerecomputedusingthemass1initialaccelerationasreference..........89ixLISTOFFIGURESFigure1.1Atension-alignedstructurecomprisedofasupportstructureandasensorsurface-takenfromJonesetal.Jonesetal.(2008)................2Figure2.1Atwodegree-of-freedommass-spring-dampersystem...............5Figure2.2TotalenergyanddisplacementsoftheUnconstrainedsystem-(a),(b);TotalenergyanddisplacementsoftheConstrainedsystem-(c),(d)..........7Figure2.3Switchedsystem:(a)Totalenergy;(b)energyassociatedwiththelow-frequencymode(s);(c)energyassociatedwiththehigh-frequencymode;(d)displace-mentsofthemasses;(e),(f)modaldisplacements.Inallofthesefiuc"andfic"denotetheunconstrainedandconstrainedstatesofthesystem.Aviewofthemodaldisplacementisshownintheconstrainedstates.8Figure2.4Vibrationsuppressionthroughenergyfunnelingfromlow-frequencymodes(LFM)intohigh-frequencymodes(HFM).....................11Figure2.5Atension-alignedstructureformedbyconnectingasupportstructure(incompression)toanarrayofhingedpanels(intension)...............12Figure2.6Aplanarelasticaarch...............................13Figure2.7Thearrayofhingedpanels............................20Figure2.8Stiffnessvariationinthetension-alignedstructureisrealizedusingtwometh-ods:(A)and(B);thesearedescribedinsection3.4................20Figure2.9Theeight-paneltension-alignedstructureusedinsimulations...........21xFigure2.10PlotofenergyforthethreecasesdiscussedinSection4.2.............23Figure2.11Plotofenergyforthethreecasessimulated....................25Figure2.12Plotsofthetransversedisplacementsofthreepointsonthehingedpanelarray(seeFig.2.9),inthecaseofmethods(A)and(B)combined............26Figure2.13Overviewofthetension-alignedthree-dimensionalstructure.8panelsareconnectedusing7hingesandsupportedbyatrussstructure............26Figure2.14Geometryofthethree-dimensionalstructure....................27Figure2.15Plotofenergyforcasesofcontrollingdifferentnumberofjoints.........28Figure3.1Apinned-pinnedbeamwithaslidingconstraint..................30Figure3.2Plotsofthesysteminfreevibration:normalizedenergy;transversedisplace-mentsampledatS=0:75ors=2:743m;materialcoordinateofthepointQincontactwiththeslider;constraintforceintransversedirection;constraintforceinaxialdirection...............................39Figure3.3Preliminaryfeedbackcontroldesign........................43Figure3.4Plotsofthesystemappliedwithdirectcontrol:normalizedenergyofthedirectcontrolresultsinsolidline,normalizedenergyoffreevibrationindashedlineasreference;transversedisplacementsampledatS=0:75ors=2:743m;sliderposition;slidervelocity;constraintforceinhorizontaldirection.......................................44Figure3.5controldesignwith.........................45xiFigure3.6Plotsofthesystemappliedwithdirectcontrol:normalizedenergyofthecontrolresultsinsolidline,normalizedenergyoffreevibrationindashedlineasreference,normalizedenergyofthedirectcontrolresultsindotted-solidlineasreference;transversedisplacementsampledatS=0:75ors=2:743m;sliderposition;slidervelocity;constraintforceinhorizontaldirectionafterthelow-pass..........................46Figure3.7Simulationofthesystemunderfreevibration(timedomain).Plotsfromtoptobottom:axialforceevaluatedontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2............50Figure3.8Simulationofthesystemunderfreevibration(frequencydomain).Plotsfromtoptobottom:axialforceevaluatedontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;displacementofmass1;displacementofmass2........................................51Figure3.9Simulationofthemassaccelerationsunderfreevibration.............52Figure3.10Simulationofthesystemenergyunderfreevibration...............53Figure3.11Simulationofthesystemunderslidingcontrolonly.Plotsfromtoptobot-tom:axialforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2..................54Figure3.12Simulationofthesystemenergyunderslidingcontrolonly.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion.....................55xiiFigure3.13Simulationofthesystemundercombinedcontrolofstiffnessvariationandforwardslidingmechanism.Plotsfromtoptobottom:axialforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2...............................57Figure3.14Simulationofthesystemenergyundercombinedcontrolofstiffnessvaria-tionandforwardslidingmechanism.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion....................................58Figure3.15Simulationofthesystemundercombinedcontrolofstiffnessvariationandreverseslidingmechanism.............................59Figure3.16Simulationofthesystemenergyundercombinedcontrolofstiffnessvaria-tionandreverseslidingmechanism.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion....................................61Figure4.1Beltdriveactuatortoprovideslidingmotion....................67Figure4.2Theslidercarriagethatconsistsofrotationalon/offmechanismandencodertomeasuretherotation.Aslidershaftistobeassembledontothecarriage...68Figure4.3Slidercontactinterfacewithforcesensors.....................69Figure4.4Assembledpartsofslidingandrotationalmechanismedonthesupporttableincontactwiththevibratingbeam......................70Figure4.5Interfaceofthereal-timecontrolanddataprocessingsoftware..........71xiiiFigure4.6Completeexperimentsystembuiltafterdesignimplementation..........72Figure4.7Experimentresultsofthesysteminfreevibration(timedomain).Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.............73Figure4.8Experimentresultsofthesysteminfreevibration(frequencydomain).Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;accelerationofmass1;accelerationofmass2......................................75Figure4.9Axialforceonslider,post-processedfromexperimentresultsofthesysteminfreevibration(timedomain)...........................76Figure4.10Axialforceonslider,post-processedfromexperimentresultsofthesysteminfreevibration(frequencydomain)........................77Figure4.11Experimentresultsofthesystemunderslidingcontrolonly.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.............78Figure4.12Axialforceonsliderintimedomain,processedfrommeasurementsoftotalforceonsliderandsliderslope...........................80Figure4.13Slidervelocityintimedomain,processedfromthemeasurementofsliderposition.......................................80Figure4.14Axialforceonsliderinfrequencydomain.....................82xivFigure4.15Slidervelocityinfrequencydomain........................83Figure4.16Experimentresultsofthesystemunderstiffnessvariationcontrolonly.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2..........84Figure4.17Experimentresultsofthesystemundercombinedcontrolofstiffnessvaria-tionandforwardslidingmechanism.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldis-placement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2..........................86Figure4.18Experimentresultsofthesystemundercombinedcontrolofstiffnessvari-ationandreverseslidingmechanism.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldis-placement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2..........................88xvCHAPTER1INTRODUCTIONTOTENSION-ALIGNEDSTRUCTURESLargespacestructuresarecontemplatedforuseasspace-basedradarsforimagingandmovingob-jectandtracking.Theseradarsconsistofalargesupportstructureandphasedarrayantennasattachedtothisstructure.Thecompletesystemhastobedesignedsothatitcanbefoldedintoacompactvolumeforeaseoftransport,andsothatthephasedarrayantennascanmaintainahighdegreeofaccuracyafterdeployment.Ahighdegreeofaccuracyisdiftoachievesincethesestructuresarelargeandsensitivetodisturbancesthatresultinvibration.Thestructurecanbemodelledanditsdeformationscanbemeasuredandcorrectedusingsensorsandactuatorsinrealtime,butsuchsystemsareextremelychallengingtoengineer.Thisisbecausethesestructuresaredesignedwithnumerousandcomplexjointsandmechanismsforfoldingthatintroducenonlin-earitiessuchasslipping,backlashanddeadband.Model-basedcontrolalsorequiresdevelopmentandofahigh-dimensionalmathematicalmodelthataccountsforthenonlinearitiesandtheintegrationofcontrolsystemhardwareintothestructure.Vibrationsuppres-sioninlargespacestructuresisachallengingproblemandapracticalsolutionneedspassiveorsemi-activecontrol,oranactivecontrolschemeusingjustfewsensorsandactuators.Tomeettheprecisionrequirementsofspace-basedradars,tension-alignedstructureshavebeenproposedbyMikulasetal.(2008)andJonesetal.(2008).Similartoabowwithastring,inatension-alignedstructure,thearrayantennasareattachedtothesupportstructureviatensionersateachend(seeFig.1.1);thesupportstructureisusedasacompressionmembertoimparttensiontothearrayantennaspinnedateachend.Thetensionintheantennaarrayhelpsmaintainbut,moreimportantly,increasesthestiffnessofthearray,whichisnecessaryfordisturbancerejection(Adleretal.,1998).Otheroftension-alignedstructuresincludeeliminationofthehighdimensionalaccuracyrequirementsofthesupportstructure,greaterxibilityindesign(sincethesupportstructureandthearrayantennascanbeseparatelypackagedanddeployed),andreduced1effectofnonlinearitiessuchasdeadbandandbacklash(Jonesetal.,2008);compensationforcreepandmanufacturingtolerancebuild-up(Winslow,1993);andincreaseinstructuraldamping(Fang&Lyons,1996)whichfacilitatesvibrationsuppression.Thetension-alignedarchitectureisequallywell-suitedforradardesignswheretheantennasareanarrayofpanelsoraxiblemembrane(Kemerley&Kiss,2000;Jeon&Murphey,2012;Footdaleetal.,2012).Toinvestigatethefeasibilityoftension-alignedarchitectures,Jonesetal.(2008),Jonesetal.(2007)studiedtheeffectoftensiononthestiffnessofalargeapertureantenna.UsingtheDARPAISAT1astherepresentativeplatform,nonlinearmethodswereusedtocomputethesystemfrequencieswithsensorsurfacesrangingfromgossamerstopaneledradars.Forafree-freesupportstructure,itwasshownthatitisnotpossibletoatension/massratiocombinationthatyieldsahigherorevenequivalentpeakfrequencytothatofthestructure.Thisimpliesthattheadditionofthesensorsurfacewillonlyreducethefundamentalfrequencyoftheoverallsystem.Thisproblemcanbealleviatedbyintroducingaloadoffsetsuchthattensionaddsbendingmomentsonthesupportstructure.However,aloadoffsetincreasesthedeformationofthesupportstructureFigure1.1Atension-alignedstructurecomprisedofasupportstructureandasensorsurface-takenfromJonesetal.Jonesetal.(2008).1InnovativeSpace-basedradarAntennaTechnology2andthetensionrequiredtoachievethesamefundamentalfrequencyofISATisclosetothebucklingload.Toeliminateproblemsrelatedtobucklingandlargedeformation,Jonesetal.(2008)proposedtointroduceintermediateconnectorsandguy-wirestoprovidecounter-tension.Thetension-alignedarchitectureproposedbyJonesetal.(2008)canbeviewedasapassivemethodforvibrationsuppressionwherethelocationoftheintermediateconnectorsandinternalstressinthestructureareoptimizedtoattainthesamelevelofstructuralstiffnessastheISATplatform.Thismaynotbesufformeetingthehighaccuracyrequirementsofspace-basedradarssincespacestructuressuchastheISATplatformarelargeandpronetolow-frequencyexcitation.Thetensionrequiredtoachievethedesiredlevelofstiffnessisalsohighandmaynotbesuitableforlong-termoperation.Toaddressthesechallenges,twocontrolstrategiesfortension-alignedstructuresareproposedinthispaper.Strategy1isbasedontheconceptofstiffnessvariation.Thismethodappliesandre-movesconstraintscyclicallysuchthatvibrationenergyisfunneledintothehigh-frequencymodesofthestructure,whereitcanbedissipatedquicklyandnaturallyduetohighratesofinternaldamping.Strategy2employsaslidermechanismwhichappliesmovingloadsonthesurfaceofthestructure.Thereisaconstraintforceonthestructureappliedbytheslider.Usingthismeasuredconstraintforceasfeedback,themotionofthesliderisprescribedsuchthatitdoesnegativeworkonthecontrolledstructureandvibrationthereforeissuppressed.Bothcontrolstrategiesrequirefewersensorsandactuatorsandeliminatestheneedforextensivecomputationsbasedonamath-ematicalmodelofthestructure,comparedtotraditionalactivecontrol,whichreliesonaccuratemeasurementbysensors,carefulcompensationbyactuatorsanddetailedmathematicalmodellingofthecontrolsystem.Severalresearchers(Onodaetal.,1991,1992;Clark,2000;Corr&Clark,2001;Ramaratnam&Jalili,2006)haveexploredstiffnessvariationasamethodforvibrationsuppression.Inalltheseworks,variable-stiffnesselementsareplacedinastateofhighstiffnessandenergyisstoredinthem.Oncethestoredenergyreachesamaximumvalue,thestiffnessoftheelementisswitchedtoalowvaluetodissipateenergy.IntheworkofDiaz&Mukherjee(2006b,2008);Issaetal.3(2009),stiffnessvariationwasachievedthroughapplicationandremovalofconstraints;andenergydissipationwasaccomplishedthroughatargetedandpurposefulenergyredistributionfromlow-frequencymodestohigh-frequencymodes.Thisthesisextendedthoseworktoasimpletension-alignedstructure.Slidingmechanismintheformofconstraintsinxiblestructureshasbeeninvestigatedbymanyresearchers,especiallyinareassuchascontactmechanicsandxiblemultibodydynamics,forexampleinresearchbyBarhorst(2004);Hong&Ren(2011).Therearealsostudieswherecon-straintforcesareusedtodeterminethestatesofthesystemSakamoto&Park(2006).However,theideaofdirectlymanipulatingtheconstraintforcetoreducesystemenergyhasnotbeenproposed.Suchanideaisdevelopedinthisthesisintoamethodthatisimplementableintension-alignedstructures.Theplanofthisthesisisasfollows.Chapter2introducesthecontrolstrategyusingstiffnessvariationforapplicationinmulti-doflinearsystemsandafimodaldisparityindex",amet-ricthatcanbeusedtodeterminetheefyofourcontrolstrategy.Bothtwo-dimensionalandthree-dimensionalmodelsarepresented.Chapter3considerstheslidingmechanismonanonlin-earbeam.Aftermodellingthestructureandsolvingfortheconstraintforce,thecontrolschemewasformulatedandvnumerically.Chapter4presentstheexperimentalstudyofthetwoproposedvibrationsuppressionstrategies.Chapter5summarizesthetwomethodsindifferentapplicationconditionandproposetheworktobedoneasthenextstep.4CHAPTER2VIBRATIONSUPPRESSIONTHROUGHSTIFFNESSVARIATION2.1Atwo-DOFillustrativeexampleConsiderthetwodegree-of-freedommass-spring-dampersysteminFig.2.1.Thetwomassesm1andm2areconnectedtoedsupportsbyspringsofstiffnessk1andk2,andtoeachotherbythespringoftime-varyingstiffnessk3(t).Thedisplacementsofthetwomassesaredenotedbyx1andx2andthespringsareundeformedwhenthemassesareintheirequilibriumi.e.,x1=x2=0.Weassume(k1=m1)6=(k2=m2)suchthatthetwomasseshavedifferentnaturalfrequencieswhenk3(t)=0.Theequationofmotionofthetwodegree-of-freedomsystemisgivenbelow264m100m2375264¨x1¨x2375+264k1+k3(t)k3(t)k3(t)k2+k3(t)375264x1x2375=26400375(2.1)Nowconsiderthreecaseswherethestiffnessk3(t)ischosendifferentlyUnconstrained:k3(t)=0Constrained:k3(t)=krSwitched:k3(t)=8><>:0ift2[ti;ti+1)krift2[ti+1;ti+2);i=0;2;4;(2.2)Itisassumedthatkrislargecomparedtok1andk2,andtimestn,n=0;1;2;;arechosensuchthatx2(tn)x1(tn)=0.ThisensuresthatnoenergyisremovedfromthesystemwhentheFigure2.1Atwodegree-of-freedommass-spring-dampersystem.5stiffnessisswitchedfromkrto0,oraddedtothesystemwhenthestiffnessisswitchedfrom0tokr.Assumingmodaldampingwithuniformdampingratioz,theequationofmotionofthesystemcanbewritteninmodalcoordinatesasfollows:264¨q1¨q2375+2z264W1(t)00W2(t)375264q1q2375+264W21(t)00W22(t)375264q1q2375=26400375(2.3)whereq1andq2arethemodalcoordinates,andWj(t),j=1;2,arethenaturalfrequenciesofthesystem.Forthethreedifferentcases,thenaturalfrequenciesaredenotedasfollows:Unconstrained:Wj(t)=wjConstrained:Wj(t)=¯wjSwitched:Wj(t)=8><>:wjift2[ti;ti+1)¯wjift2[ti+1;ti+2);i=0;2;4;(2.4)wherewj=qkj=mj,j=1;2.Theexpressionsfor¯wjarecomplicatedandarenotprovidedhere.Table2.1Parametersusedinthe2-DOFsimulationsm1(kg)m2(kg)k1(N/m)k2(N/m)kr(N/m)z1:002:002:002:00200000:001Simulationswereperformedforthethreecasesdiscussedabove,usingparametersinTable2.1andthesamesetofinitialconditions.ThenaturalfrequenciesoftheUnconstrainedandCon-strainedsystemswerefoundtobe(w1;w2)=(1:4142;1:0000);(¯w1;¯w2)=(1:1547;1:7321102)(2.5)wheretheunitsarerad/s.OneofthefrequenciesfortheConstrainedsystemwashighrelativetotheothernaturalfrequencies.Thisfrequencywasassociatedwiththerelativemotionofthetwomasses,whentheywereconnectedbythestiffspringkr.ThesimulationresultsareshowninFigs.2.2and2.3.ThetotalenergyandthedisplacementsoftheUnconstrainedandConstrainedsystemsareshowninFig.2.2.FortheUnconstrainedsystem,6thetotalenergyofthesystemdecayedveryslowly;only10.5%wasdissipatedin42.7s.Thisisbecauseoflowinternaldampingassociatedwithlownaturalfrequenciesofthesystem.FortheConstrainedsystem,thetotalenergydecayedrapidlyinitially,butslowlythereafter;35.9%wasdissipatedin42.7s.OnenaturalfrequencyoftheConstrainedsystemwashighandrapiddecayoftheenergyassociatedwiththismodecontributedtotheinitialrapiddecayofthetotalenergy.ThedisplacementsofthetwomassesoftheConstrainedsystemappearedtobeidentical.Thisisbecauseofsmallrelativemotionofthemasses,aconsequenceofhighstiffnessofthespringconnectingthem.ForboththeUnconstrainedandtheConstrainedsystemsinFig.2.2,asmallfractionoftheenergywasdissipated.Incontrast,theenergyoftheSwitchedsystem(seeFig.2.3)decayedsig-faster;83.4%wasdissipatedin42.7s.FortheSwitchedsystem,thetwomasseswereinitiallyunconstrained.Theywereconnected(constrained)bythespringatt1=11:24s,released(unconstrained)att2=21:25s,andagainconnectedatt3=32:70s.Asmentionedearlier,t1,t2andt3werechosensuchthatnoenergywasaddedtoorsubtractedfromthesystemduringtheprocessofapplicationorremovaloftheconstraint(stiffnessswitching).Att=t1andt=t3,appli-Figure2.2TotalenergyanddisplacementsoftheUnconstrainedsystem-(a),(b);TotalenergyanddisplacementsoftheConstrainedsystem-(c),(d)7Figure2.3Switchedsystem:(a)Totalenergy;(b)energyassociatedwiththelow-frequencymode(s);(c)energyassociatedwiththehigh-frequencymode;(d)displacementsofthemasses;(e),(f)modaldisplacements.Inallofthesefiuc"andfic"denotetheunconstrainedandconstrainedstatesofthesystem.Aviewofthemodaldisplacementisshownintheconstrainedstates.cationoftheconstraintcreatedahigh-frequencymodeandfunneledenergyintothismode,whereitwasdissipatedquickly;thiscanbevfromtheenergyplotsofthelow-andhigh-frequencymodes.AsinthecaseoftheConstrainedsysteminFig.2.2,thedisplacementplotsofthetwomassesfortheSwitchedsystemappeartobeidenticalwhentheywereconstrainedbythespring.Theplotsofthemodalcoordinatesshowdiscontinuitiesatthetimesofconstraintapplicationandremoval.Thisisbecausethemodalcoordinateshavedifferentfunctionaldescriptionsinthecon-strainedandunconstrainedstates.Intheconstrainedstate,thehigh-frequencymodehadasmall8amplitude(q2inFig.2.3),butitsenergycontentwasThisenergydecayedrapidlyeachtimeafterthesystemwasswitchedfromtheunconstrainedstatetotheconstrainedstate.Thiscanbeseenfromtheviewsofq2inthetimeinterval[t1;t2],andagainintheinterval[t3;42:7].Acomparisonoftheviewsofq2intheintervals[t1;t2]and[t3;42:7]alsothatswitchingresultedinfunnelingofenergyintothehigh-frequencymode.Theexampleaboveillustratesthatenergydissipationisfasterinsystemswithswitchedstiff-nessthaninsystemswithconstantstiffnessinthepresenceofmodaldamping.Thefasterrateofdissipationisnotduetodirectremovalofenergybytheactionofswitching,butduetofunnelingofenergyintothehigh-frequencymodesofthesystem.Theeasewithwhichenergycanbefunneledfromthelow-frequencymodestothehigh-frequencymodesisdiscussedinthenextsectionforageneralmulti-degree-of-freedomlinearsystem.2.2Stiffnessvariationinmulti-DOFsystemsConsidertheN-DOFlinearsystemM¨X+K(t)X=0(2.6)whereX=(x1;x2;;xN)Tdenotesthevectorofgeneralizedcoordinates,Mdenotesthemassmatrix,andK(t)denotesthestiffnessmatrix.ThestiffnessmatrixK(t)consistsofaconstantstiffnessmatrixK0andatime-varyingstiffnessmatrixDK(t)asfollows:K(t)=K0+DK(t);DK(t)=8><>:0ift2[ti;ti+1)Krift2[ti+1;ti+2);i=0;2;4;(2.7)whereKristhechangeinthestiffnessmatrixduetotheadditionofspringsconnectingpairsofgeneralizedcoordinates.Inthesimplestcasewhereasinglespringisusedtoconnectapairofgeneralizedcoordinatesxmandxn,theentriesofKr2RNNcanbeobtainedfromtheHessianof9theadditionalstrainenergy(1=2)kr(xmxn)2Kr(i;j)=8>>>><>>>>:krif(i;j)=(m;m)or(n;n)krif(i;j)=(m;n)or(n;m)0otherwisem6=n(2.8)wherekristhestiffnessofthespring,whichislargecomparedtothemagnitudeoftheentriesofK0.InEq.(2.7),tj,j=0;1;2;;arechosensuchthatthechangeinstiffnessdoesincreasethetotalenergyofthesystem.ThisisassuredbychoosingthetimetjwhenswitchingDK(tj)from0toKr(stiffnessincrease)suchthatalltherelevantrelativedisplacementsarezero.Inthesimplestcasementionedabove,whereasinglespringisused,tjischosentoswitchDK(tj)from0toKr(stiffnessincrease)suchthatxm(tj)xn(tj)=0.WhenDK(tj)isswitchedfromKrto0(stiffnessdecrease),tjcanbearbitrary.Inthisprocessofstiffnessdecrease,theremightbedirectandinstantaneouslossofenergyduetothefactthatrelativedisplacementisusuallynonzerogivenarbitrarytj.Thisdirectenergylossonlyfavorablyreducestheenergyofthesystem.Letfiandmi,i=1;2;:::;N,denotethelinearlyindependentorthogonalmodeshapesandthecorrespondingmodalcoordinatesintheunconstrainedstate.Similarly,letyiandni,i=1;2;:::;N,denotethelinearlyindependentorthogonalmodeshapesandthecorrespondingmodalcoordinatesintheconstrainedstate.Atthetimeofapplicationoftheconstraint(DKchangesfrom0toKr),thegeneralizedcoordinatesandtheirvelocitiescanbeexpressedasfollows:X(ti+1)=8><>:åNi=1mi(ti+1)fi=Fm(ti+1)åNi=1ni(ti+1)yi=Yn(ti+1)i=0;2;4;(2.9)X(ti+1)=8><>:åNi=1mi(ti+1)fi=Fm(ti+1)åNi=1ni(ti+1)yi=Yn(ti+1)i=0;2;4;(2.10)whereF=[f1;f2;:::;fN]andY=[y1;y2;:::;yN]aremodalmatricesintheunconstrainedandconstrainedstatesrespectively.UsingEqs.(2.9)and(2.10),thetransitionofthesystemfromthe10unconstrainedstatetotheconstrainedstatecanbedescribedbytherelationsn(ti+1)=Gm(ti+1);n(ti+1)=Gm(ti+1);i=0;2;4;(2.11)whereGisthemodaldisparitymatrix(Diaz&Mukherjee,2006a;Issaetal.,2008),andisgivenbytherelationG=YTMF(2.12)Thetransitionofthesystemfromtheconstrainedstatetotheunconstrainedstatecanbesimilarlydescribedbytherelationsm(ti)=GTn(ti);m(ti)=GTn(ti);i=0;2;4;(2.13)ThetransformationmatrixGistheidentitymatrixwhenKr=0,i.e.,whennostiffnessvariationisintroduced.WhenKr6=0,G(i;j)6=0forsomevaluesofiandj,i6=j.Thisimpliesthatenergywillbetransferredfromthej-thmodeoftheunconstrainedstatetothei-thmodeoftheconstrainedstate,andviceversa.Ifthefrequencyofthei-thmodeoftheconstrainedstateismuchhigherthanthatofthej-thmodeoftheunconstrainedstate,theenergytransferredfromthelow-frequencymodetothehigh-frequencymodewillbequicklydissipated.ThisfollowsfromtheassumptionFigure2.4Vibrationsuppressionthroughenergyfunnelingfromlow-frequencymodes(LFM)intohigh-frequencymodes(HFM).11ofthemodaldampingmodelwiththeuniformdampingratio.Fortheprocesstoberepeated,thesystemhastobeswitchedbackfromtheconstrainedstatetotheunconstrainedstate.Toavoidenergywfromthehigh-frequencymodesinonestatetothelow-frequencymodesintheotherstate,thesystemshouldbeheldineachstatesuflongtimesuchthatenergyinthehigh-frequencymodesisdissipated.ThisstrategyforvibrationsuppressionisexplainedwiththehelpofFig.2.4.Thesuccessofvibrationsuppressionusingstiffnessswitchingwilldependonmodaldisparitycreatedbythechangeinstiffness.Toquantifymodaldisparity,wethemetricl=Nåi=j+1N1åj=1(ij)jgijj(2.14)wheregij=G(i;j)isthe(i;j)-thentryofthemodaldisparitymatrixG.Thismetricisaweightedsumoftheprojectionsofthelow-frequencymodesintheunconstrainedstateontohigh-frequencymodesintheconstrainedstateandtheweightsarethedifferenceoftheindicesofthemodesinthetwostates.Thismetricwillbeusedtodeterminebetterlocationofconstraintsinasimpletension-alignedstructure,modeledandsimulatedinthefollowingsubsections.Figure2.5Atension-alignedstructureformedbyconnectingasupportstructure(incompression)toanarrayofhingedpanels(intension).122.3Modelofatwo-dimensionalsimpletension-alignedstructureInthissubsectionwepresentamodelofatwo-dimensionaltension-alignedstruc-ture.Thetension-alignedstructure,showninFig.2.5,consistsofaplanarelasticaarchsupportstructureincompressionandahingedpanelarrayintension.Theplanarelasticaarchisinitiallyastraightslenderrod;itisbentintoitscurvedshapebyeccentricendloadsthatmaintainequilibriumwiththetensionforcesinthepanels.2.3.1NonlineardynamicmodelofthesupportstructureThedynamicmodeloftheelasticaarchisreproducedfromtheworkbyPerkins(1990).Theelasticaarch,showninFig.2.6,isassumedtobeaslenderrodoflengthL,heldinstaticequilibriumunderthehorizontalend-loadfandmomentfd,whereddenotestheverticaleccentricityoftheend-loadf.Inadisturbedstate,apointontherodhasadisplacementof~u(s;t),wheresdenotesthearclengthalongthecenterlineofthestaticequilibriumshape,andtdenotestime.~u(s;t)canbedecomposedintoitstangentialcomponentandnormalcomponentsasfollows:~u(s;t)=ut(s;t)‹et+un(s;t)‹enwhere‹etand‹enareunitvectorsalongthetangentialandnormaldirectionsofthestaticequilibriumshape,showninFig.2.6.WefollowKirchhoff'sassumptionsforroddeformation(Dill,1992),whichare(i)rodislin-earlyelastic,(ii)strainsaresmall(althoughrotationsmaybelarge)andcross-sectionaldimensionsFigure2.6Aplanarelasticaarch13oftherodaresmallcomparedtoitslength,(iii)cross-sectionsremainplane,undistortedandnor-maltotheaxisoftherod,and(iv)thetransversestressandrotaryinertiacanbeneglected.Undertheseassumptions,thekineticenergyandthestrainenergyoftherodcanbeexpressedasfollows:PT=12ZL0r"¶ut¶t2+¶un¶t2#ds(2.15)PV=12ZL0EIk2+EAe2ds(2.16)wherer,E,AandIareconstantsanddenotethemassperunitlength,Young'smodulus,cross-sectionalarea,andareamomentofinertiaoftherod,respectively.InEq.(2.16)k=k(s;t)ande=e(s;t)arethecurvatureandtheaxialstrain.Theexpressionfork(s;t)isobtainedfromLove(1944)andthatofe(s;t)isobtainedfromPerkins&Mote(1987)k=ks+¶¶s¶un¶s+ksut(2.17)e=pEA=psEA+¶ut¶sksun+12"¶ut¶sksun2+¶un¶s+ksut2#(2.18)wherep=p(s;t)istheaxialforce,andpsandksarethestaticvaluesofpandkrespectively,inthestaticequilibriumTheworkdonebyexternalforcescanbeexpressedasWnc=f(utcosq0+unsinq0)js=0+fd¶un¶s+ksuts=Ls=0(2.19)whereq0istheangleofinclinationoftherodats=0,whichwillbedeterminedlater.SubstitutingEqs.(2.17)and(2.18)intoEq.(2.16),neglectingtermsthathavedegreethreeandhigherofvariablesutandun,andtheirspatialderivatives,andusingHamilton'sprincipledZt2t1(PTPV+Wnc)dt=0(2.20)wegetthenon-dimensionalequationsofmotioninthenormalandtangentialdirectionsPerkins(1990)¶3¶S3¶Un¶S+KUt+¶¶SP¶Un¶S+KUt+KP+1¯I¶Ut¶SKUn¶2K¶S2+PK=¶2Un¶T2(2.21)14K"¶2¶S2¶Un¶S+KUt#+¶¶SP+1¯I¶Ut¶SKUnKP¶Un¶S+KUt+K¶K¶S+¶P¶S=¶2Ut¶T2(2.22)Intheequationsabove,thenon-dimensionalvariablesareasfollows:S,sL;D,dLUt,utL;Un,unL;K,ksLP,psL2EI;F,fL2EI;¯I,IAL2;T,trL4=EI1=2(2.23)Togetherwiththeequationsofmotion,thefollowingboundaryconditionsareobtainedfromHamilton'sprinciple¶¶S¶Un¶S+KUt+KFDd¶Un¶S˙S=0+¶¶S¶¶S¶Un¶S+KUt+K+P¶Un¶S+KUt+Fsinq0dUn˙S=0+K¶¶S¶Un¶S+KUt+K2+P+1¯I¶Ut¶SKUn+PFDK+Fcosq0dUt˙S=0+¶¶S¶Un¶S+KUtK+FDd¶Un¶S˙S=1+¶¶S¶¶S¶Un¶S+KUt+KP¶Un¶S+KUtdUn˙S=1+K¶¶S¶Un¶S+KUtK2P+1¯I¶Ut¶SKUn+P+FDKdUt˙S=1=0(2.24)2.3.2StaticequilibriumofthesupportstructureThestaticequilibriumoftheelasticaarchdependsonthevaluesoffandd,oralternatively,onthenon-dimensionalvariablesFandD.Foratension-alignedstructure,FandDaredesignvariables;thevalueofFwilldependonthetensiondesiredinthehingedpanelarray,andthevalueofDwilldependonthestiffnessoftheslenderrod(elasticaarch)andthedifferenceinlengthsofthehingedpanelarrayandtheslenderrod.AssumingthatthevaluesofFandDare15provided,wedeterminethestaticequilibriumbysubstitutingUt=Un=0inEqs.(2.21)and(2.22).Thisyieldsthefollowingequations:K00+KP=0(2.25)P0+KK0=0(2.26)where(:)0denotesthederivativeof(:)withrespecttoS.SubstitutingUt=Un=0inEq.(2.24),andusingthegeometricboundaryconditions:dUn(S=1)=0;dUt(S=1)=0;dUn(S=0)=tanq0:dUt(S=0)weobtainthefollowingnaturalboundaryconditionsK=FDatS=0;1F+Pcosq0K0sinq0=0atS=0(2.27)Aclosed-formsolutiontoEqs.(2.25),(2.26)and(2.27)involvesellipticintegralsofthekindandcanbefoundinPerkins(1990).ThesolutionsK(S),P(S),andq0determinetheequilibriumandthepre-stressinthis2.3.3LineardynamicmodelofthesupportstructureWeusetheRaleigh-RitzmethodRao(2007)toobtainthelineardynamicmodeloftheelasticaarchaboutitsstaticequilibriumTowritethedifferentialequations,wesubstitutetheequilibriumvaluesofP=P(S)andK=K(S)obtainedfromthesolutionsofEqs.(2.25),(2.26)and(2.27)intoEqs.(2.17)and(2.18),whichyieldsthenon-dimensionallinearvibrationequationsWeusetheRaleigh-RitzmethodRao(2007)toobtainthelineardynamicmodeloftheelasticaarchaboutitsstaticequilibriumTowritethedifferentialequations,wesubstitutetheequilibriumvaluesofP=P(S)andK=K(S)obtainedfromthesolutionsofEqs.(2.25),(2.26)and(2.27)intoEqs.(2.17)and(2.18),whichyieldsthenon-dimensionallinearvibrationequations¶3¶S3¶Un¶S+KUt+¶¶SP¶Un¶S+KUt+KP+1¯I¶Ut¶SKUn=¶2Un¶T2(2.28)16K"¶2¶S2¶Un¶S+KUt#+¶¶SP+1¯I¶Ut¶SKUnKP¶Un¶S+KUt=¶2Ut¶T2(2.29)TosolveEqs.(2.28)and(2.29),weneedtogobacktotheenergyform.Tothisend,wesub-stitutetheequilibriumvaluesofP=P(S)andK=K(S)intothenon-dimensionalversionofEqs.(2.17)and(2.18),andthensubstitutetheresultsinthenon-dimensionalformofthekineticandstrainenergiesinEqs.(2.15)and(2.16).NeglectingtermsthathavedegreethreeandhigherofvariablesUnandUtandtheirspatialderivatives,wehavethefollowingexpressionsforthenon-dimensionalkineticandstrainenergies¯PV=12Z108>>><>>>:terms1and2z}|{K2+P2¯I+terms3and4z}|{2K¶¶S¶Un¶S+KUt+2P¶Ut¶SKUn+¶¶S¶Un¶S+KUt2+P+1¯I¶Ut¶SKUn2+P¶Un¶S+KUt2)dS(2.30)¯PT=12Z108<: ¶2Un¶T2!2+ ¶2Ut¶T2!29=;dS(2.31)Notethat¯PVand¯PTarerelatedtoPVandPT,respectively,bytherelations¯PV=LEIPT;¯PT=LEIPTInEq.(2.30),terms1and2oftheintegrandarefunctionsofSalone,andnotafunctionoftime.Thesameistrueforterms3and4sinceavariationoftheintegralofthesetermscanbeshowntobezero.ThefourtermsofEq.(2.30)thereforeresultinconstantstrainenergy,whichdoesnotcontributetothevibrationofthesystem.WeassumeUnandUttobeoftheformUn(S;T)=Vn(S)eiwT;Ut(S;T)=Vt(S)eiwT(2.32)17whereVn(S)andVt(S)arethemodeshapes.Themodeshapesarediscretizedasfollows:Vn(S)=åiWn;i(S)Yi=Wn(S)Y(2.33)Vt(S)=åiWt;i(S)Zi=Wt(S)Z(2.34)whereWn(S)andWt(S)arevectorsofknownshapefunctions.Theyareconstructedusingpiece-wisepolynomials(cubicandlinearrespectively),standardinelementdiscretizations,withdiscontinuitiesatnodes.YandZarevectorsofnodaldegreesoffreedom(seeEq.(2.38)below)associatedwithVnandVt.SubstitutingEqs.(2.32),(2.33)and(2.34)intoEqs.(2.30)and(2.31),werewritethenon-dimensionalkineticandstrainenergiesasfollows:¯PV=ei2wT2YTZTKA264YZ375+Const(2.35)¯PT=ei2wT2w2YTZTMA264YZ375(2.36)whereConst=12Z10ˆK2+P2¯I+2K¶¶S¶Un¶S+KUt+2P¶Ut¶SKUndS(2.37)istheconstantstrainenergyassociatedwiththestaticequilibriumdiscussedbefore.ThemassandstiffnessmatricesMAandKAoftheelasticaarch(supportstructure)areassociatedwiththegeneralizedcoordinateXAXA=eiwTYT...ZTT=u`n;q`A;;uin;qiA;;urn;qrA...u`t;;uit;;urtT(2.38)whereunandutarethetranslationaldegrees-of-freedomandqAistherotationaldegree-of-freedomofeachnode,and`,randidenotetheleftend-node,rightend-nodeandi-thnode,respectively,of18theelasticaarch.NotethatelementsofYandZneedtobeconsistentwiththegeometricboundaryconditionsVn(S=1)=0;Vt(S=1)=0;Vn(S=0)=tanq0:Vt(S=0)Theaboveboundaryconditionswillbechangedwhentheelasticaarchisassembledwiththehingedpanelarray.2.3.4HingedpanelarraymodelThearrayofhingedpanelsisshowninFig.2.7.Itwasmodeledusingastandardelementmethod.Weusedtwo-dimensionaltwo-nodeframeelementswiththreedegreesoffreedomateachnode:twotranslationalandonerotationaldegreesoffreedom.Ageometricstiffnessmatrixwasaddedtothestandardframestiffnessmatrixtomodeltheeffectoftensionf.Ahingebetweentwopanelsistreatedasanodeintheelementmodel.Theleftandrightelementsofthehingenodehaveindependentrotationsbuthavecommontranslations.Thedegrees-of-freedomofthehingedpanelarrayaredenotedbyXP=2664x`;y`;q`P;;xk;yk;qkP|{z}nodekonpanelarray;xk+1;yk+1;qk+1P|{z}node(k+1)onpanelarray;xr;yr;qrP3775T(2.39)wherexandyarethetranslationaldegrees-of-freedomandqPistherotationaldegree-of-freedomofeachnode,and`,randkdenotetheleftend-node,rightend-nodeandk-thnode,respectively,ofthehingedpanelarray.ForthegeneralizedcoordinatesXP,themassandstiffnessmatricesareassembledasMPandKP.Thehingedpanelarrayisthenassembledwiththeelasticaarchbyconnectingtheirendsto-getherusingpinjoints.Inthemodelling,thatistoassembleMAwithMPandassembleKAwithKP.Afterassembly,theendnodesofthetwosubstructuressharetranslationsintheplanebutmaintainindependentrotationaldegreesoffreedom.19Figure2.7Thearrayofhingedpanels2.3.5MethodsofstiffnessvariationStiffnessvariationdescribedbyEq.(2.7)isrealizedintheassembledtension-alignedstructurebytwomethods.ThesetwomethodsaredepictedinFig.2.8andaredescribedbelow:(A)Therotationsoftwoadjacentpanelsattheircommonhinge,qkPandqk+1P,areconnectedbyarotationalspringoftime-varyingstiffness.(B)Nodeiontheelasticaarchandnodejonthepanelarrayareconnectedbyatranslationalspringoftime-varyingstiffness.Method(A)canbeimplementedbyplacinganelectromagneticbrakeatthehingeoftheadjacentpanels.TurningonthebrakewillpreventrelativerotationbetweentheadjacentpanelsandwillbeequivalenttoconstrainingthedegreesoffreedomqkPandqk+1Pbyarotationalspringofveryhighstiffness.Turningoffthebrakewillreleasethedegreesoffreedomandwillbeequivalenttosettingthespringstiffnesstozero.Method(B)canbeimplementedbyconnectinganddisconnectinganelasticbarbetweenapointonthearchandapointonthepanel.Thesetwopointswillbechosentocoincidewithnodesoftheelementmodelforthepurposeofsimulation.Figure2.8Stiffnessvariationinthetension-alignedstructureisrealizedusingtwomethods:(A)and(B);thesearedescribedinsection3.4.20Thestiffnessofthetension-alignedstructurecanbevariedusingmultiplespringsofthetypedescribedinmethod(A)and/ormethod(B).Sinceeachofthesespringscanbeinoneoftwostates,thetension-alignedstructurewillhavemultiplestiffnessstates.Inthenextsection,wherewepresentsimulationresults,thestiffnessofthestructurewillbeswitchedcyclicallybetweentheloweststiffnessstateandthehigheststiffnessstateviaintermediatestiffnessstates.Thelowestandhigheststiffnessstatesareasthestateswiththelowestandthehighestfundamentalfrequency.2.3.6NumericalsimulationThematerialandgeometricpropertiesofthetension-alignedstructureareprovidedinTable2.2.Thestructureismadeofaluminumandthedampingratioofallmodesisassumedtobez=0:001.ThepanelarrayiscomprisedofeightpanelsofdimensionsLpbh;thesedimensionsareshowninFig.2.9.Eachpanelismodelledusing10beamelements.Thesupportstructure(elasticaarch)isinitiallyastraightrodofradius0:04mandlengthˇ8:00m.Itismodelledusing80elements.Theeccentricityoftheloadappliedtothesupportstructureis0:008m.Thetensioninthehingedpanelarraywasassumedtobe1000N.Thisislessthan5%ofthebucklingloadofthestraightrodwithfree-freeboundaryconditions.Inthissubsection,wesimulatethebehaviorofthestructurewithoutcontrolandthestructurecontrolledusingtwodifferentmethodsofstiffnessvariation.Wetakeaninitialconditionwherethesecondjointofthehingedpanelarray(seeFig.2.9)wasdisplacedverticallyby0:01m(1%ofthelengthofthepanelarray)andreleased.Thet25Figure2.9Theeight-paneltension-alignedstructureusedinsimulations.21Table2.2PropertiesofSimulatedTension-AlignedStructureMaterialAluminumYoung'smodulusE69109PaDensity¯r2700kg/m3Dampingratioz0:001Panelnumber8PanellengthLp1:000mPanelareabh0:500m0:015mRadiusofsupportrodr0:040mLengthofsupportrodL8:000mapprox.Eccentricityofconnectiond0:008mTensionf1000Nmodesofthestructureweresimulated;thesedonotincludetherigid-bodymodes.Theenergyofthetension-alignedstructureisshowninFig.2.10forthreedifferentcases,asdescribedbelow:1.Unconstrainedstructure(nocontrol)undergoingfreevibration,2.Constrainedstructure(nocontrol)withhigh-stiffnessrotationalspringinjointsJ1,J3,J4andJ6-seeFig.2.9.Usingmethod(A)ofstiffnessvariation,thestiffnessoftherotationalspringsisactivatedwhentheadjacentpanelsarealigned.Therotationalspringsareactivatedattheearliestpossibleopportunityinasequentialmannerandarekeptintheirhighstiffnessstate.3.Controlledstructurewithswitchedstiffnessusingmethod(A).Thehigh-stiffnessrotationalspringsinjointsJ1,J3,J4andJ6areactivatedsequentiallywhentheiradjacentpanelsarealignedandtheirstiffnessarethensettozerosimultaneously.Theprocessisrepeated92timesinthesimulationperiodof180sec.ItisclearfromFig.2.10thattheenergiesoftheunconstrainedstructureandtheconstrainedstruc-turedecayslowlycomparedtothestructurewithswitchedstiffness.After180sec,theuncon-strainedstructureandtheconstrainedstructurehaveˇ22:5%oftheirinitialenergyleft;incontrast,thestructurewithswitchedstiffnesshasˇ0:4%ofitsinitialenergyleft.Althoughvibrationen-22ergyisdissipatedthroughinternaldampinginallthreecases,thestructurewithswitchedstiffnesshasahigherrateofenergydissipationsinceiteffectivelyfunnelsenergyfromthelow-frequencymodestothehigh-frequencymodes.Forthestructurewithswitchedstiffness,thejointsarereleasedsimultaneously,notsequen-tially,toreducethetimerequiredforeachcycleofconstraintapplicationandremoval.Insimu-lations,wherehigh-stiffnessspringsareusedtoconstrainthejoints,simultaneousreleaseofthejointscausesresidualenergystoredinthespringstovanish.Thisdiscontinuouschangeintheen-ergyisnotthemainmechanismofenergydissipation.Anevaluationofthisenergyoverallcyclesindicatesthatitdoesnotexceed0:1%ofthetotalenergyatitsinitiallevel.Thismeansthebulkoftheenergyisdissipatedduetoenergytransferfromlow-frequencymodestohigh-frequencymodes.Inpracticalimplementation(Issaetal.,2008),whereelectromagneticbrakesmaybeusedtoconstrainthejoints,releaseofthebrakeswillnotresultindirectlossofenergy(sincebrakesdonotstoreenergy)butfacilitateenergytransfertothehigh-frequencymodeswheretheywillbedissipatedquickly.Figure2.10PlotofenergyforthethreecasesdiscussedinSection4.2.Theratesofenergydecayoftheunconstrainedstructureandtheconstrainedstructureareal-mostidentical.Sincethestructurehasmanydegrees-of-freedomandactivatingthespringsinfourjointsonlymakesitmarginallystifferthantheunconstrainedstructure.ThiscanbevfromTable2.3,whichshowsthesixnaturalfrequenciesoftheunconstrainedandconstrained23structures.Table2.3Firstsixnaturalfrequenciesoftheunconstrainedandtheconstrainedtension-alignedstructureinrad/s.Unconstrainedw1w2w3w4w5w63.8206.9999.12913.14415.85920.213Constrained¯w1¯w2¯w3¯w4¯w5¯w63.9967.50211.09628.04237.87571.666Thestructurewithswitchedstiffness,wherestiffnessisvariedusingmethod(A),hasafasterrateofenergydecaythantheuncontrolled(unconstrainedandconstrained)structures.Inordertofurtherimprovetheefyofthecontrolusingswitchedstiffness,weneedtoinvestigatedifferentmethodsofstiffnessvariation.Applyingmethods(A)and(B),againusing25modes,thebehaviorofthestructurewassimulatedforthesameinitialconditionthatwasusedinthepreviouscase.Theenergyofthetension-alignedstructureisshowninFig.2.11forthefollowingthreecases:1.Unconstrainedstructureundergoingfreevibration2.Structurewithswitchedstiffnessusingmethod(A).Thehigh-stiffnessrotationalspringsinjointsJ1,J3,J4andJ6areactivatedsequentiallywhentheiradjacentpanelsarealignedandtheirstiffnessarethensettozerosimultaneously.Theprocessisrepeated92timeswithinthesimulationperiodof180sec.3.Structurewithswitchedstiffnessusingmethods(A)and(B).Thehigh-stiffnessrotationalspringsinjointsJ1,J3,andJ4areactivatedsequentiallywhentheiradjacentpanelsarealigned.Thisisfollowedbyconnectinganelasticbar(high-stiffnesstranslationalspring)betweenapointontheelasticaarchandapointonthepanelarray(seeFig.2.9)inamannersuchthatnoenergyisaddedtothestructure.Thestiffnessofallfourspringsarethensettozerosimultaneously.Theprocessisrepeated105timeswithinthesimulationperiodof5180sec.24ItisclearfromFig.2.11thattheenergyoftheuncontrolledstructuredecaysslowlycomparedtothestructurewithswitchedstiffness.After180sec,theuncontrolledstructurehasˇ22:5%ofitsinitialenergyleft;incontrast,thestructurewithswitchedstiffnessusingmethod(A)hasˇ0:4%ofitsinitialenergyleft.Forvibrationsuppressiontoˇ0:4%energylevel,methods(A)and(B)combinedrequires76secascomparedto180secrequiredbymethod(A).Accordingly,forthecaseofmethods(A)and(B)combined,thetransversedisplacementsofthreepointsonthehingedpanelarrayFig.2.12,whichclearlyshowsthesuppressionofvibration.Figure2.11Plotofenergyforthethreecasessimulated.Theefyofvibrationsuppressionusingstiffnessvariationcanbemuchimprovedbycom-biningmethods(A)and(B).Thisimprovementineffectivenesscanbeunderstoodbyexaminingthemodaldisparitymatricesforthetwocasesandcomparingtheirmodaldisparityindices.ThemodaldisparityindicesforthesetwocaseswerecomputedaslA=22:15;lAB=40:48(2.40)SincelABisgreaterthanlA,stiffnessvariationcombiningmethods(A)and(B)ismoreeffec-tivethanmethod(A)intransferringenergyfromthelow-frequencymodestothehigh-frequencymodes.Modaldisparityprovestobeaeffectivemeasuretoevaluatethemagnitudeofstiffnessvariationforthepurposeofvibrationsuppression.25Figure2.12Plotsofthetransversedisplacementsofthreepointsonthehingedpanelarray(seeFig.2.9),inthecaseofmethods(A)and(B)combined.2.4Modelofathree-dimensionaltension-alignedstructureFigure2.13Overviewofthetension-alignedthree-dimensionalstructure.8panelsareconnectedusing7hingesandsupportedbyatrussstructure.Athree-dimensionalmodelwasbuilttovtheeffectivenessofthestiffnessvariationmethodinamorerealisticstructure.Thismodeltakesthedesigndataofthetrussstructurefrom26theISATprojectandwesetallmaterialasaluminum.17trusscellswerebuilt.Eachcellhasthedimensionof0:5mandthetrussstructurehasthetotallengthof8:5m.Thetrussfunctionsasasupportstructureandprovidestensionforthehingedpanelarraywheretheantennaismounted.Thereare8panelshingedtoformanarrayandeachpanelhasthedimensionof1:00m0:50m0:01m.TheoverviewofstructureisshowninFig.2.13.Thecombinedstructureisclampedatoneendandfreeatanother.Thetensionlevelissettobe200N.ThegeometryofthestructureisdepictedinFig.2.14.Figure2.14Geometryofthethree-dimensionalstructure.ANSYSwasusedfortheelementmodelling.Thetrussstructurewasmodelledusinglinkelementsthathavetwonodesandthreedegreesoffreedomateachnode.Theplatesweremodelledusingshellelementsthathavefournodesandsixdegreesoffreedomateachnode.ThetotalnumberofDOFis648,and25modeswereusedinthedynamicsimulation.Onlymethod(A)wasusedinthismodelfortheapplicationofstiffnessvariation.Thecontrollogicwassimilartothetwo-dimensionalmodel.Resultsofvibrationsuppressionwereverycon-sistent,asshowninFig.2.15.Whenfourhingeswereused,theenergyplotresemblestheresultsofthetwo-dimensionalmodel.Asthenumberofcontrolledhingesincreased,vibrationsuppres-sionbecamemoreefThisisbecausethemagnitudeofthemodaldisparityincreasesasthenumberofcontrolledhingesincreases.Inallcasesofcontrolledhingesused,stiffnessvaria-tionsasacontrolmethodhasshownitsefyinvibrationsuppressioninthethree-dimensionaltension-alignedstructuremodel.27Figure2.15Plotofenergyforcasesofcontrollingdifferentnumberofjoints.28CHAPTER3VIBRATIONSUPPRESSIONUSINGASLIDINGMECHANISM3.1SlidingmechanismdescriptionSlidingmechanismintheformofconstraintsinxiblestructureshasbeeninvestigatedbymanyresearchers,especiallyinareassuchascontactmechanics(Popov,2010;Fischer-Cripps,2007)andxiblemultibodydynamics(Bauchau,2010;Wittbrodt,2006).Agoodexamplethathasbeenstudiedfrequentlyisthequickreturnmechanism(Barhorst,2004)whereaninvertedslidercrankisconnectedwithaxiblefollower.UsuallyHamilton'sprincipleisusedtogetherwithLagrangemultiplierstoformulatetheequationsofmotionofthesystem.ConstraintforcescanbefoundbysolvingfortheLagrangemultipliers.Finiteelementmethodsareverypowerfultoolsfornumericalsimulationsofthesesystems,whereArbitraryLagrange-Euler(ALE)descriptionsareusuallyneededtoallowelementnodestomoveinthematerialcoordinatesystem(Hong&Ren,2011).Thedynamicsofconstrainedsystems,suchastheslidingconstraintdiscussedabove,areusu-allydescribedbyDifferentialAlgebraicEquations(DAE)inordertoobtaintheconstraintforcesandtosolveforthesystemstates.Theideaofcontrollingthemotionofaslidingconstraintinaxiblestructureforthepurposeofvibrationsuppressionisnewandhasnotbeenexploredintheliterature.Thisstudyconsidersasimplestructureofanonlinearbeamwithpinned-pinnedboundaryconditionsandaslidingconstraintwhichisfrictionless.Assumingthattheconstraintforcecouldbemeasured,astraightforwardcontrolstrategyisdevelopedtodonegativeworkonthesystemandsuppressthevibrationenergyofthebeam.Amathematicalmodelofanonlinearbeamwillbepresentedinthefollowingsection.Asimplefeedbackcontrolschemeforthepurposeofvibrationsuppressionwillbepresentedaswell.Thenacontroldesignthatreducesthebandwidthrequirementoftheactuatorwillalsobeintroduced.293.2EquationsofmotionforanonlinearbeamwithasliderSimilartoSection2.3,wederivetheplanarvibrationmodelofthenonlinearbeamfollowingtheworkbyPerkins(1990)ontheelasticaarch.ConsideraslenderbeamshowninFig.3.1,heldinequilibriumunderhorizontalendloadf,wherefcanbeeithertensileorcompressiveyetlessthanthebucklingload.Thebeamisinitiallystraightandkeepsbeingstraightwithfapplied.ItismeasuredtohavelengthLunderf.Bothendsofthebeamaresubsequentlypinnedtotheground;thiscreatespre-stressinthebeam.Africtionlesssliderisassumedtoconstrainthemotionofthebeam;itslocationisedinspacethatinitiallycoincideswiththemidpointofthebeam.Thesliderrestrictsthepositionofthematerialpointonthebeamincontactwithit,butdoesnotrestricttheslopeofthebeam.Sincethesliderisedanddoesnotmovewiththebeam,materialcanwthroughit.ApointQonthebeaminstantaneouslyincontactwiththeconstraintcanbedescribedusingmaterialcoordinates=sq,wherecoordinatesismeasuredintheinitialequilibriumusingLagrangiandescription(materialcoordinate).Duetoadisturbance,thebeamstartstodeform.Agivenpointonthebeamhasdisplacementof~u(s;t),wheretdenotestime.~u(s;t)canbedecomposedintoanaxialcomponent~ut(s;t)whichisalwaysintheX-directionandatransversecomponent~un(s;t),whichisalwaysintheY-direction.Notethattwosetsofcoordinatesareusedhere,namelythematerialcoordinatesystemsandtheX-Yframe.Figure3.1Apinned-pinnedbeamwithaslidingconstraint.30TheslidingconstraintscanbedescribedasCx=ut(s=sq;t)+sq(t)L=2=0(3.1)Cy=un(s=sq;t)=0(3.2)whereEq(3.1)statesthatthematerialpointQincontactwiththesliderisinitiallymeasuredassqintheinitialequilibriumion,andthatpointQhasanaxialdisplacementofutinordertokeepincontactwiththeslider;Eq.(3.2)statesthatanymaterialpointincontactwiththeconstraintshouldneverhavetransversemotion.WeuseKirchhoff'sassumptionsforbeamdeformation(Dill,1992).(i)Thebeamislinearlyelastic.(ii)Strainsaresmall(althoughaccumulatedrotationsmaybelarge),andthecross-sectionaldimensionsofthebeamaresmallcomparedtoitslength.(iii)Cross-sectionsremainplanar,undis-tortedandnormaltotheaxisofthebeam.(iv)Thetransversestressandrotaryinertiaarenegli-gible.Inlightoftheseassumptions,withtheconsiderationthatthebeamremainsstraightinitsequilibriumthenonlinearaxialstraincanbewrittenasPerkins&Mote(1987)e=f=EA+¶ut¶s+12"¶ut¶s2+¶un¶s2#(3.3)ThecurvaturekcanbeexpressedasinLove(1944)k=¶2un¶s2(3.4)whereA,E,andIdenotethecross-sectionalarea,Young'smodulus,andareamomentofiner-tiaofthebeamrespectively.ThevirtualworkdonebyinternalelasticforcesduetoanyvirtualdeformationcanbewrittenasZWsd[e]dW=ZL0EAed[e]ds+ZL0EIkd[k]ds(3.5)wheresanderepresentgeneralstressandstrainrespectively.Wisthedomainofinterest,whichincludesallmaterialpointsonthebeam.31WeusetheprincipleofvirtualdisplacementandD'Alembert'sprinciple,togetherwithLa-grangemultiplierstodescribethedynamicsofthebeaminthefollowingform:ZWsd[e]+(P+r¨u)d[u]dW=d[lC](3.6)wherevectorPistheexternalforce,uisthedisplacement()isthederivativewithrespecttotimet,lisavectorofLagrangemultipliers,andCisthevectorofconstraintexpressions.ThetermontherighthandsideofEq.(3.6)canbeexpandedaslC=lxCx+lyCy(3.7)wheretheexpressionsforCxandCyaregiveninEqs.(3.1)and(3.2).SinceLagrangemultipliershavethephysicalinterpretationsofconstraintforces,lxandlyrepresenttheconstraintforcesinaxialandtransversedirections,orX-andY-directionsrespectively.Equation(3.6)statesthatthebeamisindynamicequilibriumiftheexternalvirtualworkdonebytheappliedforces,includingtheconstraintforces,isequaltotheinternalvirtualworkdonebytheforcesduetoanyvirtualdeformationthatthekinematicboundaryconditions.Here,theinternalvirtualworkisextendedwithaninertialtermusingD'Alembert'sprinciple.Thekinematicboundaryconditionsforconstraintsareincludedontherighthandsideoftheequation.TheseconstraintconditionsdescribedbyEqs.(3.1)and(3.2)couldberecoveredifcollectingthevariationsofLagrangemultipliers.Inourproblem,Pwillonlybeusedinsettingupthestaticproblemtocreatetheinitialcondi-tion.Asfarasthedynamicproblemisinconcern,wesimplysetexternalforcestobezeroP=0(3.8)SubstitutingEqs.(3.3)and(3.4)intoEq.(3.5),thensubstitutingtheresultingequationtogetherwithEqs.(3.1),(3.2),(3.7)and(3.8)intoEq.(3.6),andusingthefollowingnon-dimensionalquan-tities:S,sL;SQ,sqL;Ut,utL;Un,unL;K,kLF,fL2EI;¯I,IAL2;T,trL4=EI1=2;L,lL2EI(3.9)32wegettheenergyformoftheequationsofmotionZ10¨Und[Un]dS+Z10¨Utd[Ut]dS+Z10¶2Un¶S2d"¶2Un¶S2#dS+Z10 F+1¯I¶Ut¶S+12¯I¶Ut¶S2+12¯I¶Un¶S2!¶Un¶Sd¶Un¶SdS+Z10 F+1¯I¶Ut¶S+12¯I¶Ut¶S2+12¯I¶Un¶S2!1+¶Ut¶Sd¶Ut¶SdS=ˆLy¶Un(S=SQ)¶S+Lx¶Ut(S=SQ)¶S+1d[SQ]+Un(S=SQ)d[Ly]+Ut(S=SQ)+SQ1=2d[Lx]+LydUn(S=SQ)+LxdUt(S=SQ)(3.10)IntegratingEq.(3.10)bypartsandcollectingliketermsofindependentvariationsyieldthedifferen-tialformofequationsofmotionandboundaryconditions.ThePDEsarewrittenasthefollowing:fromcollectingtermsinvolvingd[Un]¶4Un¶S4F¶2Un¶S2+32¯I¶Un¶S2¶2Un¶S2+12¯I¶2Un¶S2¶Ut¶S2+1¯I¶Un¶S¶Ut¶S¶2Ut¶S2+1¯I¶2Un¶S2¶Ut¶S+1¯I¶Un¶S¶2Ut¶S2+LyDSQS=¶2Un¶T2(3.11)fromcollectingtermsinvolvingd[Ut]F+1¯I¶2Ut¶S2+32¯I¶Ut¶S2¶2Ut¶S2+1¯I¶Un¶S¶Ut¶S¶2Un¶S2+12¯I¶Un¶S2¶2Ut¶S2+3¯I¶Ut¶S¶2Ut¶S2+1¯I¶Un¶S¶2Un¶S2+LxDSQS=¶2Ut¶T2(3.12)fromcollectingtermsinvolvingd[SQ]Ly¶UnS=SQ;T¶S+Lx¶UtS=SQ;T¶S+Lx=0(3.13)togetherwiththerecoveredconstraintequationsinnon-dimensionalformfromcollectingtermsinvolvingdLxanddLy33UnS=SQ;T=0(3.14)UtS=SQ;T+SQ1=2=0(3.15)whereinEqs.(3.11)and(3.12)D()istheDiracdeltafunction.Eqs.(3.11)-(3.15)areafullsetofequationsofmotiondescribingthedynamicsofanonlinearbeamwithaedslider.Ifdampingistobeaddedtothesystem,Eq.(3.6)needstoberewrittentoincludetheviscousdampingterms,suchasZWsd[e]+hsd[e]+(P+r¨u)d[u]dW=d[lC](3.16)wheretheaddedtermhsd[e]expressesthattheinternaldampingforcescontributingtothevirtualworkduetoanyvirtualdeformationofthestructure.Italsostatesthatthedampingforceisproportionaltothedampingcoefhandthestressrates.StartingfromEq.(3.16),andrepeatingtheprocessofnondimensionalizationandvariationde-scribedabove,onecangetanothersetofequationsofmotionwithdampingpresent.Theprocessisrepetitiveandisomittedhere.3.3Numericsimulationoftheslidingmechanismofanonlinearbeam3.3.1FiniteelementdiscretizationusingadaptivemeshTheniteelementmodelofoursystemisestablishedintheframeworkoftheArbitraryLagrange-Euler(ALE),followingtheworkofHong&Ren(2011).TodiscretizeEq.(3.10)usingaelementmethod,wechoosetoputaslidingnodeonthebeaminthesamepositioncoincidingwiththeconstraintthattakesintoaccounttheslidingnatureoftheconstraint.Theslidingnodedoesnotmoveinabsolutespaceaslongastheconstraintised.Thisspecialslidingnode,therefore,isdescribedusingEuleriandescription.Asforothernodes,wheretheLagrangiandescriptionisused,34nodesareattachedtocertainmaterialpointschosenatthebeginning,andmaterialcoordinatesofthosenodeswillremainthesameastimeprogresses(Spencer,1980).Becauseoftheexistenceofthespecialslidingnode,twoneighboringelementsthatsharethisnodebecomevariable-lengthelements,whileotherelementsarestillregular.Inthissense,themeshingschemeofthestructureisadaptiveasthestructuredeformsandchangesitscontactpoint.Itisnecessarytoderiveexpressionsforvelocityandaccelerationofanymaterialpointinavariable-lengthelement,which,comparedtoregularelements,aremorecomplicated.Considerastandard2-node,6-DOFplanarframeelementdescribedbymaterialcoordinatesofthetwonodes(Se1;Se2)and6nodaldisplacements(Uen1;U0en1;Uet1;Uen2;U0en2;Uet2),wherethesuper-scripteindicateselement-wiseorlocalnumberingisused.IntheframeworkofALEdescription,both(Se1;Se2)and(Uen1;U0en1;Uet1;Uen2;U0en2;Uet2)canvarywithtime.UsingHermitepolynomialsforshapefunctionsofdisplacementinthenormalortransversedirectionUen(S;T),wegetUen(S;T)=NTeS;Se1(T);Se2(T)qen(T)(3.17)whereNe=0BBBBBBB@2x33x2+1x32x2+x2x3+ex2x3x21CCCCCCCA;qen=0BBBBBBB@Uen1U0en1Uen2U0en21CCCCCCCA,0BBBBBBB@ae1be1ae2be21CCCCCCCA(3.18)andx=(SSe1)=(Se2Se1).TheshapefunctionsNearefunctionsofbothmaterialcoordinateSandnodelocations(Se1;Se2),whichmeansNearefunctionsofSandT.ThenodaldisplacementsqenarefunctionsoftimeTonly.Similarly,choosingLagrangepolynomials,thetangentialoraxialdisplacementUet(S;T)canbewrittenasUet(S;T)=RTeS;Se1(T);Se2(T)qet(T)(3.19)35whereRe=0B@(Se2S)=(Se2Se1)SSe1)=(Se2Se1)1CA;qet=0B@Uet1Uet21CA,0B@ce1ce21CA(3.20)TheshapefunctionsRTearefunctionsofSandT.ThenodaldisplacementsqetarefunctionsoftimeT.DifferentiatingEqs.(3.17)and(3.19)twicewithrespecttotime,theaccelerationscanbede-rivedas¨Uen=NTe¨qen+¶NTe¶Se1qen¨Se1+¶NTe¶Se2qen¨Se2+2¶NTe¶Se1qenSe1+2¶NTe¶Se2qenSe2+¶2NTe¶Se12qenSe12+2¶2NTe¶Se1¶Se2qenSe1Se2+¶2NTe¶Se22qenSe22(3.21)¨Uet=RTe¨qet+¶RTe¶Se1qet¨Se1+¶RTe¶Se2qet¨Se2+2¶RTe¶Se1qetSe1+2¶RTe¶Se2qetSe2+¶2RTe¶Se12qetSe12+2¶2RTe¶Se1¶Se2qetSe1Se2+¶2RTe¶Se22qetSe22(3.22)SubstitutingEqs.(3.17)-(3.22)intoEq.(3.10)givesustheelementversionofthedynamicsystem.Ifthebeamismeshedwith2melements,therearemelementstotheleftandmelementstotherightoftheconstraint.Thusnodem+1countedfromtheleftendistheslidingnodeonthebeamandisedinspace.ThisspecialnodehasthematerialcoordinateSQthatvarieswithtime,whileothernodeshaveedmaterialcoordinates.Nowwecanusegeneralizedvariablesq,avectorwith6m+4entrieswrittenasq=(a1;a2;:::;a2m+1;b1;b2;:::;b2m+1;c1;c2;:::;c2m+1;SQ;Lx;Ly)T(3.23)36Z10¨UnUnqd[q]dS+Z10¨UtUtqd[q]dS+Z10¶2Un¶S2¶¶2Un¶S2¶qd[q]dS+Z10 F+1¯I¶Ut¶S+12¯I¶Ut¶S2+12¯I¶Un¶S2!¶Un¶S¶¶Un¶S¶qd[q]dS+Z10 F+1¯I¶Ut¶S+12¯I¶Ut¶S2+12¯I¶Un¶S2!1+¶Ut¶S¶¶Ut¶S¶qd[q]dS=ˆLy¶Un(S=SQ)¶S+Lx¶Ut(S=SQ)¶S+1d[SQ]+Un(S=SQ)d[Ly]+Ut(S=SQ)+SQ1=2d[Lx]+LydUn(S=SQ)+LxdUt(S=SQ)(3.24)First,variationofthetermsinEq.(3.10)iscarriedoutbytakingderivativeswithrespecttoq.ThentheelementversionofthedynamicsystemisgivenasEq.(3.24).WiththehelpofLagrangemultipliers,wecantreatallvariablesinqindependentwitheachother,thenbycollectingcoefofarbitraryd[q],Eq.(3.24)givesriseto(6m+6)nonlinearequations.Amongthem,bycollectingcoefofd[Lx]andd[Ly],twoconstraintequationsarerecoveredasCx=cm+1+Sm+11=2=0Cy=am+1=0(3.25)Geometricboundaryconditions,pinned-pinnedtwoends,giveUn(S=0)=Ut(S=0)=Un(S=1)=Ut(S=1)=0ora1=c1=a2m+1=c2m+1=0(3.26)Theycancelfourequationsoutfrom(6m+6).Intotal,(6m+2)equationsareobtainedre-gardingq.TheycanbewrittenasM(q;T)¨q=F(q;q;T)(3.27)37whereMisthegeneralizedmassmatrixandFisthegeneralizedforcevector.Ifdampingispresent,onecanfollowthewholeprocessstartingfromEq.(3.16),thenobtainadiscretizeddynamicsystemwiththeformverysimilartoEq.(3.27).Theprocessisnotrepeatedhere.Itisworthmentioningthatsincethetwoconstraintequationsexpressedareincluded,Eq.(3.27)isessentiallyaDifferen-tialAlgebraicEquation(DAE)system,featuredbythegeneralizedmassMbeingsingular.Thisrequiresspecialnumericsolverstobedescribedlater.Uptothispoint,ithasbeenshownthatthesystemissuccessfullydiscretizedusingvariable-lengthelements(adaptivemeshing).Table3.1Propertiesandgeometryofthesimulatedbeamwithaslidingconstraint.MaterialAluminumYoung'smodulusE69109PaDensity¯r2700kg/m3Dampingcoefh105secBeamlengthL3:66mBeamcrosssectionarea38:1mm1:57mmPre-loadf0NAnumericsimulationoffreevibrationisgiventodemonstratetheeffectivenessofourmath-ematicalmodel.GeometryandmaterialpropertiesusedinthesimulationarelistedinTable3.1.Inthenumericmodel,thebeamwasmeshedusing20frameelements.Aslidingjoint,nodenum-ber11countedfromtheleftend,wasplacedonthebeamatthesamepositionwheretheslidingconstraintwaslocated,withthematerialcoordinateSQasavariabletobesolved.Therewere10elementstotheleftand10elementstotherightoftheslidingnode.Elementsnumber10and11werevariable-lengthelements,whileotherelementswereregularelementswhichhadnodeswithedcoordinatesallocatedatthebeginning.Initialconditionswerecreatedbyapplyinga5NtransverseforcepositiveinY-directiononthebeamatS=0:75ors=2:743m.Eq(3.27)wereformulatedandnumericallyintegratedbytimetosolveforq(T),q(T)astimeprogressed.ThisprocesswasdoneusingMATLABsolverode15s.ode15sisavariableordersolverbasedonthenu-mericaldifferentiationformulas(NDFs).Optionally,itusesthebackwarddifferentiationformulas38(BDFs,alsoknownasGear'smethod)thatareusuallylessefode15swaschosenasthenumericsolverbecauseitissuitableforsolvingadifferentialalgebraicproblem,i.e.,massmatrixbeingsingular.Formoredetails,refertoShampine&Reichelt(1997)andShampineetal.(1999).TheresultsofthefreevibrationsimulationareshowninFig.3.2withdimensions.Figure3.2Plotsofthesysteminfreevibration:normalizedenergy;transversedisplacementsam-pledatS=0:75ors=2:743m;materialcoordinateofthepointQincontactwiththeslider;constraintforceintransversedirection;constraintforceinaxialdirection.TheoscillatorybehaviorofthenonlinearbeamcanbeclearlyobservedfromFig.3.2.ThefrequencyofthetransversedisplacementunsampledatS=0:75ors=2:743misapproximatelyhalfofthefrequencyofthematerialcoordinatesqwhichcorrespondstothepointQincontactwiththeslider.Thefrequencyoftheconstraintforcelyintransversedirectionisstronglyassociatedwiththetransversevibrationun(S=0:75),whilethefrequencyoftheconstraintforcelxinaxialdirectionisstronglyassociatedwithsq.Constraintforcesalsocarryhighfrequencycontent.This39isduetothefactthatconstraintforceisrelatedtohigherdegreesofderivativesofunandutwithrespecttos.Thetotalenergyofthesystemwasobtainedbycombiningkineticandstrainenergiesandthennormalizedbytheinitialenergylevel.Energydecaysslowlyduetothepresenceoflightmaterialdamping.3.3.2Numericissueofcomputingtheconstraintforce:Lagrangemultipliervs.penaltymethodIthasbeenshownintheprevioussubsectionthattheslidingdynamicscanbesolvedusingthenumericschemethatemploystheLagrangemultiplierthatgivesrisetoanalgebraicproblem.Thisleadstoasystemofextremelystiffequations(singularmassmatrix)andmakesthesolutionprocessverycostlyandtimeconsuming.Aspecialimplicitsolverthatemploysbackwarddifferentiationformulashadtobeusedbutshoweditsinefy.Thenumericexampleshownintheprevioussubsectiontookover24hourstocompletewithoccasionalmanualintervention.Consideringthatthesizeoftheexampleproblemisactuallysmall(64unknownsintotal),thistimeconsumptionistoohigh.TheauthoralsoexploredthenumericschemeusingthepenaltymethodinsteadofLagrangemultiplierstosolvefortheconstraintforces.Inthisstructuralproblem,thepenaltymethodes-sentiallywasappliedbyhandlingtheconstraintwithlargesprings.Inprinciple,anytwodegreesoffreedomthataresupposedtoobeythekinematicconstraintofbeingtiedtogetherrigidlyareinsteadconnectedusingalargespring.Thentheconstraintforcecanbederivedfromtheinternalelasticforceofthespringgiventhestiffnessofthespringandtherelevanttwodegreesoffreedomsolved.Stiffnessofthespringshouldbechosenbyconsideringthetrade-offbetweenthecon-straintaccuracyandthenumericexpense.Thatistosay,ifthespringstiffnessischosentobeverylarge,thentheconstraintandtheconstraintforcecouldbesimulatedaccurately,butnumericallytheproblemmaybetoostiffanddiftosolve.Ontheotherhand,ifthespringstiffnessisnotsetlargeenough,thenumericproblemcouldbesolvedmoreeasily,buttheconstraintmodelinglosesitsaccuracy.40TheofusingthepenaltymethodusuallycomefromthefactthatinsteadofusingaspecialimplicitnumericsolverforDAEs,ageneralexplicitsolvercanbeusedwithaedtimestep.Thismakestheproblemsolvingprocessmorepredictableintermsoftimeconsumption,givenachosentimestepthatguaranteesthenumericstability.However,afterexperimentingwiththepenaltymethodonthisproblem,theauthorfoundnoapparentadvantageofitintermsofnumericexpense.ThereforethenumericsimulationspresentedinthefollowingallusedLagrangemultipliersandDAEformulations.3.4Feedbackcontroldesignoftheslidermotion3.4.1PreliminarydesignoftheslidermotioncontrolThecontrolstrategyusingslidingmechanismisbasedontheideaofnegativework.Thisideasimplyexploresthatenergylossofthebeambefacilitatedbynegativeworkdonebytheconstraintforceappliedbytheslider.Whenthesliderisedinspace,noactualworkcouldbedonebytheconstraintforceandthetotalenergyofthesystemisonlydissipatedthroughmaterialdamping,asshowninsimulationresultsinthefreevibrationcase.Sinceonecanmeasure(inpractice)orcompute(insimulation)theaxialconstraintforceLx,itcanbeusedasfeedbackinthecontrolscheme.Insteadofusingaedconstraint,theschemeprescribestheslider'sX-directionmotionXCtobeoppositetothedirectionofLxtodonegativework.ThisstrategycanberealizedbyreplacingthestaticconstraintinEq.(3.1)withamovingoneasfollows:VC=XC=¶Ut(S=SQ;T)+SQ(T)¶T(3.28)TheconstraintinEq.(3.2)remainsthesame.ThiscontrolschemeisshowninFig.3.3.BasedonpreviousworkdonebyNudehietal.(1992)andIssaetal.(2010),wechoosetheLyapunovcandidateasV1=Etotal(q;q;T)(3.29)41TheoriginofV1correspondstothestaticequilibriumstateofthebeam.Itisobviousthatwhenthebeamstaysinstaticequilibrium,i.e.,q=q0andq=0,thereisV1=0.Inthestaticstateq0hasallentrieszerobutSQdependingonthesliderposition.ThechangeoftotalenergycanonlybecausedbydampingandtheworkdonebytheaxialconstraintforceLxiftheX-directiondisplacementXCofthesliderisprescribedbythecontrolschemewhiletheY-directiondisplacementisalwayskeptaszero,namelyEtotal=Edamping+XCLx(3.30)whereEdamping0.Toimplementthiscontrolscheme,thematerialboundaryneedstobesetsinceinpracticetheslidercanonlyoperatewithinacertainrangeofthebeam.InordertorealizeSlowerSQSupper,thesliderpositionisfurtherprescribedasXC=u=8><>:h(y1=Lx)ifSlower<SQ<Supper0otherwise(3.31)SubstitutingEqs.(3.30)and(3.31)intoEq.(3.29)yieldsV1=8><>:Edampinguy1=Edampingy1h(y1)0ifSlower<y2=SQ<SupperEdamping0otherwise(3.32)ForthechoiceofcontrolinputuinEq.(3.31),ithasbeenshowninEq.(3.32)thatitleadstoV10.UsingLaSalle'sTheorem(Khalil,2002),onecanclaimthattheoriginisasymptoticallystable.WenowinvestigatetheefyofthecontroldesigninEq.(3.31)bysimulation.Thebeamisagainmeshedusing20elements,withthesamepropertiesandgeometryshowninTable3.1.Setmaterialboundariesslower=1:33mandsupper=2:33mforoursimulation.Inimplementation,thematerialrange[slower;supper]forcontrolisdeterminedbythestrokelengthoftheslidingdevice.Thespeedlimitofthesliderissettobe1m/sec.Withthesameinitialconditionsappliedasinthefreevibrationcase,resultsareshowninFig.3.4withdimensions.42Figure3.3Preliminaryfeedbackcontroldesign.AsshowninFig.3.4,attheendof12sec,thetotalenergyofthesystemwaseffectivelydis-sipatedto1:0%ofitsstartinglevel,whilethetransversedisplacementatS=0:75wasreducedto17%.Mosthighfrequencyvibrationwassuppressed,asthehistoryofun(S=0:75)shows,leavingthebeamwithslowly-varyingresidualtransverseoscillationsthatshouldbefairlyeasytocopewithinapplication.Therangeofmotionforthesliderturnedouttobewithin0:2m,whichsuggeststhatarelativelysmalltraveldistancesufforthepurposeofvibrationsuppressioninthisdesign.Asthevibrationenergywasdissipatedandthedisplacementofthebeamwasreduced,constraintforcelxinX-directiondecreasedrapidly.Thisisbecausethegeometricnonlinearitywhichgivesrisetolxbecomeslessasthereducedamplitudeofundecouplestheinter-actionbetweenunandutintheaxialstrain.Onepotentialproblemofthiscontroldesignisthatbecauseofthehighfrequencycomponentsinlx,thepositionoftheslideroscillatestoofastinordertofollowthechangeoflx,asonecanseefromtheplotofxc.Thismayexceedtheband-widthoftheactualactuator.Thepotentialissueinimplementationleadstoourfeedbackcontroldesign,whichistobeintroducednext.43Figure3.4Plotsofthesystemappliedwithdirectcontrol:normalizedenergyofthedirectcontrolresultsinsolidline,normalizedenergyoffreevibrationindashedlineasreference;transversedisplacementsampledatS=0:75ors=2:743m;sliderposition;slidervelocity;constraintforceinhorizontaldirection.3.4.2controldesignAsstatedinlastsubsection,whentheslideristryingtofollowthechangeofaxialconstraintforceLx,itmayoscillatetoofastthatthehardwarerequirementexceedsthebandwidthoftheactuator.Totacklethisproblem,weintroduceaLow-PassFilter(LPF)intothefeedbackloop,whichmeasuresLxasitsinputandgivesasignalzastheoutput.ThisLPFischosentobeorderwiththetimeconstantt.InsteadofusingLxdirectlyinthecontrollaw,weusethesignalzinthefeedbackcontroldesign.ChoosinganewLyaponovcandidate,wecanprovethatthestabilityofthesystemcanberetained.ThecontrolschemeisshowninFig.3.5.ChooseanewLyapunovcandidateas44Figure3.5controldesignwith.V2=Etotal+tZz0h(x)dx(3.33)TheoriginofV2correspondstothestaticequilibriumstateofthebeam,i.e.,q=q0andq=0,whereq0hasallentrieszerobutSQdependingonthesliderposition.UseEq.(3.30)wecanobtainthatwhenSlower<y2<Supper,thefollowingholds:V2=Etotal+th(z)z=Edampingy1u+h(y1z)=Edampingzh0(3.34)Andwheny2=SQSlowerory2=SQSupper,thereisV2=Edamping0.ThereforeusingLaSalle'sTheorem,againwecanclaimthattheoriginisasymptoticallystableforthecontrolsystem.Followingistheinvestigationoftheefyofthecontroldesignbysimulation.Meshing,materialproperties,geometricandcontrolparameters,plusinitialconditionswerecho-45sentobethesameasinthepreliminarycontroldesign.Forthelow-pass,timeconstanttwassimulatedas40ms.ResultsofsimulationareshowninFig.3.6withdimensions.Figure3.6Plotsofthesystemappliedwithdirectcontrol:normalizedenergyofthecon-trolresultsinsolidline,normalizedenergyoffreevibrationindashedlineasreference,normalizedenergyofthedirectcontrolresultsindotted-solidlineasreference;transversedisplacementsam-pledatS=0:75ors=2:743m;sliderposition;slidervelocity;constraintforceinhorizontaldirectionafterthelow-pass.Addingthelow-passisatrade-offbetweencontrolefyandactuatorbandwidth.AsshowninFig.3.6,theslidervelocityxcdrivenbytheactuator,switchedlessfrequentlycomparedwiththepreliminarydesignresults,especiallyattheearlystage(before2sec).Thiseasestherequirementfortheactuatorbandwidth.Thecostforthisisthatthevibrationsuppressionprocessbecomeslessefnt.Attheendof12sec,9:7%ofthetotalenergyofthesystemremained.Toachievethesamelevelof1:0%suppression,comparedtothecontrolwithouta,the46designobviouslyrequireslongertimeofthesliderapplication.Insummary,uptothispoint,usingtheprincipleofvirtualdisplacementandD'Alembert'sprinciple,wehasmodeledatwo-dimensionalnonlinearbeamwithpinned-pinnedboundarycon-ditionsandafrictionlessslidingconstraint.Avariable-lengthelementmethod(adaptivemeshing)intheframeworkofALEdescriptionwasemployedtodiscretizethedynamicsystemandtosolvetheequationsofmotionnumerically.Withtheaxialconstraintforcesolvedatanymoment,theslidermotionwasprescribedtodonegativeworkonthestructuresothatvibrationcanbesuppressed.Numericalsimulationresultswerepresentedtodemonstratetheeffectivenessofthiscontrolstrategy.Tomeetthebandwidthrequirementoftheactuator,anonlinearwasplacedinthefeedbackloopandasymptoticstabilityoftheequilibriumwasestab-lishedusingLyapunovstabilitytheory.Thismethodemployingaslidingmechasimforvibrationsuppressioninthexiblestructurehasbeenshownitsgreatpotential.3.5CombiningslidermotionwithstiffnessvariationWithtwovibrationcontrolstrategiesdeveloped,thissectionlooksintothecombinationofthetwo.Inageneralsense,astheslidermovesandchangesthecontactpointwiththebeam,thestiffnessofthesystemvaries.Butthisvariationoccurscontinuously.Inthiscontextthestiffnessvariationisreferredtoasthemechanismthatvariesthestiffnessofthestructureinadiscontinuousway.Stiffnessvariationinthefollowingisperformedonthesliderbytreatingtherotationofthesliderasanon/offjoint.Whenthisjointisactivated,therotationofthesliderised,aswellastherotationofthecontactedpointofthebeam.Numericallyitisimplementedbyformulatingaconstraintontherotationaldegreeoffreedomattheslidingnodeonthebeammodel.Forsuchstiffnessvariationtobeeffective,thebeamstructureneedstohavemoderatedampingpresent,whichisverydifferentfromthesystemwithlightdamping.Inthelattercase,asshowninprevioussections,energydissipationcannotutilizethedampingmechanism.Ithastorelyonthedirectenergyreductionmeansthroughslidingmotion.Withmoderatedampingpresent,asshownin47Table3.2Properties,geometryandotherparametersofthebeamsystemusedinthevalidationsimulation.MaterialAluminumYoung'smodulusE69109PaDensity¯r2700kg/m3Dampingcoefh103secBeamlengthL3:66mBeamcrosssectionarea38:1mm1:57mmPre-load(tensile)f10NAddedmass10:66kgAddedmass20:66kgChapter2,energyredistributioninthefrequencydomainduetostiffnessvariationallowsthehighfrequencyenergytobedissipatedrapidly.Iftheslidingmotiondesignedsofarisreferredtoas"forwardsliding"intermsofthedirectionofthemotionprescribed,onecaneasilyimaginethatwithlightdamping,theslidingshouldnotbeappliedinthereverseway,forthereasonthattherewouldbeenergycontinuouslyaddedbytheslidingmotion.Suchenergywillaccumulatewithnoeasyoutletthroughmaterialdamping,eveniftheenergyexistsinhighfrequencyform.Ifreverseslidingisapplied,thesystemwillmostlikelygounstableorfailstructurallyinreality.However,ifmoderatedampingispresent,thecombinationofthereverseslidingandstiffnessvariationbecomespossible,fordampingmaybeenoughtodissipatetheenergyaddedbytheslider.Therefore,acombinationinvolvingreverseslidingwillalsobeinvestigated.3.5.1SimulationofthesystemundergoingfreevibrationTable3.2liststheproperties,geometryandparametersusedinthefollowingsimulation.ThegeometryofthebeamwassettobeidenticalasintheexperimentstobepresentedinChapter4.SowasthematerialpropertiessuchasYoung'smodulusanddensity.Therewerelumpedmassesaddedtothebeamtodecreasethenaturalfrequenciesofthestructure.Mass1waslocated48ats=0:95m(1=4ofthebeamlength).Mass2waslocatedats=2:74m(3=4ofthebeamlength).Thedampingcoefh=103secwastheresultoftrialstoapproximatethedecayrateobtainedfromtheexperimentresultsofabeamstructuresubjectedtomoderatedamping(Chapter4).Thedampingcoefusednowiscantlyhigherthanthevalueh=105secusedinearliersimulations.ResultsofthefreevibrationcaseareshowninFig.3.7.Theinitialconditionsofthefreevibrationandofallsimulationcaseswerecreatedbydisplacingmass1(locatedats=0:91m)by11:2mm.Afterbeingreleasedfromitsinitialdisplacement,thebeamunderwentfreevibration.Duringthisprocesstherewasnoslidingmotion,butthesliderwasallowedtorotatefreely.FivequantitiesareshowninFig.4.7.Theyare:1)axialforceevaluatedonslider,appliedbythebeam.2)slopeoftheslider,indicatingtherotationaldisplacementatthecontactpoint.3)positionoftheslider,measuredfromleftendofthebeam,indicatingthelineardisplacementoftheslider.4)transversedisplacementofmass1.5)transversedisplacementofmass2.Resultsoftheaxialconstraintforceandsliderslopewerebasicallyobtainedasthesameor-derofmagnitudecomparedtoexperimentresults(tobepresentedinChapter4).Sliderslopeanddisplacementsofthemassesallshowdominantfrequencycontentin2˘3Hzrange.Alowfre-quencyenvelopeorbeatingbehavior,wascaptured.ThebeatingwillbeanalyzedlaterindetailinChapter4.Dominantfrequencycomponentoftheaxialconstraintforcewastwiceofthedomi-nantfrequencycomponentintransversevibration,consistentwithsimulationresultsearlier.Lowfrequencycomponentsuperimposedintheaxialconstraintforcecanbeestimatedas0:2Hz.ThiscanbevbyaFastFourierTransform(FFT)analysisshowninFig.3.8.Mass1,directlydisplacedintheinitialcondition,hadthedisplacementthatstartedatthelevelof0:011m.At10sec,mass1displacementmagnitudedecreasedtoˇ0:006m,55%oftheinitiallevel.Mass2,notdirectlyexcitedbythedisturbance,hadtheinitialdisplacementof0.Asthevibrationpropagatedfrommass1andreachedmass2,itsdisplacementstartedtoincreaseandvary.At10sec,mass2alsohadthedisplacementofˇ0:006m,estimatedbyobservingtheneighboringbeatingpeaks.Vibrationwasevenlydistributedonthetwosidesoftheslideronthebeam.49Figure3.7Simulationofthesystemunderfreevibration(timedomain).Plotsfromtoptobottom:axialforceevaluatedontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2.50Figure3.8Simulationofthesystemunderfreevibration(frequencydomain).Plotsfromtoptobottom:axialforceevaluatedontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;displacementofmass1;displacementofmass2.51Alsoevaluatedaretheaccelerationsoftwomasses,asplottedinFig.3.9.Mass1hadtheaccelerationthatstartedatthelevelof7:4m/s2.At10sec,mass1accelerationdecreasedtoˇ1:2m/s2,16%oftheinitiallevel.Mass2hadtheinitialaccelerationof0.At10sec,mass2alsohadtheaccelerationofˇ1:2m/s2,theplotwasextendedto12secinordertobetterrevealtheaccelerationlevelasitmaybecoveredinbeats.Figure3.9Simulationofthemassaccelerationsunderfreevibration.AsystemenergyhistorywasevaluatedandisplottedinFig.3.10.Thesystemenergywascomputedbysummingthepotentialenergyandkineticenergyofthebeamwithaddedmasses.Theplotshowsthatthetotalenergystartedat15:3mJanddecreasedto2:1mJat10sec,with14%remaining.Thismeans86%oftheenergywasdissipatedthroughmaterialdamping.Thesystemhasmoderatedampinglevelthatiscomparabletowhatwasobservedinexperiments(seeChapter4).Withareasonablymodelestablishedabovethroughfreevibrationevaluation,thecombinationofcontrolmethodswereinvestigatedinthefollowing.52Figure3.10Simulationofthesystemenergyunderfreevibration.3.5.2SimulationofthesystememployingslidingcontrolonlyFirstisthecaseemployingslidingmechanismonly.ResultsofsimulationwereproducedusingthesameinitialconditionasinthefreevibrationcaseshowninFig.3.11.Theslidermovedatthemaximumvelocityof1m/s.Astrokelengthof1:76mwasassumed.Thislengthsetthematerialboundaryrightbeforethesliderreachesthetwomasses.Theslidervelocitywassettobezeroasithitthematerialboundaryandcontinuedwiththedirectionofattemptingtogoovertheboundary.Samealgorithmwasapplied,asshowninFig.3.3.OnecanclearlyseetheuppermaterialboundarybeingreachedinFig.3.11.Formostofthetimeinthe10secinterval,theaxialforceonthesliderwasinpositiveX-direction,ortowardstherightendofthebeam.Displacementofmass1startedatthelevelof0:011m.At10sec,mass1displacementde-creasedtoˇ0:007m,64%oftheinitiallevel,highercomparedtothefreevibrationcaseinwhichthevaluebecameˇ0:006m.Thisdefeatsthepurposeofvibrationreduction.Oritcanbeun-derstoodasthedirectenergyremovalbytheslidercouldnotcompetewiththemoderatedamping53Figure3.11Simulationofthesystemunderslidingcontrolonly.Plotsfromtoptobottom:axialforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2.54rate.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialdisplacementof0.Aftersomeoscillation,mass2displacementdecreasedbackto0,whichshowedtheeffectofvibrationsuppression.Similarly,accelerationsresultswereextracted.ResultsarelistedtogetherinTable3.3forcomparison.Thevibrationwasnoteffectivelysuppressedbytheslidingmechanismalone.Aconjecturewasthenmadethatthedirectenergyremovalratebytheslidermaynotbefastenoughtocreateameaningfuldifferencecomparedtothefreevibrationcase.Basedonthisconjecture,thefollowingevaluationwasperformed.Figure3.12Simulationofthesystemenergyunderslidingcontrolonly.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion.TheenergyofthesystemareplottedinFig.3.12.Thesolidlineshowsthetotalenergyofthesystem.Thedashedlinerepresentstheenergychangedirectlyduetotheslidingmotion.Thetotalenergyofthesystemstartedfrom15:3mJanddecreasedto1:7mJat10sec.Thetotalenergywasreducedby89%,slightlybetterthanthefreevibrationcase(86%reductionthroughmaterial55dampingonly,nocontrol).Whatisinterestingisthatevaluationoftheenergychangedirectlyduetoslidingmotionshowsthatthesliderkeptreducingtheenergyofthesystem(therebynegativevalueforenergychange).At10sec,theslidingmotionreducedthesystemenergyonlyby3:1mJ,whichaccountedfor20%ofthetotalenergyat0sec.Thisshowsthatthemainportionoftheenergydecaywasduetomaterialdamping(69%),notduetothedirectenergyremovalbysliding.Directenergyremovalbyslidingwasnotefcomparedtomaterialdamping.3.5.3SimulationofthesystememployingslidingcontrolcombinedwithstiffnessvariationUsingtheconsistentparametersandinitialconditions,thesimulationcontinuedwiththecaseusingthecontrolmethodcombiningstiffnessvariationandforwardslidingmechanism.Itisreferredtoas"forwardsliding"forthedirectionoftheslidingmotionisthesameasthealgorithmoriginallydesignedinSection3.4,asopposedto"reversesliding",whichwillalsobeinvestigated.Inthefollowingsimulation,theslidingmotionwasindependentfromtheactivating/deactivatingoperationforstiffnessvariation,meaningthattheslidercouldstillmoveregardlessoftherotationalon/offjointstatus.Stiffnessvariationwasactivatedanddeactivatedcyclically.Insimulation,theactivationoccurredexactlyattheinstanceswhenthesliderslopepassed0.Itwaskeptactivatedfor0:1sec,thenwasdeactivatedandreleasedtheslopeforanother0:1sec.Afterthattherotationalon/offjointwaitedforthenextinstanceofactivationandthecyclewenton.ResultsofthiscaseareshowninFig.3.13.Theaxialforceonthesliderhadapositivevalueformostofthetime.Itisconsistentwiththefactthattheoverallslidermovedinapositivedirectiontowardstherightendofthebeam.After5sec,theslidingmotionstoppedduetothematerialboundary.Stiffnessvariationwasactivatedanddeactivatedcyclicallythroughoutthewholeprocess,asinthesliderslopeplot.Thedisplacementofmass1startedatthelevelof0:011m.At10sec,mass1displacementdecreasedtoˇ0:006m,55%oftheinitiallevel,notmoreeffectivecomparedtothefreevibrationcase.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialdisplacementof0.Aftersomeoscillation,mass2displacementdecreasedbackto0,whichshowedthevibrationsuppressioneffect.Similarly,accelerationsresults56Figure3.13Simulationofthesystemundercombinedcontrolofstiffnessvariationandforwardslidingmechanism.Plotsfromtoptobottom:axialforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;displacementofmass1;displacementofmass2.57wereextracted.ResultsarelistedtogetherinTable3.3forcomparison.Figure3.14Simulationofthesystemenergyundercombinedcontrolofstiffnessvariationandforwardslidingmechanism.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion.Moreinsightcanbegainedfromtheenergyplot.AsshowninFig.3.14,thetotalenergyofthesystemstartedfrom15:3mJanddecreasedto1:2mJat10sec.Thetotalenergywasreducedby92%,alittlebetterthanthefreevibrationcaseinwhichtherewas86%reductionthroughmaterialdampingonly,nocontrol.Evaluationofenergychangedirectlyduetotheslidingmotionshowsthatthesliderkeptreducingtheenergyofthesystem(therebynegativevalueforenergychange).Atthe10sec,theslidingmotionhadreducedthesystemenergyonlyby1:3mJ,whichaccountedfor9%ofthetotalenergyat0sec.Thisagainshowsthatthemainportionoftheenergydecaywasduetomaterialdamping(83%),facilitatedbymaterialdamping,andcannotbeattributedtothedirectenergyremovalbysliding.Inanefforttoimprovetheperformanceusingthecurrentacompletelynewcombinationwasthenexplored.Thiscombinationalsoinvolvesstiffnessvariationandsliding58Figure3.15Simulationofthesystemundercombinedcontrolofstiffnessvariationandreverseslidingmechanism.59mechanism.Butinsteadofslidingaccordingtothealgorithmpresentedbefore,theslidingmotionwascarriedoutinthereversedirection.Theoretically,ifslidinginthereversedirection,alwaysoppositetowhattheoriginalalgorithmsuggestsinanymoment,slidercontinuouslyaddsenergyintothesystem.Thisseemstobehurtingthepurposeofvibrationsuppression.Buttherearepotentialofdoingso.Ithasbeenobservedandtheconjecturehasbeenvalidatedthatthroughslidingapproach,thedirectenergyremovalrateonlymaynotbefastenoughtocreateameaningfuldifferencecomparedtothefreevibrationcase.Themainportionofenergydecaywasstillduetothecontributioninternalmaterialdamping.Thehopeofapplyingreverseslidingisthatifthesliderkeepsmovingtowardsthesidewithhigheraxialforce,itmaydramaticallyreshapetheenergyspectruminthefrequencydomain.Theslidingmotionwilldirectlyaddsomeenergy.Butifitcreatesamuchhighermaterialdampingrate,theresultsofthetradeoffmayyieldfasterenergydecayoverall.Therefore,thecaseofcombiningreverseslidingandstiffnessvariationwassimulated.Simu-lationresultsareshowninFig.3.15.Axialforceontheslideralmostalwayshadapositivevalue.Theslidermovedalmostmonotonicallyinnegativedirectiontowardstheleftendofthebeam,foritwascontrolledinthereversedway.Itcanbeunderstoodasthatthesliderwasalways"movingagainstthewave",towardsonepartofthebeamthatexertingmoreaxialforce.After3sec,slidingmotionwasbarelyobservedduetothematerialboundary.Stiffnessvariationwasactivatedanddeactivatedalternativelythroughoutthewholeprocess,asinthesliderslopeplot.Dis-placementofmass1startedatthelevelof0:011m.At10sec,mass1displacementdecreasedtolessthan0:0001m,whichislessthan1%oftheinitiallevel,showingvibrationreduc-tion.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialdisplacementof0.Aftersomeoscillation,mass2displacementdecreasedto0:0007m,7%comparedtotheinitiallevelofmass1.Similarly,accelerationsresultswereextracted.ResultsarelistedtogetherinTable3.3forcomparison.Themechanismoftheenergychangeismorerevealingfromtheenergyplot.AsshowninFig.3.16,thetotalenergyofthesystemstartedfrom15:3mJanddecreasedto0:05mJat10sec.The60Figure3.16Simulationofthesystemenergyundercombinedcontrolofstiffnessvariationandreverseslidingmechanism.Solidlineshowsthetotalenergyofthebeamsystem.Dashedlinerepresentstheenergychangedirectlyduetotheslidingmotion.totalenergywasreducedby99:7%,showingsuccessfulvibrationreductioncomparedtothefreevibrationcase.Evaluationofenergychangedirectlyduetotheslidingmotionshowsthatthesliderkeptaddingenergytothesystem(therebypositivevalueforenergychange).At10sec,theslidingmotionhadadded1:6mJ,whichaccountsfor10%ofthetotalenergyat0sec.Thismeansthatmaterialdampingpracticallydissipated110%oftheinitiallevelofthesystemenergy.Apparently,byapplyingreverseslidingcontrolthesliderkeepsmovingtowardsthesidewithhigheraxialforce.Thisindeeddramaticallyreshapestheenergyspectruminfrequencydomain.Theslidingmotioncouldaddsomeenergy(10%).Butitcreatesamuchmorematerialdamping,whichisfacilitatedbystiffnessvariation.Theneteffectismuchfasterenergydecayoverall.Inthiswayvibrationseemstobesuppressed.Combinationsofthetwocontrolstrategiesprovidetheinsightofhowarealstructurewiththesimilaranddimensionmaybehaveundercontrol,asthestructureinexperiments61Table3.3Comparisonofperformanceofdifferentcontrolstrategies.Allrelativedisplacementsandaccelerationsarecomputedusingthemass1initialstatusasreference.ControlConditionMass1Mass2Mass1Mass2SystemRelativeRelativeRelativeRelativeEnergyAccelAccelDisplacemtDisplacemtRemainedat10secat10secat10secat10secat10secFreeVibration55%55%16%16%14%SlidingOnly64%0%14%0%11%ForwardSlidingandStiffnessVariation55%0%16%1%8%ReverseSlidingandStiffnessVariation1%6%0%7%<1%usuallyhavemoderatedampingpresent.Thenextchapterpresentstheexperimentalstudyofasystemwithtwocontrolstrategiesimplementedandapplied.62CHAPTER4EXPERIMENTALSTUDYOFVIBRATIONSUPPRESSIONTHROUGHSTIFFNESSVARIATIONANDSLIDINGMECHANISM4.1ExperimentdesignAnexperimentalsetupispresentedinthischapter.Thecontenttobeshownstartsfromconcep-tualizationanddesigninSection4.1,followedbyimplementationandoperationinSection4.2.Section4.3showstheresultsofthisexperimentalplatformthatvalidatethevibrationsuppressionstrategiesproposed.4.1.1Tension-alignedstructureThebasicstructureunderstudyfollowsthedesigndepictedinFig.3.1.Thesystemshouldconsistoftwoparts:1)Axibletension-alignedbeamstructure.2)Arelativelyrigidsupportstructure.Furthermore,therearethefollowingconsiderationswhendesigningthesystemonthestructurallevel:1.Thebeamundertensionshouldbexibleandlight-weight.Thesefeaturesmakethebeamexposedtonoticeablevibrationproblemduetoexternaldisturbances,meanwhilesensitivetocontrolforcefromtheactuators.Sincelow-frequencyvibrationisthemaintargetinthisstudy,thereisatrade-offbetweenxibilityandlight-weight.Tomanufactureastructurewithlownaturalfrequencies,thestiffnessneedstobeaslowasstructuralstabilityallowsxibility),whilethestructureneedstocarryareasonableamountofmass.Ontheotherhand,theamountofmassofthebeamshouldbekeptsmall(light-weight)toallowactuatorswithreasonabledrivingpowertobeabletochangethedynamicsofthebeam.2.Atthetwoendsofthebeam,thereneedstobeastableandeasy-to-realizetensiondevicethatimpartsenoughtensiontothebeam.Afterthepre-tensionisapplied,thereoughttobea63lockingmechanismonthistensiondevicethatmaintainsaconstantboundaryconditionforthebeam(pinned-pinned).Thisistoguaranteethatwhilethebeamisundergoingvibration,thetensiondevicedoesnotaddunexpecteddynamicstothesystem.3.Becauseonecontrolstrategyinvolvesaslidingmechanism,thesurfaceofthebeamshouldbehardandsmoothonwhichtheslideroperates.Alsoalthoughtheslidingmotionisdesignedtobealongtheaxialdirectionofthebeam,deviationfromtheaxisisalmostunavoidable.Therefore,thebeamneedstohaveenoughwidthtotoleratethemotionerror.Butthewidthcannotbetoolargetoproduceanytorsionaldisplacementthatisnotaccountedfor.4.Thesupportstructureshouldberigidenoughsothatthevibrationofthetension-alignedbeamdoesnotinteractwiththesupportstructure.Ideally,onewantsthebeamtobedirectlygroundedthroughthetensiondevicetoeliminateanyunwanteddynamicsandconcentratethestudyonlyonthebeamitself.Butinreality,thesystemincludingacomplexslidingactuatorneedstobemountedonaplatform-likesupportstructure.Onethenhastodesignastructurethathasthecomparablesizeofthebeamandbemuchmorerigidsothatthefromthevibratingparts(beam,actuators)onthesupportstructureareminimized.4.1.2StiffnessvariationmechanismStiffnessvariationasourproposedvibrationcontrolstrategywasimplementedintwowaysinsimulations.Onewaywastousearotationalon/offjointthatcanbefairlyeasilyrealizedonacontinuousbeam.Thisistouseaclampingdevicethathasanon/offlockingmechanism.Theotherwayistoconnectthebeamtothesupportstructurethroughatranslationalspringoftime-varyingstiffness.Thisisratherexpensivetoconstructanddiftoengineer.Theexperimentdesignchoosestherotationalon/offjointasthemeansforstiffnessvariation.Inaddition,asdescribedinSection.4.1.3,thelocationofthejoint,i.e.,thecontactpointwiththebeam,canbevariedormovedonthesurfaceofthebeam,alongthebeam'saxialdirection.This64furthervariesthestiffnessofthebeam.ThisdiffersfromthesituationinSection2.3orinSection2.4,wheretheon/offjointsareedatthelocationsofthediscretehingesbetweenpanels.4.1.3SlidingmechanismThedesignoftheslidingmechanismconsidersthefollowingfactors:1.Thesliderneedstohaveenoughstrokespantocoveraportionofthebeam'stotallength.Italsoneedstohavefrequencybandwidthandspeedtocopewiththebeamdynamics.Thisisduetofactthatthecontrolalgorithmdictatesthattheslidingmotioncloselyfollowsthechangeintheconstraintforce.Asthenumericsimulationreveals,thisrequiresthebandwidthoftheslidertobetwicecomparedtothebeam'stransversefrequencythatistobesuppressedafterexcitedbyexternaldisturbances.2.Thesliderneedstobealwaysincontactwiththebeamsurfaceasitmoves.Notethatthebeamsurfaceortheslidingmotionwillnotbeperfectlysmoothorstraight.Itisnecessarytocreateaslidercontactinterfacethatisnotabsolutelyrigidbuthassomecompliancetotoleratetheseimperfections.Otherwiseitispossiblefortheslidertodamagethesurfaceofthebeamorevenworsetodestabilizethebeambytheslidingitself.4.2Experimentimplementation4.2.1AnonlinearbeamInordertoexplorenonlinearityinthebeamvibration,whichisessentialtovalidatethevibrationsuppressionstrategythroughslidingmechanism,thebeamoughttobebothxibleandslender.FlexibilityrequireslowYoung'smodulusofthematerialtobeused.Thatallowsthebeamtoundergorelativelylargevibrationdisplacementcausedbymoderatedisturbance.Slendernessisneededbecausegeometricnonlinearitystemsfromlargetranversedisplacementcomparedtothebeam'sthickness.Inotherwords,thebeamneedstobethinenough.Consideringthebeam65surfacewillbeconstantlyexposedtotheslidercontact,itispreferabletousemetalmaterialtoresistthewearfordurabilityconsideration.ThebeamwasthereforechosentohavethepropertiesandgeometryasshowninTable4.1.Thebeamwasestimatedtohavea20mmdisplacementastheinitialcondition,thisisover10timesitsthickness,whichprovidesgeometricnonlinearity.Table4.1Propertiesandgeometryofthebeamusedinexperiment.MaterialAluminumYoung'smodulusE69109PaDensity¯r2700kg/m3BeamlengthL3:66mBeamcrosssectionarea38:1mm1:57mmThebeamwasconnectedtothesupportstructurethroughthetensiondeviceusingapinned-pinnedboundarycondition.Thisboundarycondition,comparedtootherboundaryconditions,hasthexibilitythatkeepsthenaturalfrequencyofthebeamlow.4.2.2AddedmassLumpedmasseswereaddedonthebeamtolowerthenaturalfrequency.Twomasseswereattachedonthebeam.Mass1,withtheweightof0:33kg,waslocatedats=0:95m(1=4ofthebeamlength).Mass2,withtheweightof0:32kg,waslocatedats=2:74m(3=4ofthebeamlength).4.2.3TensiondeviceThetensiondeviceorthetensioneremployedacrankmechanismtostretchthebeamatitsoneend.Thecrankrotationwasconvertedtolineardisplacementthroughashafttrackwhichwasaxiallyalignedwiththebeamandapairofbushingsmountedonit.Aforcesensorwasconnectedbetweenthecrankandthebeamtomakesureadesiredtensionlevelwasachieved.Afterthepre-tensionwasapplied,thetensiondeviceprovidedalockingmechanismthroughfasteningscrewstothetrack66whichnolongerallowedanyaxialmovementofthebeam.Thismaintainedaconstantboundaryconditionforthebeam(pinned-pinned).Thebeaminevitabilityslackedafterinitialmountingduetogravity.Thetensiondeviceturnedouttobeextremelyimportantintheexperimentsetupinmaintainingthestraightnessofsuchalongbeam.4.2.4SlidingmotioncontrolSlidingmotionwasprovidedbytheThomsonTolloM50beltdriveactuator,depictedinFig.4.1.ThisactuatorwaspoweredbyaKollmorgenAKM23Ebrushlessservomotorthrougha5:1gearreducer.ThemotioncontrolwasimplementedbyaKollmorgenAKD-P00306programmable.Figure4.1Beltdriveactuatortoprovideslidingmotion.674.2.5Rotationalon/offmechanismThesliderwasamovingjointincontactwiththebeamsurface.Itextendedfromthecontactpointdownwardswithashaftandsatonacarriagedrivenbytheslidingactuator.Thecarriagehadarotationalon/offmechanismontheslidershaft.Thiscontrollableon/offmechanismwasrealizedbyaelectromagneticbrakethatwasmountedtothecarriage.Alsoinstalledinthecarriagewasanencoderwhichtheslidershaftgoesthrough,asshownFig.4.2.Theencodermeasuredtherotationoftheslidershaftaspartofthefeedbackinformation.Figure4.2Theslidercarriagethatconsistsofrotationalon/offmechanismandencodertomeasuretherotation.Aslidershaftistobeassembledontothecarriage.4.2.6SlidercontactinterfaceThecontactinterfacebetweenthesliderandthebeamusedballplungertoprovidesomecom-pliance.Ballplungersarecylindricalpartswithrotationalspheresatthetipwhichminimizethefrictiononthebeamsurface.Thespheresareloadedwithspringsinsidethecylindricalbodywhichallowslimitedlineardisplacement.Onepairofballplungersprovidedtwocontactpointoneachsideofthebeam.Intotalfourballplungerswereusedandcreatedfourcontactpoints.Oneachsidethetwocontactpointswereseparatedby76mmsothatthisportionofthebeamhadcontrollableslopetoimplementstiffnessvariationforthepurposeofvibrationsuppression.68Theballplungersweremountedonafork-likestructurethroughtwoplates.ThetwoarmsoftheforkstructurewereequippedwithforcesensorsmadeofstraingaugesasshowninFig.4.3.Theforceinthenormaldirectionofthebeamsurfaceatthecontactpointswasalsopartofthefeedbackinformation.Figure4.3Slidercontactinterfacewithforcesensors.TheassembledbeamwiththeslidinginterfacethatsitsonthecarriageoftheslidingtrackisdepictedinFig.4.4.4.2.7MeasurementforfeedbackinformationandthesystemstatusThecompletesetoffeedbackinformationwasmeasuredthroughthefollowingsensorsystems:1)encoderontheslidershaftmeasuringsliderslopeorrotationangle;2)straingaugesmeasuringtheforceontheslider,appliedbythebeamwiththedirectionnormaltothebeamsurface;3)belt-drivenactuationsystemmonitoringsliderposition.Amongthese,1)and2)werefurthercombined69Figure4.4Assembledpartsofslidingandrotationalmechanismedonthesupporttableincontactwiththevibratingbeam.andprocessedtoderivetheaxialforceappliedbythebeamtotheslider.ThisforceisequalinmagnitudetotheconstraintforceinX-direction,i.e.,Lx.Besidesfeedbackmeasurement,twoaccelerometerweremountedonthetwoaddedmassesrespectivelytomonitorandrecordthesystemstatus.4.2.8DataacquisitionMeasurementsobtainedfromsensorswereacquiredusingadSPACEDS1104controllerboard.ThecontrollerboardprovidedA/DandD/Aconverters,aswellasdigitalI/Ointerface.ItisprogrammablefromSimulinkblockdiagramenvironment.TheDS1104boardwasconnectedwithaPCthroughPCIbus.704.2.9Real-timecontrolanddataprocessingalgorithmReal-timedataacquiredandtransmittedfromthecontrollerboardatthesamplingrateof1kHzandwasprocessedinthePCbyacontrolsoftwarewritteninaSimulinkenvironment.ThesignalwofthealgorithmfollowedthedesignshowninFig.3.5.Therewasalsoagraphicuserinterfaceforcontroloperation.Theinterface,asshowninFig.4.5,wasredintheControlDesksoftwaretoolprovidedalongwiththedSPACEcontrollerhardware.Figure4.5Interfaceofthereal-timecontrolanddataprocessingsoftware.4.2.10InitialdisplacementholderTocreateconsistentinitialconditionsforeachexperimentforcontrolstrategyvalidation,asimpleinitialdisplacementholderwasdevisedtocreateastaticdisplacementinitialcondition.Itfunctionsinawaysuchthatthematerialpoints=0:95m(adjacenttomass1)wasdisplacedby20mmand71heldbythedevice.Whentheexperimentstarted,thedevicewasmanuallyreleasedandtriggeredthesystemtooperate.Thebeamstartedtovibrateandthecontrolmechanismstartedtotakeaction.4.2.11CompletesystemTheoverallsystembuiltafterdesignimplementationisshowninFig.4.6.Figure4.6Completeexperimentsystembuiltafterdesignimplementation.4.3Experimentresults4.3.1FreevibrationTheexperimentinvestigationstartsfromthefreevibrationcase.Thebeamwasgiventheinitialconditionthatthematerialpoints=0:95m(adjacenttomass1)wasdisplacedby20mm.After72Figure4.7Experimentresultsofthesysteminfreevibration(timedomain).Plotsfromtoptobot-tom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplace-ment;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.73beingreleasedfromitsinitialdisplacement,thebeamunderwentfreevibration.Duringthisprocesstherewasnoslidingmotion,butthesliderwasallowedtorotatefreely.Fivequantitiesweredirectlymeasuredandrecorded,asshowninFig.4.7.Theyare1)totalforcesensedonslider,appliedbythebeamwiththedirectionnormaltothebeamsurface;2)slopeoftheslider,indicatingtherotationaldisplacementatthecontactpoint;3)positionoftheslider,measuredfromleftendofthebeam,indicatingthelineardisplacementoftheslider;4)accelerationofmass1;5)accelerationofmass2.Thefollowingobservationcanbemade.Thesliderpositionstayedconstantsincenoslidingmotionwasallowed.Quantitiessuchastotalforceonslider,sliderslopeandaccelerationsofthemassesallexhibitedadominantfrequencycomponentaround2:5Hz.Alowfrequencyenvelope,typicallyreferredtoasbeats,wasformulatedontheamplitudeofthedominant2:5Hzvibration.Theenvelopecanbecountedas4˘5secforitshalfperiod.Highfrequencycomponentsappearedatthebeginningofthedynamichistory,whichcanbeclearlyseeninthe1secofthesliderslopeplot,andthe2secofthemassaccelerationplots.Thehighfrequencydynamicswerethendissipated,leavingasmootherdynamiccurveastimeprogressed.Asthemostrelevantquantitiestotransversevibrationofthebeam,themassaccelerationscanbeevaluatedasfollows.Mass1,locatedlessthan30mmtothedisturbancesourcepoint,startedatthelevelof10m/s2.Itroseto15m/s2around0:3sec.At10sec,mass1accelerationdecreasedtoˇ2m/s2,20%oftheinitiallevel.Mass2,notdirectlyexcitedbythedisturbance,hadtheinitialaccelerationof0.Asthevibrationpropagatedfrommass1andreachedmass2,accelerationofmass2increasedtoitsmaximumvalueof8m/s2around0:5sec.At10sec,mass2alsohadtheaccelerationofˇ2m/s2,asignindicatingthatthevibrationenergyisevenlydistributedontwosidesofthebeam.Tobetterunderstandthefreevibrationdynamicsofthesystem,aFastFourierTransform(FFT)analysiswasperformedtoinspectthefrequencydomain.TheanalysiswasperformedonfourquantitiesoutoftherecordedvequantitiesinFig.4.7.Theconstantsliderpositionwasnotanalyzedsincetheresultwouldbetrivial.ShowninFig.4.8,vibrationinformationrecordedbythe74Figure4.8Experimentresultsofthesysteminfreevibration(frequencydomain).Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;accelerationofmass1;accelerationofmass2.75sensorshasbeenconvertedinfrequencydomain.Forallfourquantities,onecantellfromtheplotsthattheenergymainlyexistsinthefrequencyrangebetween2Hzand3:5Hz.Interestingly,allfourplotsshowtwoneighboringfrequencypeakscenteredaround2:5Hz.Thedifferencebetweenthetwopeaksis0:2˘0:3Hz.Thisexplainsthebeatsapparentinthetimedomain,showninFig.4.7,forthebeatsareindeedcausedbysuperimposingtwofrequenciesthatareclosetoeachother.Andthehalfperiodofthebeats,measuredbeforeas4˘5sec,agreeswiththedifferencebetweenthefrequencypeaks(1=0:25Hz=4sec).1Figure4.9Axialforceonslider,post-processedfromexperimentresultsofthesysteminfreevibration(timedomain).Thekeyinformationneededforthepurposeofvibrationcontrolistheaxialforce.Thisquantitywasderivedfrommeasurementsbymultiplyingthetotalforceonthebeamandthesliderslope.Fig.4.9showsaplotoftheaxialforcethroughpost-processingthemeasurementdataofthefree1Giventwosinewavesofunitamplitudeandclosefrequenciesoff1andf2,whenaddedto-gether,thecombinedwaveiscos(2pf1t)+cos(2pf2t)=2cos(2pf1f22t)cos(2pf1+f22t),wherethebeatingfrequencyisjf1f22j.76vibrationcase.Theprocessedaxialforceclearlyexhibitstwofrequencycontents,onearound0:2Hzandtheotheraround5Hz.ThiscanbevbytheFFTanalysisshowninFig.4.10.DifferentfromquantitiesinFig.4.7,thelowfrequencyvibration(0:2Hz)oftheaxialforceexistedindependently,anddidnotresultfrombeating.Instead,thisfrequencywassuperimposedonthehigherfrequencycomponentof5Hz.Thisisveryimportantforthecontrolschemetobeapply,asthelowfrequencyiseasyforthemechanicalpartofthecontrolsystemtoreactandtofollow.Figure4.10Axialforceonslider,post-processedfromexperimentresultsofthesysteminfreevibration(frequencydomain).Thehigherfrequencyaround5Hz,istwiceofthefrequencyofthequantitiesinFig.4.7.Thisagainvthatwhentheaxialandtransversevibrationsarenonlinearlycoupled,theaxialvibrationdoublesthefrequencyofthetransversevibration.77Figure4.11Experimentresultsofthesystemunderslidingcontrolonly.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.784.3.2PerformanceofcontrolemployingslidingmechanismonlyThevibrationcontrolmethodappliedwasthecontrolschemeusingslidingmechanismonly.Again,vequantitiesweredirectlymeasuredandrecordedintheexperiment,asshowninFig.4.11.TheyarepresentedinthesameformasinFig.4.7.Vibrationofthebeamandcontrolledmotionofthesliderstartedafterreleasefromtheinitialdisplacementi.e.,materialpoints=0:95m(adjacenttomass1)displacedby20mm.VariationofthetotalforceonslideexhibitedmoderatebeatingbutnotasobviousasinFig.4.7.Thehalfperiodofthebeatingamplitudeenvelopecanbecountedasˇ1:5sec.Thispatternalsovaguelyappearedinthesliderslopeplot.Likewise,vibrationofperiodˇ1:5seccanbeobservedinthesliderpositionplot,correspondingtoamotionfrequencyof0:6Hz.However,thisbeatingbehaviorcanhardlybedetectedintheaccelerationplotsofthetwomasses,wherethesingledominantfrequencyisbetween2Hzand3Hz.Itcanbeobservedthatthefrequencyofmass1accelerationdecreasedslightlyfromˇ3Hzinthe1sectoˇ2:3Hztowardstheendof10sec.Thisdecreaseinfrequencycanbeexplainedbythefactthattheeffectivebeamlengthconnectedtomass1increasedastheslidermotionoverallwasmovingtowardsmass2.Highfrequencycomponentsappearedinthe1secofthemass1acceleration,andthe2secofthemass2acceleration.Thehighfrequencydynamicswerethendissipated,leavingasmootherdynamiccurveintheaccelerationplotsastimeprogressed.Accelerationofmass1startedatthelevelof10m/s2.Itroseto16m/s2around0:3sec.At10sec,mass1accelerationdecreasedtoˇ4m/s2,40%oftheinitiallevel.Comparedtothefreevibrationcaseinwhichthevaluebecame2m/s2,heretheaccelerationlevelislarger,defeatingthepurposeofvibrationreduction.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialaccelerationof0.Asthevibrationpropagatedfrommass1andreachedmass2,accelerationofmass2increasedtoitsmaximumvalueof4m/s2around0:4sec.At10sec,mass2hadtheaccelerationofˇ1:5m/s2,slightlydecreasedcomparedtothefreevibrationcase(2m/s2at10sec).79Figure4.12Axialforceonsliderintimedomain,processedfrommeasurementsoftotalforceonsliderandsliderslope.Figure4.13Slidervelocityintimedomain,processedfromthemeasurementofsliderposition.80Theslidermovedatthemaximumvelocityof1m/s.Aspanlengthof0:55mwasused,whichsetthematerialboundaryonthebeamfortheslidingmotion.Theslidervelocitywassettobezeroifithitthematerialboundaryandcontinuedwiththedirectionofattemptingtogoovertheboundary,asthealgorithmdepictedinFig.3.5.OnecanclearlyseetheuppermaterialboundarybeingreachedinFig.4.11.Theaxialforceontheslider,whichdirectlydeterminedtheslidermotioninthealgorithm,wasextractedandpresentedinFig.4.12.Formostofthetimeinthe10secinterval,theaxialforceonthesliderwasinthepositiveX-direction,ortowardstherightendofthebeam.Thiscanbeunderstoodasthevibrationwaspropagatingoutfromtheinitialdisturbance;thewavehadthetendencyofpushingthesliderawayfromthesourceofdisturbance,asthedesignintentofthealgorithm.SlidervelocityisplottedinFig.4.13.Astheslidervelocitywascontrolledtomoveinthesamedirectionastheaxialforceontheslider,itreducedtheenergythatwedthroughtheslidertotheothersideofthebeamwheremass2islocated.Thisexplainswhytheaccelerationonmass2wasmuchsmallerthattheaccelerationofmass1,andingeneralsmallerthanthesamequantityinthefreevibrationcase.Thisalsogavethesliderlesschancetomovebackinnegativedirectiontothemass1side,becausetheenergywasmostlykeptonthemass1side,leavingthemass2sidenotenoughenergytocompete.AsproposedinSection3.4,thisschemeemploysalowpassinthefeedbackloop,toeasethebandwidthrequirementofthemechanicalhardware.Knowingfromthefreevibrationanalysisthatmostofthevibrationenergyofthesystemstaysbelow10Hz(Fig.4.8andFig.4.10),thelow-passlterusedthetimeconstantoft=0:01sec,correspondingtothebandwidthof12pt=16Hz.PostFFTanalysisofaxialforceonsliderwasperformedandpresentedinFig.4.14,andtheanalysisonslidervelocityinFig.4.15.InbothFFTresultstheDCvalueswereomitted.Theenergyofaxialforcewaslumpedintwofrequencyranges,onebelow1Hz,theotherintherangeof5˘7Hz.Sincethebandwidthoftheslidervelocitywasreasonablysettobe16Hz,thesliderhadnoproblemeffectivelyreactingtothefeedbackinformation.Althoughthecontrolschemewascarriedoutasdesigned,thevibrationdoesnotseemtobeeffectivelysuppressedbytheslidingmechanism.Thiscouldbyexplainedasthatthedirectenergy81Figure4.14Axialforceonsliderinfrequencydomain.removalratebytheslidermaynotbefastenoughtocreateameaningfuldifferencecomparedtothefreevibrationcase.Othercontrolschemesneedtobeapplied.4.3.3PerformanceofcontrolemployingstiffnessvariationmechanismonlyThecontrolstrategyemployingstiffnessvariationwasinvestigated.Thismechanism,asourproposedvibrationcontrolstrategyinChapter2,wasimplementedhereintheexperimentbyaddingontotheslideraclampingmechanismthathasanon/offlockingcapability.Thisclampingdevice,implementedusinganelectromagneticbrake,canconstraintheslopeortherotationaldisplacementaroundthecontactpoint.Bylocking/releasingtheclampingdevice,aon/offjointisthencreatedsimilartothemethoddescribedinSection2.3,butwiththedifferencethatthejointworksonacontinuousbeamanditcanchangeitspositionbysliding.Thethresholdfortheon/offclampingbraketobeactivatedwassettobe1,meaningthatthe82Figure4.15Slidervelocityinfrequencydomain.brakewasactivatedandappliedarotationalconstraintwhentheencodermeasuredthattheslopeatthecontactpointfellbelow1.Foreachaction,theclampingbrakewaskeptactivatedfor0:05sec,lockingtheslopeduringthistimeinterval.Itwasthendeactivatedandreleasedtheslopeforanother0:05sec.Afterthattheclamperwaitedforthenextinstanceofactivationandthecyclewenton.Inthisexperimentthesliderhadnotranslationalmotion.Therefore,thecontrolemploysstiffnessvariationmethodonly.ResultsareshowninFig.4.16.Briefpausescanbeseeninthesliderslopeplot,astheresultofactivatingtheclamper.Theimmediateeffectthatactivatingtheclampereachtimehadwasthegenerationofhighfrequencyvibrations.Thiscanbeseenintheaccelerationplots.AsexplainedinChapter2,theenergyofthehighfrequencyvibrationwasshiftedfromlowfrequencyvibration.Theredistributionofvibrationenergyinfrequencydomainistheresultsofstiffnessvariation.Thevibrationsuppressionwasenhancedbyinternallydampingoutthehighfrequencyvibrationata83Figure4.16Experimentresultsofthesystemunderstiffnessvariationcontrolonly.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accel-erationofmass2.84fasterrate.Astheoutcome,theaccelerationofmass1startedatthelevelof10m/s2.At10sec,itdecreasedtoˇ1m/s2,10%oftheinitiallevel.Comparedtothefreevibrationcaseinwhichthevaluebecame2m/s2,heretheaccelerationlevelwassuppressedbyanextra10%.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialaccelerationof0.Asthevibrationpropagatedfrommass1andreachedmass2,accelerationofmass2startedtoincreaseandvary.At10sec,mass2hadtheaccelerationofˇ2m/s2,almostthesamecomparedtothefreevibrationcase.Performanceofcontrolemployingthestiffnessvariationmechanismshowedamoderatevibra-tionsuppressioneffectonmass1butnoeffectonmass2.Consideringonlyoneon/offjointwasusedinthestructure,thisresultisnotsurprising.Butroomforimprovementneedstobeexplored.4.3.4PerformanceofcontrolcombiningstiffnessvariationandslidingmechanismAsdescribedbefore,stiffnessvariationwasimplementedusingaclamperonthesliderthatcanbeactivated/deactivatedasanon/offjoint.Inthefollowingexperiment,theslidingmotionwasindependentfromtheactivating/deactivatingoperationforstiffnessvariation,meaningthattheslidercouldstillmoveregardlessoftheclamperstatus.Theslidermotionwasonlydeterminedbythealgorithmthattakesthecomputedaxialforceasthemaininput.Usingthesamesettingofstiffnessvariationandslidingcontrol,theresultsofcombiningthetwomethodsareshowninFig.4.17.TheslidingmotionhasthedirectioncontrolledbythealgorithmoriginallydesignedinSection3.4,thereforereferredtoas"forwardingsliding",tobedifferentiatedfromthe"reversesliding",whichwillbeinvestigatedlater.Itisinterestingtoseefromthesliderpositionplotthattheslidermotionwentoninapositivedirectionalmostmonotonicallyuntilitreachesthematerialboundaryandstoppedonitafterjust2sec.Onecanunderstandthisbecausethevibrationenergyisnotevenlydistributedonbothsidesoftheslider.Themotionofslidingplustheon/offstiffnessvariationprovidedbytheclampersomehowblockedthevibrationpropagationoverthecontactpointfromthesideofmass1tothe85Figure4.17Experimentresultsofthesystemundercombinedcontrolofstiffnessvariationandforwardslidingmechanism.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.86sideofmass2.Thatexplainsthatthemass2sideofthebeamlacktheenergytocompetewiththemass1sideandtopushthesliderbackinnegativedirection.Thestiffnessvariationbytheclamperstillcontinuedtobeappliedafterthetranslationalslidingmotionstopped.Theaccelerationofmass1startedatthelevelof10m/s2.At10sec,itdecreasedtoˇ1:5m/s2,15%oftheinitiallevel.Comparedtothefreevibrationcaseinwhichthevaluebecame2m/s2,heretheaccelerationlevelwassuppressedbyanextra5%.Mass2,notdirectlyexcitedbytheinitialdisturbance,hadtheinitialaccelerationof0.Asthevibrationpropagatedfrommass1andreachedmass2,accelerationofmass2staredtoincreaseandvary.At10sec,mass2hadtheaccelerationofˇ0:8m/s2,suppressedbyanextra12%comparedtothefreevibrationcase.Performanceofcontrolcombiningstiffnessvariationandtheforwardslidingmechanismshowamoderatevibrationsuppressioneffectonbothmass1andmass2.Inanefforttoimprovetheperformanceusingthecurrentexperimentalthecombinationofstiffnessvariationandreverseslidingwasalsoexplored.Thisstrategyalsocombinesstiffnessvariationandtheslidingmechanism.Butinsteadofslidingaccordingtothealgorithmpresentedbefore,theslidingmotionwascarriedoutinthereversedirection.ResultsareshowninFig.4.18.SincethealgorithmwasthesliderpositionplotshowstheslidingistakingtheoppositemotioncomparedtoFig.4.17.Themomentafterthebeamwasreleasedfromtheinitialcondition,thesliderquicklymovedtowardsmass1,wheretheinitialdisturbanceoccurred.Thesliderstoppedafterˇ3secwhenitreachedthematerialboundary.Thestiffnessvariationbytheclamperstillcontinuedtoapplyafterthetranslationalslidingmotionstopped.TheeffectofvibrationsuppressionwasBothmass1andmass2accelerationsdecreasedtoˇ0:5m/s2,comparedto2m/s2inthefreevibrationcase.Thisgivesthebestresultsamongallcontrolstrategies.Thisshowsthatthenewstrategyofcombiningstiffnessvariationandreverseslidingeffectivelypushedthevibrationenergyintohighfrequencyregionandcreatedmuchfasterenergydecayratethroughinternalmaterialdamping.Onecanobservefromtheaccelerationplotsthathighfrequencycomponentscontinuouslyshowedupandthecurveswereneversmooth.87Figure4.18Experimentresultsofthesystemundercombinedcontrolofstiffnessvariationandreverseslidingmechanism.Plotsfromtoptobottom:totalforcemeasuredontheslider,appliedbythebeam;sliderslope,orrotationaldisplacement;sliderposition,measuredfromleftendofthebeam;accelerationofmass1;accelerationofmass2.88Table4.2Comparisonofperformanceofdifferentcontrolstrategies.Allrelativevalueswerecomputedusingthemass1initialaccelerationasreference.ControlConditionMass1RelativeAccelMass2RelativeAccelat10secat10secFreeVibration20%20%SlidingOnly40%15%StiffnessVariationOnly10%20%ForwardSlidingandStiffnessVariation15%8%ReverseSlidingandStiffnessVariation5%5%Theenergyredistributionwasveryeffectivethatthetotalenergyincludingtheenergyaddedbyreverseslidingitselfwasdissipatedalltogetherquicklyenough.AcomparisonofallexperimentcasesaresummarizedinTable4.2.Inthetablethepercentagewascomputedusingthemass1initialaccelerationof10m/s2as100%reference.4.3.5DiscussionExperimentsshowvariousvibrationsuppressioneffectsusingdifferentcontrolmethods.Themosteffectivemethod,whichcombinesstiffnessvariationandslidingmechanism,waspredictedinChapter3.Thismethodappliesstiffnessvariationthroughaclamperwhilecontrolstheslidertomoveinareverseway.Itconstantlyaddsenergydirectlybutalsoexciteshighfrequencyvibrationinthesystem.Overall,throughinternaldampingenhancedbystiffnessvariation,theneteffectisthatthedecayrateofthesystemvibrationishigherthaninothercasessuchasforwardsliding.InvestigationwithnumericsimulationinChapter3revealedtheimportantrelationshipbe-tweenthetwoenergyremovalmechanisms.Thesetwomechanismsare1)directenergyreductionthroughslidingand2)indirectenergyreductionthroughmaterialdamping.Itwasvintheexperimentthatgiventhemoderatedampingrateofthesystem,directenergyremovalthroughslidingmotiondidnotperformasthedominantfactorofvibrationsuppression.Themajorpor-tionofenergydissipationhastorelyoninternalmaterialdamping.Tomaximizethedamping89effect,slidingmotioncanbeusedasasecondaryroletofacilitatedampingprocess.Togetherwithstiffnessvariationimplementedusingtheon/offclamperwhichdoesnotchangeenergydirectly,sliding,especiallyreversesliding,effectivelyreshapestheenergydistributioninthefrequencydo-main.Itkeepsfunnelingtheenergyintohigh-frequencydomainwheredampingeffectivelyturnsthevibrationenergyintoheat.ThisscenarioissimilartothesystempresentedinChapter2,wherethedampinglevelisalsomoderate.InChapter2vibrationsuppressionisachievedthroughstiffnessvariationalone,atthepriceofemployingmultipleactuators.Asthenumberofcontrolledhingesorspringsdecreases,vibrationsuppressiongetslessandlessefAsintheexperiment,withslidingmechanismadded,actuatorsandsensorsarealllumpedononeslidingdevice.Withasimilarlymoderatedampinglevel,thesamevibrationsuppressiontargetcanbeachievedusinglessactuatorsbutmoresophisticatedsensordesign,signalprocessingandcontrollogic.Itisfairlyclearthatthecontrolstrategyneedstobeformulateddifferentlytowardssystemswithdifferentdampinglevels.Forlightlydampedstructures,itisbettertousedirectenergyre-movalmethodsuchassliding,asshowninChapter3.Formoderatelydampedstructures,dampingfacilitatedbyslidingand/orstiffnessvariationseemstobeabetterchoice.Forheavilydampedstructures,whicharenotwithinthescopeofthisthesis,traditionaltreatmentusingpassivemethodsuchasoptimizingthedamperlocationshouldsufasanaffordableandpracticalsolution.90CHAPTER5SUMMARYANDFUTUREWORKTension-alignedstructureshaveshowngreatpotentialofprovidingmassefy,easeoftrans-portandahigh-degreeofaccuracy.Suchfeaturesarehighlyvaluableinspace-basedapplications.Tostudythevibrationsuppressionstrategyinsimpletension-alignedstructures,modelsofstruc-turaldynamicswerebuiltwithcarefulconsiderationofthetension'sroleinthestructuralvibrationbehavior.Analysisshowedthatthetensionenterthevibrationdynamicsmainlythroughgeometricnonlinearity.Severalmodelsillustratedinthisstudyallsharethesimilarbow-stringliketopol-ogy.Theyarecomprisedofasupportstructuresincompression,whichprovidetensiontosensorsurfacesviatheconnectionsatends.Intheexamplesused,thesupportstructurevariesfromabso-lutelyrigidobject,toacurvedlydeformedrod,andfurthertoathree-dimensional,morerealistic,truss-likespacestructure.Thesensorsurfaceundertensionrangesfromasimplebeamtohingedpanelarrays.Twodifferentcontrolstrategieswereproposedandappliedonthosestructuremodels.Thestrategyisvibrationsuppressionthroughstiffnessvariation.Thissemi-activemethodsequentiallyappliesandremovesconstraintsinthestructure.Theprocessvariesthesystemstiffnessandfunnelsvibrationenergyfromlow-frequencywheredampingrateislowtohigh-frequencymodeswhereinternalmaterialdampingrateisnaturallyhigh.Byleveragingmaterialdamping,energyisthenindirectlybuteffectivelydissipated.Thesecondstrategy,anactivemethod,employsaslidingmechanismwheretheconstraintforceismeasuredinrealtimeandusedasfeedbacktoprescribethemotionoftheslideritself.Theprescribedslidermotionkeepsdoingnegativeworktothestructuretherebyvibrationenergygetsdirectlyremoved.Thisstrategyusesastraightforwardideaandveryfewsensors.Itrequiresmoresophisticatedactuatorsandmorepowerfulreal-timecomputation.Numericresultsshowthatthetwostrategiesworkeffectivelytoachievethevibrationsup-91pressiongoal.Thestiffnessvariationmethodwasdemonstratedonbothtwo-dimensionalandthree-dimensionalnumericmodelswheremoderatematerialdampingwaspresented.Comparedtofreevibration(nocontrol),thecontrolmethodprovidedfasterenergydecay.Theconceptofmodaldisparitywasinvestigatedintheprocesstofurthertheunderstandingofthere-lationshipbetweenstiffnessvariationandcontrolefy.Thestrategyusingslidingmechanismwasvalidatednumericallyusingaone-dimensionalnonlinearbeammodelwherelightdampingwaspresented.Severalandsignalprocessingtechniqueswerestudiedtoaddresstheprac-ticalissuessuchaslimitedactuatorbandwidth.Overallthedesignideaofdoingnegativeworkbytheslidertothestructuremettheexpectationofsuccessfulvibrationsuppression.Furthermore,aexperimentplatformwasbuilttotestbothcontrolstrategiesinarealisticenvironment.Itwasfoundthattoachievethebestvibrationsuppressionresultsinamoderatelydampedbeamstructure,oneshallcombinebothcontrolstrategiesinawaysuchthatthestiffnessvariationmethodservesasaprimaryrole,facilitatedbytheslidingmechanism.Thiscombinationdeliveredthemosteffectiveresultbyleveragingthestructuraldampingtothemost.Allthenumericandexperimentalstudiesledtotheconclusionthatvibrationcontrolstrategyneedstobeformulateddifferentlytowardssystemswithadifferentdampinglevel.Forlightlydampedstructures,itisbettertousedirectenergyremovalmethodsuchassliding.Formoderatelydampedstructures,dampingfacilitatedbystiffnessvariationand/oraslidingmechanismseemstobeabetterchoice.Forheavilydampedstructures,whicharenotwithinthescopeofthisstudy,traditionaltreatmentusingpassivemethodsshouldbeanaffordableandpracticalsolution.Inthenumericinvestigationoftheslidingmechanism,thecontactproblemshowedtobeachallengeforitcostagreatdealofcomputationtimeandpower.ThisnumericissuecomesfromtherootthatifthecontactproblemisformulatedusingLangrangemultipliers,aDAEsystememergesandneedstobetackledusingspecialsolvers.Ontheotherhand,iftheproblemisformulatedusingapenaltymethod,averystiffODEsystemthenneedstobesolved.Bothseemtobeexpensivenumerically.Tofastenthesimulationprocessoftheslidingmechanism,moreunderstandingofnumericmethoddealingwiththecontactproblemneedstobegained.Inthefuturestudyabetter92numerictoolisessentialtoproducemoreinvestigationresultsbyconstructingmorescenarios.Intheexperimentalstudy,acomprehensiveapproachwastakentocombinetwostrategiesindifferentwaystothemostefcombinationforvibrationsuppression.However,thesearchwasstilllimitedandnotexhaustive.Itcouldonlybeconcludedthatthebestresultssofarcamefromthecombinationthatwasthebestamongthecombinationsthatweretested.Itisexpectedinthefuturestudythatanaccurateandmathematicalmodelcouldbedevelopedtocharacterizethesystemdynamicsunderbothcontrolmethods,especiallytoaccuratecapturethecontactbehavioranddampingbehavior.Inthatwayanoptimalcontrolmethodcombinationsearchcouldbeperformedandthecontrolresultscouldbewellpredicted.Morestudiesneedtobeconductedandbetterunderstandingneedstobegainedtodevelopreliable,robustandpracticalvibrationsuppressionsolutions,fortension-alignedstructurestohavegreaterapplicationinspacemissionsandinother93BIBLIOGRAPHY94BIBLIOGRAPHYAdler,A.,M.M.Mikulas,J.M.Hedgepeth,M.Stallard&J.Garnham.1998.Novelphasedarrayantennastructuredesign.InIEEEaerospaceconference,AspenCO.Barhorst,A.A.2004.Ontheefyofpseudo-coordinates.part1:Movinginteriorconstraints.InternationalJournalofNon-LinearMechanics39(1).123Œ135.Bauchau,O.A.2010.Flexiblemultibodydynamics.Springer.Clark,W.W.2000.Vibrationcontrolwithstate-switchingpiezoelectricmaterials.JournalofIntelligentMaterialSystemsandStructures11(4).263Œ271.Corr,L.R.&W.W.Clark.2001.Energydissipationanalysisofpiezoceramicsemiactivevibrationcontrol.JournalofIntelligentMaterialSystemsandStructures12(11).729Œ736.Diaz,A.R.&R.Mukherjee.2006a.Modaldisparityenhancementthroughoptimalinsertionofnonstructuralmasses.StructuralandMultidiscilinaryOptimization31(1).1Œ7.Diaz,A.R.&R.Mukherjee.2006b.Topologyoptimizationproblemincontrolofstructuresusingmodaldisparity.ASMEJournalofMechanicalDesign128(3).536Œ541.Diaz,A.R.&R.Mukherjee.2008.Optimaljointplacementandmodaldisparityincontrolofxiblestructures.ComputersandStructures86(13-14).1456Œ1462.Dill,E.H.1992.Kirchhoff'stheoryofrods.ArchiveforHistoryofExactSciences44(1).1Œ23.Fang,J.&G.J.Lyons.1996.Structuraldampingoftensionedpipeswithreferencetocables.JournalofSoundandVibration194(3).891Œ907.Fischer-Cripps,A.C.2007.Introductiontocontactmechanics.Springer.Footdale,J.N.,S.K.Jeon&T.W.Murphey.2012.Staticshapeandmodaltestignofadeployabletensionedphasedarrayantenna.In53rdAIAAstructures,structuraldynamics,andmaterialsconference,.Hong,D.&G.Ren.2011.Amodelingofslidingjointonone-dimensionalxiblemedium.MultibodySystemDynamics26(1).91Œ106.Issa,J.,A.R.Diaz&R.Mukherjee.2009.Energydissipationindynamicalsystemsthroughsequentialapplicationandremovalofconstraints.ASMEJournalofDynamicSystems,Mea-surementandControl131(2).Issa,J.,A.R.Diaz,R.Mukherjee&S.W.Shaw.2008.ModaldisparityanditsexperimentalvJournalofSoundandVibration311(3-5).1465Œ1475.Issa,J.,R.Mukherjee&S.W.Shaw.2010.Vibrationsuppressioninstructuresusingcableactua-tors.ASMEJournalofDynamicSystemsMeasurementandControl132(3).95Jeon,S.K.&T.W.Murphey.2012.Fundamentaldesignoftensionedprecisiondeployablespacestructuresappliedtoanx-bandphasedarray.In53rdAIAAstructures,structuraldynamics,andmaterialsconference,.Jones,T.C.,H.Bart-Smith,M.Mikulas&J.Watson.2007.Finiteelementmodelingandanalysisoflargepretensionedspacestructures.JournalofSpacecraftandRockets4(1).183Œ193.Jones,T.C.,J.J.Waston,M.Mikulas&H.Bart-Smith.2008.Designandanalysisoftension-alignedlargeaperturesensorcraft.In49thAIAAstructures,structuraldynamics,andmaterialsconference,SchaumburgIL.Kemerley,R.T.&S.Kiss.2000.Advancedtechnologyforfuturespace-basedantennas.InIEEEMTT-Sinternationalmicrowavesymposiumdigest,vol.2,717Œ720.BostonMA.Khalil,H.2002.Nonlinearsystems.PrenticeHall.Love,A.E.H.1944.Atreatiseonthemathematicaltheoryofelasticity.DoverPublications.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