An Experimentally-Based Procedure for Aeroservoelastic Model Identification and Control Synthesis for Morphing and Flapp...

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Title:
An Experimentally-Based Procedure for Aeroservoelastic Model Identification and Control Synthesis for Morphing and Flapping Wings
Physical Description:
1 online resource (204 p.)
Language:
english
Creator:
Love,Robert D
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
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Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Lind, Richard C
Committee Members:
Ukeiley, Lawrence S.
Ifju, Peter
Ho, Jeffrey

Subjects

Subjects / Keywords:
aeroelastic -- aeroelasticity -- aeroservoelastic -- aeroservoelasticity -- aerospace -- aircraft -- analysis -- basis -- control -- design -- dynamics -- experimental -- feedback -- feedforward -- flapping -- flexibility -- flexible -- flight -- frequency -- hydroelasticity -- identification -- interdisciplinary -- mav -- mechanics -- model -- morphing -- optimization -- performance -- periodic -- processing -- robustness -- sensor -- shape -- signal -- structural -- time -- uav -- wing -- wings
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Morphing and flapping wings are enabling technologies for vehicles of the future. Vehicles with morphing and flapping wings will have greater mission capability and flexibility thereby enabling more autonomy, will be substantially more maneuverable, will be able to fly maintaining stability in the presence of larger gusts and will weigh less than current vehicles by eliminating repetitious control effectors. The dynamics of morphing and flapping wing vehicles are inherently aeroservoelastic since the interaction of aerodynamics, structural flexibility, and structural dynamics are critical to performance and will be altered by any control effectors. However, current aeroservoelastic modeling and control strategies are not sufficient to realize the full range of benefits offered by wings which change shape substantially. Most vehicles attempt to eliminate aeroservoelastic dependencies with aircraft design or decrease their effects with some form of control. Yet these aeroservoelastic dependences may be harnessed to provide substantial benefits for morphing and flapping wings. This dissertation reviews historical examples of morphing and flapping wings and the aeroservoelastic phenomena present which may affect their performance. The work then measures and identifies nonlinear behaviors in the aeroservoelastic dynamics present for morphing and flapping with time-frequency analysis for a variety of wings. A model of the morphing and flapping wings as a function of each control effector is formulated. These models capture the nonlinear behavior and are a basis from which to compute the deflection in response to any available control command. This work then identifies a model of the aeroservoelastic dynamics for morphing and flapping flexible wings based on experimentally obtained data. Lastly, the work defines a feedforward and feedback control synthesis which may be used to control the aeroservoelastic models which have been identified. The models are used to track a desired wing shape for a realistic morphing and flapping wing, thereby leveraging aeroservoelastic effects to provide performance benefits for vehicles with morphing and flapping wings.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Robert D Love.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: Lind, Richard C.

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UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2011
System ID:
UFE0043309:00001


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ANEXPERIMENTALLY-BASEDPROCEDUREFORAEROSERVOELASTICMODELIDENTIFICATIONANDCONTROLSYNTHESISFORMORPHINGANDFLAPPINGWINGSByROBERTD.LOVEADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011RobertD.Love 2

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Tomyfamilyandfriends,whosecontinualandunconditionalloveandencouragementhavegivenmewingstoyhigherthanIeverimagined 3

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ACKNOWLEDGMENTS ThankstotheAirForceOfceofScienticResearchandtheUniversityofFlorida,whomadethisworkpossiblebysponsoringmeasapartofamulti-universityresearchinitiativeunderFA9550-07-1-0547andanAlumniFellowship.IwouldliketothankDr.RickLindformentoringmethroughouttheprocessofgraduateschoolandforprovidingfocusandmotivationwhereverandwhennecessary.ThankstoDr.PeterIfju,Dr.LarryUkeiley,andDr.JeffreyHoforbeingonmycommitteeandfortheirwillingnesstoshareideasandresourcesthroughouttheprocess.SpecialappreciationisextendedtoPinWuandBradLaCroixattheUniversityofFlorida,BretStanfordofWright-PattersonAirForceBase,andKevinShortellefromSystemDynamicsInternationalInc.fortheirassistanceinobtainingexperimentaldata.TheirwillingnesstoletmetostandontheirshouldersprovidedanexampleofrealcollaborationandamodelforwhoIneededtobecomeinordertobeasuccessfulengineer.ThanksalsotomycolleaguesSankethBhat,BrianRoberts,BaronJohnson,DanielGrant,DavidEaton,RyanHurley,JoeKehoe,SeanRegisford,MujahidAbdulrahim,DongTran,StephenSorley,AbrahamPachikara,JudBabcock,RobertoAlbertani,JohnSaxon,AhmedJorge,SangminOhandCrystalPasiliaointheFlightControlLabandErikSallstromattheUF-REEFfortheirwillingnesstoshareandsharpenideasaswellastoparticipateinthejoysandburdensassociatedwithgraduateschool.Thankstomyteachersandthescientistswhohavegonebeforemeandwhoinvestedtheirtimeandenergyinsharingtheirknowledge.Thanksalsotothemanyotherfriendswhohavedirectlyorindirectlyimpactedmylife.ThankstoCynthiaGoddardforyourloveandstandingbesidemethroughthefunanddifculttimes.ThanksalsotoDad,Mom,Shelley,andLauraforyoursupportandlove.Nomanisanislandandwithoutthesupportofalltheseindividualsthisworkwouldnothavebeenpossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1Motivation .................................... 15 1.2ProblemStatement ............................... 16 1.3Contributions .................................. 16 1.4MajorAssumptions ............................... 17 1.5DissertationOutline .............................. 18 2WING-SHAPEMODIFICATIONHISTORY ..................... 19 2.1TraditionalWings ................................ 19 2.2MorphingWings ................................ 20 2.3FlappingWings ................................. 28 2.4OtherApplications ............................... 34 3AERO/HYDROSERVOELASTICSYSTEMS .................... 39 3.1BackgroundforAero/hydroservoelasticSystems .............. 39 3.1.1SystemClassications ......................... 39 3.1.2AeroservoelasticDesignOverviewsforTraditional,Morphing,andFlappingWings ............................. 41 3.2TheoreticalandComputationalAnalysis ................... 42 3.2.1DecoupledSystemAnalysis ...................... 42 3.2.1.1Fluiddynamics ........................ 43 3.2.1.2Solidmechanics/elasticityandmaterialselection ..... 45 3.2.1.3Structuraldynamics ..................... 48 3.2.1.4Rigid-bodydynamics .................... 49 3.2.1.5Control ............................ 50 3.2.2CoupledSystemAnalysis ....................... 51 3.2.2.1Aero/hydroelasticity ..................... 51 3.2.2.2Servoelasticity ........................ 54 3.2.2.3Aero/hydroservoelasticity .................. 54 3.3ExperimentalApproachesforAnalysisofAeroservoelasticSystems ... 58 3.3.1FluidDynamics-ParticleImageVelocimetry ............. 59 3.3.2SolidMechanics/Elasticity-MechanicalTesting ........... 59 5

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3.3.3StructuralDynamics-LaserDopplerVibrometry .......... 59 3.3.4DynamicsandControl-FlappingMechanism ............ 60 3.3.5DigitalImageCorrelation:HoveringTests .............. 60 4SENSORSANDSIGNALPROCESSING ..................... 66 4.1BackgoundforSignalProcessingandSignalClassications ........ 66 4.1.1DeterministicSignals .......................... 66 4.1.2Non-deterministicSignals ....................... 67 4.2SignalAnalysis ................................. 67 4.2.1Time-DomainAnalysis ......................... 68 4.2.2Spectral(Frequency)DomainAnalysis ................ 68 4.2.3Time-FrequencyAnalysis ....................... 69 4.2.4ProbabilisticSignalsAnalysis ..................... 71 4.3SensorSelectionandPlacement ....................... 74 4.4SignalCharacteristics ............................. 77 4.4.1MorphingSignals ............................ 77 4.4.2FlappingSignals ............................ 78 4.4.2.1Signalcapture-digitalimagecorrelation ......... 78 4.4.2.2Periodicbehavior ...................... 79 4.4.2.3Periodicphenomena-timeandfrequencyanalysis .... 81 4.4.2.4Periodicnon-sinusoidalfeatures-waveletanalysis .... 83 4.4.2.5Time-varyingfeatures .................... 85 4.4.2.6Variationsinstructuraldynamics .............. 86 4.4.2.7Variationsinwingdesign,appingkinematicsandwinglocation ............................ 87 5WING-SHAPEMODELSANDSYSTEMIDENTIFICATION ........... 89 5.1BackgroundforSystemModelIdentication ................. 89 5.1.1ModelClassications .......................... 89 5.1.2SystemIdentication .......................... 92 5.2DecoupledWing-ShapeModels ........................ 93 5.2.1FluidDynamicModels ......................... 93 5.2.2SolidMechanicsModels ........................ 94 5.2.3StructuralDynamicModels ...................... 96 5.2.4Rigid-bodyDynamicModels ...................... 98 5.2.4.1Singledegree-of-freedommorphingandappingmodel 98 5.2.4.2Generalizedwing-shapemodel ............... 100 5.3CoupledWing-ShapeModels ......................... 104 5.3.1Aero/hydroelasticModels ....................... 104 5.3.1.1Staticaero/hydroelasticmodels ............... 104 5.3.1.2Dynamicaero/hydroelasticmodels ............. 107 5.3.2Aero/hydroservoelasticModels .................... 107 5.3.2.1Coupledmodelsofsmartstructures ............ 108 5.3.2.2Flexiblerobotmanipulators ................. 108 6

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5.4OtherModelingMethods ............................ 109 5.4.1AdaptiveandLearning-BasedModels ................ 110 5.4.2SurrogateModels ............................ 110 5.4.3HybridModeling ............................ 110 5.5ModelErrorAnalysis .............................. 111 5.6AeroservoelasticModelsofMorphingWings ................ 111 5.7ProposedModelforMorphingWings ..................... 112 5.7.1MorphingWingModelBasisSelection/SystemIdentication .... 114 5.7.2MorphingWingModelLimitations ................... 114 5.8AeroservoelasticModelsofFlappingWings ................. 115 5.9ProposedModelforFlappingWings ..................... 119 5.9.1AeroservoelasticBehaviorsofFlappingWings ............ 120 5.9.2FlappingWingModelBasisSelection ................. 123 5.9.3FlappingWingSystemIdentication/OptimizationMethod ..... 123 5.9.4FlappingWingModelLimitations ................... 125 6WING-SHAPECONTROLAPPROACHES .................... 126 6.1BackgroundforWingShapeControl ..................... 126 6.2ControlSynthesis ................................ 127 6.2.1Openvs.Closed-LoopControlApproaches ............. 127 6.2.2LinearControlApproaches ...................... 127 6.2.2.1Linearparametervaryingapproaches ........... 128 6.2.2.2Robustcontrolapproaches ................. 128 6.2.3NonlinearControlApproaches ..................... 128 6.2.4AdaptiveControlApproaches ..................... 128 6.2.5OtherControlApproaches ....................... 129 6.3SensorFeedback ................................ 129 6.4ControlActuation ................................ 130 6.5FeedforwardControl .............................. 131 6.6Closed-LoopControlApproachforWing-ShapeControl .......... 133 6.6.1ControlArchitecture .......................... 133 6.6.2ControlUpdateLaw .......................... 134 6.6.3ControlSynthesis ............................ 135 6.6.3.1Hand-tuning ......................... 135 6.6.3.2Optimization ......................... 136 6.6.4ControlApproachIssues ........................ 136 7EXAMPLE1:PIEZOELECTRICMORPHINGWINGS .............. 138 7.1BackgroundforMorphingWingSystemIdenticationandControl ..... 138 7.2PiezoelectricActuators ............................. 139 7.2.1PiezoelectricActuatorFabrication ................... 139 7.2.2PiezoelectricActuatorResponseCharacterization ......... 140 7.2.2.1Timeresponse ........................ 142 7.2.2.2Frequencyresponse,shakerexcitation .......... 143 7

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7.2.2.3Frequencyresponse,sinusoidaldwellresponses ..... 144 7.2.2.4Modalresponse,internalexcitation ............ 145 7.3PiezoelectricallyActuatedComposite-MembraneWing ........... 147 7.4ModelIdentication ............................... 149 7.4.1ModelDiscretization .......................... 149 7.4.2BasisSelection ............................. 150 7.4.3MorphingSystemIdentication .................... 151 7.4.4MorphingModelEvaluation ...................... 151 7.5PiezoelectricFeedforwardControlDesign .................. 152 8EXAMPLE2:FLAPPINGWINGMICROAERIAL-VEHICLES .......... 155 8.1BackgroundforFlappingWingSystemIdenticationandControl ..... 155 8.2WingFabrication ................................ 156 8.3FlappingMechanismDesign ......................... 157 8.4StructuralDynamics .............................. 158 8.5SignalCapture ................................. 160 8.6ModelIdentication ............................... 164 8.7ConsiderationstoDetermineDesiredFlappingMovement ......... 166 8.8FeedforwardControlResults ......................... 167 8.8.1FeedforwardControlIdentication ................... 167 8.8.2FeedforwardControlRobustnessandErrorAnalysis ........ 169 8.8.3LimitationsofFeedforwardControlMethod .............. 170 8.9Closed-LoopControl .............................. 170 8.9.1StabilityforClosed-LoopController .................. 171 8.9.2EffectofSensorPlacementonClosed-LoopPerformance ..... 171 8.9.3EffectofTimeDiscretizationonClosed-loopPerformance ..... 171 8.9.4Closed-loopPerformancewithRespecttoWingLocation ...... 172 8.9.5Closed-loopResponsePerformanceEvaluation ........... 173 8.9.5.1Effectsofparametricuncertaintyonperformance ..... 174 8.9.5.2Effectsofdisturbancesonperformance .......... 175 8.9.5.3Effectsofnoiseonperformance .............. 177 9CONCLUSIONSANDFUTUREWORK ...................... 179 9.1Conclusions ................................... 179 9.2FutureWork ................................... 180 REFERENCES ....................................... 182 BIOGRAPHICALSKETCH ................................ 204 8

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LISTOFTABLES Table page 2-4HistoricalExamplesofVehicleswithShapeChangingWings .......... 23 2-1MissionScenariosforModernAircraftandUnmannedAerialSystems/TypesofAircraftDesignations[ 52 64 198 ] ........................ 35 2-2BenetsofMorphingParameters .......................... 36 2-3TrendsinDesignParametersDuetoWing-ShapeChanges[ 158 ] ........ 36 2-5SelectedMorphingResearchfromAcademia ................... 37 2-6ParametersNecessarytoCharacterizeFlappingWings ............. 38 2-7OverviewofAcademicResearchofRepresentativeFlappingOrganisms .... 38 3-1ClassesofCoupledSystemsinContinuumMechanics .............. 63 3-2DesignVariablesforVehicles ............................ 64 3-3Reduced-OrderModelOverviewforAeroservoelasticSystems ......... 64 3-4MeasurementsUsefulforModelIdentication ................... 65 4-1DistributedSensorSetupsforControlFeedbackforMorphingandFlappingWings ......................................... 76 4-2CrossCorrelationsandErrorsforFollowing3PeriodicCycles,Wing-3(BattensParallel) ........................................ 81 4-3ConclusionsSummary:ParametersWhichAffecttheTime-FrequencyResponseofAFlappingWing .................................. 88 5-1RationaleforInclusionofBasisCoefcients .................... 124 7-1Macro-FiberComposite(MFC)PiezoelectricBeamCharacteristics ....... 139 8-1WingCharacteristics ................................. 156 8-2ExperimentallyMeasuredDiscretizationofFlappingParameters ........ 164 8-3ModelingErrorsforWing-4(DiagonalBattenWing)AcrossDesignSpace ... 166 8-4EvaluationofFeedforwardControlRobustnesswithRespectto(c,!c) .... 170 9

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LISTOFFIGURES Figure page 2-1TheGreatFlightDiagram:RelationshipofWingLoading,GrossWeightandCruiseSpeedofFlyingVehicles[ 238 ] ....................... 29 3-1Fluid-Structure-Control-TemperatureCouplingsOrganizedwithRespecttoTimeDependenceofForces ............................ 40 3-2CoupledBehavior:Fluid-Structure-Control[ 93 ] .................. 41 4-1SignalClassications[ 29 ] .............................. 66 4-210HzSinusoidTimeHistory,FastFourierTransformandWaveletAnalysis(Frequencyvs.TimeandMagnitudevs.TimeViews) .............. 71 4-3ExampleMorletMotherWavelet .......................... 72 4-4PeriodicNon-SinusoidalDeectionsofaFlexibleFlappingWing ........ 79 4-5ExamplesofCross-correlationofPeriodicSignals ................ 80 4-6OverviewofTimeHistory,FrequencyandTime-FrequencyBasedSignalAnalysisforFlexibleFlappingWings ............................. 82 4-7TemporalNatureofFlappingDeections ...................... 84 4-8ComparisonofSteady-StateFlappingandSweepAcrossFlappingFrequencies 86 5-1StandardModelingProcedure[ 197 ] ........................ 89 5-2CartesianandCylindricalCoordinateSystemSpecication ........... 99 5-3CoordinateSystemforGeneralizedMorphingWingModel ............ 101 5-4ComparisonofRigidWingBodyMovement,FullWingDeectionandDeformationfromUnaccountedforAeroservoelasticEffects .................. 121 5-5RelativeSizeofContributionsofAeroservoelasticPhenomenatotheFinalFlappingDeectionsasaFunctionofWingLocation ............... 122 6-1Closed-LoopControlArchitecture .......................... 134 7-1LaserDopplerVibrometerSystemSetupforGroundVibrationTestingofPiezoelectricBeam ......................................... 140 7-2OverviewofPrintedCircuitBoardUsedforPiezoelectricActuation ....... 141 7-3TimeResponseofUni-3PiezoelectricActuatorto10VStepInput ........ 143 7-4ModalResponseofPiezoelectricBeamwithShakerExcitation ......... 144 10

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7-5SinusoidalFrequencyResponseofUni-3andModeShapeforApplied22.6Hz 145 7-6FrequencyResponse,CoherenceandModeShapesofPiezoelectricBeamsUnderDifferentExcitationMethods ......................... 146 7-7Piezoelectrically-ActuatedComposite-MembraneWingAloneandInstalled .. 147 7-8GroundVibrationTestSetupForPiezoelectrically-ActuatedComposite-MembraneWing .......................................... 148 7-9FrequencyResponse,CoherenceandModeShapesofthePiezoelectrically-ActuatedComposite-MembraneWing ....................... 148 7-10DigitalImageCorrelationMeasurementsofStaticDeectionsofPiezoelectrically-ActuatedComposite-MembraneWing ....................... 149 7-11MorphingWingExperimentalDiscretization .................... 150 7-12ModelPredictionforaPeriodicCycleofMorphingDeections .......... 152 7-13ComparisonofMorphingDeectionsDesiredandFeedforwardSimulation ... 154 8-1FlappingWingsInvestigated ............................ 156 8-2SingleDegreeofFreedomFlappingMechanism,DesignedandBuiltbyWuforFlappingWingExperimentation[ 278 ] ...................... 158 8-3LDVExperimentalSetup ............................... 159 8-4FrequencyResponseforVariationsinShape ................... 160 8-5FrequencyResponseforVariationsinStructure .................. 161 8-6DICExperimentalSetup ............................... 162 8-7WingSamplingMappingRectangularGridontoWing .............. 163 8-8ExperimentallyDeterminedCoefcients(*)A1(,!)A11(,!)OverlaidwithInterpolatedSurfacesA1(,!)A11(,!) ...................... 164 8-9ComparisonofExperimentallyMeasuredFlapping(MeshedWing)toModelPredictedFlapping(SmoothWing)AcrossKinematicSpace ........... 165 8-10FeedforwardControlResult:LowFrequency,HighAmplitudeFlapping ..... 168 8-11FeedforwardControlResult:HighFrequency,HighAmplitudeFlapping ..... 169 8-12AverageClosed-loopControlErrorsforOut-of-PlaneResponseswithRespecttoSensorMeasurementLocation .......................... 172 8-13EffectofTimeDiscretizationonClosed-LoopControlPerformance ....... 173 11

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8-14ErrorswithRespecttoLocationontheWing(a)FeedforwardControlResult(b)Closed-loopControlResult ........................... 174 8-15Signal-BasedComparisonsofDesired,Feedforward,andClosed-loopControlDeections ...................................... 175 8-16FullWingResponsesClosedLoopControl .................... 176 8-17ComparisonofRobustnessofFeedforwardandClosed-loopPerformancewithRespecttoParametricUncertainty ...................... 177 8-18ComparisonofDisturbancesontheFeedforwardandClosed-loopControllerPerformance ..................................... 178 8-19ComparisonofNoiseontheFeedforwardandClosed-loopControllerPerformance 178 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyANEXPERIMENTALLY-BASEDPROCEDUREFORAEROSERVOELASTICMODELIDENTIFICATIONANDCONTROLSYNTHESISFORMORPHINGANDFLAPPINGWINGSByRobertD.LoveAugust2011Chair:RickLindMajor:AerospaceEngineering Morphingandappingwingsareenablingtechnologiesforvehiclesofthefuture.Vehicleswithmorphingandappingwingswillhavegreatermissioncapabilityandexibilitytherebyenablingmoreautonomy,willbesubstantiallymoremaneuverable,willbeabletoymaintainingstabilityinthepresenceoflargergustsandwillweighlessthancurrentvehiclesbyeliminatingrepetitiouscontroleffectors.Thedynamicsofmorphingandappingwingvehiclesareinherentlyaeroservoelasticsincetheinteractionofaerodynamics,structuralexibility,andstructuraldynamicsarecriticaltoperformanceandwillbealteredbyanycontroleffectors.However,currentaeroservoelasticmodelingandcontrolstrategiesarenotsufcienttorealizethefullrangeofbenetsofferedbywingswhichchangeshapesubstantially.Mostvehiclesattempttoeliminateaeroservoelasticdependencieswithaircraftdesignordecreasetheireffectswithsomeformofcontrol.Yettheseaeroservoelasticdependencesmaybeharnessedtoprovidesubstantialbenetsformorphingandappingwings. Thisdissertationreviewshistoricalexamplesofmorphingandappingwingsandtheaeroservoelasticphenomenapresentwhichmayaffecttheirperformance.Theworkthenmeasuresandidentiesnonlinearbehaviorsintheaeroservoelasticdynamicspresentformorphingandappingwithtime-frequencyanalysisforavarietyofwings.Amodelofthemorphingandappingwingsasafunctionofeachcontroleffectoris 13

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formulated.Thesemodelscapturethenonlinearbehaviorandareabasisfromwhichtocomputethedeectioninresponsetoanyavailablecontrolcommand.Thisworkthenidentiesamodeloftheaeroservoelasticdynamicsformorphingandappingexiblewingsbasedonexperimentallyobtaineddata.Lastly,theworkdenesafeedforwardandfeedbackcontrolsynthesiswhichmaybeusedtocontroltheaeroservoelasticmodelswhichhavebeenidentied.Themodelsareusedtotrackadesiredwingshapeforarealisticmorphingandappingwing,therebyleveragingaeroservoelasticeffectstoprovideperformancebenetsforvehicleswithmorphingandappingwings. 14

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CHAPTER1INTRODUCTION 1.1Motivation Humankindcontinuestostrivetocreatevehicleplatformsofincreasingcomplexitytogaininformationaboutadifferentlocationortotransportthemselvesorsomecargotoadifferentlocation.Vehicledesignsarecategorizedbythetypeofspacewheretheywillbeused(submarines,boats,cars,aircraft,spacecraft)andbyhowmuchhumancontrolisrequiredtooperatethevehiclewhileitperformsthedesiredtask(manned,unmanned,human-in-the-loop).Thedesignsofsuchvehiclesarelimitedbyhuman'scurrentunderstandingoftheworld,theirabilitytoimagineadifferentworld,thephysicalconstraintsandvariabilityimposedbytheworldaroundus,thematerialresourcesavailableforouruse,andtheanalysistechniquesavailablefordesignerstopredicthowthevehiclewillperform.Practicalengineersusethematerialsandforcesofnaturetobenethumansbyattemptingtodesignthebestvehiclespossiblewhilemitigatingtheselimitations. Vehicleswhichexperiencecouplingbetweencontrol,structural,anduiddynamicforcesarecategorizedasaeroservoelastic.Traditionalproceduresforvehicleswhichexperienceaeroservoelasticcouplingattempttodecoupletheseforcesforeaseofdesignandanalysis.Incaseswheredesignisunabletoeliminatedangerousaeroservoelasticeffects,designersareforcedtominimizeoreliminatethedetrimentaleffectsoftheseforceswithsomeformofautomaticwing-shapecontrol.Thecomputationalandtheoreticaltechniqueswhichdoexisttopredictcoupledaeroservoelasticeffectsaretimeconsumingorextremelycomplex.Thereforethestandardanalysistechniquesaredifculttousetoprovideaprioriasinsightfordesignersaswellastoprovideamodelwhichmaybeusedforwing-shapecontrolofavehicleinreal-time. 15

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1.2ProblemStatement Aeroservoelasticeffectsarepresentinmorphingandappingwingsandmaybeusedtosubstantiallybenetvehicleperformanceinvehicleswhichusethesetypeofwingstomove.Thesebenetsmayincludeenhancedgusttoleranceandmaneuverabilityforthevehicleaswellasaerodynamicandcontrolefciency.Thereforemodeling,analysis,andcontroldesignmethodswhichcaptureaeroservoelasticeffectspresentinthesewingsandprovidewing-shapecontroltoleveragethoseeffectsareneeded.Anefcientlystructured,experimentallybasedmodelwhichincludesalltheaeroservoelasticeffectsaffectingthewingmaysupplythemeanstocontrolthewingsuchthataeroservoelasticeffectsprovideperformancebenetsforthevehicle. 1.3Contributions Thisworkwillexaminethestate-of-theartinsystemidenticationandcontrolofsystemswithcoupledcontrol,structuralanduiddynamics.Thepurposeofthisworkistocontributea(1)time-frequencyspecicunderstanding,(2)experimentally-basedmethodologyforsystemidenticationand(3)controlsynthesisforaeroservoelasticsystems.Referencestoavailable,relevantpapersbytheauthorandcolleaguesarespeciedforeachcontribution,whileopensourcecodefortheworkisarchivedattheauthor'swebsite[ 128 ].Specically,thecontributionsinclude: 1. Thetime-frequencydependenciesofanaeroservoelasticmorphingwingandappingwingwillbeexperimentallymeasuredandanalyzed.Theperiodicityevidentinmostappingsignalswillbedemonstratedandlocationsintimeandfrequencywherethesignalsexhibitedtime-frequencydependentornonlinearbehaviorwillbeidentied[ 132 133 ].(Ch. 4 ) 2. Theeffectsofwingdesignandactuationmethodonthestructuraldynamicsofcomposite-membranemorphingandappingwingswillbeexamined[ 133 ].(Ch. 7 8 ) 3. Theeffectsofthestructuraldynamics,kinematicparametersandappingdeectionsontheaerodynamicperformanceofthewingwillbeexaminedforaappingwing[ 279 ]andadesirablebutnotoptimalappingmovementwillbeidentied.(Ch. 8 ) 16

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4. Ageneralizedgeometricalmodelofashape-changingwingwithtemporal,spatialandfrequencydependencieswillbedened.Themodelwillbeusedtoformabasisfortwoaeroservoelasticmodels:(a)apiezoelectrically-actuatedmorphingcomposite-membranewingmodeland(b)aappingcomposite-membranewingmodel[ 129 130 133 ].(Ch. 5 ) 5. Atechniqueforidentifyinganaeroservoelasticmodelofwingdeectionsfromexperimentaldatawillbedeveloped.Themodelwillincorporateallaeroservoelasticphenomenapresentintheexperimentaldata.Inaddition,themodelsidentiedwillaccountforthenonlinear,timeandfrequency-varyingdynamicsthatareexperiencedbyexiblemorphingandappingwings[ 130 ].(Ch. 5 ) 6. Afeedforwardcontrolstrategywhichusestheidentiedmorphingandappingmodelswillbesynthesized.Thefeedforwardcontrollerwillbeusedtocommandadesiredwing-shapeforamorphingandaappingwing[ 130 ].(Ch. 6 ) 7. Aclosedloopcontrolstrategywhichusestheidentiedmorphingandappingmodelsandfeedforwardcontrolsynthesiswillbedevelopedwhichmatchesthedesiredwing-shapemovementforamorphingorappingwing.Theperformanceandrobustnessoftheclosed-loopcontrolsynthesiswithrespecttothefeedforwardcontrolsynthesis,parametricuncertainty,noiseanddisturbanceswillbeexaminedfortheappingwing.(Ch. 6 ) 1.4MajorAssumptions Severalbroadandimportantassumptionsaremadewhicharegenerallyapplicabletotheproblemsolvingapproach.Theseassumptionsarelistedhere,whileassumptionsspecictothemathematicaldetailsofthemodelingprocessorcontrolsynthesiswillbeaddressedastheyarepresentedinthiswork. 1. Thesystemidenticationapproachassumesthatallofthedynamicsofaeroservoelasticwingsmaybecapturedbyanalyzingthekinematicsofthemorphingorappingwingatdiscretevaluesoftheactuatorsasmeasuredwithlaserdopplervibrometryorhigh-speed,high-resolutionphotography. 2. Themass,inertia,andstiffnesspropertiesofthewingandaerodynamicloadingsonthewingareassumedtobeunknown. 3. Adesiredwing-shapemovementisassumedtobeknowntoachievesomedesirablevehiclemaneuveratsomedesirablelevelofperformance.Forexample,adropinappingamplitudefortherightwingresultsinarightturnwhilemaintainingcruisevelocityatadesirablelifttodragratio.Inotherwords,therelationshipbetweenwingmovementandvehicleperformanceisknown. 17

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4. Themorphingwingmodelassumesthemorphingactuatorswillmoveonlyatcertainspeciedintervalsintimeorwillmaintainsomeperiodicfrequencyforatleastonecycle. 5. Theappingwingmodelassumesthatthewingmovementisperiodic.Thereforeinputcommandsmaybesentatthebeginningofeachcycle.Assuch,thefeedforwardcontrolmaynotbecapableofrespondingtochangesinthesystemwhicharemadefasterthantheperiodofasingleappingcycle. 6. Thevehicleisassumedtohavesomearrayofsensorswhichmaybeusedtoobtainthepositionofthewing.Thecontrolstrategywillassumethatfeedbackabouttheshapeofthewingispresentfromseverallocationsonthewing. 7. Thevehicleisassumedtohavethecomputationalpowertoobtaindata,evaluatetheidentiedmodel,andcommandthecontrolactuationwithinthelengthofasingleperiodofmorphingorapping. 1.5DissertationOutline Chapter 1 describestheproblem,theapproachrequiredtoaddressit,andexaminesthecontributionsofthework.Chapter 2 givesthehistoricalcontextforhowwing-shapehasbeenmodiedtoenhancevehicleperformance.Chapter 3 introducesthecomplexitiesofcoupleduiddynamicandservoelasticbehavior,explainsphenomenapresentinthesesystems,andintroducesexperimentaltechniquesandmodelingproceduresforthesesystems.Chapter 4 examinesthesignalprocessingrequiredtounderstandthenatureofaeroservoelasticproblems,specicallybyusingtime,frequencyandtime-frequencyanalysis.Chapter 5 explorescurrenttechniquesusedtoprovidemodelsforsystemswithcoupledcontrol,structuralanduiddynamicsandpresentstheapproachesusedtoidentifyamodelforthesesystems.Chapter 6 introducespreviouscontrolapproachesforthesesystemsandpresentstheapproachthatwillbeusedtocontroltheproposedmodel.Thedissertationreviewspriorworkandprovidesreal-worldexamplesoftheproposedmodeling,systemidenticationandcontrolapproachwhenappliedtoamorphingwinginChapter 7 andaexibleappingwinginChapter 8 .Chapter 9 presentsabriefprojectsummaryandconclusionsdrawnfromthecontributionsofthisdissertation. 18

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CHAPTER2WING-SHAPEMODIFICATIONHISTORY 2.1TraditionalWings Fromthestartofaviationhistory,theWrightbrothersdesignedtheirwingstohavecoupledexibilityandcontroldynamicssincethepilotmaygainanintuitivegraspofhowtwistingthewingaffectstheaircraftdynamics.Asaircraftdesignprogressed,thewingshavebeenmadenearlyrigidandautomaticcontrolsystemshavebeenimplementedusingcontrolsurfaceswithmostlydecoupledeffects.Theresultwasthetraditionalsetofcontrolactuators:aileronstocontrolroll,elevatorstocontrolpitch,andaruddertocontrolyaw.Automaticcontrolsystemsenablesubstantialperformancebenets,butrelyonapreciseknowledgeoftheaircraftsystem.Thereforedesignersformostmodernaircrafthavedecoupledtheeffectsofcontrolactuationandwingexibilityforeaseofdesignanalysisandrecentlytoeasethedevelopmentofautonomoussystems.Thedownsideisthattheserigid-wingdesignsaddsubstantialweighttotheaircraftbyrequiringseparatecontrolsurfaces,limittheabilityoftheaircrafttoperformawidevarietyofmissions,anddonottakefulladvantageoftheabilitiesofautomaticcontrol.Asaviationhistoryhasprogressed,designershavestartedtousewingswithincreasingamountsofexibilityandcoupledcontrolactuationtogainperformancebenetsbyeliminatingstructuralweight.Researchintoxedwingmicroair-vehicles(MAVs)hasindicatedthatgusttolerancemaybeimprovedsubstantiallywhensomeexibilityisintroducedintothewing'strailingedge[ 99 ].WhiletheliteratureonhowtodesignMAVswithxedwingsissubstantial[ 150 151 ],aerodynamicconstraintsduetoincreasinglylargeviscousuidforcesasReynoldsnumberdecreasessuggestthattraditionalxedorhelicopterwingswillnotalwaysprovidethemostaerodynamicallyefcientsolutionsatReynoldsnumberslessthan10000. ThemissionsandclassicationsforvariousmodernaircraftarespeciedinTable 2-1 asaccumulatedfromthecurrentUSmilitarydesignations[ 198 ]andUnmanned 19

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AircraftSystems(UAS)Roadmap[ 52 64 ].Itisclearthattraditionalaircraftareprimarilydesignedtoperform1missionswithahighdegreeofefciency.Whenthecosttodevelopamodernaircraftreachesintothebillionsofdollars,increasingtheabilityofanaircrafttoperformmultiplemissionsbyincorporatingvariablewinggeometrymustalsobeconsideredinthedesignprocess.Inaddition,unmannedairsystemswillbeexpectedtoperformawidevarietyofmissions.Inthefuture,morphingwingswillenableaircrafttoeffectivelyaddressmultiplemissionsfromtheoptionsspeciedinTable 2-1 .TheobjectioncouldberaisedthatthelowerunitpriceofUASwilljustifythedevelopmentofmultipleaircrafttodospecializedtasksinateambasedenvironment;however,ateamconsistingofrelativelyinexiblebuthigherperformanceUASwillhaveatacticaldisadvantageagainstamoreexibleteam.Militariesofthepasthavesucceededbasedontheirabilitytoadaptquicklytochangingbattleeldscenarios.MorphingUASwillgiveamilitaryasignicanttacticaladvantageinthefutureduetotheirabilitytorespondtochangingconditionsfasterandmoreeffectively. 2.2MorphingWings Manydenitionsofamorphingwingexistbutusuallymorphingreferstolargechangesinwing-shape,accomplishedwithactuatorscontrolling1or2degrees-of-freedom.Thetermsvariablegeometryorpolymorphousaircraftareanalogous[ 260 ].NASA'smorphingprojectdenesamorphingaircraftas:efcient,multi-pointadaptabilitythatincludesmacro,micro,structuraland/oruidicapproaches[ 141 ].DARPAdenesmorphingasaplatformthatis:(1)abletochangeitsstatesubstantially(ontheorderof50%morewingarea),(2)toadapttochangingmissionenvironmentstoprovideasuperiorsystemcapabilitynotpossiblewithoutreconguration,and(3)usesadesignthatintegratesinnovativecombinationsofadvancedmaterials,actuators,owcontrollersandmechanismstoachievethestatechange[ 271 ].Forthisworkmorphingwillbeconsideredtobewing-shapechangeswhichonlymoveatcertainintervalsintime,whileappingwilldescribewingswhicharecontinuouslyarticulatedintime. 20

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Thebenetsofmorphingaretypicallyto(1)improveightperformancetoexpandtheightenvelope,(2)replaceconventionalcontrolsurfacesforightcontroltoachieveperformance,stealthorreducedweight,(3)reduceddragtoimproverangeor(4)toreduceorcontrolutter[ 74 ].Themainchallengestoimplementmorphinginclude:distributedhigh-powerdensityactuation,structuralmechanization,exibleskins,smartmaterialsandcontrollawdevelopment[ 185 ].TheprimaryresearchgoalsfortheDARPAmorphingaircraftstructures(MAS)programwere:(1)responsivenessasmeasuredbydeploymentspeed,(2)agilitytoattackfastmovingairandgroundtargetsand(3)persistencetodominatelargeareasforlongtimeperiods[ 260 ]. Thedesignermustbalancetheadditionalcomplexityofmorphingwiththeperceivedbenetsofincreasedperformanceormissionexibilitytodetermineifmorphingisjustiable.Missioncapabilityisdictatedprimarilybygeometry[ 204 ],thereforevehicleswithmorphingwingsaretraditionallycategorizedbythetypeofactuationusedtomovethewing.Changingaspectratio,wingarea,wingtwist,wingsweeparethemostwidelyusedsincetheseenabletheaircrafttoperformmostdesirablemissionseffectivelywithoutintroducingtoomuchweighttotheaircraft.GoalsforpercentchangesinmorphingparameterswereestablishedfortheMorphingAircraftStructures(MAW)programas200%aspectratio,50%wingarea,50%wingtwist,and20owingsweep[ 204 ].Forcomparison,traditionalaircraftvarywingareasbylessthan5percent[ 259 ].AsummaryofphysicalparameterswhichmaybemorphedandtheirperceivedbenetsaresummarizedinTable 2-2 .Table 2-3 fromNicolai[ 158 ]alsoinvestigatestheaffectofvariablegeometryonparameterswhichdriveaircraftperformancemetrics.CD0isthedragcoefcientatzerolift,Kscalesthecoefcientofdragduetolift,CListheliftcoefcientchangewithrespecttoangleofattackandCLmaxisthemaximumliftcoefcient.Informationabouttheeffectivenessofthesetrendsmaybefoundinanytypicalaircraftdesigntextbook[ 158 184 ]whileanindepthexplanationofeachparameter'susefulnessformorphingandsupportingpaperswas 21

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givenbySoa[ 220 ].Mostinvestigationsofthesetrendsassumesteadyaerodynamicswherethewingmorphingoccursslowlyandthewingsarelarge. Tradestudiesusingvariousoptimizationtechniquesareincreasinglyreliedupontodeterminetheidealwing-shapeforagivenmission.Aconceptualdesignoptimizationacrosswingloading,thrusttoweightratio,wingthicknesstochordratio,taperratioandaspectratiowasperformedbyRoth[ 189 ]andFrommer[ 75 ]whilesizingequationsforthemorphingwingweightandvehiclesizeweredevelopedforconceptualdesignersbySkillen[ 217 218 ].Joshioptimizedvariouswingandairfoilgeometriesforgivenightmissionsegments(takeoff,climb,cruiseatvariousaltitudes,dash,endurance,turning)usingtheFirebeedroneasabaseline.Optimizationhasalsobeenusedtoexaminethetopologyofthewingandplaceactuatorsforadesiredtypeofmorphing[ 100 ].Otheroptimizationframeworkshaveincorporatedexperimentalwind-tunnelteststogiveinsightintohowtooptimizemorphingperformanceinrealtimewithdeectionandangleofattack[ 176 ]orwingangleofattack,camber,andreex[ 36 ]. Althoughthedesigntechniquesandanalysismethodsrequiredtounderstandmorphing-wingvehiclesarenotfullydeveloped,wingswhichchangetheirshapetoinuencethevehicle'smovementorperformancearecommon.EarlyexamplesofmorphingwingsincludetheMAK-10byIvanMakhonine,whichwasatelescopingwingaircraftdevelopedinthe1930'swhileothersfollowedwiththeG-11C-1,BaksaevLIG-7andfs-29sailplane[ 260 ].Nikitin-ShevchenkodevelopedtheIS-1whichwasabi-planewhichmorphedintoamonoplanetooperateathigherspeeds[ 260 ].SweptwingswereverycommonstartingwiththedesignoftheMesserschmidtP-1011andconstructionoftheBellX-5aircraft.Aplethoraofsweep-wingaircraftincludingthePterodactylIV,GrummanXF10FJaguar,F-111,Su-24,MIG-23,MIG-27,PanaviaTornado,RockwellB-1A,TupolevTuf-22MBackreandTu-160Blackjack[ 61 260 ]weredeveloped.Severalunderwatervehicleshavebeendevelopedwhichcanmorphtheirwings,andinspiredbytheCommonGuillemot,researchersareinvestigatinghowtodesignvehicleswhichcan 22

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bothyandswim[ 125 ].ROVs(RemotelyOperatedVehicles),UAVs/UASs(UnmannedAirVehicles/UnmannedAirSystems)andUUVs/UUSs(UnmannedUnderwaterVehicles/UnmannedUnderwaterSystems)atallscalesshouldprovidetheimpetusfornewmorphingdesignssincethecomputationalpowerandtheoreticalunderstandingrequiredtocontrolthesevehiclesarebecomingmorereadilyavailable.Thetrendtowardmorphingwingsshouldexpandasunmannedsystemscontinuetoproliferateandbecomemoreautonomousifactuatorsizemaybelimitedandthenecessaryfeedbackcontroltechniquesaredeveloped.Issuesexperiencedwithhigh-altitudelongendurance(HALE)aircraftwithhighlyexiblewingshasmotivatedNASAtoprioritizethedevelopmentofaeroservoelasticanalysistechniques[ 228 ]whilemilitarylabsareextendingtheirresearchintobiomimicry.Theseresearcheffortsarehelpingalleviatethebarrierofunderdevelopedaeroservoelasticandhydroservoelasticanalysistechniqueswhichmaybeapplicabletomorphingandappingvehicles.AbriefphotographicreviewofrepresentativevehicleswithshapechangingwingsisshowninTable 2-4 Table2-4.HistoricalExamplesofVehicleswithShapeChangingWings Year/RefOrganization/VehicleShapeChangeDescriptionPhoto ExperimentalMannedAircraft 1903 WrightFlyer TwistAngle 1951 BellX-5 SweepAngle 23

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Table 2-4 .Continued. Year/RefOrganization/VehicleShapeChangeDescriptionPhoto1964 NorthAmericanAviationXB-70 WingSpan 1984 GrummanX-29 PassiveBend-TwistAngle 1986 AFTIF-111MAW Camber,SweepAngle 2002 BoeingX-53AAW WingTwistAngle ProductionMannedAircraft 1955 VoughtF8UCrusader WingIncidenceAngle 1964 GeneralDynamicsF-111 SweepAngle 1970 GrummanF-14 SweepAngle 1974 Rockwell/BoeingB-1 SweepAngle 24

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Table 2-4 .Continued. Year/RefOrganization/VehicleShapeChangeDescriptionPhotoExperimentalUnmannedAirVehicles 1983[ 178 ] NASAHiMAT Camber,AeroelasticTailoring 2001 NASAI2000 Inatable-WingSpan/Area 2002[ 14 ] NextGenMFX-2 SlidingSkinSpan/WingArea 2002 LockheedMartinAgileHunter FoldingWingSpan/Area 2002 RaytheonUAV WingSpan/Area 2003[ 138 ] VirginiaTechBetamax WingSpan/Area 2003 AerovisionsDroid WingSpan/Area 2006[ 3 4 ]UniversityofFloridaUrbanStuntPlaneWingDihedral,TwistAngle 25

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Table 2-4 .Continued. Year/RefOrganization/VehicleShapeChangeDescriptionPhoto2007[ 80 ]UniversityofFloridaEbonyThunderWingSweep,TwistAngle 2008 WageniganUniversityRoboSwift WingSweepAngle 2010 AerovironmentNanoAirVehicle FlappingDihedralAngle ProductionUnmannedAirVehicles 2010 PrioriaMaveric PassiveTwistAngle ExperimentalMannedUnderwaterVehicles 2009 BogusBatoid Dihedral/PassiveAngleofAttack ProductionUnderwaterVehicles 2010 InnespaceSeabreacherX AngleofAttack UnmannedUnderwaterVehicles 2007 NewYorkUniversityAQUA Twist/AngleofAttack 26

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Year/RefOrganization/VehicleShapeChangeDescriptionPhoto2008 FestoAquaRay Dihedral/AngleofAttack 2008[ 119 ] MIT/WHOIFinneganUUV Dihedral/AngleofAttack 2009 FestoAquaPenquin Dihedral/AngleofAttack 2010[ 131 ] UniversityofFloridaSolarRay Dihedral/PassiveAngleofAttack 2010[ 249 ]NextGenAeronauticsCuttleshChord-DependentDihedralAngle Severalhelpfulreviewsofacademicworkonmorphingvehicleshavebeenperformedrecently.In2005and2006,Siegler[ 202 ]andNeal[ 155 ]summarizedmostoftheindustrialmorphingprojectsalongwithseveralfromacademia.In2010,Soasummarizedacademicmorphingworkcategorizedaccordingtothemorphingparameterexamined[ 220 ].VehicleswithmorphingfromacademiaareincludedinTable 2-4 whilethosewhichwerenotimplementedonavehiclearesummarizedinTable 2-5 .Itshouldbenotedthatseveraloftheresults[ 3 4 83 ]wereimplementedonhighlyexiblewingsormicro-airvehicles.Thereforethebenetsimpliedformorphingmaynotextendtoaircraftwithhighwingloadings. Undercertainconditionsaeroservoelasticdependenciesmaybeneglectedformorphingwings.Thewingdynamicsandinertialeffectsneedtobeconsideredformorphingwingswhenthemorphingisquickoriflargeamountsofmassaremoved.Whenthemassismovedslowly,morphingmaybetreatedwithquasi-steadyanalysis.Morphingvehiclesalsogenerallydonotneedtoaccountforunsteadyaerodynamic 27

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effectsunlesstheReynoldsnumberisbelow106orthemorphingoccursquickly.Wingexibilitywillalsobelessimportantunlessthewingisdesignedtobehighlyexibleduetosevereweightconstraints,orthemorphingoccursquickly.Insummary,allcoupledeffectsbecomemoresignicantoncethewingismovedfaster.Sinceappingwingsmovequickly,appingwingsmaybeconsideredamorecomplexsubsetofmorphingwingswhereaeroservoelasticanalysisismandatory. 2.3FlappingWings Theearliestattemptstosystematicallydesignaircraftinthelate1800swereinspiredbybirdsandthereforeoftenincludedappingwings.Formanyyearsappingremainedanelusivedreambutapowered,mannedornithopterwasrealizedbyDelaurierin2006.Forthemostparthowever,aerodynamicconsiderationsdemandthatappingbeappliedonsmaller,unmannedvehiclesasisclearlyevidentfromthegreatightdiagram,whichsummarizestherelationshipbetweenwingloading,grossweightandcruisespeedofyingvehiclesasshowninFigure 2-1 [ 238 ].Thesesmallervehicleshavethepotentialtoproliferateastheymeetincreasingmarketdemandforeffectiveandwidelydistributedmobilesensingtoaddressawidevarietyofapplications.InpartbyexaminingFigure 2-1 ,appingisidentiedasanenablingtechnologyforappingwingmicroair-vehicles(FWMAVs)withspanlessthan15cm[ 151 ].Flappingwingshavealsocomeunderintenseresearchscrutinybecauseofthepotentialadvantagestheyoffertodesignersatthisscale,anincreasedfocusonbio-inspireddesign,andthelackofsufcientdesignknowledgeforappingvehicles.Naturaliersexhibitaerodynamicefciency,gusttolerance,andmaneuverabilityfarbeyondthatofman-madevehicles.Thisprovideshopefordesignersthatthedifcultiesfacedbyxedwingvehiclesandhelicoptersatthesesmallscalesarenotinsurmountableandmaybesolvedwithefcientappingwingvehicles. Qualitatively,appingvehiclesmovetheirwingsquicklytogeneratecirculationoverthewingandtherebygeneratelift,whilethemovementofthewingsheds 28

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Figure2-1. TheGreatFlightDiagram:RelationshipofWingLoading,GrossWeightandCruiseSpeedofFlyingVehicles[ 238 ] counter-rotatingvorticestoproducethrust.Aquasi-steadyaerodynamicmodelisoftensufcientforvehiclesthesizeofbirdsandevenhummingbirds,especiallywheninforwardight.However,itisnotsufcientforinsect-sizeanalysis,whichrequiremodelingtheaerodynamicswiththeunsteadyNavier-Stokesequations.TheKnoller-Betzeffectdescribeshowbirdsexploitthepropulsivenatureofpitchingandplungingwhiletheirforwardspeedprovidestheneededlift.Unsteadylift-enhancementmechanismsforappingarenecessarytoachievesufcientaerodynamicperformance 29

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sinceinsectwingsachieveliftcoefcientsintherangeof0.6
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owdescriptionsandwingresponseinformationmaybenecessarytoensuretheaeroservoelasticnatureofappingwingsisfullycharacterized. Aninterpretationofthesimilarityparametersgivesvehicledesignersaplacetostartwhenconsideringwhichtypeofwingmovementtouseaftergrossweighthasbeenconsideredandaninitialwingsizedetermined.AthighReynoldsnumbers(above106),inertialforcesdominateviscousforces,thusfavoringxedwingvehiclesandhelicoptersforaerodynamicefciency.MorphingandappingvehiclesmightbeabletogainadvantagesathigherReynoldsnumbersbysavingweightthroughconsolidationofcontrolsurfaces,butacarefulanalysiswouldbeneededtoensurethelimitednumberofcontrolsurfacescouldstabilizeallaircraftightdynamics.AtlowReynoldsnumbersaround104105,viscouseffectsbecomesignicantandxedorspinningwingsloseaerodynamicefciency,justifyingappingasamoreattractivesolutionandexplainingwhybirdsandbatsoperateat1041,theowcanbeconsideredquasi-steady,whileJ<1willgenerateunsteadyow,requiringdifferenttypesofanalysis[ 91 ].Therstbendingandrsttorsionalmodalfrequenciesanddampingswillindicatethestructuraldynamicparametersandarenecessarytocharacterizetheappingsincetheyareoftenlowenoughtobeexcitedduringapping.AnalternativeapproachtocharacterizethewingmovementistousetwoloopareasforasingleperiodofappingasmeasuresofwingexibilityinthespanwiseandchordwisedirectionsasdenedbyWu[ 275 ]. 31

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Flappingcanbebenecialduetohigheraerodynamicefciency[ 92 ],delayedstall[ 211 ],increasedmaneuverability[ 92 94 ]andenhancedgusttolerance[ 94 235 ]whencomparedtoxedwingaircraftandhelicopters.Theabilitytoincludeliftproduction,thrustproductionandcontrolactuationinasinglestructuremayalsodecreasethevehicleweightenablingincreasedefciency. Manyobstaclesremainbeforeanefcientlydesignedappingvehiclecanberealized.Flappingmaybeefcientforforcegenerationofbothliftandthrust;however,theappingmotionisgenerallymorecomplexthanthemotionformorphingwings[ 11 257 ].Unsteadyaerodynamicandhydrodynamicanalysismayberequiredtopredictthrustproduction,butcalculationoftheunsteadyowiscomputationallyexpensive.Thestructureisrequiredtohavesufcientexibilitysuchthattime-varyingaerodynamicloadingsonthestructuredonotconictwithabilityofthewingtoproduceliftandthrust.Inadditiontothe6degrees-of-freedomofthevehicledynamics,thecontrolactuatorsmustquicklyalterthewingdihedral,sweepandtwistacrosstime,makingitdifculttodescribethekinematicbehaviorofaappingvehicle.Thekinematicmovementofthewingsaredescribedasrigid-wingdeections,whilewingexibilityintroducesadditionaldeformationsthroughouttheappingcycle,producingthenaldeectionsofthewing.Inertialeffectsalsocontributetowingmovementssincethewingsusuallymovequiterapidly.Lastly,thesewingsoftenundergostructuralresonance.Theresultingdynamicsareinherentlyaeroservoelasticsincetheaerodynamicsandstructuraldynamicsaretightlycoupledalongwithanycontroleffectors.Assuch,theightdynamicsaredependentupontherelativedeectionsacrossthewingsoccurringthroughouttheappingcycle. Natureprovidedtheinitialinspirationformentowanttoy,butthusfarmancontinuestostruggletomatchnature'selegantdesignsinmannedandunmannedaerialvehicles.Formanyyears,researchershavereviewedmanyofnaturesiersincludingincludingbirds[ 19 136 ],bats[ 164 ],andinsects[ 192 272 ].Underwater,batoids, 32

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commonlyknownasrays,utilizesimilarprinciplestoswimandareincreasinglybeingconsideredforthedesignofbothmannedandunmannedunderwatervehicles[ 46 54 169 188 ].TheNavalResearchLab[ 180 ]andaDepartmentofDefenseMulti-UniversityResearchInitiative(MURI)areinvestigatingbatoidandshinspiredvehicledesigns[ 146 147 ]sincepectoralmotionwasshowntobemoreefcienttomaneuveratlowspeedsthanbodyandcaudalnundulation[ 32 ]andmaybemoreeffectivethanthrusters[ 180 ]. AbriefsummaryofquantitativepapersexaminingthekinematicsofappingwingedorganismsisprovidedinTable 2-7 ,focusingontherelationshipbetweenbodymass,appingfrequency,appingamplitude,bodyangleandforwardspeed.Ingeneralbodyangledecreasesandstrokeplaneangleincreaseswithincreasingightspeed[ 66 269 ].Strokeamplitudedecreaseswithincreasingightspeedbuttheliteratureislessconsistent[ 66 269 ].Noconsistenttrendhasbeenfoundbetweenappingfrequencyandightspeed,butmostinsectshavearelativelyconstantappingfrequency,suggestingtheparametermaybedenedbythestructuraldynamicsoftheirwings.Insectsinnaturevaryseveralotherkinematicparameterstoeffectcertainmaneuvers.Asymmetricchangeinstrokeamplitudeisusedtoclimbordivewhileanasymmetricchangeproducesrollrotation.Asymmetricchangeinstrokeoffsetproducespitchrotation.Anasymmetricrotationtimingchangeyieldscoupledyawrollrotationwhileasymmetricchangeyieldspitchrotation.Anasymmetricchangeinangleofattackchangestheforwardthrustdirection[ 235 ]. Articialappingvehiclesarecurrentlymuchlesscomplexthannature'sexamples.Thereforeresearchwiththesesystemsmayclarifyhowvehicleperformancechangeswithrespecttokinematicorelasticparameters.Importantexamplesfromindustryandacademiaincludethe:Caltech/AerovironmentMicrobat[ 177 ],UniversityofTorontoMentor[ 283 ],Cybird,DelyII,Kolibri,HarvardMicroroboticFlyingInsect(MFI)[ 18 280 ],WowweeDragony,iFlyVamp,TakahashiOrnithopterandAerovironmentNanoAirVehicle.Allofthesesystemshaveexiblewingswhichundergolargeactuationand 33

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thereforeencountercouplingbetweenunsteadyaerodynamics,controlactuation,wingexibilityandstructuraldynamics(especiallyfortheMFI). Tradestudiesinvestigatingtheeffectsofplanformshape,wingstiffnessandkinematicbehavioronwingperformancesuchashavebeeninvestigatedforappingwingsarequitelimitedduetothecomplexityinmodelingtheaerodynamicsinvolved,butseveralhavebeenattemptedusingsurrogatemodelstoinvestigateplungeamplitude,angularamplitude,andphaselag[ 246 ]andexperimentallybymeasuringthethrustwithrespecttowingstiffness,appingamplitudeandappingfrequency[ 275 ].Anumericalstudyofappingwingssuggestedthatanangleofattack>0.3andpassivefeatheringangle>14.9o,denedbytheanglefromtheinitialwingcongurationtolineformedfromtheleadingedgetodeectedtrailingedge,werenecessaryfortheappingtoproducethrustinsteadofdrag[ 282 ].AnotherstudyincorporatingasomewhatrigidwingsuggestedthatL=DandCLincreasewhenappingamplitudeisincreasedfrom30-90degrees[ 109 ].Thelargesensitivityofperformancewithrespecttothenalwingmovementsuggeststhatacoupledaeroservoelastictechniquewhichpreciselydenesthewingmovementwithrespecttosomeappliedcontrolactuationisneededtoensureefcientappingperformance. 2.4OtherApplications Lookingbeyondtheproblemsathandwillgiveinsightintohowthesecoupledsystemsmaybeanalyzedandwhereinsightsgainedmayalsobeapplied.Inadditiontovehiclewings,servoelasticsystemswithcoupleduiddynamicsarebeinginvestigatedforuseinwindturbinebladesforenergygeneration,roboticssystemswithexiblestructures,vibrationisolationforspacecraft[ 249 ]andactivestructuresincludingtensegrity[ 146 ],piezoelectric[ 250 ]orshape-memory-alloy-basedstructures[ 168 ].Thereforeanaeroservoelasticcontrolschemewillndmanyapplicationsbeyondjustmorphingandappingwingvehicles. 34

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Table2-1. MissionScenariosforModernAircraftandUnmannedAerialSystems/TypesofAircraftDesignations[ 52 64 198 ] MissionProle(USMilitaryDesignationLetter)Well-KnownExamples Attack(A)AttackTargetsA-4Skyhawk,A-6Intruder Bomber(B)BombTargetsB-1BLancer,B-2Spirit Cargo(C)TransportCargoC-5AGalaxy,C-17AGlobemaster,CH-3BSeaKing,CH-47DChinook Drone(D)MultipleMissionsDC-130AHercules Electronic(E)CommunicationsandSensingE-3ASentry Fighter(F)Air-to-AirCombatF-14ATomcat,F-15AEagle,F-16AFalcon,F-18AHornet,F-22Raptor,F-35LightningII Glider(G)PilotTrainingTG-1A SearchandRescue(H,SAR)HC-130HHercules,HH-3ASeaKing Tanker(K)In-airRefuelingKC-135AStratotanker Polar(L)PolarOperations,OftenSki-EquippedLC-130FHercules Multi-Mission(M)MultipleMissionsMV-22Osprey Observation(O)SurveillanceOV-1AMohawk Patrol(P)AirspacePatrolP-3AOrion DroneandTarget(Q)UnmannedAerialSystems,Multi-purposeRQ-1APredator,RQ-3ADarkstar,RQ-4AGlobalHawk,MQ-9Reaper Reconnaissance(R)AerialPhotographySR-71A Anti-Submarine(S)SubmarineTracking/AttackS-2ETracker,S-3AViking Trainer(T)PilotTrainingT-37BTweet,T-38ATalon Utility(U)SurveillanceandMultipleMissionsU-2DragonLady VIP(V)PresidentialTransportVC-25AAirForce1 Weather(W)InvestigateWeatherPatternsWP-3AOrion Experimental(X)ExperimentalAircraftX1-A Prototype(Y)PreliminaryProductionAircraftYF-22A Lighter-Than-Air(Z)Blimps/Dirigibles,ObservationZPG-3WAirship IndoorReconnaissance,Lethal,Non-lethal,Communication,SwarmingFlapping/Nano/MicroAirVehicle PersonalISR,Lethal,Cyber/ElectronicWarfare(EW),CounterUAS,Auto-sentrySmallUAS(WaspIII/Raven) SuppressionofEnemyAirDefense(SEAD)SmallUAS(Raven)-MediumUAS(MQ-9) Intelligence,Surveillance,Reconnaissance(ISR),CommunicationsRelay,SignalsIntelligence(SIGINT)TierIISTUASUAS/Air-LaunchedSUAS(ScanEagle) ElectronicAttack(EA),CloseAirSupport(CAS),Counter-Air/MissileDefenseMQ-Ma,MQ-Mb,MQ-Mc InformationIntegration,AirInterdiction,AeromedicalEvacuation/PersonnelRecoveryMediumSizeUAS Electro-optical,Infrared,SyntheticApertureRadar(EO/IR/SAR)High-AltitudeLong-Endurance(HALE)Aircraft 35

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Table2-2. BenetsofMorphingParameters WingMorphingParameterDescription/PotentialAdvantages DihedralAngleControlRollingMoment,SuppressDutchRoll,ControlSizeofBoundVortexforFlapping SweepAngleReduceAdverseEffectsofTransonic/SupersonicFlow,ImproveLateralStability,ControlRollingMomentduetoSideslip TwistAngle(AngleofAttack)PreventWingStall,ObtainEfcient(Elliptical)LiftDistribution,ChangeCLmaxandCL IncidenceAngletoFuselageMinimizeFuselageDragforCruise,EliminatePitchUpforLanding AirfoilCamberIncreaseorDecreaseAirfoilLiftandDrag AspectRatio(AR)IncreaseWingEfciency,IncreaseEnduranceorManeuverability,IncreaseFlappingWingThrust TaperRatioTailorLiftDistribution AreaIncreaseGrossWeightofAircraft,DecreaseWingLoading SpanChangeAspectRatio,ReduceDragforHigherSpeed SpanwiseStiffnessControlWingBending,ThrustforFlapping TorsionalStiffnessControlWingLiftDistribution,ThrustforFlapping Table2-3. TrendsinDesignParametersDuetoWing-ShapeChanges[ 158 ] IncreasesinParameterofLeftColumnYieldsRelativeChangeinParametersinTopRow,RelativeChangeRangesfromaLargePositiveChangetoLargeNegativeChangeExpressedBy:*..."...)]TJ /F3 11.955 Tf 11.96 0 Td[(...#...+,)]TJ /F1 11.955 Tf 12.62 0 Td[(IndicatesNoEffect ParameterCD0SubsonicCD0SupersonicKCLCLmaxWingWeightWingVolume AirfoilThicknessRatio*"## AspectRatio*#""*+ Camber**#" LeadingEdgeRadius*#"" TaperRatio#*"+*+ WingSweep#*++(Aft)(Fwd)* 36

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Table2-5. SelectedMorphingResearchfromAcademia Year/Refs.MorphingParameterMorphingDescriptionMorphingBenet 2003[ 60 117 135 ]Hyper-EllipticWing:TipSweep/TipDihedralAnglesDiscontinuous,+/-90o"TipAngle:"L/DRatio 2003[ 135 ]Hyper-EllipticWing:TipSweep/TipDihedralAnglesContinuous,+/-90o"TipAngle:"L/DRatio 2003[ 33 34 ]AspectRatioTelescopingSpar,Discontinuous,2.54.85"AR:"L/D 2003[ 137 ]SweepAngleDiscontinuous,0)]TJ /F3 11.955 Tf 11.95 0 Td[(45o"SweepAngle:CL#CLmax",Stall" 2004[ 181 182 ]TipSweep/TipDihedralAnglesCompliantTrussesActuatedbyTendonsOctohedralUnitCellsSupportedbyCablesFormHyper-EllipticWing-Shape:"L/DRatio 2004[ 3 ]WingTwistAngleTorqueRodAttachedtoWingTipIncreasedRollPerformance,GustTolerance 2004[ 4 ]WingDihedralAngleServosatWingRootandMidWingIncreasedRoll,StallPerformance 2004[ 45 ]WingArea,CamberInatableWing(NasticStructure)/RollControlwithPiezoelectricControlActuatorsMissionFlexibility:HighSpeedDash/LongEnduranceLoiter,WingSmoothing 2005[ 13 ]Flexural/TorsionalStiffnessRotatingInternalSpars,ShiftingChordwiseSparLocationsWingMayTwistFromAerodynamicLoads 2005[ 83 ]TwistAngleTorqueRodAttachedtoWingTipRib15%"L/DRatioforMorphingvs.ElevonWing,#DragforGivenMoment 2007[ 250 ]WingCamberPiezoelectricActuatorIncreasedRollAuthority,#Weight,#PartCount,#PowerConsumption 2007[ 89 ]WingStiffnessConventionalControlSurfaces,PiezoelectricTabDecreasedDrag,GustLoadAlleviation 2010[ 252 ]WingTwistInternalScrewforContinuousTwistAppliedat4Ribs"Twist"L/DRatio,EspeciallyforLowerAOA 37

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Table2-6. ParametersNecessarytoCharacterizeFlappingWings ParameterEquationNomenclatureMeaning ReducedFrequency(k)k=2!Lref 2Uref!=appingfrequency,Lref=chordlength,U=forwardvelocityRatioFlappingWingSizetoFlappingMovement ReynoldsNumber(Re)Re=UrefLref =density,=dynamicviscosity,Lref=chordlengthRatioInertialtoViscousForces StrouhalNumber(St)St=2!ha U!=appingfrequency,=appingamplitude(distance),U=forwardvelocityRatioOscillatingSpeedtoForwardSpeed(TypeofPropulsiveEfciency) AdvanceRatio(J)J=Uref 2!haU=forwardvelocity,!=appingfrequency,ha=appingamplitude(distance)RatioForwardSpeedtoOscillatingSpeed(TypeofPropulsiveEfciency) StructuralModesAllmodes<=4!m1,m2,m3...Frequency-DependentBehavior LoopArea,PhaseAngleTipDeection/Chordvs.FlapAngleAreaorAngle(ConvexHull(wtip cvs.))wtip=tipdeection,=appingamplitude(angle)SpanwiseStructuralCompliance LoopArea,TwistAnglevs.FlapAngleAreaorAngle(ConvexHull(twistvs.))twist=wingtwistangle,=appingamplitude(angle)ChordwiseStructuralCompliance Table2-7. OverviewofAcademicResearchofRepresentativeFlappingOrganisms ByColumn:Reference,OrganismName,OrganismMass,WingMass,WingLength,FlappingFrequency,FlappingAmplitude,CruiseSpeed,ReynoldsNumber Year/Refs.OrganismM(g)mw(mg)l(mm)f(Hz)(o)Ucruise(m/s)Re 2008[ 211 ]FruitFly0.0020.009632001492130 2008[ 211 ]Bumblebee0.1750.9131501204.51200 2000[ 211 269 ]Hawkmoth1.57994492510034200 2008[ 211 ]Hummingbird8.45888523149811000 1990[ 66 67 95 ]Butteries0.26739.811103310000 1992[ 88 ]CownoseRay2501240.690000 2006[ 53 ]MantaRay45000.521000000 38

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CHAPTER3AERO/HYDROSERVOELASTICSYSTEMS 3.1BackgroundforAero/hydroservoelasticSystems 3.1.1SystemClassications Mechanicsdescribeshowmatterbehaveswhensubjectedtophysicalforcesanddisplacements.Engineersusuallysubdividetheproblemsfoundinmechanicsintopiecesforeaseofmathematicalanalysis.Theyisolatethemostimportantphenomenaandthenanalyzethemtopredictandcontrolthesystembehavior.Allaircraftexperienceaeroservoelasticcoupling,butthecoupledeffectsmaybeaddressedseparatelysincethecouplingisofsmallermagnitude.Morphingandappingsystemshavetime-varyingforceswheretheuiddynamics,controldynamics,structuraldynamics,andsolidmechanicsaretightlycoupledandoflargemagnitude,sosuperimposingtheeffectsofeachdecoupledanalysiswillbeinsufcienttopredictthesystemresponse.Thereforetheapproachesusedtoanalyze,predictandcontrolthesesystemsfallintoadifferentclassofsystemswheredifferentapproachesmayberequired. Figure 3-1 demonstratestheclassesofanalysisusedtoanalyzesystemswithdecoupledorcoupleduid,structure,controlandtemperaturedynamicsdrivenbyunsteadyorsteadyforces.Termsspecictouidorgaseoussystemsareontheleftwhiletermsusedforsolidsystemsareontheright.Systemswithexternally-appliedforcesareontopwhilesystemswithinternalforcesareonthebottom.Steadyanalysisiskepttowardsthemiddle,whileunsteadyanalysisislistedatthefartopandfarbottomofthediagram.Itshouldbenotedthatclassifyingasystemastime-invariantortime-varyingarenotthesameassteadyorunsteady.Atime-invariantsystemwillhavethesameresponsetothesameinputatt1andt2,whileatime-varyingsystemwillnotifthesystemstatesremainconstant.Systemsusingsometypeofexternalcontrolarelistedgoingintothepage,whilesystemswithsomeformoftemperaturevariationarelistedcomingoutofthepage.Eachclassicationofsystemanalysisusedinthediagram 39

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inFigure 3-1 isdenedinTable 3-1 .Thisdiagramisusefultoclarifywhichassumptionsarebeingmadeabouttheanalysistypeforanygivensystem.Notethatforthisdiagram,macroscopicelectromagneticforcesareneglectedbecausetheyareoflessimportancetomorphingwingdesign. Figure3-1. Fluid-Structure-Control-TemperatureCouplingsOrganizedwithRespecttoTimeDependenceofForces Manysystemsintherealworlddonotneedtobeanalyzedwithafully-coupledapproach.TheconventionalaeroservoelasticVenndiagramshowninFigure 3-2 [ 93 152 ]isolatesthemostcommonanalysisclassicationsrelevanttowingsmovinginairwithcoupledaerodynamics,controlactuationandstructuralmechanics.ThisdiagramisasmallersubsetofthegeneralcasepresentedinFigure 3-1 wherethe 40

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time-dependentnatureoftheforcesinvolvedandwhatthoseforcesareappliedtoareleftmoreambiguous. Figure3-2. CoupledBehavior:Fluid-Structure-Control[ 93 ] 3.1.2AeroservoelasticDesignOverviewsforTraditional,Morphing,andFlappingWings AnalysisofthemovementofavehiclemustaccountfortheimportanteffectsshowninTable 3-2 whetheracoupledordecoupledapproachisused.Coupledapproachesaremoreoftenrequiredformorphingandappingwings. Traditionalaircraftdesigndecouplesaerodynamic,controldynamic,structuraldynamicandsolidmechanics(wingexibility)byiteratingbetweenthem.Forexample,adesignermightrstobservehowmuchliftisnecessarybasedonaguessoftheaircraftweight.Thepropulsionsystemandairfoilarechosenandthesteadyaerodynamicloadingsonthewingarecalculated.Theknownpressuredistributionfromtheaerodynamicsallowsthedesignertodesignasufcientwingstructurebasedonknownmaterialproperties,loadingandfatigueconstraintswhileaccountingforanytemperatureproles.Thecontrolsystemisthendesignedforsomedesiredvehicleperformancewhilepreventinganydetrimentalightdynamicandstructuraldynamiceffects.Iftheaircraftweightistoolargeattheendoftheprocess,theprocessisrepeated.The 41

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complexitiesofthesevehiclesandtheinterdependenceofeachareaofdecoupledanalysisnecessitatesusingthisiterativedesignprocedure.Optimizationroutinesarecommonlyusedtofacilitatedesigndecisionsanddecreasedesigniterations. Analysisofamorphingwingcanignoremanyofthecouplingsbetweenwingexibilityandaerodynamicsifthemorphingisrelativelyslowandthewingisrelativelyinexibleirrespectiveofthetime-varyingnatureoftheuiddynamics.However,thetime-varyingnatureofthemorphingactuationcausestheaerodynamicstochangeasthemorphingoccurs.Thecouplingbetweenaerodynamicsandmorphingmaybeanalyzedbyassumingtheaerodynamicsaresteadyatsomediscretizedvalueofmorphingifthewingmorphsslowlyenoughcomparedtotheairspeed.Thisquasi-steadyassumptionallowsdesignerstouseparameter-varyingcontroltechniqueswheretheaerodynamicsaredecoupledfromthewingkinematics.Thereforethedesignprocessmayproceediteratively,similartothatofatraditionalwing. Flappingwingsnecessitateacompletelycoupleddesignandanalysisprocesssinceappingvehicleshaveunsteadyaerodynamicswhicharetightlycoupledwiththewingexibility,structuralandcontroldynamicsifthewingcannotbeconsideredrigid.Ingeneraltemperatureislessofaconcern,soaero-orhydroservoelasticanalysisissufcientforappingvehicles. 3.2TheoreticalandComputationalAnalysis Usingtheoryandcomputationtopredictthebehaviorofawingisoftenthebestmethodforwingdesign.However,thesetechniquesmustbecapableofaccuratelypredictinghowthesystemwillbehaveortheywillonlymisleadthedesigner.Decoupledandcoupledtechniquesusedtopredicttheperformanceofawingaresummarizedinthissection. 3.2.1DecoupledSystemAnalysis Traditionalwinganalysisreliesondecouplingtheaerodynamics,solidmechanics,structuraldynamics,rigid-bodydynamics,controldynamicsandthermodynamics. 42

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Understandingtheassumptionsmadeineachofthesedecoupledapproachesisessentialtounderstandinghowacoupledaero-orhydroservoelasticmodelisconstructed.Relevantconceptsfromeacheldwillbeaddressedbrieybeforemovingtocoupledapproachessincethedecoupledanalysisgivesinsightintothedesignrequiredformorphingandappingwingvehicles. 3.2.1.1Fluiddynamics TraditionalaerodynamicsforastandardaircraftsuchasaCessna172relieson2Dincompressible,steadyowassumptionsformostaircraft.Inviscidornegligibleviscousforcesandlaminarorsmoothowassumptionsarecommonlymade.Thepressuredifferentialbetweenthetopandbottomsideoftheairfoilcreatescirculationaroundthewingtogeneratelift.Whensteadyowisassumed,theaerodynamicsaretimeinvariant,soanairfoiltypeandthicknessforthewingisspeciedasoneoftherstpartsoftheinitialiterationofanaircraftdesign.Thepressuredistributionatagivenvelocityofowmaybedetermined.Forconsistentcomparisons,theReynoldsnumber,lift,dragandmomentcoefcientsareallexamined.Thedragpolarfortheairfoilhelpsdeneanairfoilwithanefcient(high)lifttodragratioatthedesiredcruisevelocity.Anairfoilwithdesirableliftcurve,Cl,maximumliftcoefcientCLmaxandlargestallangleofattackmustalsobeselected.Lastly,theairfoilmustalsohaveanacceptablemomentcoefcienttoensurethattheaircraftwillnotexperiencepitchdivergence.Theliftanddragarethencalculatedaswellasthewingloadingdistribution. Manycircumstancesmaycomplicatemattersfortheaerodynamicist.Separatedowbubbles,boundarylayerseparation,andowreattachmentdescribehowtheowmayormaynotbelaminarovertheairfoil,dramaticallychangingtheliftanddragforagivenairfoil.Turbulenceintheairowbefore,during,andafteritowsovertheairfoilalsosubstantiallyimpactperformance.LowReynolds'numberconsiderationscomeintoplaywhereviscouseffectsbecomemoreimportantthaninertialforcesontheuid.Inaddition,compressibilityeffectssuchasshockwavesmayneedtobeconsidered. 43

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Thegoverningequationsforbasicowincludethecontinuity,momentumandenergyequationandaerodynamicforcesmaybeexpressedasbodyorsurfaceforces[ 15 ].Newton'ssecondlawmaybeusedwiththeseequationstoformthegeneralformoftheNavier-Stokesequationstodescribeuidow.Asaresultofthecomplexmathematicsrequiredforaerodynamicsproblems,wingaerodynamicloadingsarenowalmostuniversallyanalyzedwithcomputationaluiddynamics(CFD)modelswhichareoftensubjecttostrictboundaryconditionsandassumptions.CommercialofftheshelfsoftwarepackagesarewidelyavailabletoperformCFDanalysis.TheapplicabilityofCFDtechniquesandlimitationswasreviewedrecentlybyShang[ 207 ].Hydrodynamicsapproachesmayusesimilar,butslightlydifferentapproaches.Forexample,RankinesourcesorthetimedomainGreenfunctionmayhelpanalyze3Dhydrodynamics,butReynoldsAverageNavierStokes(RANS)methodsforincompressibleowincludingsmoothedparticlehydrodynamics(SPH)approachandmovingparticlesemi-implicit(MPS)schemehavealsofoundapplications[ 90 ].Radiation-diffractiontheorymaybeusedifa3Dhydrodynamicresponseisrequired. Thedistinctionbetweensteady,quasi-steadyandunsteadyowassumptionsisimportanttonoteformorphingwingssincetheincreasefromasteadyowtounsteadyapproachmaybeanorderofmagnitudemorecostlyintermsofcomputationrequired[ 200 ].Thequasi-steadyassumption,thataerodynamicforcesonawingatsomeairvelocityandangleofattackaretimeinvariantforsomeperiodoftime,maybeusedformorphingwingswhicharenotmovingquickly.Prandtl'slifting-linetheoryhasbeenextendedforwingsmorphingcurvatureorchorddistributionstogivegreaterefciencybychangingthelifttodragratio[ 265 ].Aquasi-steady3Dvortexlatticemethodwasusedtoanalzyemorphing[ 12 ].Forunsteadyaerodynamiceffectsonanoscillatingwing,Theodorsen[ 239 ]orGarrick's[ 77 ]theorymaybeusedtocalculatetheliftandmomentforthinairfoilsandrigidwings.However,theassumptionsrequiredbyTheodorsenandGarrick'smethodsareoftentoorestrictivetomakethemaccurate 44

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meanstopredictunsteadyforces.PanelmethodssuchasthedoubletpanelmethodusedbyNiksch[ 159 ]arealsopopular.Petersprovidedareviewofthetechniquesusefultoanalyzeunsteadyowfordeformableairfoilsin2008[ 173 ]. Forappingwings,quasi-steadytechniqueshavebeendeveloped,butanunsteadyapproachmayberequiredtopredictaerodynamicperformance.Rozhdestvenskyreviewedappingincludingaquaticorganismsin2003[ 190 ],whileAnsarireviewedaerodynamicmodelingforappingwingsin2006[ 17 ].Asemi-empiricalmodelbySaneandDickinsonfromtheRoboyexperimentshasbeenwidelyusedtopredictliftanddragcoefcients[ 194 ],butwhilethetime-averagedliftpredictionsmaybematched,unsteadyliftanddragandquasi-steadydragcoefcientsdifferedfromexperimentsbyafactorof3ormore[ 91 ].Whenappingissufcientlycomplexthattheunsteadysolutionisrequired,theNavier-Stokesequationsmayneedtobesolved.AzumaextendedtheworkofTheodorsen[ 239 ]andGarrick[ 77 ]tondunsteadyforcesandmomentsbasedonthrust,liftandmomentcoefcients[ 19 ].Forhover,lowReynoldsnumber,incompressibleandlaminarowmaybeassumed[ 17 ].TheseassumptionshelpedZbikowskiandAnsaripredicttheforcesinhoverforaquasi-three-dimensionalwingusinganarrayof2Dairfoils[ 16 ].Severalrecentworksforappingindicatethatspanwiseowmaybeimportantforkeepingtheleadingedgevortexboundtothewing.Thereforeforsomeappingsystems,a3DunsteadyNavier-Stokessolutionmaybeneeded.Severalpapershaverecentlyexaminedthecoupleduid-structureproblemforappingwithunsteady3Dow[ 49 ]. 3.2.1.2Solidmechanics/elasticityandmaterialselection Oncetheaerodynamicproleissomewhatestablishedforatraditionalaircraftdesign,thepressuredistributionacrossthewingmaybespecied.Thestructureofthewingmaythenbedesignedtoensurethatthewinghassufcienttorsionalandbendingstiffness.Aircraftwingsareusuallyconstructedofassemblieswheredifferentelementsaredesignedtocarryprimarilyonetypeofload.Thistypeofstructuraldesignresultsin 45

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wingboxeswith1mainsparsdesignedtoresisttheaerodynamicbendingloadswhileribspreventthesparfrombuckling.Wingskinsandsparwebsaredesignedtohandlethetorsionalandshearloadings. Thestandardequationsofelasticityforstressandstrainmaybeusedfordesignwhenusingisotropicmaterials,althoughorthotropicoranisotropicmaterialsmayaddsubstantialdesignfreedomandcomplexitytothedesignprocess[ 229 ].Thesimple(Bernoulli-Euler)beamequationorTimoshenkobeamtheoryforslenderthinwalledbeamsmaybeusedtoanalyzebending,whiletorsioncanbediscussedwiththeconceptsoftorqueandshearow[ 229 ].Whilethesesolidmechanicsapproachesarecommonlyusedforconceptualdesigns,aniteelementanalysis(FEA)isoftenneededtoanalyzethecomplexwingassembliesforaircraft. Strengthandstiffnesstoweightratiosaswellastheabilitytotailormaterialpropertiesareimportantfactorsinmaterialselectionsincewingweightmustbeminimized.Inrecentyearsthishasledtoanincreaseduseofcompositematerials,suchascarbonberwithapolymermatrix,inthewingboxassemblyandwingskins.Thehighperformanceandabilitytotailorthesematerialshasmotivatedtheiruse,whiletheiranisotropicbehaviorhasrequiredthedesignerstocarefullytailorthestructuraldeformation.Corrosionresistance,fatiguelife,anddurabilityarealsoimportantconsiderations.Aluminum,steelandtitaniumalloysdesignedforstrengthandlongevityarestillcommonlyusedforlargeaircraft.Forsmalleraircraftandmicroairvehicles,balsa,maple,spruceandplywoodarecommonalongwithvariousplastics,foams.Monokote,wood,nylon,siliconeandlatexarecommonlyusedforwingskinsatthesesmallerscales.Materialsdesignedformacroscaleapplicationsfromthenanoscaleuparenotyetcommon,butshouldprovidesubstantialbenetsforstructuraldesignsinthefuture. Theweightofthewingcomponentswhicharemovedisvitallyimportantformorphingwingdesignssinceitstronglyinuenceshowpowerful,andthereforeheavy, 46

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theactuatorsneedtobe.Thereforefortraditionalmorphingstructureswheretheactuationisatasingleposition,thereisastrongincentivetotailorthewingstructuretothemorphingonacasebycasebasis.Theaerodynamicloadsatasinglepositioninthemorphingareusuallywellknownsothedesignisverysimilartotraditionalwingdesignwhichspeciesamaximumwingloadingconditionforeachpartofthewingandthenbuildsthestructurebasedonthoseloads.Thewingsmusthaveenoughrigiditytohandleaeroelasticloadsaswellastheeffectofdetrimentalaeroservoelasticforces,butthewingsuseanactuationmethodwhichrequireslowenergyinput. Thekinematicbehaviorofthemorphingwingisfacilitatedbythewingstructureandactuatorsmaybedistributedthroughoutthewing.Materialsincludingthermoplasticpolyurethanes,copolyesterelastomer,shapememorypolymerandwovenelastaneyarnswereinvestigatedbyKikutaforuseinmorphingwings[ 110 ].Strainbasedmorphingtechniquesincludingpre-stressedpiezoelectricmaterialshavebeenighttestedonahobbysizedaircraft[ 250 ].However,itshouldbenotedthatmorphingwithstrain-basedactuatorsmaynotprovideenoughshapechangeandthereforeperformancebenetstojustifytheirinclusioninamorphingwing,especiallyforwingswithlargewingloadings.Activematerialsincludingshapememoryalloyshavebeeninvestigatedformorphingbecausetheyallowtheactuatorstructurestocarryaload,similartotensegritystructureswhenformulatedtoprovidemovement[ 145 ].However,someresearchersdonotthinktensegritystructureswillbeabletoprovidesufcientlylargeactuationformorphingwingssincethestructurecanonlyexistatcertaincongurationsandmayevencollapseasthetensegritystructuregainsmobility[ 227 ].Inaddition,themathematicstondthepositionsthestructurewillnotcollapsearequitecomplex[ 264 ].Variablegeometrytrusses(VGT)changethelengthofeachlinktoprovidemovementandmaybeabettercandidateforamorphingstructuresincetheycansupportloadsatmorepositionsthantensegritystructures[ 264 ].A1-DOFVGThasbeendevelopedforapplicationonamorphingwingtominimizesomeoftheissues 47

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withdistributedactuators[ 227 ].Furtherdevelopmentofmorphingshouldrevealifthesetypesofstructureswillprovidesufcientactuationorifthewingstructureswillneedtobedesignedasmechanismswithmanydegrees-of-freedom. Wingexibilityhasbeenshowntobebenecialforapping[ 54 87 149 215 275 279 ],althoughtoomuchspanwiseexibilitymaybedetrimentaltothegenerationofthrust[ 87 ],whilechordwiseexibilityisgenerallybenecial[ 143 ].Substantialwingexibilityandresultantdeformationsareobservedinmanyinsects[ 55 56 273 ]wherewingrigiditymaybecontrolledbytheamountofbloodowingthroughveinsinthewing.Flatwingswherestiffcarbonberbattensreinforceexiblemembranesarecommonlyused,substantiallyincreasingthecomplexityofthestructuralanalysisthatisrequired[ 275 ].Piezoelectricmaterialsalsohavecomeunderincreasedscrutinyforusewithappingandmorphingwings[ 160 250 ]buttheiruseisoftenlimitedbyinducedstraintotheprimarywingstructure,highvoltagerequiredforadaptivematerialactuation,materialdurability,fabricationdifcultyandstrictlimitsonwingweight.Shapememoryalloyshavealsobeeninvestigated,butarelimitedbytheirrelativelyslowresponserates,hysteresisandnonlinearresponse. 3.2.1.3Structuraldynamics Inadditiontohavingsufcientstructuralstrength,thestructuraldynamicsofthewingcandetrimentallyaffectthevehiclesbehavior,insomecasesenoughtocausethevehicletofailcatastrophically.Structuraldynamiceffectsareusuallydescribedbymodeshapesatsomeresonantfrequencies,butmayalsoincludearesponseduetoinitialconditions,aforcedresponseinthetimedomainandafrequencyresponseprole[ 93 ].Ifaforcingoccursattheresonantfrequencyofthestructure,thestructurewillvibratewithsomedenedshapeatasubstantiallylargeramplitudethanatsimilarfrequencies.Forabeam,thereisusuallyaprimarybendingandtorsionmode,followedbybendingandtorsionmodeswithloweramplitudesathigherfrequencies. 48

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TheRitzandGalerkinmethodsoraniteelementanalysisofthestructuremaybeperformedtondaninitialestimateofthestructuralmodeshapesandcorrespondingfrequencies[ 93 ].However,theforcesthatthewingwillexperiencearenotyetknown.Groundvibrationtestingmayidentifythefrequency,dampinganddynamicresponsecharacteristicsofthestructureaswellasthemodeshapesatcertainfrequenciesbyshakingthestructureatsomeinputfrequencyandmeasuringtheoutputresponse.Flighttestingmaybeusedtorenethestructuraldynamicmodelandensurethattheaerodynamicforcesandactuatorloadingsdonothavefrequencycomponentswhichwouldexciteundesirableordisastrouswingvibrations. Workonhowtoutilizethestructuraldynamicsofmorphingwingstobenetthevehicledesignisrare.Whenmembranestructuresareusedonsmalleraircraftthedetailedmembranedynamicsrequiremorecomplexapproaches.Forappingwingstructures,thestructuraldynamicsofarticial[ 133 ]andnaturalwingshavebeenanalyzed[ 165 ].Thefrequencydependentnatureofappingnecessitatesastructuraldynamicanalysisofthewingssincetheshapeandamplitudeoftheappingwillbehighlydependentonifthewingisbeingappedataresonantfrequency. 3.2.1.4Rigid-bodydynamics Traditionaldynamicsofanaircraftrelyon6-Degree-of-Freedommovementwhichisdenedbyavehiclepositionandorientation(heading)in3-dimensionalspace.Thedynamicsaredescribedsystematicallybywritingtheequationsofmotionreferencedtoaninertialreferenceframeandwilldescribetheposition,velocityandaccelerationofthevehiclewhichissubjectedtoasetofforces.Inpractice,theequationsareoftenlinearizedaboutthetrimconditionofsteadylevelightwhereliftequalsweightandthrustequalsdragassumingsmalldisturbancestofacilitateanalysis.Airowvelocity,vehiclevelocityandacceleration,andangularvelocitiesandaccelerationsfortheheadinganglesareimportanttoassessthevehicleperformance.Flightdynamicmodes 49

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suchasphugoid,short-period,roll,spiral,anddutchrollmustalsobeconsideredtoensurethattheaircraftremainsstablethroughoutthevehicle'sightenvelope[ 70 ]. Theposition,orientationandcorrespondingratesofwingmovementmustalsobeconsideredforthewingswhenanalyzingmorphingorappingwingedvehicles.Thecomplicationinvokedforthesesystemsissubstantial,sinceanadditionalcoordinatetransformationisrequiredtogobetweenthewingandbodyframeandwingexibilitymayfurthercomplicatematters.Themorphingandappingwillcausechangesinthecenterofgravitylocation,thusinuencingthevehicledynamics.Inaddition,ifthewingismovingquicklyorhasasignicantweight,winginertiamayplayasubstantialroleintheoveralldynamics.Itisworthnotingthatthetimeandfrequencyvaryingnatureofthedynamicswhenmorphingorappingmustbeconsideredinthedesignprocess.Sincealargenumberofassumptionsareusuallymadetokeepthedynamicsofthewingandvehiclerelativelysimplistic,theresultsoftenarenotbroadlyapplicableandaeroservoelasticcouplingsarecommonlyneglected.ManymodelsofmorphingandappingwingdynamicsexistdespitethecomplexitiesandwillbeexaminedinfurtherdetailinChapter 5 3.2.1.5Control Atthemostbasiclevel,acontrollerforatraditionalaircraftattemptstomatchthedynamicresponseofthevehiclewithsomedesireddynamicresponse.Anopenlooporclosedloopcontrolschememaybeused.Theopenloopcontrollersimplycommandstheactuatorstoperformsomeactionwhichpriorknowledgehaspredictedwillresultinadesireddynamicvehicleresponse.Theclosedloopcontrollerusessometypeofsensortomeasuresomesignalindicativeofthedynamicresponseofthevehicle,comparesthatsignaltothedesireddynamicresponseofthevehicle,andattemptstominimizetheerrorusingtheaircraft'sactuators.Linear,nonlinear,optimal,robust,andadaptiveapproachestocontrolsynthesisarecommonlyusedtoday.Theabilitytosenseandadapttothecurrentstateofthevehicleorsystemisanimportantadvantage 50

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gainedfromclosedloopcontrolapproaches.Forlinearsystems,proportional,integralandderivative(PID)gainsareusuallyappliedtotheerrorsignalwithaPIDcontroller.Typicallythecontrolgainsappliedtotheerrorsignalaremodied(scheduled)duringightsinceaircraftchangeweightbyasignicantamountthroughouttheirmissionasfuelburns.Controlisusuallyoneofthelastareasconsideredintraditionalaircraftdesign,althoughplacementofactuators,sensors,ightcomputersandbatteriesmaybecomeaprimarydesignrequirementasautonomoussystemsprioritizethesecomponentsinthefuture.Aircrafthandlingqualities,vehiclemaneuverabilityandactuatorperformanceefciencyaresubstantiallyaffectedbythecontroldesign. Commercialautopilotsforconventionalaircraftdesignshavebecomesowidespreadastobeavailabletohobbyists.However,thedynamicsandcontrolofmorphingandappingightissignicantlymorecomplexandthereforecontrollerimplementationsarestillrare.ControlofmorphingandappingightisinherentlyinterdisciplinaryandcurrentapproacheswillbeaddressedindetailinChapter 6 3.2.2CoupledSystemAnalysis 3.2.2.1Aero/hydroelasticity Coupleduid-elasticanalysisconsiderscouplingbetweentheuiddynamicloadingsandthestructuredeformationofthevehicleorwingcausedbytheuid-inducedloadings.Approachesmaybeorganizedintostaticapproaches,wheretheuidisassumedtobesteadyowandtheuiddynamicloadingsareconstantordynamicapproaches,wheretheuiddynamicloadingschangeovertime.Itisalsoimportanttonoteiflinearornonlinearanalysisisbeingused.Linearanalysiscombinesdecoupledanalysistechniqueswithsuperpositiontogetestimatesoftheaeroelasticforces.However,couplingbetweenforcescancreatenonlinearitieswhichrenderthisapproachinaccuratewhenlargestructuraldeectionsorlargechangesinaerodynamicforcesasafunctionoftimearepresent.Animportantreviewofaeroelasticanalysisfromthestart 51

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ofaviationhistorywasprovidedbyMukhopadyay[ 152 ],followedbyanotherreviewbyFriedmann[ 73 ]. Staticaeroelasticphenomenaincludedivergence,controlsurfacereversalandlossoflifteffectiveness.Divergenceoccurswhendeformationdependentaerodynamicforcesexceedtheelasticlimitsofthewingstructuralmaterial,possiblycausingalossoftheaircraft.Controlsurfacereversaloccurswhenthedeformationofthecontrolsurfacecausedbyaerodynamicloadspreventsthecontrolsurfacefromcontrollingtheaircraftasdesired.Lifteffectivenessreferstoadecreaseinthedesiredaerodynamicperformancecausedbyaerodynamicloadingandcorrespondingwingdeformation. Dynamicaero/hydroelasticityisassociatedwiththeinteractionbetweenaerodynamic,elasticandinertialforces.Dynamicaeroservoelasticphenomenaincludeutter/dynamicinstability[ 72 ]aswellasresponsestovariousdynamicloadingsmodiedbyaeroelasticeffectscausingbuzz,buffet,gustresponsesandotherdynamicresponses.Inmanyofthesecases,thenonlinearaerodynamicsinteractwiththestructuraldynamicstocauseproblems.Liftingsurfaceutteroccurswhentheliftingforceonthewingdrivesastableorunstablestructuraldynamicresponse.Stalluttercausedbyseparationoftheairowfromthewingasitvibratesmayalsooccur.Stalluttercausesthegoverningequationstobecomenonlinearandisoftenseeninturbojetcompressorsorhelicopterblades[ 93 ].Propellerwhirluttermayalsooccurwhenthewingstartstoutterduetoapropeller-nacelle-winginteraction.Complicationsmayalsobegeneratedbyexternalstoresonthevehicle,causingutterfromenginenacellesorweaponsmounts.Otherimportantexamplesareseenwhenwavesslamintoshipsandregionsofturbomachinesastheyencounterow. Hydroelasticproblemswererecentlyreviewedingeneral[ 48 ]andspecicallyfortheirapplicationstoships[ 90 206 ].Hydroelasticproblemsarelesscommonsincevehiclesgenerallymovethroughthewatermoreslowlyandstructuresareoftenmademorerigidtowithstandunderwaterpressures.However,aspringingresponsecaused 52

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byhullbending,aswellasawhippingresponsecausedbybowandsternslammingareimportant,especiallyforlargerships.Beamtheoryiscommonlyusedtoassessshiphulls,althoughnonlinearapproachesarealsobeingconsidered. Fortraditionalaircraftwherethestructureencounterssomeutterintheightenvelope,reduced-ordermodelsoftenrelyonmodaldescriptionsofthedynamics.Themodelsusuallyassumethatthestructuralresponsemaybespeciedintermsofthesuperpositionofafewmodeswithspeciedfrequenciesanddampingsontotherigidbodydynamics.Acontrollermaythenbeimplementedtoreducetheeffectofutteratthesefrequencies,possiblyimplementingnotchlterstodoso.Tostart,theoryforunsteadyaerodynamicsandtheoryorniteelementanalysisforthestructuraldynamicswillbecalculated.Thenvibrationtestsandwindtunneltestswillbeperformedonascalemodel.Next,groundvibrationtesting,auttermodel,andighttestingareusedtonalizetheutterbehaviorasthedesigndevelops[ 226 ].Computationalaeroelasticity(CAE)toolsarealsobecomingcommon,whereCFDtoolsarecoupledwithstructuraldynamicstools.WhileCAEhasbeenverysuccessful,manyofthetechniquesarestillbeingdeveloped. Bothmorphingandappingaircraftcouldexperienceaeroelasticeffectssimilartotraditionalaircraft.Itislikelythatsimilaranalysisasthatusedforutterandcontrolsurfacereversalorlossofeffectivenesswouldbeusefultoanalyzemorphingandappingforcircumstanceswheretheseeffectshadcontributedsubstantiallytowingdeectioncomparedtotherigid-bodywingmovement.However,bothclassesofvehicleswouldalmostcertainlyneedtobeanalyzedwithaeroservoelastictechniquessincetheircontrolactuationissotightlycoupledwiththeuiddynamics,exibility,andcontroldynamicspresentforthesewings.Jae-Sunghasanalyzedtheaerodynamicandstaticaeroelasticcharacteristicsforavariable-spanmorphingwing[ 20 ]whileDeBreukeraddressedsweepmorphing[ 42 ]. 53

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3.2.2.2Servoelasticity Servoelasticproblemscombineissuesfoundincontrolsystemsandexiblestructures.Roboticssystemswithexibilityinthestructureoractuatorscommonlyrequireservoelasticanalysis.Theliteratureisquiterobustforexibleroboticsincludingarecenttextbook[ 186 ]andliteraturereview[ 68 ].Thetechniquesusedforservoelasticsystemsmayhelpgiveinsightformorphingandappingvehiclesiftheaerodynamicsarelinearandsteady(i.e.thewingsmovedslowlyanddidnotmovesignicantly).Theeldmayalsobeusefultogaininsightintotheinuenceofjointandlinkexibility.However,theimportanceoftheuiddynamicswouldnecessitatetheaerodynamicforcestobesuperimposedinsomerespect.Thereforetheapplicabilityofthesetechniquesmaybelimited. 3.2.2.3Aero/hydroservoelasticity Aeroservoelasic(ASE)effectsresultfromtheinteractionbetweentheuiddynamics,structuraldynamicsandcontroldynamics.Commonaeroservoelasticphenomenaincludelimitcycleoscillations(LCO)andgustresponseproblems.Limitcycleoscillationsaresustainedvibrationswithlimitedamplitudewhichmaybecausedbynonlinearitiesintheaerodynamics,structureorvehiclecontrolsystem,especiallytheappearanceanddisappearanceofshockwaves,vorticesandseparatedowregions[ 200 ].BaldelliattemptedtounderstandthenonlinearitiesforLCO'sincludingLCOonsetandoscillationamplitude[ 22 ].Flutterhasoftenlimitedhighperformanceaircraftandaeroservoelasticapproachestotailorthewingstructureandcontroleffectors[ 243 261 ]insteadofaddingweighttoincreasethewingstiffness[ 187 ]havebeensuccessful.Suppressinginteractionsbetweenthecontrolsystemandstructuralmodes[ 171 ]hasalsobeeninvestigated.Inaddition,theaccelerometersandgyroscopesintheaircraftbodymaydetecttheaircraftmotionandthemotionintheexiblemodes,therebyfeedingbackincorrectinformationintotheightcontrol 54

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system[ 274 ].Notchltersmayhelpcorrecttheseinaccuraciesbydecreasingthesystemresponseatcertainfrequencieswherewingexibilityisknowntobeaproblem. AeroservoelasticeffectshavebeenobservedonsystemsincludingtheB-52E,YF-16,F/A-18,X-29AandActiveFlexibleWing[ 73 ].TheF-18HighAlphaResearchVehiclewiththrustvectoring,NationalAerospaceplane,B-52Pegasus,NationalAerospacePlane,SR-71HypersonicLaunchVehicleandHighSpeedCivilTransportalsohaveneededaeroservoelasticanalyses[ 38 ].TheF-16andF-18experienceddynamicinteractionsbetweentheiraeroelasticeffectsandthecontrolsystems,whiletheX-29wouldhavebeenunstablethroughoutasignicantportionofthedesignightenvelopeifcontroltechniqueshadnotbeenabletoaccountforaeroservoelasticeffects.TheDronesforAerodynamicandStructuralTesting(DAST)programusedtheFirebeedronedesignatedARW-1andARW-2toresearchutter[ 57 154 ].Aeroservoelasticanalysishasfacilitatedtheexpansionintothedesignspacemadepossiblebyunmannedaerialvehicles(UAVs),whereithasbeenappliedtomicroair-vehicles(MAVs)withhighlyexiblewings[ 3 ]andvehicleswithhighaspectratiowings[ 134 172 216 ].Adaptiveorsmartmaterialsincludingshapememoryalloys(SMA),piezoelectricmaterials,andshapememoryorelectro-activepolymersarealsounderintenseresearchscrutinyforvehiclesutilizingactivematerialsforcontrolsurfaces[ 28 81 111 140 ]andformorphingsystems[ 220 ].Aeroservoelasticeffectsonhypersonicvehicleshavealsobeeninvestigated[ 161 ].Thesenewapplicationsfurthermotivatedevelopmentofaeroservoelasticmodelingandcontrol[ 156 ]. Aplethoraofanalysismethodsareavailableforaero-andhydroservoelasticsystemssincedesignersmixvariousdelitymodelsforeachanalysisarea.Modelsofaeroservoelasticity(ASE)areoftengeneratedusinghigh-delitynonlinearcomputationalapproaches,butlinearandreduced-ordermodelshavealsobeenappliedsuccessfully.Systemidenticationtechniqueshavealsobeenappliedtotmodelstoexperimentaldata. 55

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AeroservoelasticeffectsmaybeaddressedbythelinearframeworksuchastheAutomatedStructuralOptimizationSystem(ASTROS),buttheseapproachesarelimitedtoapproachingproblemswherecontroldeectionsaresmallandgeometricaldiscontinuitiesfromthecontrolsystemdonotgenerateaerodynamicnonlinearities[ 200 ].Correctionfactorsformatchingpressuresordownwasheintheunsteadyaerodynamics,airfoilsectionpropertiesforliftandmomentcoefcients,andtotalforces,momentsandpressuresforagivenangleofattackorcontrolsurfacedeectionhavebeenexamined[ 266 ].Rationalfunctionshavebeenusedtoapproximatethereduced-frequencydependentunsteadyaerodynamicforcecoefcientswithfrequencyorLaplacetransformsincludingtheInteractionofStructures,AerodynamicsandControls(ISAC)code[ 5 6 ].Thiscodeindicatessuccessfullysuppressingutterusingadynamicpressurerootlocusplotandpreventingthesystempolesfrommovingintotherighthandplaneatahigherdynamicpressureastheywouldnormally[ 226 ].Theseapproachesdoincreasethesizeofthestatevector,indicatingthebalancemodelersmustconsiderbetweenmodelingaccuracyandlimitingthenumberofmodelcoefcients.Anactivelycontrolled,full-spanaeroelasticwindtunnelmodelofanadvancedtaillessghtermadefortheActiveFlexibleWing(AFW)programwhichlinkedtheISACcodewiththedoubletlatticeunsteadyaerodynamicmethodwhileamodiedWoodwardcodewasusedtondtheunsteadyaerodynamicsforsupersonicconditions.TheBenchmarkActiveControlsTechnology(BACT)programalsoinvestigatedmodelingandcontroldesignforawindtunnelmodelwithtransonicutter,shockinducedinstabilitiesanddynamicvortex-structureinteractionformodelvalidation[ 226 ].Severalstudieshaveintroducedcontrolmodelingtolineardoublet-latticecodesthatarealreadyusedforutterprediction.TheseincludestheAnalogandDigitalAeroservoelasticityMethod(ADAM)[ 162 ],FlexibleAircraftModelingUsingStateSpace(FAMUSS)[ 174 ]andStructuralAnalysisRoutines(STARS)[ 84 ]computerprograms.Theseprogramsusetheleastsquares(LS),matrixPade(MP)andminimumstate(MS)methods.When 56

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combinedwithenforcingorrelaxingequalityconstraints,thesemethodshavebeenincludedintheISACcode[ 38 ].Thesedoublet-latticebasedapproachesgeneratelinearfrequency-responsemodels;however,theyrequireadetailedtheoreticalmodelofthesystemphysics.Matchedltertheoryandrandomprocesstheoryhavealsocontributedtoassessingaeroservoelasticgustproblems[ 161 ].Anopen-sourcecodeASWingisalsoavailable[ 65 ]whichhasbeencomparedwithNASTRANresults[ 127 ] Nonlinearaeroservoelasticapproachesarealsocommonsincethetransonicregimebringsnonlinearaerodynamicsandstructuraldeformationsarenotalwayssufcientlysmall.Fortransonicconditions,theComputationalAeroelasticityProgram-TransonicSmallDisturbance(CAP-TSD)codewasusedfortheAFWprogram.CFL3Dandnonlinearmatchedltertheoryhavealsobeenused[ 226 ].ZonaTechnology'sZAEROcodealsohasmatchednumericalpredictionsandexperimentaluttertests[ 38 ].Wagner'sfunctiongavethestepresponseinangleofattackwhileTheodorsen'sfunctionanalyzedsinusoidalresponses.However,nonlinearaerodynamicsmethodsrequireCFDanalysisandquicklybecomecomputationallyexpensive.Volterra,neuralnetworksandradialbasisfunctionshavebeeninvestigatedtohelpspeedtheanalysis[ 226 ]. Reducedordermodelinghasbeenshowntobebenecialtocharacterizeandmodelmanyofthecomplexdynamicsinaeroservoelasticsystems.Inthesimplestform,areducedordermodeldenesabasissuchthatthebehaviorofthesystemmaybedescribedasacombinationofcomponentsofthatbasis.Thisapproachisusedinlinearcontroltheoryandstructuraldynamicsthroughmodalanalysis.Theentiresystemmaybemodeled,orreduced-ordermodelsforeachtypeofdecoupledanalysismaybecombinedtoproducethenalresult.Theminimumstateapproachmaybesufcientinsomecircumstances.AquickreferenceofcommonlyusedreducedordermodelsaresummarizedinTable 3-3 [ 226 ].Bothlinearandnonlinearkernelsaregeneratedfromaxednumberofnumericalsimulationsofthesystemtoagenericimpulseandarethensuperimposedwithconvolutiontomodelthesystemresponsetoanarbitraryinput[ 200 ]. 57

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Waveletshavebeenshowntobeusefultoestimatemodalparametersforreduced-ordermodelsofaeroservoelasticsystems[ 120 ]. Identicationofcomplexandreduced-orderaeroservoelasticmodelsfromightdatahasalsobeeninvestigated.Volterrabasedreduced-ordermodels(ROMs)arecommonlyidentiedtoformaeroelasticmodels,eveninthepresenceofaerodynamicandstructuralnonlinearities[ 153 ],wherethehigher-orderkernelsrepresenttheamountofnonlinearityinthesystem[ 226 ].Ablock-orientedapproach[ 23 ]andmatch-pointapproach[ 21 ]wereformulatedthatconsideredcertainnonlinearitiesintheaeroservoelasticdynamics.Anapproachtoestimatemodelsandassociateduncertaintywasformulatedusingarobustmini-maxschemethatevaluatedwavelet-basedmodalparameters[ 40 41 ]andtopredictlimitcycles[ 121 ]. Recenthydroservoelasticanalysishasbeenusefulforoatingwindturbines[ 103 ]andappingairfoils[ 47 ].However,theareaissubstantiallylessresearched. Aeroservoelasticanalysisformorphingandappingsystemsbecomeshighlycomplex,especiallywhenlargerigidbodymovementsandstructuraldeformationswhichmaybeinvolved.DetailsandcommonapproachesforsystemidenticationandmodelpredictionparticularlyapplicabletomorphingandappingwillbeaddressedinmoredetailinChapter 5 3.3ExperimentalApproachesforAnalysisofAeroservoelasticSystems Experimentalresultsallowtheresearchertocaptureallofthecouplingthatmayoccurduetocoupleduiddynamicandservoelasticeffectswithoutneedingtomathematicallymodelallofthecomplexitiescausedbycoupling.Experimentalresearchislimitedinthesensethattheexperimentscannotbeguaranteedtohaveobservedalltheeffectsthatareofthemostimportanceandexperimentsmayprovetoodangeroustobeappliedforhuman-sizedvehicles. Manymeasurementmethodsexistthatmightbeusedtocharacterizetheaeroservoelasticnatureoftheexiblemorphingorappingwingsystemandtherebyprovideabasisfor 58

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modelidenticationandcontroldesign.SeveralarelistedinTable 3-4 withtheiruniqueadvantagesanddisadvantages. 3.3.1FluidDynamics-ParticleImageVelocimetry Manyexperimentaltechniquesareavailable,butparticleimagevelocimetry(PIV)providesquantitativeresultsbyseedingtheowwithparticles,illuminatingitwithastronglightandimaginglightscatteredbytheparticlesastheymove[ 44 ].Assuch,PIVisusefulasanalternativetowell-establishedhot-wireandlaseranemometryandthetechniquehasbeenreviewedmanytimes[ 7 44 ].PIVhasbeenusedmanytimestoanalyzetheaerodynamicsofexiblewingsforMAVs[ 95 ],morphingwings[ 106 ]andappingwings[ 210 221 279 ]. 3.3.2SolidMechanics/Elasticity-MechanicalTesting Whileexperimentaltechniquesmaybehelpful,formostcasesatthemacroscale,computationaltoolsarenowsufcienttoanalyzethedeformationwhenloadingsareknownforagivenstructure.Somestructuresarestillcomplexenoughthatmodelingbecomesextremelydifcultduetoanisotropicmaterialproperties,compositematerialdesignsandbondingirregularities.Measuringthedeformationofastructureunderapointloadordistributedloadwithstraingaugesorsomeothermeasurementtechniqueisrelativelystraightforward.Thisapproachhasbeenusedforexible,morphingandappingwings[ 279 ]. 3.3.3StructuralDynamics-LaserDopplerVibrometry Modalanalysisiscommonlyusedtoassessthestructuraldynamicsofasystemandisgenerallyarststepinpredictingandworkingtowardcontrollingvibrationsoranalyzingthestructuraldynamicsofasystem.Inaddition,modalanalysismaybeusedtomeasureastructure'svibrationpropertiesinordertocomparethemwithatheoreticalorniteelementmodel,toproduceamathematicalmodelofacomponentwhenitisdesiredtointegrateintoapre-existingstructure,topredicttheeffectsofchangesintheoriginalstructure,ortodeterminethedynamicforcesormaterialpropertiesina 59

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structure[ 71 ].Thethreemainissuesforanyexperimentalmodalanalysisinclude(i)measurementofthefrequencyresponsefunction,(ii)excitationmethodsand(iii)modalparameterestimationthroughcurvetting[ 201 ]. LaserDopplerVibrometry(LDV)isatechniquethatisoftenusedtoassessthemodalcharacteristicsofasystem[ 223 ].TheLDVsystemusingasinglebeamlasertoassessthemodalcharacteristicsofasystemwasdescribedin1992[ 222 ].TheLDVsystemshinesalaserontothestructureasitisvibratingundersomeexcitationandmeasuresthescatteringofthelightreectedbackbythestructure.ThephaseshiftofthereectedlightwavecausedbytheDopplereffectisproportionaltotheobject'svelocityandthebeamwavelengthandthereforemaybeusedtogeneratethefrequencyresponseofthestructure[ 9 ].LDVhasbeenusedonstructuresaslargeasbridges[ 79 ]tostructuresassmallascarbonnanotubes[ 30 ]. 3.3.4DynamicsandControl-FlappingMechanism Theinputsignaltotheappingmechanismmaybemeasuredtondtheinputsignalwhileallmechanismeffectsmaybeobservedbytrackingthemotionwheretheactuatormeetsthewing.Asimplesinusoidalsignalmaybedesired,butloosejointsorgearsandexibilityinthemechanismcomponentsmakeitdifculttocommandpurelysinusoidalmotion.Anexperimentallybasedmodeloftheactuatormaybeneededforsuccessfulcontrolsynthesisforbothmorphingandappingwingswhenactuatornonlinearitiesarepresentandtheshapeofthewingneartheactuatorisimportant. 3.3.5DigitalImageCorrelation:HoveringTests DigitalImageCorrelation(DIC)isanon-contacttechniquethatwasoriginallydevelopedattheUniversityofSouthCarolina[ 230 ]todeterminedeections,deformations,andstrainsofastructure.DICiswelldescribedandestablishedtechniqueusinginexperimentalmechanics[ 51 231 ].DIChasbeenusedpreviouslytomodelthedeectionsofexiblewings[ 224 ]andappingwings[ 277 ]andshowssignicantadvantagescomparedwithprevioustechniquesusedtoinvestigateappingwings 60

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includingstripsmodeling,projectedlaserline,andphotogrammetry[ 254 ].DICistheonlyexperimentaltechniqueexaminedwhichhastheabilitytoanalyzethecoupledeffectspresentintheaeroservoelasticwingmovement. DICexaminesmultipleimagesforrelationshipsbetweenregionsinthoseimagesandthendescribesthatrelationship.Typically,spatialandtemporalrelationshipsofstructuresaredeterminedwithDIC.Arandomspecklingpatternisappliedtothestructure.Twoormorecamerasarecalibratedandthelocationofthestaticstructureisdeterminedwithstereo-triangulationtechniques.Thecamerasthenusetemporaltrackingtoattempttocorrelatearegionofspecklesbetweensubsequentframes,specifyingacross-correlationcoefcientsuchastheoneseeninEquation 3 wherethein-planedisplacementeld(u,v)isspeciedwithf(x,y)beingthegraylevelatcoordinatex,yfortherstimageandg(^x,^y)thegraylevelvalueatcoordinate(^x,^y)forthesubsequentimage.Thefandgaretheaveragegrayscalevaluesfortheregionand(u,v)arethedisplacementcomponentsforthesubsetcentersinthe(x,y)directions[ 166 ]. C(u,v)=mPi=1mPj=1[f(xi,yj))]TJ /F8 7.97 Tf 6.62 1.77 Td[(f][g(^xi,^yj))]TJ /F8 7.97 Tf 6.75 0 Td[(g] s mPi=1mPj=1[f(xi,yj))]TJ /F8 7.97 Tf 6.62 1.77 Td[(f]2s mPi=1mPj=1[g(^xi,^yj))]TJ /F8 7.97 Tf 6.75 0 Td[(g]2^x=x+u+@u @xdx+@u @ydy^y=y+v+@v @xdx+@v @ydy(3) Acalibrationpoint(region)isspeciedtoinitiatethecorrelationanalysis.ThispointmustbetrackedacrossframesfortheDICsystemtodescribedeectionswithinreasonableerrorbounds.TheDICsystemusesthecorrelationprocesstosamplethedisplacementofmanypointsonthestructuresimultaneouslyinthreedimensions(whereapointrepresentsaregionofspeckles).Asignalmaybeformedbytrackingthedisplacementofanyoftheseregions.Inadditiontothedeections,thedeformationsofthestructuremaybeobtainedbysubtractingtherigid-bodymotionsofthestructure.Thesesignalsmaythenbeanalyzedwithstandardsignalprocessingmethods.The 61

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processingmethodsusedbyDIChavebeenestimatedtoenablesub-pixelresolutionfordeections[ 199 225 ]whentheDICsystemisabletocorrelatesubsequentregionsofspeckles.Forthewingsexaminedinthispaper,theDICresolutionenabledsub-millimetermeasurementerrors.Therefore,theDICtechniqueisusefultoanalyzethecoupledeffectsofmorphingandappingwithorwithoutanappliedfreestreamow.DICresultsformorphingandappingsystemswillbeexaminedinmoredetailinChapters 4 7 ,and 8 62

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Table3-1. ClassesofCoupledSystemsinContinuumMechanics AnalysisTypeDescription(AnalysisUsedfor:) UnsteadyFluidDynamicsTime-VaryingMovementofFluid DynamicFluidelasticityTime-VaryingMovementofSystemwithCoupledFluid-Structure Time-VaryingRigid-BodyDynamicsTime-VaryingMovementofRigidStructure FluidDynamicsFluidMovingatSteadyState FluidStaticsStationaryFluidatSteadyState FluidMechanicsMovingandStationaryFluid RheologyPlasticFlowofMatterinLiquidFormorSoftSolidPlasticFlow StaticFluidelasticityCoupledFluid-StructureMovementUsingSteadyStateAerodynamics SolidMechanicsElastic(Flexible)SolidsUnderLoad(CharacterizedbyStresses,StrainsandtheRelationshipBetweenThem) StructuralDynamicsBehaviorofStructuresUnderDynamicLoading StaticsBehaviorofStructuresinStaticEquilibriumUnderStaticLoad ViscosityFluidsUnderLoad(InternalResistancetoFlowCharacterizedbyShearandTensileStress) DynamicFluidplasticityPlasticFlowofMatterUnderDynamicLoading ViscoelasticityMaterialswithBothElasticandPlasticCharacteristics PlasticitySolidsUndergoingPermanentDeformation ControlSteadyExternalForcingBehaviors Time-VaryingControlUnsteadyExternalForcingBehaviors FlowControlSteadyExternalForcingBehaviorbyMovingFluid Aero/HydroservoelasticitySystemswithCoupledFluid,ControlandFlexibleDynamics AdaptiveMaterialsSystemsWhereMaterialMayExertaTime-VaryingForceontheSystem ServoelasticitySystemswithCoupledControlandFlexibleDynamics TemperatureInternalEnergyofMaterial EquilibriumThermodynamicsApproximatelySteadyStateSystems(ConstantEnergy) Non-EquilibriumThermodynamicsNon-steadyStateSystems(ChangingEnergy) Aero/hydrothermoelasticitySystemsWhereFluidFlow,HeatingandStructureFlexibilityMustBeConsidered StatisticalThermodynamicsThermodynamicBehaviorinTermsofBehaviorofMoleculesandAtoms ThermoelasticitySystemswithCoupledMechanicalandThermodynamicProperties Aero/hydrothermoservoelasticitySystemswithCoupledFluidFlow,Thermodynamic,ElasticityandControlDynamics 63

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Table3-2. DesignVariablesforVehicles CategoryRelatedDesignVariables AerodynamicIncomingFlowOrientation,Chordwise/SpanwisePressureDistribution,LeadingEdge,TrailingEdge,andWingTipVortexLocationandStrength,ShockWaveLocationandStrength ControlControllerStability,PerformanceandTypeofFeedback,ActuatorTypeandControlAuthority(ControlSurfacesandEnginePerformance),SensorTypeandPerformance,FlightMechanics,ActuatorandSensorSpecicVariables(ex:forFlowControlorAdaptiveMaterials) DynamicStaticandDynamicWingAngles(Dihedral,Sweep,andTwist(Feathering)AnglesandAngleofAttack),Amplitude,Frequency,DampingofWingDihedral SolidMechanicsPartWeightandLoading,MaterialProperties,ChordwiseandSpanwiseStiffness,BendingandTorsionalStrength,Inertia,StressandStrainDistributions,FatiguePerformance StructuralDynamicModalFrequencies,ModalDamping,TimeandFrequencyResponse ThermodynamicVehicleTemperatureDistributionandHeating,ThermalProtectionTypeandPerformance Table3-3. Reduced-OrderModelOverviewforAeroservoelasticSystems DecoupledAnalysisCategoryROMTypeResearchers FluidDynamicsRational/PadeApproximationJones,Theodorsen ProperOrthogonalDecompositionLumley,Holmes Multiresolution/WaveletsFarge,Wickerhauser,Lind StructuralDynamicsModalSynthesisMeirovitch,Bisplinghoff ComponentModeSynthesisCraig KrylovMethodsWilson,Craig ControlTheoryHankelApproximationPartington BalancedRealizationMoore ComponentCostAnalysis,q-MarkovCoverSkelton OptimalProjectionBernstein 64

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Table3-4. MeasurementsUsefulforModelIdentication ParameterAdvantagesDisadvantagesTestingMethod DeectionsCompletemodel:includesallaeroservoelasticcouplingincludinginertialeffectsandaeroelasticeffects,easytomeasure,leastimportantcontributionslteredoutSmallereffectsonstructurearenotmodeledthoughmaysubstantiallyaffectperformanceDIC,Photogram-metry DeformationsMayallowtoisolateimportantaeroelasticeffectsandthereforebetteraccountforaerodynamicphenomenathatareoccurringDifculttoisolate(relyonrigidwingassumption),maynotsufcientlycaptureaerodynamiceffects,maybeverysmallDIC StructuralDynamicsDeectionsEasytomeasureDonotaccountforlargeamplitudeoscillationsinappingnormostoftheaeroelasticeffectsgeneratedforlargeamplitudeappingDIC,LDV AerodynamiceffectsCapturephenomenamostimportantforvehicleperformanceUnsteadyphenomenamakemeasurementconsistencyextremelyquestionableandcomputationallydifcultPIV 65

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CHAPTER4SENSORSANDSIGNALPROCESSING 4.1BackgoundforSignalProcessingandSignalClassications Ameansofobtaininginformationabouthowthewingismovingisnecessarybeforeasystemidenticationorcontrolsynthesismaybeattempted.Sensorswillbeneededatvariouslocationsonthewingtodeterminethewingmovementformorphingorappingwings.Thereforesignalsanalysisprovidesthemeansbywhichthecharacteristicsofasystemareunderstood.Theaeroservoelasticnatureoftheresponseofamorphingstructurehastimeandfrequencydependentphenomenawhichmustbeunderstood[ 133 ]. Digitalandanalogsignalclassicationsdescribethesignalaseitherdiscreteorcontinuous.Asignalwhichcanbedescribedbyanexplicitmathematicalrelationshipisdeterministic.Ifprobabilitystatementsorstatisticalmomentsmustbeusedthenthesignalisnon-deterministic.Bothdeterministicandnon-deterministicsignalsarefoundinappingandmorphingsystems.FurtherclassicationswhichwillbediscussedaresummarizedbyFigure 4-1 [ 29 ]. Figure4-1. SignalClassications[ 29 ] 4.1.1DeterministicSignals Aperiodicsignalrepeatsitselfoverregularperiodsandissinusoidalorcomplexperiodic.Asinusoidalsignalwillhaveamagnitude,A,frequency,!andphase,such 66

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thatX(t)=Asin(2!t+).AsignalisperiodicifX(t)=X(t+nT)whereTisarealnumbercalledtheperiod,nisaninteger,tisthetimeandXisthesignal[ 183 ].Thefrequencyofthesignalisequalto1=T.Complexperiodicsignalsarethosewhichhaveaperiod,butcannotbedescribedasafunctionoftime.Anon-periodicsignalmaybedescribedasalmostperiodicortransient.Signalswhicharealmostperiodicmaybereducedtoasumofsinusoids.Theseareusuallynotperiodic,butmaybeperiodicifratiosofallpossiblepairsoffrequenciesarerationalnumbers,resultinginafundamentalperiod.Transientdataaresimplyexpressedasafunctionoftimebutarenon-periodic[ 29 ]. 4.1.2Non-deterministicSignals Non-deterministic,randomorstochasticsignalsmaybecategorizedbythenatureoftheirstatisticalmoments.Signalsmaybestationary,wherethejointprobabilitydistributionofthevariableistimeandspaceinvariant,somomentslikemeanandvarianceareconstant,ornon-stationary,wherethemomentsvaryintimeorspace.Withinstationarysignals,thesignalmaybespeciedasergodicornon-ergodic.AnergodicsignalhasatimeaveragewhichisequaltotheensembleaveragewherethetimeandensembleaveragesaredenedbyEquations 4 and 4 [ 29 ]. x(t)=limT!11 TZT=2)]TJ /F5 7.97 Tf 6.59 0 Td[(T=2x(t)dt(4) =limN!11 TNXi=1xi(t1)(4) 4.2SignalAnalysis Althoughsignalprocessingmaydealwithanalysis(breakingupthesignal)andsynthesis(puttingthesignalbacktogether),themainpurposeofsignalprocessinginthisworkistounderstandthephenomenapresentinaeroservoelasticsignals.Thereforethefocuswillbeonanalysiswhichbreaksthesignalintosimplercomponents. 67

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4.2.1Time-DomainAnalysis Amultitudeoftoolsexistwhichmaybeusefultoobtain,characterize,andanalyzeatime-responsesignal.Thesetoolsincludeasensortomeasurethedata,anoscilloscopetoviewthesignal,andacomputertorecordandanalyzethedata.Whenmeasuringdata,thesamplingratemustbesettoensurethatthemeasurementisnotdistorted.TheNyquistsamplingcriterionmustbesatisedtopreventaliasingthedata.Inotherwords,measurementsmustbetakenatleasttwiceasfastasthehighestfrequencyofinteresttobeabletosynthesizetheoriginalsignalcorrectly.Inpractice,thesignalmustbesampledmuchfastertoincreasetheaccuracywithwhichtheoriginalsignalcanbereconstructedandtoallowforlteringofhighfrequencycontent. 4.2.2Spectral(Frequency)DomainAnalysis Atime-domainsignalmaybedecomposedintomultiplesinusoidalcomponentsatcertainfrequenciesasshownbyEquation 4 wherex(t)isthesignalasafunctionoftime,Ajisthemagnitudeofthesinusoidsand!jarethefrequencies. x(t)=JXj=1Ajei!jt(4) Ifsomeinformationaboutthetime-varyingnatureofthesignalisdesired,theLaplacetransformshowninEquation 4 maybeused.TheFouriertransformofthesignalasdenedbyEquation 4 willisolatethefrequencycontentofthesignal.Similarequationsaredenedfordigitalsignalsincludingthez-transforminEquation 4 anddiscretefouriertransform(DFT)inEquation 4 forcomparison[ 183 ].TheDFTisatechniquethatprovidestheuserwithfrequency-domaininformationforagivenperiodic,stationarysignal,transformingmeasurementsx(t)inthetimedomainintox(!)inthefrequencydomain.TheFastFourierTransform(FFT)usesacomputationalalgorithmtoaccomplishtheDFT,whichnormallyrequiresO(n2)operations,inO(nlogn)operationswherenisthenumberofsamplesinthesignalanalyzed. 68

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X(s)=Z1x(t)e)]TJ /F5 7.97 Tf 6.59 0 Td[(stdt(4) X(!)=Z1x(t)e)]TJ /F5 7.97 Tf 6.59 0 Td[(j!tdt(4) X(z)=1Xk=xkz)]TJ /F5 7.97 Tf 6.59 0 Td[(k(4) X(ej)=1Xk=xke)]TJ /F5 7.97 Tf 6.59 0 Td[(jk(4) Anarrowbandsignalwillhaveenergylocatedatfrequencieswithinasmallfrequencyrange,whileabroadband(orwideband)signalwillhaveenergylocatedatfrequenciesacrossabroadfrequencyrange.Itshouldbenotedthatcompositesignalsconsistingofsignalswithfrequencycomponentsthatareisolatedbutrangeacrossalargefrequencyrangemaymakesamplingdifcultandrequiremulti-ratesignalprocessingtechniques[ 183 ]. Numeroustechniquesareavailabletoenhanceunderstandingofthesignalandtoalterthecharacteristicsofagivensignal.Filtersincludinghighandlowpass,notchandbandpassltersarecommonlyusedtoremovecertainfrequenciesfromthesignal.Kalman,Chebyshev,andButterwortharemorespecializedlterswhicharecommonlyused.Filterperformanceistypicallydenedbythepassbandorstopbandripple(dB),thecornerorcut-offfrequency(Hz)andtheroll-off(dB/decade)[ 183 ]. Timeinformationislostwhenthesignalistransformedintothefrequencydomainunlessthesignalisstationary.Toobtaintimeandfrequencycontentsimultaneously,adifferentapproachisrequired[ 144 ]. 4.2.3Time-FrequencyAnalysis TheShort-timeFourierTransformandWaveletTransformareabletoobtaintimeandfrequencyinformationsimultaneously;however,thetimewindowandtherefore 69

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precisionisxedfortheShort-timeFourierTransform[ 144 ].Wavelettransformsallowtheuseoflongtimeintervalstoobtainlow-frequencyinformationandshorterregionstoobtainhigh-frequencyinformation[ 144 ].Wavelettransformstaketheformofeitheradiscretewavelettransform(DWT)oracontinuouswavelettransform(CWT).TheirusemayprovidefurtherinsightintodynamicalsystemsthanFouriertransformalone[ 96 ].Wavelettransformsrelateaninputsignaltoabasisfunctiondenedasthemotherwavelet,similartohowtheFFTrelatestheinputsignaltoasuperpositionofcosinetermswhenittransformsasignalfromthetimedomaintothefrequencydomain.However,waveletsareofniteandvariablelength,sowaveletanalysiscanidentifynearlyinstantaneousfrequencychangesinthesignalwhiletheFFTcannotbecauseitisrelatedtoaninnitetimesignal.Thispropertyofwaveletsallowsthewavelettransformtosimultaneouslydisplaythethreerelevantdimensions(time,frequency,andmagnitude)asseeninFigure 4-2 fora10Hzsinusoid.Thecombinedtime-frequencyinformationcanbeparticularlyusefulfortheanalysisofnonlinearandtime-varyingsystems[ 121 ].Althoughnotnecessaryforapuresinusoid,itisnormallyhelpfultolimitthescale(frequency)rangetobeabletocomparepeaksinthetimedomainforgivenfrequencies.Itmustbenotedthatallwaveletplotsdemonstraterelativemagnitudesofthetime-frequencycontentofthesignal. Waveletanalysisisabletogenerateatime-frequencyrepresentationofdatathatcapturesbothtime-domaincharacteristicsandfrequency-domaincharacteristics.Suchanalysisreliesonlocalizedcorrelationtoknownwaveformswithoutassumptionsonlinearityortime-invariantproperties.Thesetime-frequencymapshavebeeneffectivelyusedtoanalyzelimitcycleoscillations[ 121 ],nonlinearnormalmodes[ 107 ],neuraldynamics[ 248 ]andnonlinearoscillators[ 104 ].Waveletanalysisisusedforanalysisofthetime-varyingnatureofappingwingsignals. TheMorletwaveletseeninFigure 4-3 iscommonlyusedwhendealingwithdynamicsystems.Inparticular,nonlinearitiesandtime-varyingstructuraldynamics 70

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Figure4-2. 10HzSinusoidTimeHistory,FastFourierTransformandWaveletAnalysis(Frequencyvs.TimeandMagnitudevs.TimeViews) havebeeninvestigatedusingtheMorletwaveform[ 107 121 ].Theequationforthemotherwavelet,theMorletwaveletandthefrequencycorrespondingtothemagnitudeandscaleareseeninEquations 4 4 and 4 respectively[ 121 ]whereaisthesale,isthetimeincrementandfsisthesamplingfrequency. (t)=e)]TJ /F8 7.97 Tf 6.59 0 Td[((1=2)t2cos(!t)(4) a,(t)=(1=p a) ((t)]TJ /F4 11.955 Tf 11.95 0 Td[()=a)(4) f=(0.796fs)=(a)(4) 4.2.4ProbabilisticSignalsAnalysis Non-deterministiccomponentsofthesignalwillexistforanysensorchosentomonitorwing-shapechangesaswellasanysignalmeasuredintherealworld. 71

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Figure4-3. ExampleMorletMotherWavelet Characterizingthesesignalsisimportantformodelbasisidenticationandtoformulateanadequatecontrolstrategy.Non-deterministicsignalsdonothavecleartimeorfrequencydependentpatterns;thereforetheymustbeexaminedusingaprobabilisticapproach.Thesesignalsarecharacterizedbyaprobabilitydistributionfunction(PDF,P(x))orprobabilitydensityfunction(pdf,p(x))showninEquations 4 and 4 ,andrst-order(mean,,n=1),secondorder(variance,,n=2)andhigher-ordermomentscalculatedwithEquation 4 wherenistheintegerorderofthemoment.ThePDFisdenedbytheprobabilityassignedtoasetofpointsksatisfyingx(k)<=x.ThepdfisthederivativeofthePDFwhenthePDFiscontinuousandrepresentstheprobabilitythatfuturedatawillfallwithinsomeboundsxandx+x.Thedensityanddistributionfunctionscanbeusedtoevaluatenormality,detectmeasurementerrors,indicatenonlineareffectsandidentifyextremevalues[ 29 ].Themeanrepresentsthecentraltendencyoftherandomvariable,whilethevariancerepresentsthespreadofthedata.Themeanandvarianceareespeciallyusefulforunderstandingtheamountofuncertaintyinagiventypeofvariableorsignal.Thestandarddeviationandcoefcientofvariationarealsocommonlyusedtoexpressthespreadofthedata.Thestandarddeviationisp (x),whilethecoefcientofvariationisCOV==. P(x)=Probability[x(k)<=x](4) 72

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p(x)=limx!1[Probability[x
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isquantiedasthecrosscorrelationfunctiondenedinEquation 4 forstationarysignals. ThecorrelationcoefcientxydenedinEquation 4 lieswithin-1and1where0representsuncorrelatedrandomvariables.ThecorrelationcoefcientrequiresthecrosscorrelationcoefcientshowninEquation 4 tobecalculated.Evenifthevariablesareuncorrelated,itdoesnotimplythatthevariablesareindependent.Independentvariablesrequirethatp(x1,x2)=p(x1)p(x2). Rxy()=1 TZT0x(t)y(t+)dt(4) xy=Rxy() p Rxx(0)Ryy(0)(4) Principalcomponentsanalysis(PCA)isusedtoconvertasetofmeasurementstoasetofuncorrelatedprincipalcomponents.PCArotatesthesetaboutthemeantoaligntheaxeswiththeprincipalcomponents.PCAisusefulinbothsignalandimageprocessing,whereitmaybeusedtoidentifywhichofmultipleinputshavethemostimpactonthemeasuredoutput,anddimensionalityreductionforreducedordermodels,whereitmaybeusedtodetermineifanadditionaltermisbenecialtoincludeinthemodel.PCAmaybeperformedwithaSingularValueDecomposition(SVD)ofasetofmeasurementsX,whereXisamxnmatrixrepresentingthedatafrommexperimentswithninputvariablesresultinginX=WVTwhereWisthemxmeigenvectormatrixofXXT,isanmxnrectangulardiagonalmatrixandVTisnxn.Itishelpfultouseinputvariableswhichhavebeenscaledtohaveunitvariance. 4.3SensorSelectionandPlacement Sensorsmayneedtobedistributedacrossthewingtosensethewing-shapeoutputinresponsetotheinputsfromtheactuatorsbeforefullwing-shapecontrolcanbeachieved.Distributedsensorshavealsobeenusedforstructuralhealthmonitoringtomeasurefatiguelifeconsumedandoverloadhistory,vibrationandnoisesuppression, 74

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andimplementationsofsmartoradaptivestructuresonwings.Thereforethesensorscouldbeusedfortripleduty,helpingbuytheirwayontothewingdesignbyprovidingfeedbackduringwingfabrication,increasedightperformancefrommorphingshapecontrolandhealthmonitoringthroughoutoperation.Althoughmanyexamplesofsensorstomonitortheshapeofastructureexist,severalrepresentativeexamplesaresummarizedinTable 4-1 Numerousimplementationsofdifferenttypesofsensorshavebeenconsideredtomonitorwing-shapechangessincetheoptimalsensorarrangementwilldependonthetypeofwingmorphing.Determinationoftheneededtypeandnumberofsensorsandsensorplacementisimportanttominimizetheadditionalweightaddedtothewing.Generalapproachestosolvingwheresensorsshouldbeplacedoncompliantstructuresformaximumenergyefciencyhavebeenexaminedandappliedtoamorphingwing[ 244 ].Researchsuggeststhatthe75percentofthespanandleadingedgeareimportantlocationsforliftproducingvorticesonexibleappingwingsandthereforetheselocationsneedsomeformofsensor[ 76 ]. Researchintosensorsusedbyieshavesuggestedthatlargenumbersofcoarse-grainedsensorsmaygivesufcientinformationforhighperformancecontrol[ 284 ].Sensorsusedtomeasurewingdamage,pressuredistribution,airowangleandairspeeddistributedacrossthewingareincludedinTable 4-1 sincesensorfusiontechniqueswhichcombinedeformationandairloadingmayprovidemoreaccuratedatathanmeasurementofstructuraldeformationalone.Algorithmswhichuseneuralnetworks,fuzzylogic,andgeneticalgorithmstocombineinformationfrommanydifferenttypesofsensorshavebeenunderdevelopmentfortheJointStrikeFighter(JSF)[ 196 ]. Importantperformanceparametersforsensorsinclude:(1)Reliablility/durability,(2)Consistentresponse,(3)Lowcost,(4)HighMechanicalTolerances,(5)Abilitytohandletemperaturevariations,(6)Agingandenvironmentaleffects,(7)Sensitivityand(8)Electricalbias[ 163 ].Sensorminiaturizationandintegrationintotheoverall 75

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Table4-1. DistributedSensorSetupsforControlFeedbackforMorphingandFlappingWings Year/Refs.SensorTypeMorphingParameterImplementationComments 1996[ 114 ]AcousticEmissionStrain18in.rangeSimpleWingBox,3in.RangeComplexWingBox 1997[ 98 ]Piezoelectricsensors/actuatorsStrainTestedOnEuroghter 1998[ 213 ]FiberOpticsStrain 1999[ 113 ]WirelessPiezoelectricSensorsinCompositesStrain 2003[ 43 ]CompositeswithEmbeddedPiezoelectric(PZT)SensorsStrainAlsoUsefulforVibrationandNoiseSuppression 2004[ 157 ]HotWireAnemometeronStructurePressureDistribution 2005[ 195 258 ]ConstantVoltage/TemperatureAnemometerFlowAngle,AirspeedMayRequireExposedElectrode 2005[ 102 ]StrainActuatorsCamberHeavybutAccurate 2005[ 26 232 ]BraggOpticalFibers(Strain)CamberAccuracyMayNotBeSufcient(10%DeectionErrorisGood) 2007[ 8 ]WireNetwork(StrainGages)EmbeddedinCompositeWingStructure(ActiveFiberComposites)StrainObtainsLinear/AngularDeections,MapofStrainDistribution 2009[ 233 ]PiezoresistiveCantileverLocalAirPressureSmall,Accurate(AppliedtoButterySizeWings,Detects1Pachangeat100+HzFlapping) structurewhencombinedwithsensorfusiontechniquesmayjustifyadistributedsensingapproachwithcoarsesensorsmoreofteninthefuture.Itisworthnotingthatseveralcircuitsexistthatwillallowtheimplementationofwavelettechniquesinelectricalsystems.Thesecircuitsrelyonadiscretesetoflterstoformalterbank.Thereforea 76

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hardwareimplementationispossibleiftime-frequencyinformationisrequiredtoanalyzethewingresponse. 4.4SignalCharacteristics Bothmorphingandappingwingsarecharacterizedbycertaintypesofbehavior.Understandingthesebehaviorsbyobservingcharacteristicsignalsisnecessarybeforeproceedingtomodelingandcontrolefforts. 4.4.1MorphingSignals Signalsobtainedfrommorphingwingsmayberesponsestodiscretestepinputsfromtheactuatorswhichhavebeendistributedacrossthewing.Inthesecases,thesignalsshouldbeanalyzedintermsoftraditionaltransientstepresponsemetricssuchasdelaytime,risetime,peaktime,maximumovershootandsettlingtime.Theresponsetoasecond-ordersystemsuchasaspring-mass-damperprovidesthemostbasicexample,butamorphingsignalresponsewillbeaeroservoelasticforanon-rigidwing.Althoughthemorphingresponsemaybemodeledateachpointlocationbyaspring-mass-damper,themagnitudeoftheresponsetoamorphinginputwilldependonwherethesignalislocatedonthewing.Thedistancefromtheactuator,exibilityofthewing,andsizeoftheaerodynamicloadswillsubstantiallyaffectthenalresponse;therefore,afullaeroservoelasticmodelmayberequiredtopredictthewingresponse,especiallyifthemorphingiscontinuouslyactuatedintime. Researchincludingmeasurementsofmorphingwing-shapearerelativelyrare.Usuallytheresponsewouldbemeasuredatacertainlocationandtheeffectontheaircraftaerodynamicsbymorphingmaybemodeled[ 3 4 ],buttheeffectonthewing-shaperesponseisrarelyanalyzedindetailsincethewingsareusuallyexpectedtobehaveasrigidstructureswherethemovementiswelldenedkinematically.Aeroservoelasticmodelsofmorphingwithexiblewingsareusuallybuiltbycombiningtheresultsofseveraltypesofdecoupledanalysismethods[ 170 ]. 77

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4.4.2FlappingSignals Thedeectionsofawingwilldirectlyinuencetheightdynamicsofanyapping-wingmicroairvehicle.Signalsobtainedfrompracticallyimplementedappingsystemsareoftendescribedintermsofafewkinematicparameters.Thetemporalcharacteristicsoftheappingareequallyasimportantasthefrequencycharacteristics.Analysisofthequasi-steadydynamicsareinsufcienttocapturebothofthesetypesofcharacteristics[ 63 69 ].Flappingatfrequenciesaroundaresonancecanenhanceefciencywhilerequiringminimalenergybutwillintroducesignicantvariationsastheappingfrequencypassesthroughthatresonance[ 108 ]. Time-frequencyanalysiswouldbeimportantforthosehopingtocontroltheeffectsofresonance[ 18 101 ].Mostcurrentexperimentalandtheoreticalcontrolschemesonornithoptersrelyonchangesinappingfrequencyforcontrol[ 91 280 ].Thereforeacontrolschemeformulatedforaappingwingsystemcouldbenetfromtheabilitytoaccountfordynamicchangesintheappingfrequency.Insummary,whiletheassumptionofperiodicityisobviouslyhelpful,theabilitytoanalyzetime-variantandnon-periodicphenomenausingtime-frequencyanalysiswillhelpresearchersfurtherunderstandandultimatelydesignboththestructureandcontrolsforappingwingsystems. 4.4.2.1Signalcapture-digitalimagecorrelation DICdataisobtainedforaCapranmembranewingwithparallelcarbonberbattensandaluminumwingappingatvariousappingfrequenciesandappingamplitudesasdescribedbytheauthorpreviously[ 133 ].WingdataandreferencenumbersareincludedinChapter 8 .Representativesignalsmaybeformedbytrackingregionsofspecklesastheymovein3Dspace.Signalswerecapturedwithasamplingrateof50timestheappingfrequency,soaliasingshouldbeminimizedupto25timestheappingfrequency. 78

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4.4.2.2Periodicbehavior Out-of-planedeectionsforsteadyappingarehighlyperiodicacrossawiderangeofinputappingamplitudesandfrequenciesdespitethepresenceofaeroelasticcoupling.Figure 4-4 shows7subsequentappingcyclesofthewingtipoverlayedontopofeachothertodemonstratethisperiodicity.Variationsseenathigherfrequencyandappingamplitudesuchasthoseat35degand30HzareduetoissuesobtainingcorrelationwiththeDICsystem,notalackofperiodicityintheactualappingcycle.Theperiodmaybeidentiedusingtheautocorrelationandidentifyingthedistanceintimebetweenpeaksorcountingzerocrossingsintheoriginalsignal. Figure4-4. PeriodicNon-SinusoidalDeectionsofaFlexibleFlappingWing WingtipVerticalDisplacementsofWing-3(BattensParallel)at10,17,and35degFlappingAmplitudeand10,20,and30HzFlappingFrequency Thestationarityoftheperiodicsignalandtheextentofthenoiseinthesignalmaybeidentiedbycorrelatingsignalswitheachother.Acorrelationcoefcientmaybefoundbyndingthemaximumcorrelationwhenslidingonesignalovertimeandtaking 79

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themaximumnormalizedcross-correlationbetweenthetwosignals.Thenormalizedcross-correlationisnotsensitivetochangesinsignalmagnitude,butdoeschangeifatimelagispresentinthesignalorifthefrequenciesinthesignalarenotexactlythesameduringeachcycleofthe75%span,trailingedgelocationshownbyFigure 4-5 .Itisinterestingtonotethatanincreaseinthefrequencyby20%,C=0.8655decreasesthecorrelationmorethandecreasingthefrequencyby20%,C=0.9531,whilechangingifthesignalhasaleadoralagofthesamemagnitudedoesnotchangethenormalizedcrosscorrelationcoefcient. Figure4-5. ExamplesofCross-correlationofPeriodicSignals (a)VaryingMagnitude,(b)VaryingLag( ==4),(c)VaryingFrequency()]TJ /F3 11.955 Tf 9.29 0 Td[(20%) Thenormalizedcross-correlationcoefcientandaveragesampleerrorbetweenthesignalsmaybeusedtodemonstratehowclosethesignalsaretobeingperiodicthroughoutthekinematicrangeasshownbyTable 4-2 .Thetableisformedbyidentifyingtherstperiodiccycle,thenndingthenormalizedcross-correlationcoefcientasdescribedaboveoftherstsignalwiththefollowing3cycles.Theaveragesampleerroristheaverageabsolutevalueofthedifferenceateachsampleofthesignalinmillimeters.Thecrosscorrelationdataclearlyshowsthatalthoughthesignalsareknowntohavesubstantialenergyatmultiplefrequencies,thesignalishighlyperiodicandvarieslittleovertime.Thereforemostoftheenergyisexpectedtobefoundat 80

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integermultiplesofsomefundamentalfrequency.Variationsinthesemultiplesmaybeaffectedbythestructuraldynamicsandaerodynamicsofthewing.SeveralofthelargererrorsathigherfrequenciesandamplitudesareagaincausedbytheinabilityoftheDICtocapturethewingatmaximumdeection. Table4-2. CrossCorrelationsandErrorsforFollowing3PeriodicCycles,Wing-3(BattensParallel) RunKinematicsC1C2C3E1E2E3 10o,10Hz0.99910.99840.99740.21620.22710.3378 10o,20Hz0.99970.99960.99930.14670.18000.2456 10o,30Hz0.99990.99660.99760.11120.68240.6410 17o,10Hz0.99970.99920.99860.24290.47830.6055 17o,20Hz1.00000.99990.99970.08900.16510.3148 17o,30Hz0.99670.99000.97842.06694.21156.1976 35o,10Hz0.99630.99130.98352.52954.95137.3583 35o,20Hz1.00000.99990.99970.19830.35880.5766 35o,30Hz0.98940.99280.99351.81432.70202.2841 Therigidbodydeectionsareknowntobeperiodicforasingledegreeoffreedomapping.Therefore,becausetheexibledeectionsareobservedtobeperiodic,thedeformationsmustalsobeperiodicwithrespecttothemainappingfrequencyforsteady-stateapping.Whilethissubstantialperiodicitymaysuggestthatwaveletanalysisisnotrequired,smallervariationswithintheappingcycleandvariationscausedbychangingactuationorresonancemayrequiretime-frequencyanalysistobefullyunderstood. 4.4.2.3Periodicphenomena-timeandfrequencyanalysis AnenormousamountofdataisavailablefromtheexperimentaltestbedwhenconsideringthattheDICprovides3-axisdeectionsataround3000pointsonthewing.Thisdataiscompoundedbyaccountingforthe3differentwingsthatareanalyzedatdifferentappingfrequenciesandappingamplitudes.Theprocessofdataanalysisisfacilitatedbyextractingafewmeasurementsofparticularinterest;specically,thedeectionsfromtrailing-edgeroot,leading-edgeroot,trailing-edgemid-span,leading-edgemid-span,andwing-tiplocationsareextracted.Arepresentativesetof 81

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dataisshowninFigure 4-6 todemonstratethediversityofinformationresultingfromtime-domainrepresentation,frequency-domainrepresentation,andtime-frequencydomainrepresentation.Trendsseenareconsistentwithwhatwouldbeexpected Figure4-6. OverviewofTimeHistory,FrequencyandTime-FrequencyBasedSignalAnalysisforFlexibleFlappingWings SignalAnalysisforWing-3(ParallelBattens)at+/-17degand20HzFlappingFrequency,a)FullLength,TimeResolved3DDeectionPlotwithRespecttoReferenceWingandBoxBoundingMaximumDisplacements,b)FullTimeHistoryofVertical()-574()-575()]TJ /F1 11.955 Tf 41.64 0 Td[()andLateral()Deections,c)DiscreteFourierTransformofVertical()-574()-575()]TJ /F1 11.955 Tf 41.64 0 Td[()andLateral()Deections,d)WaveletRepresentationofVerticalDeection,e)WaveletRepresentationofLateralDeection,f)3CycleZoomforTimeHistoryofVertical()-286()-288()]TJ /F1 11.955 Tf 34.76 0 Td[()andLateral()DeectionsforRootChordLeadingEdge(Row1),RootChordTrailingEdge(Row2),Mid-SpanLeadingEdge(Row3),Mid-SpanTrailingEdge(Row4),andWingTip(Row5) foraappingwing.Forthetimehistory,asexpected,themagnitudesincreasegoingfromtheroottotipandthemagnitudeofverticaldeection(appingupanddown)ismuchlargerthanlateraldeection(pointonwingmovestowardstheroot).Thepeak 82

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frequencyofthelateraldeectionisapproximatelydoublethefrequencyofverticaldeectionsincethetimesignaloscillatesbetween0andamaximumnegativevalueintime.Thisfrequencydoublingresultsfromthepointonthewingmovinginthenegativelateraldirectiononboththeupstrokeanddownstroke.Whenlookingatthemagniedtimehistory,itisalsopossibletonoteavariationinthepeakmagnitudesintimeforthemid-winglateraldeection,withanapparentlagatthetrailingedgeofthemid-wingindicatedbyasharperpeakinboththeverticalandlateraldeectionsignals.Thefrequency-domaindataclearlyidentiestheprimarypeaksforlateraldeectionstobeequaltotheappingfrequencyof20Hz,whiletheverticaldeectionshaveastrongpeakattherootattheappingfrequencyof20Hz,graduallyreducingasthesamplesmovetowardthewingtip.TheFourierTransformoflateraldeectionsdemonstratesthatthereisapeakatboth20Hzand40Hzforthemidplane.Incontrasttothemidplane,energyatthewingtipisalmostexclusivelycenteredat20Hzforlateraldeectionsand40Hzforverticaldeections.Theseresultsareconrmedwiththewaveletanalysis,wherethemainenergyofthewaveletplotscorrespondstothesepeakfrequencies.Comparingthewaveletanalysisofverticaldeectionstolateraldeectionsrevealsthedistinctionbetweenenergytightlycenteredat20Hzforverticaldeectionswhiletheenergyisdispersedmoreevenlyacrossfrequenciesforlateraldeections.Sincethiscaseisconsideredsteadystateapping,thereislessinformationavailablefromthewaveletanalysisthatisnotavailablefromtheFourierTransformandtimehistorycombined. 4.4.2.4Periodicnon-sinusoidalfeatures-waveletanalysis Thetemporalnatureoftheappingdeectionshavingperiodicnon-sinusoidalfeaturescorrelateswithtime-varyingfrequencycontent.Suchafeatureisnotedbyexaminingthemid-spanstationalongthetrailingedgeofWing-1whileappingat30Hzwithanamplitudeof+/-10deg.TheresultingdeectionsareshowninFigure 4-7 alongwithfrequency-domainplotsandwaveletplotsfor0.1sstartingat0.2s.Starting 83

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Figure 4-7 at0.2sfocusestheplotonthenatureofthesteadystateappingandeliminatestheinitialfrequencyshiftingofthewavelettransformattime0swhichmaybeseeninFigure 4-2 (a)(b)(c) (d)(e)(f) Figure4-7. TemporalNatureofFlappingDeections ResponseofWing-1at10degFlappingAmplitudeand30HzFlappingFrequency:(a)Time-DomainRepresentationinVerticalDirection()-1482()-1483()]TJ /F1 11.955 Tf 63.35 0 Td[()andLateralDirection(),(b)WaveletRepresentationofLateralDirection,(c)WaveletRepresentationofVerticalDirection,(d)Frequency-DomainRepresentationinVerticalDirection()andLateralDirection()-995()-997()]TJ /F1 11.955 Tf 51.71 0 Td[(),(e)WaveletRepresentationofLateralDirection,(f)WaveletRepresentationofVerticalDirection Thedeectionsinthelateraldirectionhaveespeciallynotabletime-varyingcomponents.Thetime-domaindeectionsclearlyshowabeatingphenomenonofpeakswithperiodicmagnitudes.Thefrequenciesofthesepeaksareshowninthefrequency-domainplot;however,thetemporalnatureofthebeatingisonlyseeninthewaveletmaps.Thepeaksofpositivedeectionshowalargevalueat0.22sfollowedbyasmallervalueat0.23swithanotherlargevalueat0.25sinthetime-domain 84

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plots;correspondingly,thewaveletmapshowssimilarchangesinpeakmagnitudeatthesetimevalues.Thenatureoftheappingisnotedbytheverticaldisplacementwhichshowsthelargedeectionat0.22scorrespondstothewingcrossingthecenterlocationduringtheupstrokewhilethesmalldeectionat0.23scorrespondstothewingcrossingthecenterlocationduringthedownstroke.Thisasymmetryisnotedinboththetime-domainresponsesandtheassociatedtime-frequencymapsbutnotinthefrequency-domainplots.The90Hzcomponentoftheresponseisanotherfeatureforwhichthewaveletmapprovidesadditionalinsightascomparedtoonlyafrequency-domainplot.Inthiscase,thewaveletmapsinFigure 4-7 indicateatemporalvariationinmagnitudeforthis90Hzenergy.Atemporalrelationshipisclearbetweenthebeatingofthe60Hzcontributionandthe90Hzcontribution. 4.4.2.5Time-varyingfeatures Asetofdataisgeneratedthatconsidersthedeectioncharacteristicsinresponsetodifferentapping;specically,thedeectionsinFigure 4-8 reectresponsetoasine-dwellappingat30Hzandsine-sweepappingfrom0Hzin0.5s.ThisdatacorrespondstoWing-1atanamplitudeof+/-10deg. Thetransientnatureofthesweepclearlyaffectstheresponseasnotedbythedramaticdifferenceintheresponsesat0.5sbutsimilarityintheresponsesat0.8s.Theincreasetoappingfrequencystopsat0.5sbutthedeectiontakesanother0.2stosettletothenalsteady-statevalue.Thistemporalnatureofthedecaycannotbenotedinthefrequency-domainplotsbutisevidentinthewaveletmaps. Additionally,theresponselosessomeofitssinusoidalnatureduringthesweep.Thespreadofenergyacrossmultiplefrequenciesisindicativeofaperiodicsignalthathassomelevelofnon-sinusoidalcharacteristics.Thedecayinmagnitudeafterthesweepfrom0.5sto0.7sisaccompaniedbyadecayinthespreadofenergyacrossfrequenciesand,consequently,adecreaseinnon-sinusoidalcomponentstotheresponse.Thesinusoidalnatureandassociatedtemporalvariationsoftheresponseare 85

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(a)(b)(c)(d) (e)(f)(g)(h) Figure4-8. ComparisonofSteady-StateFlappingandSweepAcrossFlappingFrequencies ResponseofWing-1at10degFlappingAmplitudeforSteadyFlappingat30HzFlappingFrequency(a-d)andChirpbetween0-30HzFlappingFrequency(e-h)forVerticalDeection()-2492()-2493()]TJ /F1 11.955 Tf 87.49 0 Td[()andLateralDeection():(a,e)Time-DomainRepresentations,(b,f)Frequency-DomainRepresentations,(c,g)WaveletRepresentationofVerticalDeection,(d,h)WaveletRepresentationofLateralDeection alsofeaturesthatcannotbeascertainedfromafrequency-domainplotbutareevidentinthewaveletmaps. 4.4.2.6Variationsinstructuraldynamics Thestructuraldynamicsofawinghaveadirectinuenceontheappingresponse.Substantialenergyaround42HzisobservedinFigure 4-8 (e-h)atabout0.45sdespitetheinputappingfrequencyneverincreasingbeyond30Hz.TherstbendingmodeofWing-1is42HzasshowninTable 8-1 .Thisresultsuggeststhattheresonantfrequency 86

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ofthewingneedstobeconsideredformodelingandcontroldesignforappingeveniftheappingfrequencyneverreachestheresonantfrequencyofthewing. 4.4.2.7Variationsinwingdesign,appingkinematicsandwinglocation Thenatureofappingdeectionanddeformationsignalsasafunctionofwingdesign,appingfrequency,appingamplitude,chordwiseandspanwiselocationhasbeeninvestigatedindetailbytheauthor[ 133 ].Measurementsfromasetofwingsclearlyindicatethatsuchvariationshavebothtime-domaincharacteristicsandfrequency-domaincharacteristics.Ineverycasewhereexibilityeffectsareexpectedtoincrease(increasesinappingfrequency,amplitudeanddistancefromtheappingactuationpointattheleadingedgewingroot),theratioofhighfrequencyenergytoenergyattheinputappingfrequencyincreases.Thissuggeststhatthehighfrequencycontentinthesignalsmaybeattributedtoaeroelasticeffects.Waveletmapsprovidesasinglepicturewhichencompassesthetime-varyingorfrequency-varyingnatureofthesecharacteristics. Inaddition,thetime-frequencymapsresultingfromwaveletanalysisareshowntoindicatethetemporalvariationsinfrequencycontentassociatedwiththedeections.Thesevariationsmaynotbeclearbysimplyobservingthetimehistoryorfrequencyresponse.AsummaryofthesevariationsidentiedaresummarizedinTable 4-3 .Ineachcase,thewaveletmapsprovideinformationthatdistinguishesfeaturesrelatingthetime-domaincharacteristicsandfrequency-domaincharacteristics. Chapter 4 demonstratesthatwaveletanalysisisavaluabletoolforanalyzingthetime-frequencycharacteristicsofaappingwingandhasfocusedoncaseswheretime-frequencydependentbehaviorispresent.Chapter 4 doesnotattempttorelatethetime-frequencydependentfeaturesidentiedtoaerodynamicperformance.Theaerodynamicimplicationsofthetimeandfrequency-varyingdeectionsofthesewingstothrustperformancearediscussedinfurtherdetailbyWuandcolleagues[ 278 279 ]. 87

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Table4-3. ConclusionsSummary:ParametersWhichAffecttheTime-FrequencyResponseofAFlappingWing CategoryConclusion PeriodicNon-SinusoidalFeaturesOut-of-planesignalsaremostlyperiodicforstiffwingsasseenforWing-1inFigure 4-4 .Out-of-planesignalsmayhavetime-varyingfrequencycomponentswhenstructuraldynamicsorinertialeffectsarepresentasseeninFigure 4-7 Time-VaryingFeaturesChanginginputappingfrequenciesmayexcitestructuralmodeseveniftheinputappingfrequencyremainsbelowtheresonancefrequencyofthemode,especiallyiftheinputappingfrequencyishalfthefrequencyofthestructuralmodeasseeninFigure 4-8 .ThewaveletrepresentationisidealforobservingthedecayinginuenceofanexcitedstructuralmodeasseeninFigure 4-8 andtoobservetime-varyingfrequencycontent.Time-frequencyanalysiswillbeespeciallyusefulwhenthekinematicparametersarevariedcontinuouslytocontroltheappingvehicle. StructuralDynamicsTherstresonantfrequencyofthewingmustbeconsideredformodelingandcontrolpurposessinceinputatthisfrequencyandathalfofthisfrequencymaydramaticallyincreasethemagnitudeoftheappingresponseasseeninFigure 4-8 FlappingAmplitudeTheratioofhighfrequencyenergytoenergyattheinputappingfrequencyincreaseswithappingamplitudeforthelateralappingresponse. FlappingFrequencyTheratioofhighfrequencyenergytoenergyattheinputappingfrequencyincreaseswithappingfrequencyforthelateralappingresponse.Non-periodicfeaturesaremoreprevalentatappingfrequenciesbelow10Hz. WingSpanTheratioofhighfrequencyenergytoenergyattheinputappingfrequencyincreasesmovingfromthewingroottothewingtipforthelateralappingresponse. WingChordTheratioofhighfrequencyenergytoenergyattheinputappingfrequencyincreasesmovingfromthewingleadingedgetotrailingedgeforthelateralappingresponse 88

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CHAPTER5WING-SHAPEMODELSANDSYSTEMIDENTIFICATION 5.1BackgroundforSystemModelIdentication 5.1.1ModelClassications Systemidenticationiscommonlyusedtobuildmodelsofdynamicalsystemsforuseincontroldesign.AnoverviewofmodelingisprovidedinFigure 5-1 [ 197 ].Modelingforaircraftwithmorphingandappingwingsrequiresunderstandingdynamicmodeling,aerodynamictheory,andightcontroltheory[ 204 ].Thecomplexityofthesemodelsmaystillbeunderstoodbyviewingthesystemintermsofmultipleinputandoutputsignals.ThusmanyoftheconceptsbuiltinChapter 4 arerelevant. Figure5-1. StandardModelingProcedure[ 197 ] Amodelmaybeexpressedasawhite-box,gray-box,orblack-boxmodel.Whiteboxindicatesthatthetheoryiswellunderstoodandbuiltfromrstprinciples.Agray-boxorsemi-empiricalmodelusesinsightintothesystemandexperimentalinformationtobuildamodelwhileablack-boxhasnotheoreticalinformationwhichisforcedonthemodel.Systemmodelsmaybedescribedasdeterministiciftheyarewelldenedor 89

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describetheuncertaintypresentinthemodelwithsomeclearmathematicaldescriptionsuchasinrobustcontrolwhereparametricuncertaintyisconsidereddeterministicallyormodelingdisturbancesorfeedbackasagaussianrandomvariable.Stochasticmodelsarebuiltusingsomeformofestimatorofthesystem'slikelyresponseandattempttominimizetheerrorstatisticallybasedonthechangesobservedinthedistributionsofthestatesinthemodel.Assuch,thesystemmodelmaybelinearornonlinearwhilestillbeingstochastic[ 139 ]. Systemsmayalsobecategorizedbywhatvariablesinuencethesystem,particularlyifthesystemistime-invariantortime-varying.Atimeinvariantsystemisnotafunctionoftime.Inotherwords,ifaninputx(t)producesanoutputy(t),thenatalatertime,t+tthesameinputx(t+t)willproducethetime-shiftedoutputy(t+t)ifthestatesremainthesame.Atime-varyingmodelmaybereducedtoaquasi-steadymodeltoexpeditecontroldesignbyformingthemodelaroundacertainequilibriumconditiony(tn)wherethesystemisnotchangingrapidly,butthissimplicationmustbejustiedsincetheassumptioncouldcausealargelossininformationaboutthesystem.Systemsareclassiedascontinuousordiscreteintime. Systemsarealsocharacterizedbyhowmanyinputandoutputvariablesareusedinthemodel.Classicationsrangefromsingle-input,singleoutput(SISO)systemstomulti-input,multi-output(MIMO)systems.Asystemmodelmaybeparametricifithasnitedimensionwhileitisnon-parametricifithasinnitesystemdimension.Non-parametricmodelsmaybebuiltfromimpulseorstepresponses,frequencyresponses,correlationfunctions,andspectraldensities[ 112 ]. Inadditiontothenatureofhowthemodelisbuilt,thesystemmaybeclassiedbythenatureofthemathematicalfunctionsinvolved.Systemsareoftensummarizedintermsofannthorderdifferentialequationrelatinginputtooutput.Thesysteminputisspeciedwhiletheoutputmeasurementsincludethesystemresponsetotheinputandexternaldisturbances.Statevariables,whichsummarizejustenoughinformationfor 90

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atleastlineartime-invariantsystemstopredictthefutureresponseofthesystem,maybeusedtoexpresstheoutput,y,intermsoftheinput,u,andstate,x,eachofwhichmayconsistofvectororscalarquantities[ 112 ].Astatevariablemodelformulationmayexpressanthorderdifferentialequationintermsofncoupledrst-orderdifferentialequations,evenforsystemswithmultipleinputs.Thereforestatevariablemodelsarecommonlyused.Modelsalsorangefromasimplelinearmodeltotime-varyingnonlinearmodelsusedfordynamicallycomplexproblems. LinearmodelsoffunctionsarecommonlyusedandmustsatisfythepropertiesofsuperpositionshownbyEquation 5 andscalingshownbyEquation 5 f(x1+x2)=f(x1)+f(x2),x1
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Realworldsystemsarealmostalwaysnonlinear.Astochastic,time-varyingsystemwhichincludesdisturbancesw(t)isformulatedinEquation 5 forcomparisonwithEquation 5 _x(t)=f[x(t),u(t),w(t),t],y(t)=h[x(t),u(t),t](5) Anonlinearsystemmaybelinearizedaroundacertainstatexnifthestateisnotchangingrapidly.Forexample,thestandardequationsofmotionforanaircraftareshowninEquation 5 forlatercomparisonswheremistheaircraftmass,Iistheinertiatensor,aretheEuleranglesoftheaircraft'sattitudewithrespecttoearthxedaxes,Visthetranslationalvelocityand!istheangularvelocity,anduisthecontrolvectorusuallyconsistingofthrottlepositionandcontrolsurfacedeections.AppliedforcesfromgravityFG,thrustFTandaerodynamicsFAareincludedwithmomentsfromthrustMTandaerodynamicsMA[ 112 ].Thedependenceofthevehicle'smovementonthemorphinganglesoftheaircraftwingmaybeinferredfromEquation 5 m_V+!xmV=FG()+FT+FA(V,!,u,),I_!+!xI!=MT+MA(V,!,u,)(5) 5.1.2SystemIdentication Systemidenticationusesaprioriknowledgeaboutthesystemwheretheinputsarespecied,asystembasisormodelstructureisspecied,andsomerandommeasurementnoiseisassumed.AcostfunctionJ[Zn,Yn()]isspeciedasafunctionofthemeasurementsZnandmodeloutputsYn()tondthemodelM().Parameterestimationforthetermsinthemodelbasisanderrorsforeachparametermaybeusedtoformthenalmodelwithvariousmethods.Whenidentifyingtheparameters,equationerrorandoutputerrormethodsaremostoftenused.Theequationerrormethoduseslinearregressionandaleast-squarescurvettondtheunknownaerodynamicparametersforEquation 5 .Theoutputerrormethodndstheunknownparametersby 92

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minimizingthesumoftheweightedsquaredifferencesbetweenthemeasuredaircraftoutputsandmodeloutputs.Thisproblemisnonlinearsincetheunknownparametersarefoundintheequationsofmotionandmustbeintegratedtondthestates.Findingthecorrectparametersmaybeexaminedusingamaximumlikelihoodapproachortransferringintothefrequencydomain.Sometimestheparametersfoundinmaybecorrelatedinwhichcaseadifferentapproachormodelrenementmayberequired.Lastly,themodelasafunctionoftheparameterswillbevalidatedbyrepredictingthemodeloutputsandcomparingtheresulttotheexperimentaldata. Thebasisassumedforthemodelhasasignicanteffectonthenalmodel.Whilearichbasiswithmanyparametersmayenhancethemodeldelity,itwillalsomakeidenticationandpredictionmoredifcultsincetherearemoredegreesoffreedominthemodelbasis.Theappropriatebasisisdeterminedbythedesireddelityofthemodelandwhichdynamicphenomenamustbeunderstood. 5.2DecoupledWing-ShapeModels Thetheorybehindmodelswhichapproachaircraftsystemswithadecoupledperspectivewhichmayberelevanttotheformulationofanalaero-orhydroservoelasticmodelareincludedinthissection.Referencesareprovidedforfurtherdetailssinceonlyacursoryglanceatthefundamentalsofeachareaisprovided.ThereaderisalsoreferencedtoChapter 3 whichprovidesamorebroad,butlessmathematicallydetailedoverviewofthetreatmentoftheseproblems.Beammodelswillbeinvestigatedindetailsincemanytypesofwingsmaybemodeledascantileverbeams. 5.2.1FluidDynamicModels Thethree-dimensional,unsteadyformoftheNavier-Stokesequationsmaybeusedtodenehowthevelocity,pressure,temperatureanddensityofauidbehave.Computationaluiddynamics(CFD)attemptstosolvetheseequationsremainextremelycomputationallyexpensive;therefore,manyapproximationsareusedevenforcomplexcasessuchasmorphingandappingwings.Quasi-steadyapproacheswhich 93

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linearizetheaerodynamicsaboutatrimconditionarecommonlyusedformorphingaircraft[ 80 ].However,asthemorphingbecomesfaster,quasi-steadyapproachesbecomeinsufcienttomakesatisfactorypredictionsofaerodynamicperformanceandstructuralexibilitymustbeincorporatedintothemodel.Forappingundersomecircumstancessolutionstothefullunsteady,3D,incompressible,laminarNavier-Stokesequationshavebeenperformed[ 123 219 ].ThereaderisreferencedtotheliteraturecitedinChapter 3 formoredetailssinceadetailedmodeloftheuiddynamicsisnotrequiredforthecurrentapproach. 5.2.2SolidMechanicsModels Themostbasicelasticmodelsdescribethestressesandstrainswhicharepresentinastructure,aswellastheabilityofthestructuretobearaload.Acontinuousmodelexpressingtherelationshipbetweenaloadandthebeamdeectionweremostcommonlyused,butdiscretemodelsusingniteelementmethod(FEM)arebecomingmoreprevalent.Forthemostsimplisticelasticcase,thestressmaybelinearlyrelatedtothestrainbyHooke'slaw,E==whereEisYoung'smodulus,isthestress,andisthestrain. Abeammodelmaybeusedtoapproximatethedeectionofanaircraftwingandmaybeusefulasarstapproximationformorphingandappingwingsiftheaerodynamicloadingsarewellknowninadvance.Euler-BernoullibeamtheoryshownbyEquation 5 expressesthebeamdeection,w(x),atsomeposition,x,intermsofsomeappliedloadq(x)andassumessmalldeections,thatthebeamissubjectedtoonlylateralloads.ErepresentsthemodulusofelasticityandIrepresentsthesecondmomentofinertiaofthecrosssection.Thedifferentialequationcaneasilybesolvedbyapplyingtheappropriateboundaryconditionsforthebeam,suchasthatw(x=0)=0and@w=@x(x=0)=0forthexedendand@2w=@x2(x=L)=0and@3w=@x3(x=L)=0@w=@x(x=0)=0forthefreeend.Itmaybehelpfultonotethatfora1-dimensionalbeammadeoflinearelasticmaterial,theslopeofthebeamis(x)=dw=dx,the 94

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stressis(x)=)]TJ /F7 11.955 Tf 9.29 0 Td[(zEd2w=dx2,thebendingmomentisM=)]TJ /F7 11.955 Tf 9.3 0 Td[(EId2w=dx2andtheshearforceisQ=dM=dx=)]TJ /F7 11.955 Tf 9.3 0 Td[(EId3w=dx3.Thedeectionofthetipofabeamoflength,l,withapointload,W,isshowninEquation 5 whilethedeectionofthetip,zofanevenly-distributedloadwisshowninEquation 5 withgeneralequationsshownasafunctionoflocationonthebeam,x.Theseequationsgivearoughmathematicalideaofhowawingwillbendundersomeaerodynamicloadingconditions. @2 @x2(EI@2w @x2)=q(x)(5) z=fracWx26EI(3l)]TJ /F7 11.955 Tf 11.96 0 Td[(x),x=l:z=fracWl33EI(5) z=Wx2 24EIl(2l2+(2l)]TJ /F7 11.955 Tf 11.96 0 Td[(x)2),x=l:Wl3 8EI(5) Timoshenkobeamtheorymaybeusedtoanalyzethebeamdisplacementsifsheardeformationandrotationalinertiaareofinterest.ThedeectionsofthebeammaybedescribedwithEquation 5 wherethedisplacementsu,v,waredenedasafunctionoftheinitiallocationsonthebeam,x,y,zasu(x,y,z)=)]TJ /F7 11.955 Tf 9.3 0 Td[(z (x),v(x,y,z)=0andw(x,y,z)=w(x), istheslopeofthebeam,AisthecrosssectionalareaandistheTimoshenkoshearcoefcient. @2 @x2(EI@ @x)=q(x,t),@w @x= )]TJ /F3 11.955 Tf 22.13 8.09 Td[(1 AG@w @x@ @x(5) Theaerodynamicloadsarerarelyknowninadvanceduetoaeroelasticcouplingforexiblewings.Evenmorecomplexmodelswhichincludetheanisotropyanddetailedshapeofthebeammaynotbesufcienttopredictwingdeectionsbecauseofthetime-varyingnatureofthewingmovement.Amoreadvancedmodelwhichaccountsforthestructuraldynamicsandkinematicsofthemorphingandappingisrequiredalthoughtheprinciplespresentedherearehelpfulforunderstandinghowthenalmodel 95

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behavessinceitroughlydescribestherelationshipbetweentheloading,thestiffnessofthewing,theinertiaofthewingandthewingdeection. 5.2.3StructuralDynamicModels AnintroductiontomodelingthestructuraldynamicsofabeamoftenstartswithdeningthemodalsolutionforastringwithmovementdenedbytheonedimensionalwaveequationshowninEquation 5 T@2v @x2=m@2v @t2(5) Usingaseparationofvariablesgivesa2ndorderordinarydifferentialequationwhichmaybesolvedtogivethecharacteristicequationforthestringwithasetofeigenvaluesassolutionsthatsatisfyEquation 5 withnalsolutionofthedifferentialequationshownbyEquation 5 .Herethemodaldisplacements,viaredeterminedbythetensioninthestring,T,themassm,asafunctionofthelocationonthestring,x,andtime,t. Asin(l)=0,i=i l(5) vi(x,t)=[Aisin(ix)][Cisin(p (T=m)it)+Dicos(p (T=m)it)](5) Thewaveequationmaybeexpressedasasumofaninnitenumberofshapeswithwhichthestringmayvibratewhereeachhavesomenaturalfrequency!i=i=lsqrt(T=m)andsatisfytheprincipleoforthogonalitywhichstatesthatvectorapproximationsofthefunctionsareorthogonalortheinnerproductequalszero. Asimilarapproachmaybeusedforbeams,althoughthesolutionvariesbasedonthetheboundaryconditionsstated.ThebeamdifferentialequationisshowninEquation 5 96

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(x,t)=[Asin(x)+Bcos(x)][Csin(sqrt(GJ=(Ip)))+Dcos(sqrt(GJ=(Ip)))](5) Aclamped-freebeamisoftenexaminedwhichhasasimilarformtotheequationsshownforthestring.Equation 5 representsthefree-endresponsewhereTisthestructuraltorque. T(l,t)=GJ@ @x(l,t)=0(5) Aclampedendorelasticandinertialconstraintsthatareoftenconsideredandmaybeapproximatedwithaxedboundary,atorsionalspringattheendofthebeamoraninertialelementrespectively.Thesolutionforthefree-endresponsehasmodalfrequenciesandmodeshapeswhichareshowninEquations 5 and 5 !i=(2i)]TJ /F3 11.955 Tf 11.96 0 Td[(1)=(2l)sqrt(GJ=(Ip))(5) i(x)=sin[(2i)]TJ /F3 11.955 Tf 11.95 0 Td[(1)x=(2l)](5) ThebendingmomentisproportionaltothelocalcurvatureasexpressedbyEquation 5 .Equation 5 maybeusedtogivethedifferentialequationofmotionforanonuniformbeaminEquation 5 M=EI@2v @x2(5) @2 @x2(EI@2v @x2)+M(@2v @t2)=q(x,t)(5) ThemodalfrequenciesandmodeshapesforthefreebeamaregivenwithEquations 5 and 5 whereconstantvaluesweredeterminednumerically[ 93 ]. 97

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!i=(il)2sqrt(EI=(ml4)),(il)=[1.875,4.694,7.854,10.99,14.14...](5) i=cosh(ix))]TJ /F7 11.955 Tf 9.52 0 Td[(cos(ix))]TJ /F4 11.955 Tf 9.51 0 Td[(i[sinh(ix))]TJ /F7 11.955 Tf 9.51 0 Td[(sin(ix)],(i)=[0.734,1.018,0.999,1.000,0.999](5) TheRitzandGalerkinmethodsbothprovideanothermeanstoapproximatethestructuraldynamicsofabeamwithLagrangeequationsorapartialdifferentialequationofmotion.However,thesemethodsultimatelydonotincorporatethecouplingbetweenthestructuraldynamicsandthecommandedkinematicsforappingandmorphing. 5.2.4Rigid-bodyDynamicModels 5.2.4.1Singledegree-of-freedommorphingandappingmodel Rigid-bodydynamicmodelsofwingmovementmaybedenedanalytically.Rigid-bodymodelsforbeammovementwherethebeamisxedattherootandcommandedtomoveatacertainappingamplitude,frequency,andangleofattackovertimeareformulated.Themodelrepresentsthemotionofanypointonthewinginresponsetoaspeciedactuatorinput. Themodelassumesthatawingisrotatedthroughasinusoidalmomentattheleading-edgeroot.Themodelconsidersadiscreteelementatapointonthewingasdenedbythelocationonthestationarywing.Themodelpredictsthespatiallocationofthatdiscreteelementduringtheappingcycle.Amorphingwingwhichmodiesthedihedralangle,angleofattackorfeatheringanglemaybeviewedasaspecialcaseofthismodelwherethespeedofmorphingisdenedbyspecifyingthefrequency. ThelocationofthewingduringappingisdescribedusingaCartesiancoordinatesystem.Theoriginofthissystemistherootofthewingattheleadingedge.Axesarethendenedsuchas^xextendsalongthespantowardsthewingtip,^yextendsalong 98

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thechordtowardthetrailingedge,and^zextendsperpendiculartothewingasshowninFigure 5-2 Figure5-2. CartesianandCylindricalCoordinateSystemSpecication Considertheaeroservoelasticdynamicsforanypointonthewingasparameterizedacrossthedesignspace.ThelocationofthepointassociatedwithadiscreteelementisexpressedintheCartesiancoordinatesasxalongthe^xaxis,yalongthe^yaxisandzalongthe^zaxis.Thevaluesofthecontroleffectorsarethengivenas!and. Thedevelopmentofabasiswithwhichtorepresenttheaeroservoelasticresponsebuildsupontherigid-bodydynamics.SuchdynamicsareinitiallyobviouswhenconsideringcylindricalcoordinatesasshowninFigure 5-2 .Theradiallocationofcrandthedepthofczareconstantforarigid-bodywingsoonlytherotationalcoordinateofcwillvarywithtimeasindicatedinEquation 5 Cr(x,y,z,,!,t)=p x2+z2 (5) C(x,y,z,,!,t)=sin(2!t) (5) Cz(x,y,z,,!,t)=y (5) 99

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Theexpressionofrigid-bodydeectionsinCartesiancoordinatesaregeneratedbyatransformationofthebasistoexpressthecylindricalcoordinatesofEquations 5 and 5 .Theresultinglocationsofthepointsassociatedwith(x,y,z)areexpressedasXralongthelateralaxisinEquation 5 ,YralongthelongitudinalaxisinEquation 5 ,andZralongtheverticalaxisinEquation 5 Xr(x,y,z,,!,t)=p x2+z2cos(sin(2!t)) (5) Yr(x,y,z,,!,t)=y (5) Zr(x,y,z,,!,t)=p x2+z2sin(sin(2!t)) (5) Theequationsabovemaybemodiedtodenethewing-shapeforwingswhichrapidlymodifysweeportwistangle.Ifthemorphingorappingmodies2anglessimultaneously,thenasingleadditionaltransformationmaybeusedtoexpressthemovementoftherigidwing,asshownbyEquations 5 whichdescribethewingmovementwithanappliedtwistangle.Thisisusefultoexpressthemovementofa1-DOFmechanismwherethewingpassivelychangesangleofattackora2-DOFwherethefeatheringangle,isactivelychanged. 266664Xr2(x,y,z,,!,t)Yr2(x,y,z,,!,t)Zr2(x,y,z,,!,t)377775=266664Xr(x,y,z,,!,t)Yr(x,y,z,,!,t)Zr(x,y,z,,!,t)3777752666641000cos)]TJ /F3 11.955 Tf 11.29 0 Td[(sin0sincos377775(5) 5.2.4.2Generalizedwing-shapemodel Thegeneralcasewheretheentirewingisbeingmorphedinthreedirectionsisnotaseasyasthe1Dor2Dtime-varyingmodelshowninthissection.Thereforeageometricalmodeldiscreditedwithrespecttospanwiseandchordwiselocationispresentedwhichgeneratesawingwitharbitrarily-morphedshape.ThecoordinatesystemwhichwillbeusedtodenethegeneralwingmodelisshowninFigure 5-3 100

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Figure5-3. CoordinateSystemforGeneralizedMorphingWingModel Aquaternionbasedformulationisrequiredtoexpressthetransformationsusedforthegeneralmorphingmodeltoincreasecomputationalefciency,toavoiddiscontinuities,andtoeliminatethetransformationorderdependenceinvolvedwhenusingEulerangle-basedtransformations.Arealquaternionisdenedbyasetoffourrealnumberswritteninadeniteorderwheretherstnumberrepresentsascalar,Sq,andthenextthreenumbersrepresentavectorVqinthei,j,kdirections,asshownbyEquation 5 .Equations 5 and 5 showhowtomultiplyquaternionsandndtheinverseofthequaternion.Theseequationsarenecessarytoexpressthevectorr0showninEquation 5 ,whichisthevectorrfromapointOwhichisrotatedabouttheunitvectorsbyanangle2.Thecorrespondingquaternionforthisrotationisq=cos+sin(sxi+syj+sjk).MultipletransformationswithnrotationsandthereforenquaternionsareexpressedusingtheformofEquation 5 .[ 58 ] q=Sq+Vq=d+ai+bj+ck(5) 101

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q=q1q2=d1d2)]TJ /F7 11.955 Tf 9.88 0 Td[(a1a2)]TJ /F7 11.955 Tf 9.88 0 Td[(b1b2)]TJ /F7 11.955 Tf 9.88 0 Td[(c1c2+d1(a2i+b2j+c2k)+d2(a1i+b1j+c1k)+j266664ijka1b1c1a2b2c2377775j(5) q)]TJ /F8 7.97 Tf 6.59 0 Td[(1=Conjugate(q) Norm(q)=Kq Nq=Sq)]TJ /F7 11.955 Tf 11.95 0 Td[(Vq qKq=d)]TJ /F7 11.955 Tf 11.95 0 Td[(ai)]TJ /F7 11.955 Tf 11.95 0 Td[(bj)]TJ /F7 11.955 Tf 11.95 0 Td[(ck d2+a2+b2+c2(5) r0=qrq)]TJ /F8 7.97 Tf 6.59 0 Td[(1(5) r0=qn...q2q1rq)]TJ /F8 7.97 Tf 6.59 0 Td[(11q)]TJ /F8 7.97 Tf 6.59 0 Td[(12...q)]TJ /F8 7.97 Tf 6.59 0 Td[(1n(5) Thewingisdenedbyadiscretenumberofsectionsinthespanwisedirection.ThecorrespondingquaternionsforeachsegmentofthewingaredenedbyEquation 5 wherethevectorsn=[sxi+syj+szk]isdenedasavectorfromtheoriginofeachwingsectioncoordinatesystemtotheendofthewingsection,whichdenesthequarter-chordlineofthenthwingsection.Thereforeeachsnstartswithsn=sxiwheresx=b=nwherebisthespanofthewingfromroottotipandnisthenumberofwingsectionsused.Themodelassumesthatthedistributionateachspanwisestationisdenedwithavectorofnormalizedairfoilcoordinates,cnwhicharescaledtomatchthewingtaperratioandchordthicknessateachspanwisestation,thentranslatedadistancesxtogivethedistributionforthenthsection,Cn.Therootquarter-chorddistributionc0isscaledappropriatelytogiveC0andthenrotateddenedbytheangleofattack,togiveadistributionC00.CnmayalsobeformulatedasafunctionCn(y)whichincorporateschordwiseexibilityifneeded. TheangleswhichareusedtodescribethemorphedwinginEquation 5 aredenedasthetwistangle,(x)=d=dx(x)+0,dihedralangle,(x)=d=dx(x)+0, 102

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andsweepangle,(x)=d=dx(x)+0.Theterms(0,0,0)denetherigidbodymotionwhile(d=dx(x),d=dx(x),d=dx(x))representthewingexibilityasafunctionofspanwiselocation.ThequaternionformulationinEquation 5 isappliedtoeachpointinCnusingthesethreequaternionsinEquation 5 asshownbyEquation 5 .ThepointsC0narethentranslatedtotheendpointoftheprevioussection,therebybuildingthewingbyformingsectionsfromtheroottothetip.Itisimportanttonotethatthesectionsarerotatedaroundalocalorigin,thentranslatedtotheproperlocationindividually. 266664q(x)q(x)q(x)377775=266664cos((x)=2)cos((x)=2)cos((x)=2)377775+266664sin((x)=2)sin((x)=2)sin((x)=2)377775[sxi+syj+szk](5) C0n=q(x)q(x)q(x)Cnq(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1q(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1q(x))]TJ /F8 7.97 Tf 6.59 0 Td[(1(5) Themodelmaybedescribedasafunctionoftimebymultiplyingtherigidmorphingangles(0,0,0)byafunctionwithrespecttotime(e.g.asinusoid)ifneeded,whereeachanglemightbeoscillatingatacertainfrequencyandamplitude.Higherordertermsmayalsobeaddedtodescribethewingangleswithrespecttostiffnessorinertialeffectsinherenttothewing.Thismodelmaybeusedasabasisforamodelofwingmovementbymultiplyingcoefcientsbyeachterminthefunctionsdescribingtheseangles.Acostfunctionofthedifferencebetweentheactualwingmovementandthemodeledwingmovementmaybespecied,therebyprovidingagray-boxwingmodelwhichincorporatescoupledphenomena,althoughnotexplicitly.Thiscost-functionbasedidenticationapproachmayhavecomplicationsifthecouplingbetweenphysicalphenomenamakesittoodifculttomodeleachofthelocalangleswithrespecttothecontrolinputs. 103

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5.3CoupledWing-ShapeModels Decoupledapproachesarerarelysufcientformorphingandappingsystems.Thereforethebasicsofseveralcoupledsystemmodelsareintroducedtodescribethetypicalapproachesusedforsystemswithcoupleddynamics.Animportantquestionthatmustbeansweredforthesemodelsisifsuperpositionofthedecoupledmodelsissufcientorifagray-boxormodalmodelwhichaccountsforthecoupledeffectsissufcient. 5.3.1Aero/hydroelasticModels ThegeneralizedaeroelasticequationsofmotionaregivenbyEquation 5 and 5 where[M],[D],[K]arethegeneralizedmass,dampingandstiffnessmatrices,q(t)isthegeneralizeddisplacementvector,F(t)isthegeneralizedforcevectorrepresentingthecoupledaerodynamicsandinertialloads,andw(x,y,z,t)arethestructuraldisplacementsatanypositionandtimeonthestructureexpressedasasummationofthestructuraldynamicmodes[ 200 ].Itisimportanttonotethatmodelsofvariousdelityfortheaerodynamicsandstructuraldynamicsmaystillbeconsideredwiththismodel,includinglinearanddecoupledanalyses. [M]q(t)+[D]_q(t)+[K]q(t)=F(t)(5) w(x,y,z,t)=NmodesXi=1qi(t)i(x,y,z)(5) 5.3.1.1Staticaero/hydroelasticmodels Wall-mounted,sting-mountedandstrut-mountedmodelshavebeendevelopedforaeroelasticeffectsforrigidairfoilswithexiblesupports.Forthewallmountedmodel,thesupportismodeledasatorsionalspring.UsinglinearangleofattackandspringconstantthestaticequilibriumequationisshowninEquation 5 104

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Mac+L(xo)]TJ /F7 11.955 Tf 11.95 0 Td[(xac))]TJ /F7 11.955 Tf 11.95 0 Td[(W(x0)]TJ /F7 11.955 Tf 11.95 0 Td[(xcg))]TJ /F7 11.955 Tf 11.96 0 Td[(k=0(5) IfCMac,CLareassumedconstant,thisequationcanbesolvedfortheelasticdeection,showninEquation 5 .WhenthedenominatorofEquation 5 vanishes,thewingexperiencesdivergence.Thus,adynamicpressure(indicativeofairspeed)maybespeciedorasituationwheretheaerodynamiccenterisincidentwiththepivotwheredivergencewilloccur.Since1=isproportionalto1=qanestimateofqDmaybemadeevenifvaluesofthemodelarenotknown.Forthesting-mountedandstrutmountedmodels,thestingismodeledasaexiblebeam,alteringtheequilibriumequations.However,adynamicpressurefordivergencemaystillbefound.FurtherdetailsareelaboratedbyHodges[ 93 ]. =(qScCMac+qSCLr(xo)]TJ /F7 11.955 Tf 11.95 0 Td[(xac))]TJ /F7 11.955 Tf 11.95 0 Td[(W(xo)]TJ /F7 11.955 Tf 11.95 0 Td[(xcg))=(k)]TJ /F7 11.955 Tf 11.95 0 Td[(qSCL(xo)]TJ /F7 11.955 Tf 11.95 0 Td[(xac))(5) Forexiblewings,theliftingsurfacemaybemodeledasabeamwhichisxedattheroot.ThetotalappliedmomentperunitspanMisgiveninEquation 5 whereLandMacarethedistributedspanwiseliftandpitchingmoment,gmisthespanwiseweightdistributionandNisthenormalloadfactorwhenthewingislevel.ThefundamentaltorsionrelationsshowninEquation 5 maythenbeusedtodescribetheequilibriumequationfortheexibleairfoilshowninEquation 5 byequatingtherateofchangeofthetwistingmomenttothenegativeoftheappliedtorque.TheliftcurveisoftenassumedtobeconstantalongthespanandstriptheoryisusedtogivetheequilibriumequationindifferentialformofEquation 5 whereristherigidrotationofthewingandistheelastictorsionalrotation.BoundaryconditionssuchaszerodeectionorzerotwistingmomentmayhelpsolvethisequationforasshownbyEquation 5 .Thisequationdivergeswherelapproaches=2,whereisdenedas=p qcae=(GJ).ThehomogeneousformofEquation 5 givesthe 105

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aeroelasticdivergencemodeshape,asopposedtothetwistdistributionwhichwillresultindivergenceobtainedfromthenonhomogeneousformofEquation 5 M=Mac+eL)]TJ /F7 11.955 Tf 11.96 0 Td[(Nmgd(5) T=GJd dy(5) GJd2 dy2=)]TJ /F7 11.955 Tf 9.3 0 Td[(qc2cmac)]TJ /F7 11.955 Tf 11.96 0 Td[(eqccl+Nmgd(5) d2 dy2+qcae GJ=)]TJ /F3 11.955 Tf 14.89 8.08 Td[(1 GJ(qc2cmac+qcaer)]TJ /F7 11.955 Tf 11.95 0 Td[(Nmgd)(5) =(r+ r)[tan(l)sin(y)+cos(y))]TJ /F3 11.955 Tf 11.95 0 Td[(1](5) ArelationshipmayalsobedevelopedtondthetorsionaldeformationandresultingairloaddistributionforaspeciedightconditionbyassumingthatthespanwiseliftdistributioncanbedeterminedasL=qca(r+)asshownbyEquation 5 andEquation 5 .Theseequationsmaybeusedtospecifythetotalforcesormomentsonthevehicleasafunctionofaltitudeandightcondition.Thestructuralengineercanusethemtoensurestructuralintegrityforaspeciedightvelocity,V,andloadfactor,N,toproduceaV-Ndiagram.Inaddition,Equation 5 maydenethetorsionaldeformationwhichgivesthemaximumstressinthewing,usuallyfoundintherootcrosssection. N=2GJ(l)2[aer+ccmac(1)]TJ /F4 11.955 Tf 11.96 0 Td[(l=tan(l))] ael[Wel=tan(l)+2mgld(1)]TJ /F4 11.955 Tf 11.96 0 Td[(l=tan(l))](5) r=NWle 2GJltan(l)+[1)]TJ /F4 11.955 Tf 26.47 8.09 Td[(l tan(l)][Nmgl2d GJ(l)2)]TJ /F7 11.955 Tf 13.15 8.09 Td[(ccmac ae](5) 106

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Itshouldbenotedthatsweepingthewingwilldramaticallychangethewingloadingandbendingprolessincecombinedbending-torsionwillbeobservedandtheboundaryconditionsmustbechanged.Itisnotablethattherootbendingmomentisgreaterforforwardsweepthanforbackwardsweepforagivenvalueoflift.Forwardsweepalsoincreasesdivergenceinstabilityandincreasesstructuralloads,whilebackwardssweepalleviatestheseissues.Tailoringthebend-twistelasticcouplingofthewingwithcompositesmustbeusedtopassivelystabilizeforwardsweepasseenintheX-29.Furtherdetailsonanalysisofbending-torsiondivergenceandhowtopreventitwithaeroelastictailoringareprovidedbyHodges[ 93 ]. 5.3.1.2Dynamicaero/hydroelasticmodels Dynamicaeroservoelasticmodelsdoexist,butarelesscommonsinceamodalformulationgenerallyissufcienttoaddressaeroelasticproblems.Iftheseapproachesarenotsufcient,theproblemalmostalwaysbecomesaeroservoelasticsincecontrolactuationisappliedtoinuencethesystemifthereisatime-varyingcomponent.Surecentlyprovidedanaeroelasticmodelwhichincorporatedanonlinearstrain-basedbeammodel,unsteadyaerodynamicsandasixdegree-of-freedomightdynamicsmodel[ 228 ].ThemodelisapplicabletoHALEaircraftwhichexperiencelargedeformationscausedbyaeroelasticcoupling. 5.3.2Aero/hydroservoelasticModels ThelinearequationusedforanaeroservoelasticproblemissimilartoEquation 5 [ 226 ]wherethemass,stiffnessanddampingtermsM,C,Krepresenttheeffectofthestructure,termswithrepresentthecontrolinputsandtermswithAcoefcientsdescribetheaerodynamics. [M]q+[C]_q+[K]q+[B1]+[B2]_+[B3]+[A0]q+[A1]_q+[A2]q+...=0(5) 107

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5.3.2.1Coupledmodelsofsmartstructures Adifferentapproachisrequiredtomodeltheaeroservoelasticitywhenthecontroleffectorsarepartofthewingstructure.Itisquitedifculttoformageneralmodelforthisapproachsincesomanymaterialsoractuatorlocationsmaybeused.ManyofthemodelsforsmartstructuresextendthesolutionsforbasicbeammodelsbutdonotrelyontheassumptionusedbytheEuler-Bernoulliapproachthattheplanenormaltotheneutralaxisremainsnormaltotheneutralaxisafterdeformation.AnexampleofapiezoelectricbeammodelisselectedandshowninEquation 5 whichrepresentstheequationofmotionforasmartbeamelementwithanactuationandsensingpiezoelectricpatch.MandKarethe(8x8)massandstiffnessmatricesofthegivenelement,q,q,fext,fctrl,parethevectorofdisplacementsandslopes,theaccelerationvector,theexternalforcevector,thecontrollingforcevectorandaconstantvectorofthebeamofsize(8x1).Vsisthevoltageofthesensordetectingthebeamdeection.Amodalapproachtodevelopabeammodelhasalsobeenshowntobeusefultopredictthemovementofsmartstructures[ 27 ]. Mq+Kq=fext+fctrl,y(t)=Vs(t)=pT_q(5) 5.3.2.2Flexiblerobotmanipulators Anotherexternalresearchareawhichmightgiveinsightintohowmorphingandappingmaybemodeledisexiblerobotmanipulators.Modelsforexiblemanipulatorsrequiresimilaranalysistomorphingwingssincetheyarerequiredtoperformundervariousknownloadingconditionsandtomovequicklythroughalargerangeofmotionswhilemaintaininghighlyaccurateandprecisepositioncontrol.Flexiblemanipulatorshavebeenexaminedduetotheiradvantagesincludingsmalleractuators,saferoperationduetoreducedinertia[ 68 ],largerworkvolume,higheroperationspeed,greaterpayload-to-manipulatorweightratio,lowerenergyconsumption,bettermaneuverabilityandbettertransportabilityalthoughtheyaremoredifculttocontrol 108

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duetodeformationandvibration[ 126 ].Researchersinthiseldaremuchmoreusedtodealingwithlargerkinematicmotionsthanthoseexperiencedbydesignersofaircraftwings,butarenotasusedtohavingtodevelopmodelswheretheloadsareunknown,difculttomeasureandtime-varying,astheyareformorphingandappingwings.Roboticjointswhichallowsmotioninall3directions(ex:ballandsocket,6-DOFparallelmanipulator)arerarelyusedsincetheyaredifculttocontrolprecisely.However,agreatdealofworkhasbeenperformedforrevolutejointswithexiblelinks,sincethesearecommonlyusedinpractice.AsimpleEuler-Bernoullibeammodelmaybeused,althoughroboticsystemswithexiblelinksarecontinuousdynamicalsystemswithinnitedegreesoffreedomgovernedbynonlinear,coupledordinaryorpartialdifferentialequations[ 68 ].Euler-Bernoullibeamapproximations,anassumedmodemethod(onlyaccountingfortherstseveralvibrationmodes),FEMandlumpedparametermodelsarewidespread[ 68 ]. In2008,Tokhisummarizedmodelingproceduresforexiblerobotsinto5categories:(1)Lagrange'sequationandmodalexpansion(Ritz-Kantrovitch),(2)Lagrange'sequationandniteelement(FE)method,(3)Euler-Newtonequationwithmodalexpansion,(4)Euler-NewtonequationandFEmethodand(5)Singularperturbationandfrequency-domaintechniques[ 240 ].Partialdifferentialequationbasedapproachesaregenerallynotabletoexpressthedetailsofthedynamicsandusuallyrelyonamass-spring-damperbasedapproach.Numericalapproachestodevelopmodelshavehadsomelimitedsuccess. 5.4OtherModelingMethods Manyothermethodsformodelingexist.Thesemodelsusuallyrelyonagenerallyapplicablebasisforfunctionapproximationandmaybeappliedtoidentifymodelsofphenomenathatcontaindecoupledorcoupledmechanics. 109

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5.4.1AdaptiveandLearning-BasedModels Geneticalgorithms(GAs),articialneuralnetworks(ANNs),radialbasisfunctionnetworksandmanymoreversionsoffunctionalapproximationareoftenusedtomodelthenonlinearitiespresentinroboticsystems.Thesemodelsstartwithlittleknowledgeofthesystemandareexpectedtolearnhowthesystembehavesduringoperationbasedonsomeoptimizationprocedurewhereacostfunctionrelatestheinputtothemeasuredoutput. 5.4.2SurrogateModels Surrogatemodelsapplybasessimilartothoseusedforadaptivefunctionapproximation,suchaslinearapproximations,polynomialresponsesurfaces,Krigingtechniques,supportvectormachinesandarticialneuralnetworks.Surrogatemodelsareusedtoformreduced-ordermodelsoftheoverallsystembehaviorwhichmaybeevaluatedatdramaticallydecreasedcomputationalcostcomparedtoevaluatingtheexactsystemdynamics.Thereducedcomputationallowsacomplexsystemmodeltobemoreeasilyimplementedinpracticeatthecostofhavingthesystemnotexactlyunderstandthedynamicsoftheunderlyingmodel.Incontrasttoonlineadaptiveandlearningbasedmodels,surrogatemodelsachievereducedevaluationtimesbysurfacettingtheinput-outputrelationshipsoftheexactsystematintelligentlyselectedpointsinsimulationandthenevaluatingtheapproximationfunctionduringoperation. 5.4.3HybridModeling Hybridmodelingissimilartograyboxmodeling.Hybridmodelscombinestructuralandempiricalapproacheswherethemainsystembehaviorisdescribedwithaphysicalmodelwhileunknowninternalforcesaremodeledwithsomenonlinearapproximationsuchasaneuralnetwork[ 175 ].Theythereforeobtainablendoftheadvantagesanddisadvantagesofaknownsystemmodelandrelativelyfastfunctionapproximations. 110

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5.5ModelErrorAnalysis Theamountofuncertaintyanderrorsinamodelmustbecharacterizedforatheoreticalmodel.Uncertaintymaybeeitheraleatoryorepistemicwherealeatoryisuncertaintyinherentinthesystem(i.e.disturbances),whileepistemicuncertaintyispresentbecauseofalackofknowledgeaboutthesystem(i.e.unmodelleddynamics)[ 167 ].Errorsdescribehowthemodeldiffersfromsomeidealorspeciedresult(i.e.differencebetweenspeciedvalueofmaterialstiffnessandmeanstiffnessintheactualwing),whilevariabilitydescribeshowmuchvariationispresentinthesystemwithrespecttothemean(i.e.variationinstiffnesscausedbydistributionofmaterialproperties). Modelingerrorsmayberandomerrors(variability)orbias(systematic)errors.Variabilityinasystemmustbecharacterizedforagivenmodelbeforeacontrolschememaybeformulated.Themeanandstandarddeviationoftheerrorbetweenthepredictedresponsebythemodelandthemeasuredresponsefromthesystemmaybeexaminedtocharacterizebiasandvariability.Thecoherencefunctionisusedtomeasuretheaccuracyoftheassumedlinearinput/outputmodel[ 29 ]andthereforemayhelpidentifybias. 5.6AeroservoelasticModelsofMorphingWings Chapter 5 hasshownthatmostdetailedaeroservoelasticmodelsfortraditionalwingstypicallyrelyonsmallperturbationsandexpressthewing-shapeintermsofposition,velocityandaccelerationdependenttermstopredicthowthewingwillbehave.Themodelsmayincorporatestiffness,massandinertiabutoftenrelyonlineartheoryorsuperimposedecoupledmodelingtechniquestomodelthewingmovement. Formorphingwings,thedynamicsoftenintroducetime-varyingeffects,makingthemevenmorecomplicated.A6-DOFmorphingdynamicalmodelforamorphingaircrafthasbeendevelopedextendingthetraditional6-DOFaircraftequationsofmotion[ 159 ].AmeriusedapolynomialRitzBasisFunctiontomodelthekinematicsu(x,y,z,t)=S(x,y,z)q(t)wherethedisplacementvector,u,isexpressedinterms 111

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ofthematrixofshapefunctions,S,andthevectorofgeneralizedcoordinates,q,eachofwhichisdenedasafunctionoftheoriginal(x,y,z)locationonthewingandtime[ 12 ].Amodelwasformulatedasalinearinputvarying(LPV)systemwithamorphingparameterinputforavariablecamberandvariablechord[ 35 ].Anothersimilarmodelgaveasimilarquasi-steadymorphinganalysisofwingsvaryingsweepandtwist[ 80 ].Afullscaledynamicsmodelwhichwasreducedtoalinearparameter-varyingsystemofthedynamicsandalineartime-varyingmodelofthemorphingwasdevelopedbySeiglerfortheNextGenAeronauticssubmissiontotheMASproject[ 14 203 205 ]. 5.7ProposedModelforMorphingWings Currentmodelsformorphingwingsstruggletoincorporatetheaeroservoelasticeffectspresentduringight,especiallytheinteractionscausedbythestructuraldynamics.Thisdissertationgeneratesasimpliedperiodicmodelofthemovementofamorphingwingwithmultiplepiezoelectricactuators.Theseactuatorsmaybeusedtochangethewingtwistangleacrossthespanofthewingortochangethewingchamber.Themodelproposedisamodaldecompositionofthestructuraldynamicsofthewingandthepiezoelectricactuators. Themodelofthemorphingwingthree-dimensionaldeections,[Xm,Ym,Zm]isshowninEquations 5 5 and 5 .Themodelisbuiltbysuperimposingthefrequencyresponseshapeofthewing,SW(!,x,y,z),andthefrequencyresponseshapeofthepiezoelectricactuator,SP(!,x,y,z),atsomelocationsinthree-dimensionalspace,[x,y,z].ThewingshapeSW(!,x,y,z)isdenedatalllocationsonthestaticwingshape,[x,y,z],whilethepiezoelectricshapeSP(!,x,y,z)isonlydenedforsomesubsetofthelocations[x,y,z]wheretheactuationisapplied. Themodelisafunctionofthefrequencies,[!i,!o]andamplitudes,[i,o]appliedtotheinboardandoutboardpiezoelectricactuatorsrespectively.Atimelagexpressedasafunctionofthephaseoftheperiodicmorphing,,isincludedtomodeltimedifferencesinthephasesoftheinputactuationtoeachactuator.Dampingfactors, 112

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[Wi,Wo,Pi,Po]areappliedtoscalethedeectionoutputsaccordingtotherelativemagnitudesofcontributionsofthestructuraldynamicsandthepiezoelectricactuatortotheoverallmorphingdeections. Thespanwiseandchordwisedeections[Xm,Ym]aresettozerosincenothefrequencyresponseinthiscaseisnotmeasuredinthesedirections.Theout-of-planedeectionsZmaredenedasthesuperpositionoffour1storderFourierterms.Thersttermdescribesthecontributionofthewingstructuraldynamicsatthefrequencyoftheinboardactuatorinputfrequency.Theoutputisscaledbytheactuatorinputmagnitudeandthedampingterm.The2ndtermdescribesthecontributionofthewingstructuraldynamicsatthefrequencyoftheoutboardactuatorinputfrequencyandisagainscaledtoreecttheactuatorinputmagnitudeaswellasadampingtermandinputlag,.The3rdand4thtermrepresentthestructuraldynamiccontributionsoftheinboardandoutboardactuatorsrespectively.Thedampingcoefcientsonthe3rdand4thtermswhichwouldsimulatetheloweramplitudespresentduetoincorporatingtheactuatorsintothewingstructure. Xm(x,y,z,i,!i,o,!o,,Wi,Wo,Pi,Po,SW(!,x,y,z),SP(!,x,y,z),t)=0 (5) Ym(x,y,z,i,!i,o,!o,,Wi,Wo,Pi,Po,SW(!,x,y,z),SP(!,x,y,z),t)=0 (5) Zm(x,y,z,i,!i,o,!o,,Wi,Wo,Pi,Po,SW(!,x,y,z),SP(!,x,y,z),t)=WiSW(!i,x,y,z)isin(2!it)+WoSW(!o,x,y,z)osin(2!ot+)+PiSP(!i,x,y,z)isin(2!it)+PoSP(!o,x,y,z)osin(2!ot+) (5) 113

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5.7.1MorphingWingModelBasisSelection/SystemIdentication Thesystemidenticationprocessisperformedbyseparatelymeasuringthefrequencyresponsesofthewingandthepiezoelectricactuator,[SW(!,x,y,z),SP(!,x,y,z)]throughgroundvibrationtesting.Thedeectionshapeduetothestructuraldynamicsmaybedeterminedwithanon-contactlaserdopplervibrometerorothertypeofmodaltest.Ideally,thefrequencyresponseshapeofthewingwouldbedeterminedbyinternalexcitationofthepiezoelectrics,andthereforeSW(!,x,y,z)andSP(!,x,y,z)wouldbeequal.Inthisidealcasethecombinedwingfrequencyresponseshapewouldbelikelyneedtobeformedbythecombinationoftheseparatelymeasuredfrequencyresponseshapesfortheinboardactuatorandtheoutboardactuator.Ameanstodeterminethecouplingbetweenactuatorsatdifferentvaluesofthecontrolinputsmightalsoberequired. Tofullyidentifythemodel,themodelwouldneedtobecomparedagainstmeasurementsofthemorphingdeection.Thewingmodelcouldthenbescaledappropriatelybydeterminingthecoefcients[Wi,Wo,Pi,Po]whichbestmatchedthemorphingdeections[Xm,Ym,Zm]asafunctionofthecontrolinputs[i,!i,o,!o,].Amodelofthewingdeectionsasafunctionofthewingloadingandtimewouldenhancethemodelaccuracy.Iftheloadingonthewingisperiodicandthewingmass,inertiaandstiffnesspropertiesareknown,themodeldesignwouldbestraightforward. 5.7.2MorphingWingModelLimitations Themodelassumesthemorphingdeectionsmaybetreatedasaperiodicfunctionoftimeandisformulatedasareduced-orderFourierseriesapproximationofthemorphingwingdeections.Thereforethismodelformorphingwillnotbeapplicablewhenspanwiseorchordwisedeectionsarelarge,norwhenthedeectionsarenonlinearornon-periodic.Themodelisusefulwhenthefrequencyresponseofthewingmaybemeasuredwithrespecttothemorphinginputsandwhenthemorphingisrelativelysmallinamplitude. 114

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5.8AeroservoelasticModelsofFlappingWings Forappingwings,thedynamicsbecomemorecomplicatedsincethespeedandoftenmagnitudeofactuationaregreaterthanformorphing,presentingmanyproblemsforresearchers.Inertialeffectsandaerodynamicloadingsmustbeaccountedforandwingdeectionsaretoolargetosatisfysmallangleapproximationsorbeanalyzedwithperturbationtheory.Inthesearchtogainshort-terminsightintoperformance,appingwingresearchershaveoftenreliedonkinematicmodelswithrigidwings[ 256 ].Attemptstomodelappingwithexibilityhaveindicatedthataerodynamicforcesincreasewhenexibilityisconsidered[ 118 ]thereforeexibilityeffectsmustbeincludedinmodelswhichwillbeusedforcontroldesign.Time-varyinginertiaaffectsthevehicledynamics[ 82 ]andwingdynamics,makingitquitedifculttoformulateageneralmodelforapping.Decoupledtheoreticalappingmodelsforthekinematics,structureandaerodynamicsbeenformulatedwithwidevariationsinthemodeldelityforeachtypeofphenomena[ 17 37 179 214 219 237 ].Thesemodelsareoftenusedtoofferinsightintoadesiredrangeforactuation(ex:appingfrequencies[ 255 ])ortoevaluatetheeffectivenessoftheaerodynamicmodel,butareinsufcientforcontroldesign.Coupledapproachesdoexistforapping,usingcomputationalstructuraldynamics(CSD)codeswhichanalyzebothwingexibilityandunsteadyaerodynamicssimultaneously[ 49 ]buttheseapproachesremaincomputationallyexpensive,limitingtheirabilitytoexplorethefeasiblekinematicspace. Unliketheoreticalmodels,experimentally-basedmodelsensurethatallaeroservoelasticeffectsareincorporatedintothenalmodel.Naturesyersandarticialexampleshavebeenusedtomodelapping.Anexperimentallybasedmodelwasrstformulatedforahawkmothwingbyttinga5thorderFourierseriestoappingdatatominimizetheRMSerrorbetweentheexperimentaldataandthemodelpredictionofthewingtipmovementovertime[ 268 270 ].Thismodelwasthenformulatedforageneralcaseforaexibleappingwingwherea3termFourierserieswasshowntobesufcient[ 122 211 ]. 115

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Theresultisthetraditionalmodelof1-DOFappingwhichformulatesthedihedralangleasaFourierseriesgiveninEquation 5 intermsoftheappingamplitudeisAn=p C2n+B2nandtheappingphaseisn=arctan(Cn=Bn)[ 86 124 ].Asimilarrst-orderFourierserieshasbeenusedtodeterminetheoptimalkinematicsfor1-DOFapping[ 86 ].AmodalbasedmodelwasalsoformulatedforaBlowywingasthesumofBeziercurvesbestttotheveinsoftheinsectwing[ 142 ]. =C0+2Xn=1[Cnsin(2n!t)+Bncos(2n!t)](5) TheFourierseriesformulationhasbecomethetraditionalmodelfor1-DOFapping,butitdoesnotsufcientlydescribetheaeroservoelasticresponseofappingasafunctionofthecontrolinputs.SincetheFourier-basedmodelsdonotdescribethemotionofthewingintermsofthecontrolactuators,themodelcannotbeusedtocontrolthewing-shapemovement.ArstorderFouriermodelofabatoidwingpoweredbyservosplacedalongthechordwasformulatedintermsofthecontrolactuatorsbutthemodeldidnotaccountforthewingexibility,norwasitcomparedtotheactualwingresponse[ 281 ].Inaddition,althoughtheFouriermodelcouldbeusedtomathematicallyexpressthemotionforanentirewing,itcannotpredicthowthewingisexpectedtomovebasedonthephysicalphenomenathatareoccurringsuchastheappliedactuationorstructuraldynamics.Inaddition,aFourier-basedmodelingapproachdoesnotdescribetherelationshipbetweenonepointonthewingandanother,limitingthepredictiveabilityofthemodel.Thereforeresearchershavebeenattemptingtouseothertechniquestomodeltheappingmovementoftheentirewing. Anaeroservoelasticmodelbasedonhighspeedphotographywasmadeforacarbon-berandCapranmembranewing[ 129 ]whichmodeledperiodicwingdeectionsignalswithanestedpolynomialbasisfunctionwithrespecttothecontrolactuators;however,thismodelwaslimitedbytheaccuracyofthettingprocessanddidnotpredictthefullwingmovement.Traditionalaeroservoelasticmodelsalsoexpressthemovement 116

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ofthewingintermsofaspring-mass-dampersystemwithwingstiffnessdeningthewingexibilityinsteadofcharacterizingthewingresponsedirectly.Themodalpropertiesoftheaeroservoelasticdynamicsmightprovidereduced-ordermodels;however,thenonlinearandtime-varyingpropertiesobservedduringeachappingcyclewouldviolateassumptionsofmodalanalysis[ 10 ].Therefore,thesetypesofapproachesalsoappeartofallshortofwhatisneeded. Severalapproacheswhichmodeltheaeroservoelasticresponseoftheentirewingexist.Aappingratshnwasmodeledasathree-dimensionalharmonicallyoscillatingsurface(i.e.atravelingwave)[ 54 241 ]asshownbyEquation 5 [ 54 ].Thefunctionh(x,z,t)describestheinstantaneouslocationofallpointsonthewing,h(x,z,t)wherethenon-dimensionalizedwingcoordinatesarexinthespanwisedirectionandzinthechordwisedirection,tistime,h0istheamplitudeoftheleadingedgeoscillationatagivenpointonthespanandthelinearamplitudevariationinthespanwisedirectionis.Theappingfrequencyis!andi=p )]TJ /F3 11.955 Tf 9.3 0 Td[(1whilethetravelingwavemovesrearwardswithaspeed,c,whilethelinearamplitudevariationinthechordwisedirectionisgivenasandthemeanangleofattackoverthestrokeisspeciedby.Themodelmaybeusefulasabasisfora1-DOFappingmodel,butitdoesnotconstituteageneralizedmodelforwingmovementifthereareotherformsofactuationorifthechordwiseexibilitycannotbedescribedwithatravelingwave. h(x,z,t)=z[[h0+(x+1)]exp(i![t)]TJ /F3 11.955 Tf 11.95 0 Td[((x+1)=c]))]TJ /F3 11.955 Tf 11.96 0 Td[([(x+1)]](5) Aslightlysimilarmodeldescribedthemotionofanforanunmannedunderwatervehicleunderdevelopmentwherethemotionofthetipofthewingwasdescribedbytheanglewhichsuperimposedtheinitialangle0withtheactivelyspeciedrotationbulk,anddeformationangleatsomechordwiselocationi,rel(i),asshowninEquation 5 [ 180 ].Themodelishelpfulasapossiblebasisfunctiontomodeltheshapeofeachspanwiseline.However,themodeldoesnotaccountforwingbending 117

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inthechordwisedirection,norotherformsofmorphinganglesandonlyspeciesthelocationofthewingtipasafunctionoftime.AslightlymorecomplexmodelwhichsplitsthemotionofapectoralnbetweentheupstrokeanddownstrokeandintoispanwiselinesrotatedaboutthecentralaxiswasgivenbyShoele[ 208 ]. i(t)=kbbulk(t)+kfrel(i)(t)+0(5) Themotionofanofabirdwrassewasalsocharacterizedwithphotogrammetryfrom5markersonthen.Themovementwasviewedasoscillationalonganarcinaspherewherethespherehadaradiusequaltothewinglength.Aprincipalcomponentsanalysis(PCA)wasusedtondthestrokeplane,butafullrepresentationwasnotpresented[ 253 ].Highspeedphotographyhasalsobeenusedtodevelopahighdelitygeometricalmodelofaappingbluegillsunshnusingproperorthogonaldecomposition(POD)toisolatethedominantdynamicsofthemotionandavoidthecomplexitiesinvolvedwithaprecisekinematicmodel[ 234 ].ThisPODmethodissimilartothreeotherusefulmethods:theKarhunen-LoeveDecomposition(KLD),principalcomponentanalysis(PCA)andsingularvaluedecomposition(SVD)[ 39 ].Themovementwasrstcharacterizedwith280nodesin3Dspaceandthedisplacementofeachnodewasputinamatrixandsubjectedtoasingularvaluedecomposition(SVD)todecomposethemovementintothemostimportantmodes.Thenbeatwasdecomposedinto19distinctmodes,althoughtherst3modeswhichincludedacupandsweep,rotationandexpansionandtipickaccountedfor67percentofthevarianceinthen'stotalmotionandpredicted92percentofthethrust[ 39 234 ].ThePODtechniquewasveryeffective,buttherelationshipbetweentheactuationandthewingoutputremainsambiguous,limitingtheapplicabilityofthetechniqueasamodelforcontroldesign.EvidenceofthisfactisseensinceBozkurttasalsogaveamodelexpressedinEquation 5 whichsplittheappingandfeatheringangleintosingleFourierseriesforimplementationintoaCFDcodewherencorrespondstothespanwise 118

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segmentnumber,Aisthepitchamplitudeandisthephasedifferencebetweenthepitchandheavemotion.ThisapproachwascomparedtothePODmethod[ 39 ]. h(t)=hisin(2ft),(n)(t)=A(n)sin(2ft+)(5) 5.9ProposedModelforFlappingWings Currentmodelsdonotreecttheaeroservoelasticnatureofappingwings.Researchershavestruggledtodenemetricswhichcanbeusedtodetermineaerodynamicperformancewhichareeasilyobservableandmayberelateddirectlytothecontrolactuation.Althoughcomplex,thefullwingmovementisthemostcomprehensiveandeasilyaccessiblemetricwhichmaybeusedtodeneperformance,irrespectiveofwhatphenomenahavebeenusedtogeneratethatmovement.Whilewemayultimatelybeabletocomputetheaeroservoelasticresponsecorrectlyandfastenoughtodesignaappingwing,fornowtheloadingonthewing,theactuationmethod,andstiffnesspropertiesofthewingmaybeassumedtobeunknown.Assuch,anewgray-boxprocedureisproposedtounderstandappingwhichleveragesthetheoreticalunderstandingpresentedinChapter 5 andtheexperimentaldataobtainedusingthetechniquesfromChapter 3 ExperimentalresultsasshowninChapter 4 havealreadyindicatedthebasicnatureofaeroservoelasticbehaviorforasetofappingwings[ 132 ].Undermostconditions,theappingmovementisperiodic.However,thestudyalsodemonstratedthevariationsinthetime-frequencydomainwhenconsideringchangestobothappingamplitudeandappingfrequency.Ineachcase,itisshownthatapurely-sinusoidalinputdoesnotgenerateapurely-sinusoidaloutput.Theactualdeectionsanddeformationscanbesignicantlyvariedfromtheinputandmayhavetimeandfrequencyvaryingcomponents.Thesetime-frequencydependenciesarealsoshowntobehighlydependentonthestiffnessthroughoutthestructure. 119

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5.9.1AeroservoelasticBehaviorsofFlappingWings Severalcomplicationsarisewhentryingtomodelanaeroservoelasticsystemsuchasaappingwing.Theinuenceofthestructuraldynamicphenomena,thecontrolactuators,theaerodynamiceffectsandthewingexibilityallvarysubstantiallyacrossthewing.Mechanismeffectsmayappearnearthewingroot,butarelessprevalentfurtheracrossthespanofthewing.Theresultishighlynonlinearbehaviorwithcouplingsthatarenoteasilymodeled. EvidenceofthiscomplicationisseeninFigure 5.9.1 ,whichdemonstrateshowevenoncetherigidbodymotioniseliminated,substantialwingmotionstillmustbepredicted.Itisfairlystraightforwardtouseaseriesofsinusoidstomodelthemovementofanyparticularpointonthewingandtodecouplethecommandedrigidwingeffectsofthecontrolactuators.However,evaluatingthecoupledeffectsofthecontrolinputsandtheeffectsofexibilityandaerodynamicsasfunctionsofthecontrolinputsismuchmoredifcult.Thisdifcultyiscompoundedbythefactthatthestructuraldynamicsmaydrasticallyincreasetheirrelativecontributionstowingdeformationsatacertainactuationfrequencies.Aerodynamicloadingsaretime-varyingsotheireffectsaremainlyobservedassignicantatthetopandbottomofthewingstroke.Figure 5-5 illustratesthetime,frequencyandwinglocationdependentphenomenabylistingtherelativesizeofthecontributionsofvariousphysicalphenomenatothenalappingdeectionsforvariouslocationsonthewing. Flexibilityeffectsthatcontributetowingdeformationsmustalsobeincorporatedintothemodel.ThedependenceofthewingdeectionsonwingexibilitymaybeapproximatedbyEquations 5 5 and 5 ,whichdescribesthewingdeformationsXf,Yf,Zf,thatis,themovementtosuperimposeontherigidbodydynamics.Themodelsofthedeformationsshouldbeabletoincludethecoupledeffectsofthedynamics,aerodynamics,andstructuraldynamicsonthewing.Whilenumerousaerodynamiceffectsmayhaveeffectsonthedeections,twoeffectsareclearly 120

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Figure5-4. ComparisonofRigidWingBodyMovement,FullWingDeectionandDeformationfromUnaccountedforAeroservoelasticEffects Wing-4(DiagonalBatten)at75PercentofFlappingWingSpanfor:Row1:LeadingandTrailingEdge,5Hz,+/-10deg,Row2:LeadingandTrailingEdge,20Hz,+/-17deg,Row3:LeadingandTrailingEdge,20Hz,+/-35deg recognizable,especiallyathighfrequencies:theinertialsnapandwash-in/wash-outeffect[ 277 278 ].Theseeffectsaremodeledbyasinusoidaltermattheappingfrequencyandanothersinusoidaltermattwotimestheappingfrequencywithacorrespondinglag.Resonanceoftherstbendingmodeiscommonlyseenand 121

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Figure5-5. RelativeSizeofContributionsofAeroservoelasticPhenomenatotheFinalFlappingDeectionsasaFunctionofWingLocation alsoneedstobeincludedinthemodel.Effectsthatmayalsobepresentincludetime-varyingaerodynamiceffectscausedbyleadingedgevortices,wakecapture,clap-and-ing[ 63 211 ]buttheseaerodynamicforceshavebeenshowntoplayaminorroleindeterminingwingdeectionsascomparedtoinertialeffects[ 55 ].Structuralfatigue,smallfractures,smalldelaminationsbetweencompositelayersandincompletebondingbetweenthestructureandmembranemayalsoplayarolebutfornowareneglected.Severalofthesephenomenamaybetimeorfrequencyvarying.Thetimeandfrequency-varyingnatureofthecombinationoftheseeffectshasbeenobservedwhenanalyzingwingdeformations[ 133 ].Alloftheseunknowneffectsareincorporatedintoathirdsinusoidaltermatsomeunknownfrequency.Thislastsinusoidaltermisprimarilyintendedtocapturetheeffectoftherstbendingmodeifitispresent,butgenerallyshouldhelpcapturethesinglemostdominantfrequency-varyingeffect. 122

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Xf(x,y,z,,!,t))]TJ /F7 11.955 Tf 28.56 0 Td[(c1xjsin(2!t)j (5) Yf(x,y,z,,!,t)c2sin(2!t) (5) Zf(x,y,z,,!,t)c3xsin(2!t)+c4ysin(2!t)+c5xysin(2!t+) (5) Therepresentationoftheaeroservoelasticdynamicsareformulatedasacombinationofrigid-bodydeectionsandexible-bodydeformations.Therelativemagnitudeofeachcontributionisdeterminedbyasetofcoefcients,A1,...,A11,whichvaryacrossthedesignspace.TheresultingexpressionforthelateralmotionofXisgiveninEquation 5 ,thelongitudinalmotionofYisgiveninEquation 5 ,andtheverticalmotionofZisgiveninEquation 5 X(x,y,z,,!,t)=A1(,!)p x2+z2cos(sin(2!t)))]TJ /F7 11.955 Tf 9.3 0 Td[(A2(,!)xjsin(2!t)j (5) Y(x,y,z,,!,t)=A3(,!)y+A4(,!)sin(2!t) (5) Z(x,y,z,,!,t)=A5(,!)p x2+z2sin(sin(2!t))+A6(,!)sin(2!t)+A7(,!)sin(4!t+A8(,!))+A9(,!)sin(2A10(,!)t+A11(,!)) (5) 5.9.2FlappingWingModelBasisSelection Thecoefcientsinthemodelservetoscalethecontributionsofrigid-bodydynamicsandexible-bodydynamics.Assuch,somephysicalinterpretationofeachcoefcientcanbedescribedasinTable 5-1 5.9.3FlappingWingSystemIdentication/OptimizationMethod Amodelcanbedeterminedfromexperimentaldata.Suchdatamustcontainthelocationsofpointsalongthewinginresponsetoappingwithparametersacross 123

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Table5-1. RationaleforInclusionofBasisCoefcients Coef.PhysicalIntepretation A1ShiftofRigidBodyMovementAlong^xDirectionA2AeroelasticPhenomenaContributiontoXDeectionat!A3ShiftofRigidBodyMovementAlong^yDirectionA4AeroelasticPhenomenaContributiontoYDeectionat!A5ShiftofRigidBodyMovementAlong^zDirectionA6AeroelasticPhenomenaContributiontoZDeectionat!A7AeroelasticPhenomenaContributiontoZDeectionat2!(Literaturesuggested)A8PhaseShiftofAeroelasticPhenomenaat2!(Literaturesuggested)A9MagnitudeofResonance/OtherUnknownFrequencyDependentPheonmenaA10FrequencyofResonance/OtherUnknownFrequencyDependentPheonmenaA11PhaseofResonance/OtherUnknownFrequencyDependentPheonmena thedesignspace.Inthiscase,considerthattheappingisgeneratedatasetoffrequenciesgivenby(!1,...,!n!)andamplitudesof(1,...,n).Thedatadescribesthelocationofasetofnppointsalongthewinggivenas(x1,y1,z1),...,(xnp,ynp,znp). Anoptimizationcomputesthecoefcientsneededtodescribetheaeroservoelasticdynamicsatasetofappingparameters,!jandk,fromthedesignspace.Thecoefcientsof( A1,..., A11)correspondtothebasisforthemodeledresponseof(X,Y,Z)fromEquations 5 5 .Thesecoefcientsminimizeacostfunctionthatisthenormoftheerrorbetweenthemeasureddata,( X, Y, Z)andthepredictedresponseasgiveninEquation 5 A1,..., A11=argminA1,...,A11 Zt0 npXi=1)]TJ ET q .478 w 216.67 -464.92 m 226.33 -464.92 l S Q BT /F7 11.955 Tf 216.67 -474.89 Td[(X(xi,yi,zi,!j,k,t))]TJ /F7 11.955 Tf 11.95 0 Td[(X(xi,yi,zi,!j,k,t)2+)]TJ ET q .478 w 188.31 -496.97 m 198.46 -496.97 l S Q BT /F7 11.955 Tf 188.31 -506.95 Td[(Y(xi,yi,zi,!j,k,t))]TJ /F7 11.955 Tf 11.95 0 Td[(Y(xi,yi,zi,!j,k,t)2 (5) +)]TJ ET q .478 w 188.31 -523.87 m 197.39 -523.87 l S Q BT /F7 11.955 Tf 188.31 -533.84 Td[(Z(xi,yi,zi,!j,k,t))]TJ /F7 11.955 Tf 11.96 0 Td[(Z(xi,yi,zi,!j,k,t)2dt1=2 Amodelmustbegeneratedthatisparameterizedacrossthedesignspace;however,themodelasrepresentedby A1,..., A11describesthedynamicsonlyatthen!bynsetofdiscretepointswithinthedesignspace.Thecontrollerbynecessityshouldhavethefreedomtochooseappingparametersatvaluesotherthan!1,...!n! 124

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and1,...n.Therefore,continuousfunctionsofAi(!,)mustbegeneratedforeveryi2[1,11].ArepresentativeoptimizationisgiveninEquation 5 thatsearchesoveragenericsetoffunctionsgivenasf:R2!R. Ai(!,)=argminfvuut n!Xj=1nXk=1)]TJ ET q .478 w 251.25 -100 m 259.32 -100 l S Q BT /F7 11.955 Tf 251.25 -109.98 Td[(Ai(!j,k))]TJ /F7 11.955 Tf 11.95 0 Td[(f(!j,k)2(5) 5.9.4FlappingWingModelLimitations Themodelproposedaboveisexpectedtobesomewhatlimited.Thisistruebecausethemodelisanonlinearfunctionofthekinematicparameters.Themodelmaybedescribedasgray-boxthoughsinceaphysicalinterpretationofthedataandtheinputparametersledtothebasisselection.Themodelisonlyvalidfor1Doratmost2Dmorphingorappingwingmovement. 125

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CHAPTER6WING-SHAPECONTROLAPPROACHES 6.1BackgroundforWingShapeControl Accurateandefcientcontrolofwing-shapeforaeroservoelasticsystemsisanextremelydifculttaskbecauseofthenonlinearitiesandcoupledeffectspresentinthesesystems.Aprimarychallengeforwing-shapecontrolishowtoexpressthenonlinear,coupledwing-shaperesponseasafunctionofthecontrolinputsandwingstructureinthesimplestwaypossible.Thegoalistobeabletomeasuretheresponseofagivenwingandtailorthecommandedkinematicssuchthatsomedesiredwing-shapewillbeobtained.Chapters 3 4 and 5 haveshownthecomplexitiesofmodelingthistaskandclearlyshownthatanewapproachisneededtomodelthewing-shaperesponse.Thecontrolschemeishighlydependentonthemodelingapproach,andthusfarpreviousapproachestocontrollingwing-shapehavenotprovidedthemeansnecessarytocontrollargeamplitudeaeroservoelasticresponseswithobviouscouplingswiththestructuraldynamics.ThereforeacontrolsynthesiswhichiscapableofbeingappliedtotheaeroservoelasticmodeldeterminedinChapter 5 willbeformulated. Wing-shapecontrolisnotanewidea,buthasbeengrowingrecently.AmethodbasedonpotentialeldswasusedbyDang[ 59 ].Thecontrolofexiblestructuresundertime-varyingloadingshaveoftenbeenaddressedintheeldsofroboticsandcivilstructuresinadditiontotheaerospaceeld[ 105 ]. Seiglerhasprovidedinsightintocontrolformorphingwings[ 202 ].Amulti-body,nonlinear,aeroelasticmodelandlinearcontrolsynthesishasrecentlybeendeveloped[ 285 ],butthemodeldoesnotdescribethemorphingdynamicsandlinearizesaroundthetrimcondition.Severalresearchershaveusedthistypeofapproachtoaddressmorphingaircraftwherethemorphingoccursrelativelyslowly[ 80 ]. Hoetal.attemptedtouseanadaptivecontrollawtoaccountfortheunsteadyowforappingwings[ 91 ].Whitmercreatedamodalformofthestructuraldynamics 126

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coupledwithastaticaerodynamicmodelusingthevortexlatticemethodtogivealinearaeroelasticmodelthatwascontrolledwitharobustLQGapproach[ 262 ]andanH-innityapproach[ 263 ]totrackacommandedliftandrollmoment.Manyotherattemptshavebeenmadetoformulateacontrollerforarealisticappingwing[ 62 193 280 ].However,manyofthemstilldonotincorporatetheaeroservoelasticnatureofaexibleappingwing.Thecontrolactuatorsareusuallyassumedtohaveadirectrelationshiptotheperformanceofthewing.Whilegenerallythisistrueforstandardwingswithaileronsandelevators,exibilityandstructuraldynamiceffectsaremuchmoreprevalentinappingwingsbecauseofthetimeandfrequencydependentnatureoftheirkinematics.Theseeffectscauseaeroservoelasticeffectswhichareinherentlyaccountedforifthefeedforwardcontrollerreliesonanexperimentallybasedmodelofthewingmovementinsteadofrelatingtheperformancetokinematicparameters. 6.2ControlSynthesis 6.2.1Openvs.Closed-LoopControlApproaches Openloopcontrolprovidesaninputtoaplantmodel,buthasnowayofknowingiftherearedisturbanceswhichmaybeaffectingtheoutput.Closed-loopcontrolusesfeedbackfromsomesensortoupdatethestatesofthesystemandideallydrivethesystemoutputtomeetsomedesiredoutputbyminimizingtheerrorbetweenthemeasuredoutputanddesiredoutput. 6.2.2LinearControlApproaches Traditionallinearcontroltechniquesrequiretheconstructionofasystemmodelwhichisnotafunctionofthecontrolinputs.Thesystemresponseisconstructedbysuperpositionofthesystemdynamicsandthecontrolinputs.Linearcontrolapproachesworkwhenthesystemdynamicsmaybeexpressedasalinear,time-invariantfunction. Kumardevelopedalinearquadraticregulator-basedapproachforatwo-dimensional,two-degreeoffreedommorphingstucturewhichmovesbetweennodesinagraphoftheperformanceenvelope[ 115 ]. 127

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6.2.2.1Linearparametervaryingapproaches Thelinearparameter-varying(LPV)approachisusedwhenthereissubstantialchangeintheplantmodelovertime.LPVisusedwidelysincelineartechniquesmaybeappliedandsystemscommonlychangethroughouttheiroperation,forinstance,whenanaircraftburnsfuel,therebychangingthegrossweightoverthecourseoftheight.Alookuptablemaybeusedtospecifythecontrolgainsnecessary. LPVapproacheshavebeenappliedformorphingsystems[ 155 ],buttheseapproachesassumethatmorphingoccursslowly.Baldelliet.al.usedinnerloopcontrolgainstoprovidestabilityforanaeroservoelasticmorphingwing,whileanouterlooplinearparametervaryingcontrollergaverobuststabilityandperformance[ 24 25 ]. 6.2.2.2Robustcontrolapproaches Robustcontrolattemptstoformulatetheparametricuncertaintyinanaeroservoelasticsystemsuchthatacertainlevelofperformancemaystillbeguaranteed.Heinzeevaluatedtheapplicabilityofrobustcontroltoolsforutteranalysiswithmodeluncertaintyandvariation[ 89 ].H1controland-synthesisarecommonrobustcontrolapproachesandwereinvestigatedontheBenchmarkActiveControlsTechnology(BACT)project[ 226 ].Robustcontrolapproachesarerobustwithrespecttomodelingerrors,nonlinearity,unmodeleddynamics,measurementerrorsandexternaldisturbances[ 163 ]. 6.2.3NonlinearControlApproaches Nonlinearcontrolsynthesisattemptstoguaranteesystemstabilitydespitenonlinearitiespresentinthesystem.Anonlinearapproximationofthesystemmodelisused,sothisapproachmayworkwellinthepresenceofclearsystemunderstanding,butislessapplicabletomorphingandappingwherethesystemmodelisnotcompletelyunderstood. 6.2.4AdaptiveControlApproaches Adaptivecontrolalsousesagenericfunctionapproximationtodeterminethesystemmodelonline.Thistypeofcontrolisusefulforsystemswithmanyunknown 128

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features,butoftenencountersissuesinpracticeduetothemanyassumptionswhicharerequired.Systemknowledgemaybeusedtoimprovethemodelbasisbeyondagenericbasis,sothebasicprincipleofadaptivecontrolmaystillbeusefulformorphingandappingwings.Anadaptivetechnique,StructuredAdaptiveModelInversion(SAMI),hasbeenusedtocombinefeedbacklinearization,dynamicinversionandstructuredmodelreferenceadaptivecontrol[ 247 ]toachieveoptimalwing-shapeswhiletrackinganaltitudetrajectoryintermsoftheEulerangles.AnadaptivecontrollerhasalsobeenappliedonNextGen'sMFX-2morphingaircraft[ 78 ]. 6.2.5OtherControlApproaches SRIInternationalusedahierarchicalcontrolmethodologywiththewavelettransformtocontrolenergyatspecicallyaheightofthewaveletmap[ 50 ].Thetime-frequencydependenciespresentinaeroservoelasticsystemslikemorphingandappingmakethistechniqueworthyoffurtherinvestigation. Inputshapinghasbeenusedtoaddressaroboticsystemwhichhasmultipleresonantfrequencies[ 240 ].Priorworkonexiblerobotshassuggestedthatopenloopshapecontrolcanresultinlongmove(response)time,instabilitycausedbyunsuppressedmodesandcontrollerrobustnessinresponsetoalargechangeintheactuatordynamics.Otherapproachesincludingacomputedtorqueapproachandbang-bangcontrolrelyonextremelydetailedmodelsofthesystem,whichmayresultinlargevibrationsifthesystemexperienceslargedisturbancesorthebang-banginputisnottimedcorrectly[ 240 ].Geometriccontrolwillalsobeconsideredasinspirationforthenalcontrolapproach. 6.3SensorFeedback Boththesensortypeandsensorplacementarecrucialdesigndecisionsforaircraftutilizingwing-shapecontrol.Straingaugesonlymonitorthelocaldisplacement,andthereforemaynotbethebestchoicetoprovidepositionfeedbacksincethedatamayrequiresubstantialcomputationbeforeitcanbeuseful.Itmaybepossibletousea 129

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singlepiezoelectricbattenplacedalongthechordat75percentofthespantomeasurethewingdeection.Anaccuratelybuiltmodelofthepiezoelectricresponseshouldprovidesufcientamountsoffeedbackandadownstrokeorupstrokecouldbeeasilydifferentiatedsincethevoltagemeasuredwouldbedifferentduetotheasymmetricnatureofthepiezoelectric-carbonbercomposite.Accelerationfeedbackisknowntoproduceasmootherandfasterresponsethanpositionfeedback[ 240 ],butexperienceslargerovershoot. PreviousworkbyCombeshassuggestedthattheperformanceoftheexterior1/5thofthewingmaybeusedtodescribetheperformanceofthewing[ 54 ].Wuusedtherelationshipbetweentheleadingandtrailingedgeat80percentofthespantodenethestructuraldeformation[ 275 ].Theseapproachessuggestthattrackinghowthetrailingedgenearthetipofthewingbehavesmaybekeytoobtainingthedesiredperformance.Inaddition,trackingthispointmaybesufcienttoapproximatetrackingthefullwingdeformation. 6.4ControlActuation Desirablecharacteristicsofactuatorsusedforcontrolinclude(1)Consistentresponseacrossdevicesandcycles,(2)Highmechanicaltolerances,(3)Loworpredictablehysteresis,(4)Lowcreep,(5)Largeforce/loadcarryingcapability(tradedwithdisplacementcapability)anddynamicrange,(6)High-frequencyresponse,(7)Linearitywithrespecttotheappliedeld,(8)HighCapacitance(abilitytoholdanelectricalcharge).Huberetal.providedausefulguideforactuatorselectionbasedonperformanceindicesforagivenactuatortype,includingactuatorswhichusesmartmaterials.Theseindicesincludemaximumlimitsonstresswhichtheactuatorcanachieveforagivenstrain,modulusranges,powerdensityandactuationfrequency[ 97 ]. Piezoelectricactuatorsproduceashapechangewhichisgenerallylinearlyrelatedtoanappliedvoltage.Piezoelectricityisathirdranktensorpropertysothesolidmustnothaveacenterofsymmetrytobepiezoelectric.LeadZirconate-Titanate(PZT) 130

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isthemostwidelyknownmaterial,butseveralpolymersincludingpolyvinylidenediuoridemayalsoexhibitpiezoelectricbehavior.Thesemaybeformedwiththinplates(standard),multi-layerstacks,injectionmoldedshapesandbers.Electrostrictionisasecondordereffectwhichcausesaquadraticrelationshipbetweenstrainandthesquareofelectricpolarization.Anexampleisleadmagnesiumniobate-leadtitanate(PMN-PT).Shapememoryalloysgeneratedeformationinresponsetoamartensiticphasechangeinthematerialatagiventemperature,generatinglargeforces.ExamplesincludeNickelTitaniumNavalOrdinanceLaboratory(NiTiNOL),copper-zinc-aluminumandcopper-aluminum-nickel.Magnetostrictivematerialsundergostraininresponsetoanappliedmagneticeld.Theyareoftentoobulkyforaerospaceapplications,butacommonexampleisterbiumironnavalordinancelaboratory(TERFENOL-D).Electroandmagneto-rheologicaluidswhichchangeviscositywhenexposedtoaeldarerarelyused.Shapememorythermoplasticsmayalsobecomeusefulinthefuture. 6.5FeedforwardControl Acontrolschemeisdevelopedthatconsiderstrackingofadesireddeectionforthewing.Theperformanceofaappingwingisdirectlyrelatedtothedeectionsofeachpointonthatwingacrossaappingcycle.Thecommunityhasinvestigatedavarietyofparameterizationsthatnoteperformancevariationsasafunctionofwingstructureandappingactuation;however,thefundamentalsourceofperformanceasdeterminedbythrustissimplytheaerodynamics.Essentially,thedesignofaappingwingischoosingparametersthatmatchthesetofdeectionsassociatedwithoptimalaerodynamicsandthusoptimalperformance. Apointinthedesignspaceischosenforwhichtheaeroservoelasticresponseisclosesttothedesireddeectionsof(Xd,Yd,Zd).TheresponseusesthebasisinEquations 5 5 withthecoefcientfunctionsdeterminedbyEquation 5 .Anoptimizationsearchesforthispointbyminimizingtheerrorintrackingasgivenin 131

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Equation 6 .Suchanoptimizationconsidersasetofpointsonthewingatallvaluesoftime. (!c,c)=argmin!, Zt0 npXi=1Xd(xi,yi,zi,t))]TJ /F7 11.955 Tf 11.95 0 Td[(A1(,!)q x2i+z2icos(sin(2!t)))]TJ /F7 11.955 Tf 9.3 0 Td[(A2(,!)xijsin(2!t)j)2+(Yd(xi,yi,zi,t))]TJ /F7 11.955 Tf 11.96 0 Td[(A3(,!)yi+A4(,!)sin(2!t))2+Zd(xi,yi,zi,t))]TJ /F7 11.955 Tf 11.96 0 Td[(A5(,!)q x2i+z2isin(sin(2!t))+A6(,!)sin(2!t)+A7(,!)sin(4!t+A8(,!))+A9(,!)sin(2A10(,!)t+A11(,!))2dt1=2 (6) Theresultofthisoptimizationisasinglesetofappingamplitudeandappingfrequency,(c,omegac).Thesevaluesareconstantssothewingisnotallowedtoalteritsappingparametersduringoperation.Inthisway,theparameterizationsofthemodelaroundthedesignspaceallowsthecontroldesigntobeexpressedasatwo-dimensionaloptimization. Thisoptimalpointinthedesignproducesaresponsethatmostcloselymatchesthedesiredresponse.Thisdesiredresponseisactuallygeneratedforeachpointonthewingonlyasafunctionoftime.Suchalimitationtotemporaldependencynotesthattheappingparametersusedtogeneratethatdesiredresponsearenotcriticalbutratheronlytheresultingdeectionsareofvalue.Assuch,differentwingsmightutilizevastlydifferentappingparametersbutgeneratethesameresponsewhichtracksadesiredresponse. ThegenerationofthisoptimalsetofappingparametersrequiresminimizingthenonlinearfunctioninEquation 6 .Theprobabilityofencounteringlocalminimaintheprocessindicatesthatanyappingparametersarethuslikelytobeonlysub-optimal 132

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solutions.Avarietyofstandardtechniquescanbeemployedalongwithdifferinginitialconditionstosearchthedesignspaceandestimatetheparametersassociatedwiththebestlocalminimum. Suchacontrolschemeiseffectivelyafeedforwardcompensator.Theoptimalvaluesofappingparametersarechoseninanoff-linefashionusingamodeloftheaeroservoelasticdynamics.Additionalfeedbackelementswillneedtobeincludedtoaccountfortheeffectsofdisturbancesanduncertainty;however,afeedforwardelementisaninitialaspectuponwhichtobuildamulti-elementcompensator. 6.6Closed-LoopControlApproachforWing-ShapeControl 6.6.1ControlArchitecture Atime-varying,closed-loopcontrolapproachforwing-shapecontrolisformulated.ThecontrolarchitectureisshowninFigure 6-1 .Thefeedforwardroutinebeginswithinitialguessesofthecontrolinputs,appingfrequency,!0andappingamplitude,0shownasu0.Theclosed-loopportionofthecontrollerreceivestheinputsucfromthefeedforwardcontroller,whichconsistofaappingfrequency,!candappingamplitude,c.TheseinputsareusedtogeneratethefeedforwardoutputsfromthefeedforwardcontrollerX,Y,Z,whichareexpressedinFigure 6-1 asy.Aproportionalcontrollaw,K,isusedtominimizetheerror,E,andtrackadesiredappingtrajectoryXd,Yd,ZdexpressedasyDinFigure 6-1 .Theplantmodel,PisformulatedwiththeoptimalcoefcientsA1A11foundinChapter 5 forinputsuc.Disturbancestothewingmovement,D,aremodeledasapercentagegainintheappinginputsfromuCLtouCLD.Sensornoise,N,ismodeledasapercentagegainontheoutputsignalsfromtheclosed-loopcontroller,XCL,YCL,ZCLexpressedasyCLwhichbecomeyCLNafterthenoiseisapplied.ThefeedforwardoutputyffisusedtogeneratethersterrorE,whilesubsequentcalculationsoftheerrorattimestepnusetheoutputyCLN. 133

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Figure6-1. Closed-LoopControlArchitecture 6.6.2ControlUpdateLaw TheerrorforthecontrollawisdenedinEquation 6 .TheerroriscalculatedthroughoutnstepsintimeoverarangeofNpoints.Thesignoftheerrorwillremainconsistentthroughouttheappingcyclewhenusingtheabsolutevalueofthedeections.Theabsolutevalueoftheout-of-planedeectionsaloneareusedtoformtheerrorsothatonlyoneparameterpercontrolinputmustbespecied.Asimilarapproachcouldbeappliedtothespanwisedeectionsofthewingsincetheyhaveadualpeakshapesimilartotheout-of-planedeections,butthechordwisedeectionsaretoocomplextoapplyasimilarapproach. En=mean(jZCLn)]TJ /F8 7.97 Tf 6.58 0 Td[(1j)-223(jZdn)]TJ /F8 7.97 Tf 6.58 0 Td[(1j)jN,E1=mean(jZFF1j)-222(jZd1j)jN(6) ThecontrollerKisexpressedintermsofthegains,G,t=0:T=2,G!,t=0:T=2andaparameterGinc.Thesethreenumberscontainonegainpercontrolinputandanadditionalparametertospecifythepercentagechangeinthegainsduringtheappingcycle.TheparameterGincissetbasedontheportionoftheappingcyclewheretheerroriscalculated.Timet=0:T=2correspondstopositivedeectionintheappingcycle,whilet=T=2:TcorrespondstonegativedeectionwhereTistheappingperiod.Thegainsforthenegativedeectionareincreasedbyapercentage,Ginctoaccountforthelargerwingtipsnapwhichoccursonthedownstrokeduetogravitationalforcesandthetypicallyasymmetricstructureofamembranewing.Anadditionaltwo 134

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usefuldegreesoffreedommaybespeciedtoobtainhigherclosed-loopperformancebydeningthegains[G,t=0:T=2,G!,t=0:T=2]tochangelinearlyintimebetweensomelowerandupperbounds.Thislinearapproachwillgenerategainsdenedbyasawtoothwavewherethegainsareeitherhigherorloweratthepeaksoftheupstrokeanddownstroke. Gn!=[G!,t=0:T=2],[G!,t=T=2:T=G!,t=0:T=2+G!,t=0:T=2Ginc]Gn=[G,t=0:T=2],[G,t=T=2:T=G,t=0:T=2+G,t=0:T=2Ginc] (6) Thecontrollawgeneratesthecontrolinputsn,!nforeachtimestepnintheappingcyclebasedontheproportionalgainsanderrorasshowninEquation 6 ,where[c,!c]aretheinputsdeterminedbythefeedforwardcontrolsynthesis.Theresultingclosed-loopdeectionsX,Y,Znclaregeneratedbyevaluatingtheplantmodelidentiedwith[n,!n]asdescribedattheendofChapter 5 .ThemodelparametersA1:A11determinedtobeoptimalforthegiven[c,!c]duringthefeedforwardcontrolsynthesisareheldconstantwhenevaluatingX,Y,Zncl.Acontrollerwherethen)]TJ /F8 7.97 Tf 6.59 0 Td[(1,!n)]TJ /F8 7.97 Tf 6.58 0 Td[(1couldbeusefulifthedesigneranticipateschangingthedesiredappingprolesignicantlyduringight.ThecontrollerinEquation 6 iteratesaroundthefeedforwardcontrolinputs[c,!c]andisthereforemoreusefulforsteadyapping. !n=!c+EnGn!,!1=!cn=c+EnGn,1=c (6) 6.6.3ControlSynthesis 6.6.3.1Hand-tuning Thecontrolgainsmaybespeciedbyhandtuningthegains.Allthreegaininputs,[G,t=0:T=2,G!,t=0:T=2,Ginc]maygenerallybedecreasediftheclosed-loopdeectionsovershootthepeakwingdeections.However,ifdecreasedtoomuch,theclosed-loop 135

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responsewillhavelimitedimprovementoverthefeedforwardresponse.Itisbesttoselectonecontrolinputatatimewhenhandtuningthesegainstolimitthedegreesoffreedom. 6.6.3.2Optimization AcostfunctionmaybespeciedtoobtaintheoptimalgainsforgiveninputappingparametersasshowninEquation 6 .Theoptimizationwilltunethegainstominimizethedifferencebetweentheclosed-loopappingmovementandthedesiredappingmovementateachofthentimesteps.ThedeectionsZclaregeneratedbyassuminginitialvaluesforthegainsandevaluatingtheplantmodelwithcoefcientsA1:A11determinedduringthefeedforwardcontrolsynthesisandcontrolinputsforthecurrentstepnasdenedinEquation 6 G,t=0:T=2opt,G!,t=0:T=2opt,Gincopt=argminG,t=0:T=2,G!,t=0:T=2,GincJn=mean(jZCLn)]TJ /F8 7.97 Tf 6.59 0 Td[(1j)-222(jZdn)]TJ /F8 7.97 Tf 6.59 0 Td[(1j)jN,J1=mean(jZFF1j)-223(jZd1j)jN (6) 6.6.4ControlApproachIssues Oppositiontotheideaofwing-shapecontrolformorphingandappingwingsmightstatethatifadesiredwing-shapepathwasknown,thenthestructurecouldsimplybetailoredtoachievethiskinematicprolewithminimumenergyinput.Thisideahasvalidity,butespeciallyforsmallerstructuresitisunlikelythattheexactkinematicresponseacrossmultiplefrequenciesmaybepredictedwithstructuraltailoringalonebecauseofthecoupledinuenceoftheaerodynamicsontheexiblewingstructure.Inaddition,smartstructureswithembeddedactuationwillalmostcertainlybeusedtocontrolthesewingsinthefuture.Assuch,theprobleminherentlydependsonthecontroltechniqueandanidealsolutionwillnotbedenedwithstructuralanalysisalone. Onemightalsoobjectthatnokinematicprolemaybedenedwhichisoptimal,however,bio-inspirationbymatchingtheStrouhalnumberandenergeticsbased 136

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approachesaswellasadvancingcoupledcomputationalapproachestohandleaeroservoelasticityshouldenableadesirablekinematicproletobefound.Anothersimilarobjectionisthattheairowshouldbecontrolleddirectlytoensureperformanceinsteadofcontrollingthewing'skinematicbehavior.However,tailoringthestructuralmovementisalmostcertainlyascloseaswewillbeabletoachieveuntilmicro-ornano-electricalmechanicalsystems(MEMS/NEMS)withbothsensingandactuationcapabilitiescanbeembeddeddirectlyintothestructuretocontroltheow.Thesetechnologiesappeartobeatleastseveralyearsawayandwouldrequiresubstantialdevelopmentbeforetheycouldbeimplementedonaappingwingunlesspriorresearchondeltawingvortexcontrollabilityisfoundtobeapplicable.Accelerationinbio-andnanotechnologycouldenablethetechnologymorequicklybutfornowcontrollingthemovementofthestructureappearstobethemostpractical. Lastly,thetechniquedoeshavethedisadvantagethatsomemeasurementsmustbetakenwithafairlyexpensivehighspeedcamerasystemandpost-processingthedatamaytakeawhile.Whilethisistrue,thepriceofhighspeedcamerasisdecreasingrapidlyandtheworkowmaybeautomatedifneeded.Assuch,theonlydisadvantageremainingisthatofanyexperimentally-basedprocedure:thevehicleperformance,especiallyregardingenergyefciency,maynotbepredictablepriortoproduction.Thisdisadvantagemaybemitigatedsincetheplatformcostforsmallersystemsisrelativelylow,butislikelytobeapermanentlimitation.Itshouldbenotedthoughthat,ifthedesiredwingmovementisachievedwiththeproposedcontrolscheme,thisshouldbetheonlyperformancelimitationandcouldbeaddressedwithseveralwingredesignstolikelyachieveasolutionwhichisclosetooptimal. 137

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CHAPTER7EXAMPLE1:PIEZOELECTRICMORPHINGWINGS 7.1BackgroundforMorphingWingSystemIdenticationandControl ResearchershaveimplementedmorphingstructuresacrossawiderangeoftheaircraftdesignspaceasseeninChapter 2 .Manytypesofmechanicalactuatorshavebeenproposedforuseonmorphingwings.TheserangefromthelargehydraulicjacksandpivotjointstoprovidewingsweepontheF-14totorquerodsandhobbyservostogenerateatwistangleforacompositemembranewing[ 1 2 ]. Piezoelectricactuatorshavealsobeenincorporatedintomorphingwingsatmanyscales.Anactive-bercomposite(AFC)helpedreducevibrationloadsinthetailnofanF-18ghteraircraft[ 148 ].Pre-stressedpiezoelectricactuatorshavebeenusedfortheaileronsonxedwingremotecontrolaircraft[ 250 251 ]. Piezoelectricactuatorsofferthepotentialtodesignamorphingwingwhichmitigatesturbulenceandgeneratesthrustbydynamicallycontrollingtheshapeofthewing.TheMacroFiberComposite(MFC)wasdevelopedbyresearchersatNASAtobealow-cost,highstrainactuator[ 267 ].ResearchhasindicatedthatusinganMFCforelevonsmayresultinlowerdragandwideractuationbandwidth[ 31 ].Theseactuatorsalsohavehighactuationauthorityandsensingoverawiderangeoffrequencies,althoughtheyrequirehighinputvoltages. Theinherentlyaeroservoelasticrelationshipbetweenthepiezoelectrically-actuatedwingresponseandwing-shapecontroltoobtainperformancebenetsfortheaircraftremainsunderinvestigation.ThereforeChapter 7 willdescribetheimplementationofpiezoelectricactuatorsinamembranewing,aswellastodeneaprocessforsystemidenticationandcontrolformorphingthewingtoachieveadesiredwing-shape. 138

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7.2PiezoelectricActuators 7.2.1PiezoelectricActuatorFabrication Multiplesizesofpiezoelectricactuatorsareexamined.Thepiezoelectricactuatorisattachedtoasingleunidirectionalcarbonberlayerwhichprovidesstiffness.Theattachmenttothestiffcarbonberprovideslargeractuatordeectionduetothestraininthepiezoelectriccausedbyanappliedvoltage.Oncethecompositeactuatorisfabricated,wiresaresolderedontotheleadsandtheleadsareinsulatedtopreventarchingcausedbythehighappliedvoltages.ThedimensionsforthecompositeactuatorsexaminedareshowninTable 7-1 Table7-1. Macro-FiberComposite(MFC)PiezoelectricBeamCharacteristics NameLength(mm)Width(mm)ActiveLength(mm)ActiveWidth(mm) Uni-110513855.0 UniCA-210514858.0 Uni-3105358527.5 TwocuringproceduresweredevelopedandperformedbyBradLacroix.Thersttechniquecreatesapermanentbondbyco-curingthepiezoelectricwithpreimpregnated(prepreg)carbonber,usingtheexcessepoxyintheprepregmaterialtobondtheactuatortothecomposite.Thepiezoelectricactuatorislaidontopoftheprepreg,whichisthenplacedonaatplatecoveredwithteonsheeting.Teonisplacedontopoftheactuator,breathermaterialislaidontopoftheteon,andtheentireplateisplacedinavacuumbag.Vacuumisappliedequivalenttoapproximatelyatmosphericpressure(14.7psi)andthesetupisplacedinanovenwherealinearrampfromroomtemperaturetothecuringtemperature(72oF260oF)over90minutesisappliedtominimizeinducedthermalstresses.Theovendwellsat260oFfor4hours,thendecreasesbacktoroomtemperatureoverthenext90minutes. Thesecondtechniquecurestheprepregcarbonberrst,thenbondsthepiezoelectrictothecarbonberwithcyanoacrylate(CA),allowingtheactuatortobereused.Theprepregiscuredinthesamemannerasdescribedintherstprocess 139

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withoutthepiezoelectricactuator.Aftertheprepregiscured,CAisappliedtotheactuator,whichisthenplacedonthecuredprepreg.ThesetupiscoveredinTeonsheetingtopreventadhesiontoothersurfacesandvacuumbaggedfor1houratroomtemperature.TheCAfullyhardenswithinoneday.TheCAmaybedegradedbyheatingtheactuatorto175oF,thenpeelingtheactuatoroffthecarbonber.AcetonemaybeusedtoremoveanyremainingCA,allowingtheactuatortobereused.ThismethodresultsinaslightlyheavieractuatorsubjecttothelimitationsofCAandresultsinlowerdeectionsandaweakerbond. 7.2.2PiezoelectricActuatorResponseCharacterization Thetimeandfrequencyresponsesofeachbeamareexaminedbeforetheyareincorporatedintoawingtoexaminehowthepiezoelectricresponseaffectstheaeroservoelasticresponseofthewing.Thestructuraldynamicresponseofapiezoelectricactuatormaybemeasuredwithtwoexcitationmethods:traditionalshakerexcitation(external)andexcitationbyapplyingavoltagetothepiezoelectricdirectly(internal).TheexternalandinternalexcitationsetupsareshowninFigure 7.2.2 .Inbothcases,thestructureisclampedacrossthewidthoftheactuatortoensureacantileveredboundarycondition,therebymimickingtheboundaryconditionpresentifthepiezoelectricsareusedasbattens. Figure7-1. LaserDopplerVibrometerSystemSetupforGroundVibrationTestingofPiezoelectricBeam PiezoelectricBeamwithShakerExcitation(Left),PiezoelectricBeamwithInternalExcitation(Right) 140

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Fortheexternaltests,aBaldor2lbshakercontrolledwithanAgilent33120Asignalgeneratorappliedthroughanamplierisusedtoapplyanexcitationtothecompositeactuator.Athreeinchthreadedrodhasoneendthreadedintotheshakeroutputshaftandclampedwithanutwhiletheotherendisplacedthroughtheholeintheactuatorclamp.Compressionontheclampisensuredbythreadednutsonbothsidesoftheclamp.Thestiffmountingarrangementensuresthatthemeasuredfrequencyresponseisduetothecompositeactuatoralone. FortheinternalteststhepiezoelectricswerecontrolledwithanAMD2012-CE2printedcircuitboard(PCB)madebyAMPowerSystems,andonloanfromSystemDynamicsInternational,Inc.Areference12Vpowersupplyandacontrolvoltagewhichwasadjustedbetween0VwithanAgilent33120AsignalgeneratorwereappliedtotheAMD2012-CE2togeneratethehighvoltages(-500Vto+1500V)requiredtoactuatethepiezoelectric.ThecircuitoverviewandresponsesummaryareshowninFigure 7.2.2 .Thecircuitprovidesactivedischargetoquicklyremovechargefromthepiezoelectric,therebyallowingasinusoidalexcitationsignaltobeappliedtothepiezoelectric. Figure7-2. OverviewofPrintedCircuitBoardUsedforPiezoelectricActuation AMD2012-CE2PCBwithInputDenitions(Left)andInput/OutputVoltageCharacterization(Right),PhotoandgraphcourtesyofSystemDynamicsInternational,Inc.[ 209 ] 141

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7.2.2.1Timeresponse ThetimeresponseoftheUni-3piezoelectric-compositebeamischaracterizedinFigure 7.2.2.1 byexaminingtheresponseofthepiezoelectricactuatortoa10VstepinputshowninFigure 7.2.2.1 (a).ThestepinputisgeneratedbyalowfrequencysquarewavefromasignalgeneratorandtheactuatorresponseismeasuredwiththeLDVsystem.ThecumulativevelocitymeasuredbytheLDVshowninFigure 7.2.2.1 (b)maybesummedtondthedisplacementtimeresponseshowninFigure 7.2.2.1 (c). Themaximumamplitudeofthetipdisplacementis21.9mm,thesteadystatedeectionis20.5mmandtherisetimeasdenedbythetimeittakesfortheoutputdeectiontogofrom10%to90%ofthesteadystatedeectionis0.18seconds.Thisdataimpliesthatthemaximumdeectionswouldonlyoccurifthebeamwasactuatedatlowerthan5Hz,whichwouldimplyastrictbandwidthlimitontheactuator.However,thislimitdoesnotaccountfortheresponsewhenactuatingattheresonantfrequencyofthebeam.Whenaresonantfrequencyisappliedtotheactuator,themaximumdeectionsdoincrease,evenatfrequencieshigherthanthebandwidthlimitimpliedbytherisetime.Thesettlingtimefromwhentheactuationisappliedatt=1.156secondstowhentheresponseremainswithin2%ofthesteadystatevalueis0.35seconds. ItisalsointerestingtonotefromFigure 7.2.2.1 (c)thatthedeectionsbegintorisebeforethestepresponseisapplied.Thismayindicatethatthereissomeerrorinwhenthetimingofthestepinputisrecordedandwhenthevoltageisapplied.Thereforetherisetimedenedaspreviouslyfrom10%to90%isthebestpossiblemetricavailabletoexaminetheUni-3responsewiththecurrentsetup.Itshouldalsobenotedthatattimet=[1.27,1.51]largespikesappearinthevelocitymeasurement.Thesearelikelyduetomeasurementerrorsandmayhavecausedthesteadystatedeectiontobeslightlylargerthantheactualdeection. 142

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(a)(b)(c) Figure7-3. TimeResponseofUni-3PiezoelectricActuatorto10VStepInput (a)VoltageStepInputtoAMD2012-CE2PCBFromSignalGenerator,(b)VelocityMeasuredbyLDVSystem,(c)DeectionDeterminedwithPost-Processing(),RiseTimeDataRangeOverlay(),SettlingTimeRange,StartStepInputtoTimeSettledWithin2%(-.-), 7.2.2.2Frequencyresponse,shakerexcitation Thefrequencyresponseofthepiezoelectricactuatorisimportanttomeasuresincetheactuatormaybeoscillatedacrossmultiplefrequenciesforthenalcontrollerdesignandtheresponseatresonantfrequencieswillgeneratedeectionmoreefcientlywithrespecttotheenergyinputtothesystem.Inaddition,thefrequencyresponsemaybeusedtoidentifyabasisforareducedordermodelwhichpredictshowtheactuatorchangesshape. Theresultingfrequencyresponse,coherenceandmodeshapesofthepiezoelectricbeamundershakerexcitationareshowninFigure 7.2.2.2 .Allthreetestswereperformedwhereasweepfrom10Hzover400mswasappliedbythesignalgeneratortotheshaker.5responseswereaveragedateachof45pointsonthebeamtogivethenalresponse.800measurementsinthefrequencyspectrumgavearesolutionof2.5Hz.Whiletherstbendingmoderemainedat27.5Hzirrespectiveoftheappliedvoltage,thehigherfrequencymodeschangedfrequencysubstantiallydependingonthe 143

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voltageapplied.Thisisimportanttoobserve,sinceitrevealsthatthedynamicresponseofthepiezoelectricactuatorisdependentonappliedvoltageandthecurrentshapeofthebeam.Assuch,theactuatorresponsemustbecharacterizedwithvoltageapplieddirectlytothepiezoelectricandmaynotbeaccurateifonlytestedwithsingle-pointshakerexcitation. Figure7-4. ModalResponseofPiezoelectricBeamwithShakerExcitation StaticVoltagetoUni-3PiezoelectricBeam,-1.6V(left),0V(center),5V(right) 7.2.2.3Frequencyresponse,sinusoidaldwellresponses TheUni-3actuatorisrsttestedbyinputtingasinedwellacrossarangeoffrequenciestogenerateanapproximatefrequencyrangefortheLDVtestandidentifytheimportantstructuralmodes.Thefrequencywhichvisuallyappearedtogeneratethelargestdeectionislikelytherstbendingmodeofthebeam.Therstbendingmodalfrequencymaybeidentiedas22.6Hzbygeneratinganinputfrequencyandobservingchangesintheamountofdeectionasshownbythelaserreectingoffthesurfaceoftheactuatortip.800Hzofbandwidthwasmeasuredwith3200fouriersamplestogivearesolutionof0.25Hz.Thefrequencyresponseandmodeshapeat22.6Hz 144

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wasmeasuredtobe22.5HzwiththeLDVasshowninFigure 7-5 .ThisexampleshowstheLDVmaycapturethepiezoelectricsystemresponseaccuratelywithrespecttofrequencywithinthemeasurementresolutionwhenusingsine-dwellexitationandthatthesignalgeneratorandAMD2012-CE2areabletocommandtheactuatortomovewiththedesiredinputfrequency.Contributionsfromharmonicsarepresentinthesine-dwellresponseatmultiplesofthecommandedfrequency.Similartestingwasperformedfortheshakerexcitation,withsimilarresults. Figure7-5. SinusoidalFrequencyResponseofUni-3andModeShapeforApplied22.6Hz 7.2.2.4Modalresponse,internalexcitation Toobtainthedynamicresponseofthepiezoelectricactuatoracrosstheentirefrequencyspectrum,thesignalgeneratorwasusedtoapplyasinesweepfrom10Hzoscillatingbetween0Vand5Vdirectlytothepiezoelectric.Theprintedcircuitboardusesthe5Vinputsignalfromthesignalgeneratoranda12Vconstantsignaltoactivelydissipatethechargeonthepiezoelectricandcausethebeamtovibratewiththeshapeitwouldnormallyattainateachinputfrequency.AsummarypictureoftheUniCA-2andUni-3actuatorsunderdifferenttypesofexcitationisshowninFigure 7.2.2.4 .Theresultsindicatethatthefrequency-dependentdynamicresponseofthepiezoelectricchangesdependingonthetypeofexcitationapplied.Thisresultconrmsthattoaccuratelycharacterizetheresponseofapiezoelectricactuator,internal 145

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excitationisneededsincethepiezoelectricswhenimplementedinavehiclewillbeactuatedwithinternalexcitation. SeveralofthemodeshapesinFigure 7.2.2.4 appeartobedominatedbytipdeection.Thismaybecausedbythepresenceoftheunconnectedelectrodesattheendoftheactuators.Thepresenceoftheexpectedrstandsecondbendingoreventorsionalmodesmaybeclearlyseenandcoherencevaluesforthesemodesareabove0.5,suggestingtheresultsarereasonable. Figure7-6. FrequencyResponse,CoherenceandModeShapesofPiezoelectricBeamsUnderDifferentExcitationMethods Column1:UniCA-2ExternalExcitation,Column2:Uni-3ExternalExcitation,Column3:UniCA-2InternalExcitation,Column4:Uni-3InternalExcitation ThemodalfrequencyoftheUni-3actuatorisfoundtobe26.3Hzwithinternalexcitation(Figure 7.2.2.4 ,Col.4),butthedeectionswithinternalexcitationatasinedwellof22.6Hzareobservedtobelargerexperimentally(Figure 7-5 ).Thiscomparison 146

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suggeststhattheresultsasmeasuredwithsweepexcitationmayindicatethattheresonantmodesarefoundathigherfrequenciesthantheyareinreality. 7.3PiezoelectricallyActuatedComposite-MembraneWing Apiezoelectrically-actuatedcomposite-membranewingwasdesignedandconstructedbyBradLacroixandtheauthorasshowninFigure 7.3 (a).Thewingdesignedforthemicroair-vehicle(MAV)asshowninFigure 7.3 (b),whichusesactuatorsthesizeofUniCA-1onthewingtip(outboard)andUniCA-2inthemiddleofthewing(inboard). (a)(b) Figure7-7. Piezoelectrically-ActuatedComposite-MembraneWingAloneandInstalled (a)PiezoelectricallyActuatedComposite-MembraneWing,12inWingSpan,9inRootChordwithMFCUniCA-2(b)MAVwithPiezoelectrically-ActuatedComposite-MembraneWingInstalledwithMFCUniCA-1andUniCA-2 ThewingwasgroundvibrationtestedasshowninFigure 7-8 tondthemodalcharacteristicsofthewing.Itmustbenotedthatthefuselagewasnotattachedtothewingandthatshaker(external)excitationwasused.Inthefutureinternalexcitationofthepiezoelectricsaftertheyhavebeeninstalledinontheaircraftshouldbeconsideredtoensurethemodelofthestructuraldynamicsisaccurate. Thevibrationtestresultsforthepiezoelectrically-actuatedcomposite-membranewingareshowninFigure 7-9 .The21Hzmodeseemstobeduetotheoutboard 147

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Figure7-8. GroundVibrationTestSetupForPiezoelectrically-ActuatedComposite-MembraneWing piezoelectricactuator.The29Hzmodeisthemainbendingfrequencyofthewing.The3rdand4thmodesareclearlytorsional,butitislessclearwhatiscausingthem.The51.5Hzmodeappearstobehavelikeasecondbendingmodeforthewing,whilethe57Hzmodemaybecausedbythepresenceofthesecondactuator. Figure7-9. FrequencyResponse,CoherenceandModeShapesofthePiezoelectrically-ActuatedComposite-MembraneWing 148

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7.4ModelIdentication 7.4.1ModelDiscretization Digitalimagecorrelationwasusedtoobtainthereferenceundeformedstaticwingandthefurthestpositiveandnegativedeectionspossibleforthepiezoelectricactuatorat1500Vand-500Vappliedrespectively.ThesemeasurementswereobtainedbyBradLacroixandareshowninFigure 7.4.1 [ 116 ].Theresponseisclearlyasymmetric,withpositivedeectionsbeingpronouncedatthetrailingedgeandofhighermagnitudethanthenegativedeections.Thepiezoelectricsarethereforecapableofmorphingthechamberofthewing. (a)(b)(c) Figure7-10. DigitalImageCorrelationMeasurementsofStaticDeectionsofPiezoelectrically-ActuatedComposite-MembraneWing (a)DeectionsforFullPositiveActuation(1500V)(b)DeectionsforUnactuatedReferenceWingLocation(0V)(c)DeectionsforFullNegativeActuation(-500V) Theout-of-planedeectionsoftheleftwingareobtainedwiththeLDVsystem.Thecorrectfrequencyresponseofthewingissimulatedbymappingthewingandactuatorresponses,asmeasuredwiththeLDV,ontothereferencewing,asmeasuredwiththeDICsystem.FourpointsfromthepiezoelectricactuatorresponsemeasuredbytheLDV 149

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aremappedtothewingresponsemeasuredbytheLDVasshowninFigure 7.4.1 (a-b).ThecombineddatafromtheLDVsamplinglocationsforthewingandactuatorareoverlayedonthesamplinglocationsfromtheDICsystemasshowninFigure 7.4.1 (c). (a)(b)(c) Figure7-11. MorphingWingExperimentalDiscretization (a)LDVPiezoelectricActuatorDiscretizationWithPointstoMaptoLDVWingData(b)LDVPiezoelectricActuatorMappingtoLDVWingDataDiscretization(a)LDVWingDataMappingtoDICDataDiscretizaton 7.4.2BasisSelection AFourier-seriesmodelasdescribedinChapter 5 wasspeciedtosuperimposethefrequencyresponseoftheUniCA-2piezoelectricactuatorsunderinternalexcitationandthefrequencyresponseofthefullwing.Themodelgeneratesthewingdeectionat32locationsonthewingasshowninFigure 7.4.1 (b).Thecontrolinputsfrequencyandamplitudearecommandedtoeachactuatorwheretheoutboardactuatorisassumedtohavesomelagtoenablethegenerationofspanwiseowifneededtoenhanceaerodynamicperformance.Thestructuraldynamicsofthewingareassumedtocontributetothenaldeectionswiththesameshapeaswasmeasuredforthestaticwingatthegivenactuatorinputfrequencies.However,thecontributionsofthewingstructuraldynamicsareassumedtobesignicantlylowermagnituderelativetothecontributionsfromthepiezoelectricactuators.Themodelmakesnoassumptionsofaerodynamicperformancenorwingexibility,thereforethemodelremainsarelatively 150

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simplistic.However,themodeldemonstrateshowamorphingmodelcouldbeformedwheremultipleinputfrequenciesandthewingstructuraldynamicsareofprimaryimportanceindeterminingthemorphingdeections. 7.4.3MorphingSystemIdentication ThefrequencyresponseshapesofthewingandpiezoelectricactuatorsareidentiedwiththeLDVsystemtobeSW(!,x,y,z)andSP(!,x,y,z)asshowninFigures 7-8 and 7.2.2.4 (Col.3).ThedeectionshapedataisfoundbyintegratingthevelocityresponsemeasuredbytheLDV.Thedeectionsarethennormalizedsuchthatthemaximumofthefrequencyresponseshapesofthewingandactuatoraresetequaltooneandtheresponseatallotherfrequenciesisscaledproportionally. 7.4.4MorphingModelEvaluation Systemidenticationwasperformedwithrespecttothecontrolinputs,[!i,!o,i,o,].Theinboardandoutboardfrequenciesweresettotheresonanceoftheactuator,28.5Hz,whiletheamplitudeoftheactuatorwassetto80forbothactuators.Thephaselagwassettozerosothatthetrailingedgedeectionsareinphasewitheachother.Thedampingparametersareheldconstantwith[Wi=0.1,Wo=0.1,Pi=1,Po=1],indicatingthatthewingstructuralmodesareanticipatedtobe1=10ththemagnitudeofthedeectionsgeneratedbytheactuators. Themodelresponsedatafor450oftheover11000pointsmeasuredwithDICwerelinearlyinterpolatedfromthe32pointsmeasuredbytheLDVsincethemodelevaluationonlyprovidedthemovementoffourpointsfortheactuatorsand32pointsforthewing.Thisgavethenalmodeloutputwherethewingmovementisshownat8stagesoftheperiodicmorphingcyclecorrespondingtophasesof[0,=4,2=4,3=4,,)]TJ /F3 11.955 Tf 9.3 0 Td[(3=4,)]TJ /F3 11.955 Tf 9.3 0 Td[(2=4,)]TJ /F4 11.955 Tf 9.3 0 Td[(=4]proceedingcounterclockwisearoundFigure 7-12 .Thetrailingedgedeectioncausedbytheactuatorsisclearlyevident,especiallyatthepeakoftheupstroke=2=4anddownstroke,=)]TJ /F3 11.955 Tf 9.3 0 Td[(2=4. 151

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Figure7-12. ModelPredictionforaPeriodicCycleofMorphingDeections NoattemptwasmadetovalidatetheFourierbasedmodel,sonoclaimsaremadeastothemodelcorrectness.ThepurposeofdevelopingthemodelwastodemonstratehowtheapproachesdescribedinChapters 5 and 6 forappingmayalsobeapplicabletoamorphingwingorstructurewheregroundvibrationtestingisfeasible(thedeectionsareperiodic,smallandout-of-planeonly),andsystemidenticationforthemorphingdeectionstoobtaincontroloverthestructuralshapeisdesired.Thismorphingexampleconsidersalinear,complexperiodicmodelandwhenmultiplecontrolactuatorsarepresentatdifferentlocationsonthewing. 7.5PiezoelectricFeedforwardControlDesign AcontrolsynthesismaybedeterminedusingthemodelidentiedfromtheLDVdatatogiveadesiredwing-shape.Thecontrollerisformulatedusinginputshapingwithrespecttoinputsfrequencyandamplitudeforboththeinboardandoutboardactuatorsaswellasaphaselagbetweentheactuators. Tostart,adesiredwing-shapemovementisassumedtobeknownwhichwillmitigateturbulence,generatethrustorprovidesomeotherperformancebenet. 152

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Thereforedirectlyrelatingtheshapeofthewingtotheaerodynamicperformanceofthewingwillnotbeconsidered.Inthiscaseadesiredwingshapewasspeciedbyassumingthedigitalimagecorrelationresultsofthewingatmaximumupanddowndeectionwerethepeakoftheupstrokeanddownstrokeofasinewave.TheresultingdesiredmorphingisshowninFigure 7-13 (a).Thewingmovementisshownat8stagesoftheperiodicmorphingcyclecorrespondingtophasesof[0,=4,2=4,3=4,,)]TJ /F3 11.955 Tf 9.3 0 Td[(3=4,)]TJ /F3 11.955 Tf 9.3 0 Td[(2=4,)]TJ /F4 11.955 Tf 9.3 0 Td[(=4]. Theoptimalvaluesof[i,!i,o,!o,]forthefeedforwardcontrolaredeterminedwithaleastsquaresoptimizationtominimizethedifferencebetweentheout-of-planedesiredandsimulatedmorphingdeectionsjZd(...)j)-270(jZsim(...)j.Thesimulatedwingmovementwithconstantssetas[Wi=0.1,Wo=0.1,Pi=1,Po=1]andmorphinginputsfoundtobe[i=81.09,!i=27.05,o=81.09,!o=27.05,=)]TJ /F3 11.955 Tf 9.3 0 Td[(0.00059]areshowninFigure 7-13 (b).Thesearefoundinresponsetotheinitialconditions[i=80,!i=28.5,o=80,!o=28.5,=0].Thereforetheinitialvalueslargelydeterminetheresultoftheoptimizationifthestructuraldynamicscontinuetobesetas1=10ththemagnitudeofthemorphingactuators.Thenalfeedforwardmorphingoutputhasanaverageerroratanypointintimeof0.371mmwhencomparedwiththedesiredmorphing.InbothcasesshowninFigure 7-13 ,themovementofthetrailingedgebythepiezoelectricactuatorisclearlypresent. Inthefutureitwouldbedesirabletosynthesizeafeedforwardcontrollerwhichcouldhavemultiplefrequenciescommandedtoeachactuatorandfortheactuatorinputstobeabletovaryintime.Aclosed-loopcontrollersynthesiswheremeasuredfeedbackwasavailablefromoneormorepointsonthewingwouldbelikelytoprovidebettertrackingofthewingdeectionsandrobustnesswithrespecttoparametricuncertainty,noiseanddisturbances. 153

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(a)(b) Figure7-13. ComparisonofMorphingDeectionsDesiredandFeedforwardSimulation (a)DesiredMorphingMovementand(b)ModelPredictedMorphingResponseatFeedforwardCommandedInputActuationforaPeriodicCycle 154

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CHAPTER8EXAMPLE2:FLAPPINGWINGMICROAERIAL-VEHICLES 8.1BackgroundforFlappingWingSystemIdenticationandControl Techniquesfordataanalysisthatcanextractinformationaboutappingdeectionsaresomewhatlessmaturethantheabilitytogeneratethatdata.Themodalpropertiesoftheaeroservoelasticdynamicswouldprovidereduced-ordermodels;however,thenonlinear,time-varyingpropertiesobservedduringeachappingcyclewouldviolateassumptionsofmodalanalysis[ 10 ].Modelsoftheappingdynamicsmaybeobtainedusingparticleimagevelocimetry[ 191 ],particleowvisualization[ 242 ]anddigitalimagecorrelation[ 277 278 ];however,thesetechniquesgeneraterepresentationsofthetime-averageddynamicsacrossmultiplecycles. Waveletanalysisisabletogenerateatime-frequencyrepresentationofdatathatcapturesbothtime-domaincharacteristicsandfrequency-domaincharacteristics[ 96 ].Suchanalysisreliesonlocalizedcorrelationtoknownwaveformswithoutassumptionsonlinearityortime-invariantproperties.Thesetime-frequencymapshavebeeneffectivelyusedtoanalyzelimitcycleoscillations[ 121 ],nonlinearnormalmodes[ 107 ],neuraldynamics[ 248 ]andnonlinearoscillators[ 104 ].Foraappingwing,bothtimeandfrequencycharacteristicsareimportant.Theabilitytoobtaintimedependentfeaturesareimportantsinceanalysisofthequasi-steadydynamicsareinsufcienttocapturebothofthesetypesofcharacteristics[ 63 69 ].Frequencydependentfeaturessuchasresonancecanenhanceefciencywhilerequiringminimalenergyinputandthereforearealsoimportanttounderstand[ 108 ]. Issuesrelatedtobothdesignandcontrolarefundamentaltothematurationofapping-wingvehicles.Inadditiontoobtainingreducedordermodelsthatwilldescribethewingbehavior,thetime-varyingpropertiesofawaveformassociatedwithresonancemustbeknownforlow-energyight[ 18 101 ].Ornithoptersareoftencontrolledbyvaryingappingfrequencyandassociatedwingdeections[ 91 280 ].Ineachcase, 155

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missionperformancedependsonunderstandingthetime-varyingdeectionsasafunctionoffrequency-varyingapping. 8.2WingFabrication Asetofwings,asshowninFigure 8-1 ,aredesignedfortheinvestigationofapping.TheassociateddimensionsandweightsforthesewingsarelistedinTable 8-1 ,aswellastherstbendingmodeasdeterminedwiththelaserdopplervibrometer.Eachwinghasanaspectratioof7.65andshapeassociatedwithaZimmermanplanformformedbytwoellipsesthatintersectattheirquarter-chordpoint.Variationstothewingsareintroducedbydifferencesinthematerialandstructuraldesign. Figure8-1. FlappingWingsInvestigated Wing-1(Aluminum),Wing-2(RadialBatten,Capran),Wing-3(ParallelBatten,Capran),Wing-4(DiagonalBatten,Capran),2ndRow:WingCarbonFiberLayoutsifApplicable Table8-1. WingCharacteristics WingSpan(mm)Chord(mm)Thick(mm)Weight(g)1stMode(Hz) Wing-176.526.30.320.2642Wing-275.926.30.1465Wing-377.126.50.1260Wing-47526.30.1135 Wing-1ismadeofsolidaluminumsheetthatisspraypaintedwhiteandthenspeckled. Theconstructionofallcarbonberwingsusea3-layer12kbidirectionalcarbon-berroottriangledimensionedas6mm12mmwithaninnerlayerofunidirectionalcarbon-berbattens.Thistriangleensuressufcientstiffnessattherootchordofthewing.Theleadingedgesofthecarbonberwingsstartatawidthof1.2mmandtaperto0.8mmatthetip.Wing-2andWing-3have2andahalfstripsofcarbonberforthe 156

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leadingedgewherethethirdhalfstripextendsfromtherootchord.Wing-4has2stripsfortheleadingedge.BattensforWing-2,Wing-3andWing-4aresinglestripshavingwidthof0.8mmmadefromunidirectionalpre-pregcarbonber. Thecarbon-berwingsarelaiduponaatplateandcuredfor2hoursat250oF.Afterthecarbonberstructuresarecured,themembranesareafxedwithspraygluewithoutpretensioningthemembrane.Wing2,3and4haveCapranmembranes. 8.3FlappingMechanismDesign FlappingisachievedforeachwingusingaappermechanismplatformdesignedbyWu[ 278 ],seeninFigure 8-2 .Thisdeviceutilizesaslider-crankmechanismtotransformtherotationalmotionofamotorintolinearmotionofareciprocator.Aappingmotionisthenintroducedtoawingthroughalinkagemechanism.Theappingmotionisrestrictedtoasingledegree-of-freedomrotationaboutthechord.Apairofwingsaremountedwitheachwingonoppositesidesofthemechanism;however,themotionofeachwingisconstrainedtobeequal.Theresultingmotioniswithin2-percentbiaserrortoanidealsinusoid.ThelinearappingmechanismisdrivenbyaMaxon15WbrushlessDCmotorEC16andpoweredbyastandardpowersupply.Themechanismincludesa57/13reductionratioplanetarygearhead,256countsperturnencoder,andanEPOS24controller. Thecontroleffectorsthatmaybevariedtoaltertheappingmotionareappingfrequencyandappingamplitude.TheappingfrequencycanbecontinuouslyvariedduringoperationbychangingtheinputvoltagetothemotorwithLabviewsoftwarewhiletheappingamplituderequiresadjustingtheoff-centerdistanceofacrankmodulewhichmaynotbechangedduringoperation.Flappingamplitudesrangingfromapproximately+/-0to+/-60degreesatappingfrequenciesbetween0and45Hzareattainabledependingonthewingweight.Wingsareattachedtothemechanismwithcyanoacrylateglue. 157

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Figure8-2. SingleDegreeofFreedomFlappingMechanism,DesignedandBuiltbyWuforFlappingWingExperimentation[ 278 ] 8.4StructuralDynamics Traditionalexperimentalmodalanalysismethods,especiallyformeasurement,donotlendthemselvestothetestingofappingwings.Contactmethodswhichuseaccelerometersorstraingagestodeterminethestructuralresponsearenotsuitableduetothesensitivityofmodalparameterstomasschangesandtheoveralllowweightofthewingsunderconsideration.Therefore,anon-contactgroundvibrationtest(GVT)setupwithaPolytechPSV-400scanningLaserDopplerVibrometer(LDV)isusedinthisinvestigationfortheanalysisofthewingsunderasimpleshakerexcitation. ThewingstructureisattachedtoamountthatwasscrewedintoaLingDynamicSystemsV201/3-PA25Eelectrodynamicshakertofacilitateaconsistentexcitation.Thesetupisplacedonavibrationisolatedlabtableexposedtoambientairinalargeroom.AloadcellisattachedtothestingerbetweentheshakerandthestructuretomeasuretheinputoftheshakerforincorporationinthePolytechsoftware.WhilethePolytechsoftwarecancommandmanytypesofinputexcitationswiththeshaker,for 158

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theseexperimentsonlytheburstchirp,periodicchirp,sweep,periodicsweep,andsinglefrequencysinusoidareconsidered.DICisalsoperformedinconjunctionwiththeshakersetuptoexamineDIC'sabilitytoidentifymodalinformation.ApictureoftheLDVsetupisseeninFigure 8-3 Figure8-3. LDVExperimentalSetup LDVresultshavebeenusedtoexaminetheappingwings[ 133 ].Wingsofvariousshapes,materials,stiffnessandmembraneusagewereexamined.Thewing-shapehasasignicantimpactonthemodeshapesforthesewingsasseeninFigure 8-4 ,whichshowstheLDVresultfor2aluminumwingswherebothhasathicknessof0.28mm.Whiletherstbendingshapeappearsforthemostpartasexpected,theboundaryconditioninthetopleftofbothwingscausestheshapestobeslightlyasymmetricfromtheexpectedbendingshape.TheZimmermanshapedwingshowstheexpecteduniformrstbendingshapefollowedbyasecondbendingonthetopandbottomasexpected.Therectangularwingshowsthebasicrstandsecondbendingaswellasthersttorsionalmodeasexpected.AlteringthestructureobservedmaybeseentoalterthemodeshapesasseeninFigure 8-5 below.Forthealuminumwingontheleft,theshapesappearverysimilartowhatwouldbeexpectedforarst,second,third,andfourthbendingmodeofacantileverbeam.However,bothmembranewingspresentsubstantiallydifferentshapes.Therstandsecondbendingshapesarefairlysimilartoeachother,wherethefrequenciesareslightlyhigherfortheradialbattenarrangement,suggestingthiswinghasaslightlygreaterstiffness.Thetwohigherfrequencyshapes 159

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Figure8-4. FrequencyResponseforVariationsinShape AverageFrequencyResponseFunction(Row1)andFirstThreeObservedOperatingDeectionShapes(Row2-4)forAluminumWingwithZimmermanPlanform(Column1),RectangularAluminumWing(Column2) revealtheunderlyingbattenstructure,asbothshapesfortheparallel-battenwinghavenodallinesfollowingthepatternofthebattenswhilebothshapesfortheradial-battenwinghavenodallinesfollowingthedirectionoftheradialbattens. 8.5SignalCapture TheexperimentalDICsetupisseeninFigure 8-6 .ThetwoPhantomv7cameraswithSigma28-300mm,f=3.5)-241()]TJ /F3 11.955 Tf 21.48 0 Td[(5.6lensesformtheDICsystem.Attopperformance,thesecamerascaneachstore2800picturesof800600pixelsat4800frames-per-second(fps).Accdsensorbooststhesignal-to-noiseratioandenablesshutterspeedsof1/500,000s.TwoSigma28mmf/3.5.6lensesareusedinthetests.Thehigh-speedcamerasarecalibratedwithadotgridwithpredeterminedspacingandCorrelatedSolutionsVicSnapcommercialsoftwarebetweenanyexperimentalruns 160

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Figure8-5. FrequencyResponseforVariationsinStructure AverageFrequencyResponseFunction(Row1)andFirstFourObservedOperatingDeectionShapesforWing-1,AluminumWing(Column1),Wing-3CapranMembraneWingwithParallelBattens(Column2),andWing-2CapranMembraneWingwithRadialBattens(Column3) whenthecamerasormechanismaremoved.Theappropriategridisselectedtollmostoftheactiveimageframe.Themeasurementsareinitiatedbyobtainingareferencepictureofthewingatthecentralmid-planelocationoftheappingcycle.Flappingisthenactivatedandsubsequentdeectionsareobtainedrelativetothisreferencelocation.Exposuretimesareadjustedwithanychangesincaptureframeratebutaregenerallyaround100mstoprovidesufcientlightinginthecapturedframes.Theframerateisscaledbasedontheappingfrequencysuchthatcapture50imagesarecapturedperappingcyclewhichresultsinarangebetween500-1500framespersecondforalldatapresentedinthiswork.Thedepthofeldisadjustedtoensurethatthefullappingcyclecanbecapturedinfocus. TopreparethewingsforDICanalysis,thewingsarespraypaintedwithanevencoatofwhitepaintandthenspeckledwithblackspraypaint.Themembranewings 161

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Figure8-6. DICExperimentalSetup aresimplyspeckledwithblackspraypaint.Careistakentomaintainanapproximatelyevenspecklesizewherespecklesareasneaspossible.FortheDICanalysis,themainvariablesinthisexperimentarewingtype,appingamplitude,appingfrequency,time,spanwiseandchordwiselocation.Experimentalrunsarereferencedwiththesevariables.OncethedataisobtainedwiththeVICSnapsoftware,thedataisanalyzedwithCorrelatedSolutionsVIC3DsoftwaretodetermineboththeinitialfulleldX,Y,Zcoordinatesofthereferenceframe,aswellasthedisplacementsofthewingineachdirectionforeachframecaptured.TheVIC3Dtracksthedeectionsofacalibrationpointandpointswithinaspeciedareaofinterestthroughallframesobtained.ThedatawasimportedintoMATLABandthereferencewingistranslatedtoazeropositionbymovingthemaximumYcoordinateandmaximumXcoordinateofthereferenceframetozero.ThisadjustmentformsarighthandedcoordinatesystemwiththeYaxisrunningfromtheroottotipalongtheleadingedge,theXaxisrunningfromtheleadingedgetotrailingedgeattherootchord,andtheresultantzaxisupfromthetopofthewing.Asignalisformedbytrackingthedisplacementofagivenpointrepresentingaregionofspeckles 162

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intheX,Y,orZdirectionatagivenlocationacrossframes.Theresultingsignalshaveerrorsindeectionlessthan1mm[ 85 ]despitesignicantoutofplanedisplacement. Theimagesobtainedhavearesolutionof800600pixels.TheDICsoftware(VIC3D)isabletoextractandtrackaround3000pointsonthewingsfromtheseimages.Arectangulargridof9pointsspanwiseand5pointschordwiseisspeciedsuchthatthelimitsofthegridaredenedbythemaximumdimensionsofthewing.ThegridismappedontothemeasurementlocationsonthewingbyidentifyingtheclosestpointtoeachoftherectangulargridpointsasshowninFigure 8-7 toensurethelargeamountofdataismanageable. Figure8-7. WingSamplingMappingRectangularGridontoWing Deectionsignalsineachofthe^x,^y,^zdirectionsforsteadystateappingat45pointsonthewingasshowninFigure 8-7 areobtained.AsetofmeasurementsaretakenwiththeDICsystematthesetofappingfrequenciesandappingamplitudesgiveninTable 8-2 Thebiasisremovedfromtherawdeectionsignalsandsinglecyclesoftheappingareisolatedbasedonwhenthesignalcrosseszeromagnitude.Thesignalsareforcedtostartoffgoinginthepositivedirectiontoensurethatthetrendsinsubsequentcurvetsofthetrainingdataareconsistent.Theappingdeectionshavebeenshowntobeapproximatelyperiodicwhenusingthismethod[ 129 ].Therefore,sevensubsequent 163

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Table8-2. ExperimentallyMeasuredDiscretizationofFlappingParameters FlappingAmplitudes(o),FlappingFrequencies(!,Hz) Wing+=)]TJ /F3 11.955 Tf 12.57 0 Td[(o!Hz 310f10,20,30g17f10,20,30g35f10,20,30g 410f5,10,15,20,25,30g17f5,10,15,20,25,30g35f5,10,15,20,25,30g appingcyclesareaveragedtogivetheperiodicdeectionsignalsateachvalueofappingfrequencyandappingamplitude. 8.6ModelIdentication TheappingwingmodelisdescribedasafunctionofcoefcientsA1A11,eachofwhichisafunctionof(,!).ThecoefcientsA1A11areshowninFigure 8-8 .Thesecoefcientsmaybereevaluatedasafunctionof(,!)byinterpolatingbetweenthedatapointsshownbytheinterpolatedsurfacesinFigure 8-8 generatedwithEquation 5 Figure8-8. ExperimentallyDeterminedCoefcients(*)A1(,!)A11(,!)OverlaidwithInterpolatedSurfacesA1(,!)A11(,!) 164

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Themodelisusedtopredictthefullwingdeectionsacrossthespanofthebasis.The=)]TJ /F4 11.955 Tf 9.3 0 Td[(=4t=3=(4!)locationcorrespondingtothepeakofthedownstrokefortherigidwingappingcycleisisolatedateachvalueof(,!)intheexperimentalbasis.Theabilityofthemodeltopredictthewing-shapeatthistimeisdemonstratedinFigure 8-9 .Errorscausedbydifcultyobtainingaccurateexperimentalmeasurementsbecomemoreprevalentathighervaluesof(,!)andforsignalsclosertothewingtip.Theseerrorsareseentoreducetheaccuracyofthemodel;however,sincetheoverallmodelisfoundconsideringtheentirewingmovementateachvalueof(,!),theeffectisreduced. Figure8-9. ComparisonofExperimentallyMeasuredFlapping(MeshedWing)toModelPredictedFlapping(SmoothWing)AcrossKinematicSpace Valuesof(,!)InModelTrainingSetforWing-4(DiagonalBattenWing)at=)]TJ /F4 11.955 Tf 9.29 0 Td[(=4t=3=(4!) Themodelingerrorsarequantiedastheresidualsoftheobjectivefunctionforthefullwingmatches.TheresidualvalueisthecostexpressedinEquation 5 .The 165

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residualsoftheobjectivefunctionsforeachvalueof(,!)areshowninTable 8-3 .Ingeneralthemodelingerrorsdoincreasewithincreasingvaluesofthecontrolactuation(,!)ashasbeensuggestedbyaninspectionofFigure 8-9 .Thedatapointsfor=35o,15Hzand25HzwereremovedfromthetrainingsettoincreasetheaccuracyofthemodelsincetheDICwasunabletomatchasignicantnumberofpoints,causingthelargerinaccuracyobservedinTable 8-3 Table8-3. ModelingErrorsforWing-4(DiagonalBattenWing)AcrossDesignSpace 5Hz10Hz15Hz20Hz25Hz30Hz 10o19240225122113637817o21423426637314834635o389264505171577335 8.7ConsiderationstoDetermineDesiredFlappingMovement Themainlift-enhancementmechanismsforapping-wingsincludeleading-edgevortices,fastpitch-up,wakecapture,clap-and-ing,[ 211 ],and3Deffectssuchasdelayedstalloftheleading-edgevortex,spanwiseow,apersistentdownwardjetinthewakeandtipvorticesandwingtipvortexanchoring[ 212 ]allfurthercomplicatethejobofdeterminingadesirablekinematicprole.AappingproleisoftencharacterizedbytheReynoldsNumber,Strouhalnumber,reducedfrequency,andstructuraldynamicmodes;therefore,theseparametersmayhelpdetermineadesirableappingmovement.However,thesenon-dimensionalparametersarestillformulatedintermsofthekinematicparameters,soitmaybedifculttodetermineadirectrelationshipbetweentheparametersandtheaerodynamicforcesgeneratedbytheappingwing.Assuch,itisrecommendedthataknownappingprolewhichproducesthrustbeselected.Awingmovementwhichisefcientisdeterminedbasedonmeasuredaerodynamicandfullelddeectiondata[ 279 ]. 166

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8.8FeedforwardControlResults Onceanaeroservoelasticmodelfortheappingwinghasbeendeterminedandadesiredwingmovementhasbeenselectedbyidentifyingwhatmovementprovidesefcientaerodynamicperformance,acontrolsynthesismaybeattempted. 8.8.1FeedforwardControlIdentication AdesiredappingproleisselectedfromtheexperimentaldataforWing-3(ParallelBattens)movingat(=35o,!=20Hz).Thewing-shapegeneratedfortheseinputsisknowntohaveproducedatime-averagednetthrustof1.25gasfoundbyWu[ 276 ]. Thefullwingoptimizationresultsinacommanded(c=36.2o,!c=23.2Hz).AcomparisonofthedesiredappingwingmovementtothefeedforwardappingwingmovementisseeninFigure 8-10 .Wingmovementisshownat8stagesoftheappingcyclecorrespondingtophasesof[0,=4,2=4,3=4,,)]TJ /F3 11.955 Tf 9.3 0 Td[(3=4,)]TJ /F3 11.955 Tf 9.3 0 Td[(2=4,)]TJ /F4 11.955 Tf 9.3 0 Td[(=4]proceedingcounterclockwisearoundFigure 8-10 .Thiswingisatarelativelylowerappingfrequencyandnotproducingalargeamountoflift.Thereforeexibilityeffectsandaerodynamiceffectsshouldn'tbeasmuchafactor.Thepredictedwingmovementappearstoleadthemeasuredwingmovementontheupstroke,thenlagitonthedownstroke,suggestingthatthecontrolledwingdeectionswerepredictedtohavelargerdeectionsthanthemeasuredwingmovement. AnotherdesiredappingproleisselectedfromtheexperimentaldataasWing-3movingat(=35o,!=30Hz).Thewing-shapefortheseinputshasbeenmeasuredtoproduceamuchhighertime-averagednetthrustof4.43g.[ 276 ]Inthiscase,thewingdeectionsarelargerandtheeffectsofexibilityinthewingareobservedbybendingacrosstheleadingedgeatthepeakoftheupstrokeanddownstroke. ThecommandedactuationforWing-4tomatchthedesiredprolestartedwith(0=30o,!0=25Hz)is(c=36o,!c=23Hz)andtheresultsareshowninFigure 8-11 .Theseresultsindicatethattheouterloopoptimizationcanchangethecommandedappingamplitudeandfrequencytoobtainadifferentwingmovement. 167

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Figure8-10. FeedforwardControlResult:LowFrequency,HighAmplitudeFlapping DesiredWingMovement(Wing-3at(=35o,!=20Hz))(SmoothWing),vs.CommandedWingMovement(Wing-4PredictedfromExperimentalModelatCommanded(c=36.2o,!c=23.2Hz))(MeshedWing) Theperformanceofthefeedforwardcontrollerisnotasgoodastheperformanceforthelowerfrequencyappingwherelessexibilityisevident.Thisobservationsuggeststhatthecurrentfeedforwardcontrollerwillnotbeaseffectiveathighervaluesofc,!c.Whilesomechordwisebendingisapparentintheappingcommanded,itislessthanthatobservedinthemeasuredapping.Inthiscasethemeasuredwingleadsthecommandedwingthroughouttheappingcycle,indicatingthatthecommandedwinghaspredictedsignicantlylessdeectionthanthemeasuredwing.Theseissuesmaybeabletoberesolvedifthecostfunctionsarerephrasedintermsofthedeformationsinsteadofthedeections.Someerrorsmaybeduetodifferencesintherealbasisforthedeformationsandtheoneassumedforthispaper.Issueswithlocalminimaofanobviouslynon-convexobjectivefunctionmayalsohaveskewedresults.Thelargererrors 168

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mayalsoresultfromthefactthatWing-4willnotnaturallyobtaintheexactwing-shapeswhichWing-3isabletoobtain,inpartduetodifferencesinthechordwiseandspanwisestiffnessofthesewings. Figure8-11. FeedforwardControlResult:HighFrequency,HighAmplitudeFlapping DesiredWingMovement(Wing-3at(=35o,!=30Hz))(SmoothWing),vs.CommandedWingMovement(Wing-4PredictedfromExperimentalModelatCommanded(c=36o,!c=25Hz))(MeshedWing) 8.8.2FeedforwardControlRobustnessandErrorAnalysis Somevariationwasobservedintheabilityofthelastoptimizationtoconvergewithrespecttochanginginitialconditions,especiallyiftheappingamplitudeisvaried.Thesesetsof0,!0wereconsideredforthecasewhere(=35o,!=20Hz):[(0=35o,!0=20Hz),result:(c=35o,!c=20Hz,residual=318),(0=45o,!0=20Hz),result:(c=36o,!c=23Hz,residual=149),(0=25o,!0=20Hz),result:(c=25o,!c=20Hz,residual=313),(0=35o,!0=30Hz),result:(c=35o,!c=30Hz,residual=191)]. Avalidationstepforthefeedforwardcontrolprocedureindicatedthatinputs(c,!c)matchedthedesiredwingmovementspeciedbyexperimentaldataofWing-3appingat(,!)assumingthatthecontinuousfunctioninEquation 5 intersectswitheach 169

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A1(,!)A11(,!)predictedbythemodel.TheabilityofthefeedforwardcontrollertomatchthemovementsofWing-3withdataspecictoWing-4acrossthedesignspaceof(,!)isshowninTable 8-4 .Thereisacleardecreaseinaccuracyasthecincreases,buttherelationshipto!cisnotclear. Table8-4. EvaluationofFeedforwardControlRobustnesswithRespectto(c,!c) ExperimentalInputs(Wing-3(o,!Hz))CommandedInputs(Wing-4(oc,!cHz))Residual 10,1010.0,1.1372.3 10,208.10,20.064.3 10,3013.6,30.083.0 17,1018.1,10.062.1 17,2019.0,20.055.0 17,3013.7,30.095.1 35,2036.2,23.2149 35,3036.1,25.0266 8.8.3LimitationsofFeedforwardControlMethod Thetermswhichconstitutethemodelbasismustbeassumed.Theremaybesignicanthighfrequencyinformationwhichthebasisisnotabletocapture,especiallyinformationduetohighfrequencycontent.Thecontrollerdoesnothavetheadvantageoffeedback,thereforetheapproachasstatedwouldnotbeabletohandlesignicantdisturbancesthatarenotaccountedforintheexperimentalmodel.Theusualissueswithoptimizationoverlargedesignspacesmayalsobeproblematicincludingsensitivitytotheinitialguessandentrapmentinlocalminimumsoftheobjectivefunction. 8.9Closed-LoopControl Aproportionalclosed-loopcontrollerisdesignedasdescribedinChapter 6 whichutilizesthemodelidentiedforWing-4andfeedforwardcontrollerresult.Theerroridentiedistheabsolutevalueoftheaverageerrorinmillimetersofanygivenpointatanygiventimefromthedesiredappingprole. 170

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8.9.1StabilityforClosed-LoopController Theupdateroutinefortheclosed-loopcontrollerhasbeenlimitedsuchthatthefeedforwardcontrolinputsareupdatedthroughouttheappingcycletoattempttomaintainthestableappingresponsecommandedbythefeedforwardcontroller.Stabilityhasbeendemonstratedforthissimulation.However,noguaranteesonstabilityaremadesincetheplantisnonlinearwithrespecttothecontrolinputs. 8.9.2EffectofSensorPlacementonClosed-LoopPerformance TheeffectofsensorplacementisevaluatedinFigure 8-12 .Theaverageerrorfortheclosed-loopresultisevaluatedforvarioussensorplacementsonthewingwhilemaintainingconstantcontrollergains,[G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2]andupdatingthecontrolleratn=32stages.Theresultsindicatethattheaverageerrorisrelativelysmallwhenthepointatthe75percentspan(spanindex=7)andtrailingedge(chordindex=5)isusedtoprovidefeedback.Thedeectionofthislocationisalsoknowntobeimportanttothrustgeneration[ 54 ].Thereforethislocationisselectedasthesensorlocationforallotherevaluationsoftheclosed-loopcontroller. Itshouldbenotedthatnoavailablesensormaydirectlymeasurethewingdeection.Thereforetheclosedloopperformancemaybeconsideredabestcasescenarioforthistypeofcontrolsynthesis.Astraingageorsomenewtypeofsensormayberequiredtopracticallyimplementtheclosed-loopcontrolsynthesis. 8.9.3EffectofTimeDiscretizationonClosed-loopPerformance Thenumberoftimeswhichtheoutputissampledduringtheappingcycleaffectstheperformanceoftheclosed-loopcontrolsynthesis.Figure 8-13 demonstratesthattheerrorroughlydecreasesasthediscretizationintimeincreases.Thelowesterroroccurswhenupdatingthecontrolinputs32timesperappingcycleor3220Hz=640Hz.Thereforethisupdaterateisselectedforallotherevaluationsoftheclosed-loopcontroller.Inthefuture,substantialworkmayberequiredtoachievesuchclosed-loopcontrolinahardwareimplementation.Thesoftwarecalculationsforthe 171

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Figure8-12. AverageClosed-loopControlErrorsforOut-of-PlaneResponseswithRespecttoSensorMeasurementLocation G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Wing-4Model closed-loopcontrollerarerelativelyslowwithrespecttotheapping,butasimplehardwareimplementationwithanoperationalamplierfortheproportionalgainmayenableapracticalimplementationoftheclosed-loopcontrolstrategy. 8.9.4Closed-loopPerformancewithRespecttoWingLocation Theaverageerrorforeachpointiscomparedforthefeedforwardandclosed-loopresultinFigure 8-14 .Thepointsnearesttothewingroot(25%ofpoints)arenotincludedsincetheZcomponentsofthedeectionsareclearlynotassinusoidalasthosefurtherfromthewingroot.Thegainontheappingfrequencywassettozerotohelpwithhandtuning,butalsobecausethesensitivityofthemodeltoappingamplitudewasmuchlargerthanthesensitivitytoappingfrequency.Theinputappingfrequencyremainsthefrequencyobtainedfromthefeedforwardcontroller. Theerrorclearlyincreasesinmagnitudegoingfromthewingroottothewingtipforboththefeedforwardandclosed-loopresult,buttheaverageerrorfortheclosed-loopis 172

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Figure8-13. EffectofTimeDiscretizationonClosed-LoopControlPerformance G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdge,Wing-4Model lessthan1=4ththeerrorofthebestfeedforwardresult.Thisimprovementismainlyduetotheincreasedtimediscretizationfortheclosed-loopcontrollerandincreasedaccuracytowardtherootchordandtrailingedgefortheclosed-loopcontroller.Itshouldbenotedthatsomewhatsignicanterrorsstillremainatthewingtip,where3mmonaveragemaybemissingfromtheresponsewhenitmovesapproximately+=)]TJ /F3 11.955 Tf 11.95 0 Td[(50mmtotal. 8.9.5Closed-loopResponsePerformanceEvaluation Thebestcasehand-tunedresultfortheclosed-loopisshowninFigures 8-15 and 8-16 .Figures 8-15 (a,c)demonstratethattheclosed-loopimprovesboththespanwiseandout-of-planedeectionsignals,despitetheclosed-looperrorbeingbasedonjusttheout-of-planesignals.Thisisimportantsinceonlyfeedbackonthewing'soutofplanemovementisrequired.Theclosed-loopcontrolmaybeseentobeworkingbyexaminingthepeakoftheupstrokeinFigure 8-15 (c),wherethefeedbackresponseovershootsthedesiredresponse,thencorrectsandundershootsthedesiredresponse,thentrackstheresponsewell.ThechordwisedeectionsinFigure 8-15 (b) 173

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(a)(b) Figure8-14. ErrorswithRespecttoLocationontheWing(a)FeedforwardControlResult(b)Closed-loopControlResult G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdge,Wing-4Model aresubstantiallydependentontheaerodynamicforcespresentonthewingandthemechanismmechanicsatthewingroot.Thereforenoattempthasbeenmadetomatchthem. Thevariationinappingamplitudeandconstantappingfrequency,aswellasthecorrespondinggainsarenotedinFigure 8-15 (e-f).ThedramaticincreaseinthecontrolgainforthesecondhalfofeachappingcycleareclearlyindicatedinFigures 8-15 (f).Theerrorswhicharestillpresentthroughouttheappingcycleareshownin 8-16 ,howevertheseerrorshavebeenreducedascomparedtothefeedforwardcontrol. 8.9.5.1Effectsofparametricuncertaintyonperformance Therobustnessofthefeedforwardcontrollerandtheclosed-loopcontrollers'performancetoparametricuncertaintyisshowninFigure 8-17 .Theclosed-loopperformanceisbetterthanthefeedforwardperformancewhenparametricuncertaintyhasasubstantialeffectontheoutputresponse.ThereisnovariationintheerrorduetoparametricuncertaintyatcoefcientsA1A4sincetheerroronlyconsiderstheout 174

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(a)(b)(c)(d) (e)(f)(g)(h) Figure8-15. Signal-BasedComparisonsofDesired,Feedforward,andClosed-loopControlDeections (a)UDeection,(b)VDeection,(c)WDeection,(d)AverageErrorforTimeStep,(e)Closed-loopControlInput,FlappingFrequency,(f)Closed-loopControlInput,FlappingAmplitude,(g)ControlGain,FlappingFrequency,(h)ControlGain,FlappingAmplitude,Wing-2,G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdgefor32UpdateStagesinTime,2FlappingCycles,Wing-4Model ofplane,orZ,deections.TheprimarysensitivitytotheresponseisobservedintheA5coefcient,whichislogicalsinceonlytheamplitudeisbeingusedasacontrolparameter.Figure 8-17 alsodemonstratesthattheclosed-loopresultagainhasasmallererrorinthepresenceofparametricuncertainty. 8.9.5.2Effectsofdisturbancesonperformance Disturbancessuchasawindgustmovingacrossthewingaremodeledas+=)]TJ /F3 11.955 Tf 12.15 0 Td[(50percentagechangestothecontrolinputsappingamplitudeandappingfrequency.The 175

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Figure8-16. FullWingResponsesClosedLoopControl DesiredWingMovement(Wing-3at(=35o,!=20Hz))(SmoothWing),vs.Closed-loopCommandedWingMovement(Wing-4PredictedfromExperimentalModelatFeedforwardCommanded(c=38o,!c=19.9Hz))(MeshedWing) effectsofthedisturbancesareshowninFigure 8-18 .TheaveragefeedforwardcontrollererrorshowsalargeandnearlylinearsensitivitytoappingamplitudedisturbancesasseeninFigure 8-18 (a).Figure 8-18 (a)showsthatthemodelhasverysmallvariationsduetodisturbancestotheinputappingfrequency.Thesensitivitiesduetochangesinthecontrolinputsforthefeedforwarderroraredirectlyrelatedtothesensitivityoftheidentiedmodel.Figure 8-18 (b)showssimilartrendsinthattheerrorsduetoappingamplituderesultindramaticincreasesintheerror,butchangesintheappingfrequencyhavelittleeffectontheclosed-looperror.Thereforetheclosed-loopappingisrobusttodisturbancesinappingfrequency,butnottodisturbancesinappingamplitude.Inaddition,forthiscase,varyingappingamplitudeandnotappingfrequencyasthe 176

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Figure8-17. ComparisonofRobustnessofFeedforwardandClosed-loopPerformancewithRespecttoParametricUncertainty Wing-4Model,G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdge primarycontrolactuatorappearstobesignicantlymoreeffectiveingeneratingwingdeformation. 8.9.5.3Effectsofnoiseonperformance Measurementnoiseismodeledasrandomwhitenoiseacrossallfrequencies.Themeasurementerrorsarespeciedas+=)]TJ /F3 11.955 Tf 12.43 0 Td[(50percentofthedeectionscalculatedbytheclosed-loopsimulation.TherobustnessofthecontrollertomeasurementnoiseisexaminedinFigure 8-19 .ThefeedforwardcontrollererrorsinFigure 8-19 arecalculatedwithouttheadditionofnoise.Figure 8-19 demonstratesthattheclosed-loopcontrollerisrobusttomeasurementerrorsduetonoiseintherangeof)]TJ /F3 11.955 Tf 9.29 0 Td[(20:+20percentoftheoutputsignal,butthattheaverageerrorsincreaserapidlyoutsideofthisrange. 177

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Figure8-18. ComparisonofDisturbancesontheFeedforwardandClosed-loopControllerPerformance Wing-4Model,G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdge Figure8-19. ComparisonofNoiseontheFeedforwardandClosed-loopControllerPerformance Wing-4Model,G,t=0:T=2=0.026,G!,t=0:T=2=0,Ginc=0.2,n=32,Feedback=75%Span,TrailingEdge 178

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CHAPTER9CONCLUSIONSANDFUTUREWORK 9.1Conclusions Thetime-frequencydependentaeroservoelasticnatureofamorphingandappingwinghasbeencharacterizedandrelevantfeatureshavebeenincorporatedintoageneralizedmathematicalmodelofamorphingandappingwing.Currentproceduresforanalysis,systemidenticationandcontrolhavebeensummarized.Amodelhasbeenidentiedwhichincorporatedalloftheaeroservoelasticeffectsobservedexperimentallyforsteady-state,periodicmovementofthemorphingandappingwing.Lastly,afeedforwardandclosedloopcontrollerhavebeensynthesizedforthesewingswhichenablesprecisecontrolofthewingmovement,eveninthepresenceofaeroservoelasticeffects.ThespeciccontributionswhichhavebeenachievedarelistedbelowinthesameordertheywerepresentedinChapter 1 1. Thetime-frequencydependenciesofanaeroservoelasticmorphingwingandappingwingwasexperimentallymeasuredandanalyzed.Theperiodicityevidentinsteady-stateappingsignalswasdemonstratedandlocationsintimeandfrequencywherethesignalsexhibitedtime-frequencydependentornonlinearbehaviorwereidentiedwithwaveletanalysis.Time-varyingandnonlinearbehaviorwasidentiedwhenthewingstructuraldynamicswereexcited,whentheinputfrequencywasvariedintime,andwhenwingexibilityeffectswerepresent.Therelativecontributionsoftheseeffectstothewingdeectionsweredependentonthelocationonthewingwherethedeectionsweremeasuredandtheinputactuation.(Ch. 4 ) 2. Theeffectsofwingdesignandactuationmethodonthestructuraldynamicsofcomposite-membranemorphingandappingwingswereexamined.Thehigherstiffnesswingshadhigherfrequencybendingmodesandthemodeshapeswerehighlydependentonthebattenstructureofthewing.Thesewingsarelikelytohavedeectionswhichareaffectedbythesemodessincethelowestbendingmodefrequenciesarebetween20Hzandappingathalftherstbendingfrequencywasdemonstratedtoexcitetherststructuralmode.(Ch. 7 8 ) 3. Theeffectsofthestructuraldynamics,kinematicparametersandappingdeectionsontheaerodynamicperformanceofthewingwereexaminedforaappingwingandadesirablebutnotoptimalappingmovementwasidentiedbasedontheaerodynamicperformanceofthewing.Thelargerdeectionpresent 179

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duetothecontributionoftherstbendingmodewasgenerallybenecialforaerodynamicforceproduction.(Ch. 8 ) 4. Ageneralizedgeometricalmodelofashape-changingwingwithtemporal,spatialandfrequencydependencieswasformulated.Themodelwasusedtoformulateabasisfortwoaeroservoelasticmodels:(a)apiezoelectrically-actuatedmorphingcomposite-membranewingmodeland(b)aappingcomposite-membranewingmodel.(Ch. 5 ) 5. Atechniqueforidentifyinganaeroservoelasticmodelofwingdeectionsfromexperimentaldatawasdeveloped.Themodelincorporatesalltheaeroservoelasticphenomenapresentintheexperimentaldata.Themodelsidentiedaccountforthenonlinear,timeandfrequency-varyingdynamicsthatareexperiencedbyexiblemorphingandappingwings.(Ch. 5 ) 6. Afeedforwardcontrolstrategywhichusestheidentiedmorphingandappingmodelswassynthesized.Thefeedforwardcontrollermaybeusedtocommandadesiredwing-shapeforamorphingandaappingwing,eveninthepresenceofaeroservoelasticcoupling.(Ch. 6 ) 7. Aclosedloopcontrolstrategywhichusestheidentiedappingmodelwasdevelopedwhichmatchesthedesiredwing-shapemovementforaappingwing.Flappingamplitudewasdemonstratedtobemorepredictablethanappingfrequencytogeneratewingdeections.Theperformanceandrobustnessoftheclosed-loopcontrolsynthesiswithrespecttothefeedforwardcontrolsynthesisweredemonstratedfortheappingwinginthepresenceofparametricuncertainty,noiseanddisturbances.(Ch. 6 ) 9.2FutureWork Asummaryoftherecommendedworkwhichwilllogicallyextendthisworkonmorphingandappingwingaeroservoelasticanalysis,systemidenticationandcontrolislistedbelow. 1. Ageneralizedformofthedynamicallymorphing/appingbasismaybedevelopedwherequaternion-based,time-varyingdenitionsofthetwist,dihedralandsweepanglesarespecied.Theseanglesmaybespeciedasafunctionoftimeandlocationonthewingtoprovidecoefcientsonthewingwhichmorereadilymaycapturethewingexibilityduringsystemidentication.Thisgeneralwingmovementbasis,ascomparedtoamodelbasiswhichisgeneral,butnotstructuredaccordingtothewingdynamics,couldvastlyspeedupcontroldesignsforvehicleswhichdynamicallychangewingshape. 2. Abetterwayofcapturingthewingexibilitywithoutknowingallthestiffness,mass,andinertialpropertiesmustbedeveloped.Atime-varyingEuler-Bernoullibeam 180

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modelmaybeusefulforappingwhilenodestiffnessrulesmightbesufcientforthemorphingexample.Morphingandappingmodelsbasedondeformationsorsomesimilarsubsetoftheoverallappingdeectionsmaybebenecialtohelpcapturethisexibility. 3. Experimentalvalidationofthemorphingwingmodelmayhelpunderstandtherelativemagnitudeofthecontributionsofthemorphingactuatorsandstructuraldynamicstothenalwingdeections.Experimentalvalidationoftheappingwingcontrolstrategyisalsoneeded,butwasnotpossibleforthepresentworksincethecurrentmechanismcouldnotvaryappingamplitudecontinuously. 4. Asetofdesirabledynamicwingdeectionswhichwillprovidesomeincreasedaircraftperformanceorenhancedmaneuverabilitymustbeidentiedforbothappingandmorphingwings. 5. Thechangeinmodelcoefcientswithrespecttotime-varyingcontrolactuatoreffectsmustbedetermined.Forexample,theeffectofacontinuouslyvaryingappingamplitudeorappingfrequencyonthemodelcoefcientsmayneedtobedeterminedsincetheseeffectswillsubstantiallychangethetime-frequencybehaviorofaeroservoelasticwings. 6. Amorepracticalmeansoffeedbackmustbedevelopedfortheclosed-loopcontrolsynthesisthanusingthewingdeectionsdirectly. 7. Aformoftheclosedloopcontrolwhichmayadjustthebestcoefcientsofsomegeneralwingmodelbasisinrealtimeaccordingtosomedesiredaerodynamicperformancewouldbeextremelybenecialinfacilitatingreal-worlddesigns.Thedesiredlevelofwingperformancecouldthenbeobtainedevenifthepriordesignworkhadbeenslightlymisinformed. Thereisclearlymuchmoreworktobedonebeforedynamicwingmorphingandappingwillprovideindustrywithnovelaircraftandvehicledesigns.Yettherearelocationswherethesetypesofwingsarestartingtoappear.Thedesiretoharnessaeroservoelasticeffectstogainperformancebenetsforvehiclesshouldcontinuetodriveresearchintobothmorphingandappingwingsinthefuture. 181

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REFERENCES [1] Abdulrahim,M.DynamicCharacteristicsofMorphingAirVehicles.Masters'Thesis,UniversityofFlorida(2004). [2] .ManeuveringControlandCongurationAdaptationofaBiologicallyInspiredMorphingAircraft.DoctoralDissertation,UniversityofFlorida(2007). [3] Abdulrahim,M.,Garcia,H.,Ivey,G.F.,andLind,R.FlightTestingaMicroAirVehicleUsingMorphingforAeroservoelasticControl.CollectionofTechnicalPapers-AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference3(2004):1776. [4] Abdulrahim,M.andLind,R.FlightTestingandResponseCharacteristicsofaVariableGull-WingMorphingAircraft.ProceedingsoftheAIAAGuidance,Navigation,andControlConference(2004).2004. [5] Adams,W.M.andPeele,E.ADigitalProgramforCalculatingtheInteractionBetweenFlexibleStructures,UnsteadyAerodynamicsandActiveControls.(1979).NASA-TM-80040. [6] Adams,W.M.andHoadley,S.T.ISAC:AToolforAeroservoelasticModelingandAnalysis.(1993).NASA-TM-109031. [7] Adrian,R.J.ParticleImagingTechniquesforExperimentalFluidMechanics.AnnualReviewofFluidMechanics23(1991):261. [8] Akl,W.,Poh,S.andBaz,A.WirelessandDistributedSensingoftheShapeofMorphingStructures.SensorsandActuatorsA(2007):94.Doi:10.1016/j.sna.2007.06.026. [9] Albertani,R.,Stanford,B.,Systma,M.,andIfju,P.UnsteadyMechanicalAspectsofFlexibleWings:ExperimentalInvestigationofBiologicallyInspiredMAVs.ProceedingsoftheEuropeanMicroAirVechicleConferenceandFlightCompetition(2007). [10] Allen,M.S.andGinsberg,J.H.FloquetModalAnalysistoDetectCracksinaRotatingShaftonAnisotropicSupports.ProceedingsoftheInternationalModalAnalysisConferenceXXIV(2006). [11] Altshuler,D.L.,Dudley,R.,andEllington,C.P.AerodynamicForcesofRevolvingHummingbirdWingsandWingModels.JournalofZoology264(2004):327.Doi:10.1017/S0952836904005813. [12] Ameri,N.A.,Lowenberg,M.H.,Livne,E.andFriswell,M.I.ModellingContinuouslyMorphingAircraftforFlightControl.ProceedingsoftheAIAAGuidance,NavigationandControlConference(2008). 182

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[13] Amprikidis,M.OntheUseofAdaptiveInternalStructuresforWingShapeControl.45thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference(2004). [14] Andersen,G.R.,Cowan,D.L.andPiatak,D.J.AeroelasticModeling,AnalysisandTestingofaMorphingWingStructure.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference1(2007):359. [15] AndersonJr.,J.D.FundamentalsofAerodynamics,3rdEdition.(2001). [16] Ansari,S.A.Nonlinear,Unsteady,AerodynamicModelforInsect-likeFlappingWingsintheHoverwithMicroAirVehicleApplications.PhDThesis(2004). [17] Ansari,S.A.,Zbikowski,R.,andKnowles,K.Aerodynamicmodellingofinsect-likeappingightformicroairvehicles.ProgressinAerospaceSciences42(2006).2:129.Doi:/10.1016/j.paerosci.2006.07.001. [18] Avadhanula,S.,Wood,R.J.,Campolo,D.,andFearing,R.S.DynamicallytuneddesignoftheMFIthorax.ProceedingsoftheIEEEInternationalConferenceonRoboticsandAutomation2(2002):52.Doi:10.1109/ROBOT.2002.1013338. [19] Azuma,A.TheBiokineticsofFlyingandSwimming.(2006). [20] Bae,J.,Seigler,T.M.andInman,D.J.AerodynamicandStaticAeroelasticCharacteristicsofaVariable-spanMorphingWing.JournalofAircraft42(2005):528. [21] Baldelli,DarioH.,Lind,RichardC.,andBrenner,Martin.RobustAeroelasticMatch-pointSolutionsUsingDescribingFunctionMethod.JournalofAircraft42(2005).6:1597. [22] Baldelli,DarioH.,Lind,Rick,andBrenner,Marty.Data-basedrobustmatch-pointsolutionsusingdescribingfunctionmethod.CollectionofTechnicalPapers-AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference1(2005):680. [23] Baldelli,D.H.,Lind,R.,andBrenner,M.Nonlinearaeroelastic/aeroservoelasticmodelingbyblock-orientedidentication.JournalofGuidance,Control,andDynamics28(2005).5:1056. [24] Baldelli,D.H.,Pena,R.S.,Lee,D.andCannon,B.ModelingandControlofanAeroelasticMorphingVehicle.JournalofGuidance,ControlandDynamics31(2008):1687.Doi:10.2514/1.35445. [25] Baldelli,D.H.,Pena,R.S.,Hopper,D.,Lee,D.andCannon,B.PracticalModeling,ControlandSimulationofanAeroelasticMorphingUAV.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference7(2007):6481. 183

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[26] Baldwin,C.S.,Salter,T.J.andKiddy,J.S.StaticShapeMeasurementsUsingaMultiplexedFiberBraggGratingSensorSystem.ProceedingsofSmartStructuresandMaterials(2004).Doi:10.1117/12.538091. [27] Bandyopadhyay,B.,Manjunath,T.C.andUmapathy,M.Modeling,ControlandImplementationofSmartStructures:AFEM-StateSpaceApproach.(2007). [28] Barrett,R.ActiveAeroelasticTailoringofanAdaptiveFlexsparStabilator.SmartMaterialsandStructures5(1996):723.Doi:10.1088/0964-1726/5/6/001. [29] Bendat,J.S.andPiersol,A.G.RandomData:AnalysisandMeasurementProcedures,3rdEd.(2000). [30] Biedermann,L.B.,Tung,R.C.,Raman,A.,Raman,R.,andRonald,G.FlexuralVibrationSpectraofCarbonNanotubesMeasuredUsingLaserDopplerVibrometry.Nanotechnology20(2009).3:035702.Doi:10.1088/0957-4484/20/3/035702. [31] Bilgen,O.MacroFiberCompositeActuatedUnmannedAirVehicles:Design,Development,andTesting.Ph.D.Dissertation,VirginiaPolytechnicInstituteandStateUniversity(2007). [32] Blake,R.W.TheMechanicsofLabriformMotion1:LabriformLocomotionintheAngelsh(Pterophyllumeimekei):AnAnalysisofthePowerStroke.TheJournalofExperimentalBiology82(1979):255. [33] Blondeau,J.andPines,D.J.DesignandTestingofPneumaticTelescopicWingforUnmannedAerialVehicles.JournalofAircraft44(2007). [34] Blondeau,J.,Pines,D.J.andJ.Richeson.Design,DevelopmentandTestingofAMorphingAspectRatioWingUsinganInatableTelescopingSpar.Proceedingsofthe44thAIAAStructures,StructuralDynamicsandMaterialsConference(2003). [35] Boothe,Jr.,K.E.DynamicModelingandFlightControlofMorphingAirVehicles.Master'sThesis(2004). [36] Boria,F.,Bowman,S.,Stanford,B.andIfju,P.EvolutionaryOptimizationofaMorphingWingwithWind-TunnelHardwareintheLoop.AIAAJournal47(2009):399. [37] Bos,F.M.,Lentink,D.,VanOudheusden,B.W.,andBijl,H.Numericalstudyofkinematicwingmodelsofhoveringinsectight.CollectionofTechnicalPapers-45thAIAAAerospaceSciencesMeeting9(2007):5782. [38] Botez,R.M.,Biskri,D.E.,Cotoi,I.,Hamza,D.,Doin,A.andParvu,P.MethodforFlutterAero-servoelasticOpenLoopAnlaysis.CanadianAeronauticsandSpaceJournal49(2003):179. 184

PAGE 185

[39] Bozkurttas,M.,Dong,H.,Mittal,R.,Madden,P.,andLauder,G.V.HydrodynamicPerformanceofDeformableFishFinsandFlappingFoils.CollectionofTechnicalPapers-44thAIAAAerospaceSciencesMeeting22(2006):16743. [40] Brenner,MartinJ.AeroservoelasticModelUncertaintyBoundEstimationfromFlightData.JournalofGuidance,Control,andDynamics25(2002).4:748. [41] Brenner,M.J.AeroservoelasticUncertaintyModelIdenticationfromFlightData.NASATechnicalReports(2001).NASA-TM-2001-210397. [42] Breuker,R.D.,Gurdal,Z.,Abdalla,M.andLinder,D.Energy-basedAeroelasticAnalysisofaMorphingWing.ProceedingsofModeling,SignalProcessingandControlforSmartStructuresConference6523(2007).Doi:10.1117/12.716731. [43] Bronowicki,A.J.,Dvorski,G.R.,Wyse,R.E.,Innis,J.W.,Betros,R.S.andKuritz,S.P.AdvancedCompositeswithEmbeddedSensorsandActuators(ACESA).PhaseIII-VFinalReport(1993).PL-TR-93-3017. [44] Buchhave,P.ParticleImageVelocimetry-StatusandTrends.ExperimentalThermalandFluidScience5(1992):586. [45] Cadogan,D.,Uhelsky,F.,Smith,T.andMacKusick,M.MorphingInatableWingDevelopmentforCompactPackageUnmannedAerialVehicles.AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference(2004). [46] Cai,Y.,Bi,S.andZheng,L.DesignandExperimentsofaRoboticFishImitatingCow-NosedRay.JournalofBionicEngineering7(2010):120. [47] Chaithanya,M.R.andVenkatraman,K.HydrodynamicPropulsionofaFlexibleFoilUndergoingPitchingMotion.Proceedingsofthe10thAnnualCFDSympo-sium(2008). [48] Chen,X.J.,Cui,W.C.,Wu,Y.S.andJensen,J.J.ReviewofHydroelasticityTheoriesforGlobalResponseofMarineStructures.OceanEngineering33(2006):439. [49] Chimakurthi,SatishKumar,Tang,Jian,Palacios,Rafael,Cesnik,CarlosE.S.,andShyy,Wei.Computationalaeroelasticityframeworkforanalyzingappingwingmicroairvehicles.CollectionofTechnicalPapers-AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference(2008). [50] Chou,K.C.,Guthart,G.S.,Flamm,D.S.andUeberschaer,R.M.MultiscaleApproachtotheControlofSmartStructures.IndustrialandCommericalApplica-tionsofSmartStructuresTechnologies2721(1996):94. [51] Chu,T.C.,Ranson,W.F.,andSutton,M.A.ApplicationsofDigital-ImageCorrelationTechniquestoExperimentalMechanics.ExperimentalMechan-ics25(1985):232.Doi:10.1007/BF02325092. 185

PAGE 186

[52] ClapperJr.,J.R.,Carwright,J.E.,YoungJr.,J.J.andGrimes,J.G.UnmannedSystemsRoadmap2007.(2007). [53] Clark,R.P.andSmits,A.J.ThrustProductionandWakeStructureofaBatoid-InspiredOscillatingFin.JournalofFluidMechanics562(2006):415.Doi:10.1017/S0022112006001297. [54] Combes,S.A.andDaniel,T.L.Shape,FlappingandFlexion:WingandFinDesignforForwardFlight.JournalofExperimentalBiology204(2001):2073. [55] .FlexuralStiffnessinInsectWings:I.ScalingandtheInuenceofWingVenation.JournalofExperimentalBiology206(2003):2979.Doi:10.1242/jeb.00523. [56] .FlexuralStiffnessinInsectWings:II.SpatialDistributionandDynamicWingBending.JournalofExperimentalBiology206(2003):2989.Doi:10.1242/jeb.00524. [57] Cox,T.H.andGilyard,G.B.GroundVibrationTestResultsforDronesforAerodynamicandStructuralTesting(DAST)/AeroelasticResearchWing(ARW-1R)Aircraft.NASATechnicalMemorandum(1986).NASA-TM-85906. [58] CraneIII,C.D.andDuffy,J.KinematicAnalysisofRobotManipulators.(1998). [59] Dang,P.,Subbarao,K.,Lewis,F.L.andStephanou,H.ShapeControlofFlexibleStructureUsingPotentialFieldMethod.ProceedingsofIEEEIn-ternationalConferenceonControlApplications(2008):540.Doi:10.1109/CCA.2008.4629650. [60] Davidson,J.B.,Chwalowski,P.,andLazos,B.S.FlightDynamicSimulationAssessmentofaMorphableHyper-ellipticCamberedSpanWingedConguration.ProceedingsoftheAIAAonAtmosphericFlightMechanicsConference(2003).AIAA-2003-5301. [61] Day,D.A.Variable-SweepWings.U.S.CentennialofFlightCommission(2003).Http://www.centennialofight.gov/essay/. [62] Deng,X.,Schenato,L.,andSastry,S.FlappingFlightforBiomimeticRoboticInsectsPartII:FlightControlDesign.IEEETransactionsonRobotics22(2006):789.Doi:10.1109/TRO.2006.875483. [63] Dickinson,M.H.,Lehmann,F.,andSane,S.WingRotationandtheAerodynamicBasisofInsectFlight.Science284(1999):1954.Doi:10.1126/science.284.5422.1954. [64] Donley,M.B.andSchwartz,N.A.UnitedStatesAirForceUnmannedAircraftSystemsFlightPlan2009.(2009). 186

PAGE 187

[65] Drela,M.IntegratedSimulationModelforPreliminaryAerodynamic,StructurasandControl-LawDesignofAircraft.AIAAStructures,StructuralDynamicsandMaterialsConference(1999). [66] Dudley,R.BiomechanicsofFlightinNeotropicalButteries:MorphometricsandKinematics.JournalofExperimentalBiology150(1990):37. [67] .BiomechanicsofFlightinNeotropicalButteries:AerodynamicsandMechanicalPowerRequirements.JournalofExperimentalBiology159(1991):335. [68] Dwivedy,S.K.andEberhard,P.DynamicAnalysisofFlexibleManipulatorsALiteratureReview.MechanismandMachineTheory41(2006). [69] Ellington,C.,denBerg,C.Van,Willmott,A.,andThomas,A.Leading-edgeVorticesinInsectFlight.Nature384(1996):626.Doi:10.1038/384626a0. [70] Etkin,B.andReid,L.D.DynamicsofFlight.(1996). [71] Ewins,D.J.ModalTesting,Theory,PracticeandApplication.(2000). [72] Frazer,R.A.andDuncan,W.J.TheFlutterofAeroplaneWings.(1929). [73] Friedmann,PeretzP.RenaissanceofAeroelasticityandItsFuture.JournalofAircraft36(1999).1:105.Doi:10.2514/2.2418. [74] Friswell,M.I.andInman,D.J.MorphingConceptsforUAVs.21stInternationalUnmannedAirVehicleSystemsConference(2006). [75] Frommer,J.B.andCrossley,W.A.EnablingContinuousOptimizationforSizingMorphingAircraftConcepts.43rdAIAAAerospaceSciencesMeetingandExhibit-MeetingPapers(2005). [76] Gardnier,R.,Cesnik,C.,Chimakurthi,S.andAttar,P.High-FidelityAeroelasticComputationsofaFlappingWingwithSpanwiseFlexibility.ProceedingsoftheAerospaceSciencesMeeting(2011).AIAA-2011-0570. [77] Garrick,I.E.PropulstionofaFlappingandOscillatingAirfoil.NACATechnicalReports567(1936). [78] Ghandi,N.,Ward,D.,Howard,D.P.,Neal,D.,Cooper,J.andCannon,B.AHardwareDemonstrationofanIntegratedAdaptiveWingShapeandFlightControlLawforMorphingAircraft.ProceedingsoftheAIAAGuidance,NavigationandControlConferencce(2009). [79] Gioffre,M.,Gusella,V.,Marsili,R.,andRossi,G.Comparisonbetweenaccelerometerandlaservibrometertomeasuretrafcexcitedvibrationsonbridges.ProceedingsofSPIE-TheInternationalSocietyforOpticalEngineering4072(2000):230.Doi:10.1117/12.386764. 187

PAGE 188

[80] Grant,D.ModelingandDynamicAnalysisofaMulti-JointMorphingAircraft.Masters'Thesis,UniversityofFlorida(2009). [81] Grauer,J.ActiveWingMorphingofAnOrnithopterWingUsingShapeMemoryAlloyActuators.Master'sThesis(2006). [82] Grauer,J.A.andJr.,J.E.Hubbard.InertialMeasurementsfromFlightDataofaFlapping-WingOrnithopter.JournalofGuidance,Control,andDynamics32(2009).Doi:10.2514/1.37495. [83] Guiler,R.andHuebsch,W.WindTunnelAnalysisofaMorphingSweptWingTaillessAircraft.AIAAAppliedAerodynamicsConference(2005). [84] Gupta,K.K.STARS-AnIntegratedMultidisciplinaryFinite-Element,Structural,Fluids,AeroelasticandAeroservoelasticAnalysisComputerProgram.(1997).NASA-TM-101709. [85] Haddadi,H.andBelhabib,S.UseofRigid-BodyMotionfortheInvestigationandEstimationoftheMeasurementErrorsRelatedtoDigitalImageCorrelationTechnique.OpticsandLasersinEngineering46(2007):185.Doi:10.1016/j.optlaseng.2007.05.008. [86] Hamdaoui,M.,Doncieux,S.,Chaskalovic,J.andSagaut,P.UsingMultiobjectiveEvolutionaryAlgorithmsandData-MiningMethodstoOptimizeOrnithopters'Kinematics.JournalofAircraft47(2010):1504.Doi:10.2514/1.45046. [87] Heathcote,S.,Wang,Z.andGursul,I.EffectofSpanwiseFlexibilityonFlappingWingPropulsion.JournalofFluidsandStructures24(2008):183. [88] Heine,C.MechanicsofFlappingFinLocomotionintheCownoseRay,RhinopteraBonasus(elasmobranchii:Myliobatidae).Ph.D.Dissertation,DukeUniversity(1992). [89] Heinze,S.AeroelasticConceptsforFlexibleAircraftStructures.Ph.D.Disserta-tion,RoyalInstituteofTechnology,StockholmSweden(2007). [90] Hirdaris,S.E.andTemarel,P.HydroelasticityofShipsRecentAdvancesandFutureTrends.ProcedingsoftheInstitueofMechanicalEngineersPartM:JournalofEngineeringfortheMaritimeEnvironment223(2009):305.Doi:10.1243/14750902JEME160. [91] Ho,S.,Nassef,H.,Pornsinsirirak,N.,Tai,Y.,andHo,C.UnsteadyAerodynamicsandFlowControlforFlappingWingFlyers.ProgressinAerospaceSciences39(2003):635.Doi:10.1016/j.paerosci.2003.04.001. [92] Ho,S.,Nassef,H.,Pornsinsirirak,N.,Y.,andHo,C.UnsteadyAerodynamicsandFlowControlforFlappingWingFlyers.ProgressinAerospaceSciences39(2003).8:635.Doi:10.1016/j.paerosci.2003.04.001. 188

PAGE 189

[93] Hodges,D.H.andPierce,G.A.IntroductiontoStructuralDynamicsandAeroelasticity.(2002). [94] Hong,Y.andAltman,A.LiftFromSpanwiseFlowinFlappingWings.JournalofAircraft45(2008).Doi:10.2514/1.34100. [95] Hu,Y.andWang,J.ExperimentalInvestigationonAerodynamicPerformanceofGlidingButteries.AIAAJournal48(2010):2454.DOI:10.2514/1.45156. [96] Hubbard,B.TheWorldAccordingtoWavelets.(1996). [97] Huber,J.E.,Fleck,N.A.andAshby,M.F.TheSelectionofMechanicalActuatorsBasedonPerformanceIndices.ProceedingsoftheRoyalSocietyofLondonA453(1997):2185. [98] Hunt,S.R.andHebden,I.G.Euroghter2000:AnIntegratedApproachtoStructuralHealthMonitoring.Proceedingsofthe19thICAFSymposium:FatigueinNewandAgingAircraft(1997):481. [99] Ifju,P.G.,Ettinger,S.,Lian,Y.,Shyy,W.,Jenkins,D.A.andWaszak,M.R.Flexible-Wing-BasedMicroAirVehicles.ProceedingsoftheAIAAAerospaceSciencesMeeting(2002).AIAA-2002-0705. [100] Inoyama,D.,Sanders,B.P.,andJoo,J.J.TopologySynthesisofDistributedActuationSystemsforMorphingWingStructures.JournalofAircraft44(2007):1077.10.2514/1.25535. [101] Isogai,K.,Kamisawa,Y.,andSato,H.ResonanceTypeFlappingWingforaMicroAirVehicle.ProceedingsoftheAIAAAtmosphericFlightMechanicsConference(2007). [102] Jackson,T.andLivne,E.IntegratedAeroservoelasticDesignOptimizationofActively-ControlledStrain-ActuatedFlightVehicles.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference(2005). [103] Jonkman,J.M.DynamicsofOffshoreFloatingWindTurbines-ModelDevelopmentandVerication.WindEnergy12(2009):459.Doi:10.1002/we.347. [104] Joseph,L.andMinh-Nghi,T.AWavelet-basedApproachfortheIdenticationofDampinginNon-linearOscillators.InternationalJournalofMechanicalSciences47(2005):1262.Doi:10.1016.j.ijmecsci.2005.04.010. [105] Junkins,J.L.MechanicsandControlofLargeFlexibleStructures.(1990). [106] Katam,V.,LeBeau,R.P.andJacob,J.D.SimulationofSeparationControlonaMorphingWingwithConformalCamber.ProceedingsoftheAIAAFluidDynamicsConference(2005):1. 189

PAGE 190

[107] Kerschen,G.,Peeters,M.,Golinval,J.C.,andVakakis,A.F.NonlinearNormalModes,Part1:AUsefulFrameworkfortheStructuralDynamicist.MechanicalSystemsandSignalProcessing23(2009):170.Doi:10.1016/j.ymssp.2008.04.002. [108] Khan,Z.andAgrawal,S.DesignandOptimizationofaBiologicallyInspiredFlappingMechanismforFlappingWingMicroAirVehicles.ProceedingsoftheIEEEInternationalConferenceonRoboticsandAutomation(2007):373.Doi:10.1109/ROBOT.2007.363815. [109] Khan,Z.A.andAgrawal,S.K.OptimalHoveringKinematicsofFlappingWingsforMicroAirVehicles.AIAAJournal49(2011):257.Doi:10.2514/1.J050057. [110] Kikuta,M.T.MechanicalPropertiesofCandidateMaterialsforMorphingWings.Masters'Thesis,VirginiaPolytechnicInstituteandStateUniversity(2003). [111] Kim,Hong-Il,Kim,Dae-Kwan,andHan,Jae-Hung.StudyofappingactuatormodulesusingIPMC.vol.6524.SPIE,2007,65241A.Doi:10.1117/12.715633. [112] Klein,V.andMorelli,E.A.AircraftSystemIdenticationTheoryandPractice.(2006). [113] Krantz,D.,Biermann,P.J.,Dubow,J.,Gause,L.W.,Harjani,R.,Mantell,S.,Polla,D.,Belk,J.andTroyk,P.Projectupdate:appliedresearchonremotelyqueriedembeddedmicrosensors.SmartElectronicsandMEMS3328(1999):124.Doi:10.1117/12.354261. [114] Kudva,J.N.,Lockyer,A.J.andWay,C.B.Van.StructralHealthMonitoringofAircraftComponents.AGARDLectureSeries205SmartStructuresandMaterials:ImplicationsforMilitaryAircraftofNewGeneration9(1996):9. [115] Kumar,M.,Valasek,J.andChakravorty,S.AHierarchicalControlApproachtoMorphingDynamics.AIAAInfotechatAerospaceConferenceandExhibitandAIAAUnmanned...UnlimitedConference(2009). [116] LaCroix,B.W.andIfju,P.G.UtilizationandPerformanceEnhancementsofMultiplePiezoelectricActuatorsonMicroAirVehicles.ProceedingsoftheAIAAAerospaceSciencesMeeting(2012). [117] Lazos,B.S.andVisser,K.D.AerodynamicComparisonofHyper-EllipticCamberedSpan(HECS)WingswithConventionalCongurations.Proceedingsofthe25thAIAAAppliedAerodynamicsConference(2006). [118] Liani,Evandro,Guo,Shijun,andAllegri,Giuliano.Aeroelasticeffectonappingwingperformance.CollectionofTechnicalPapers-AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference8(2007):8518. [119] Licht,S.,Hover,F.andTriantafyllou,M.S.DesignofaFlappingFoilUnderwaterVehicle.IEEEJournalofOceanicEngineering21(2004). 190

PAGE 191

[120] Lind,R.,Brenner,M.andHaley,S.M.EstimationofModalParametersUsingaWavelet-BasedApproach.NASATechnicalMemorandum(1997).NASATM-97-206300. [121] Lind,R.,Snyder,K.,andBrenner,M.WaveletAnalysistoCharacterizeNon-linearitiesandPredictLimitCyclesofanAeroelasticSystem.Mechani-calSystemsandSignalProcessing15(2001):337. [122] Liu,H.,Ellington,C.P.,Kawachi,K.,Berg,C.VanDen,andWillmott,A.P.AComputationalFluidDynamicStudyofHawkmothHovering.JournalofExperimentalBiology201(1998):461.Http://jeb.biologists.org/cgi/reprint/201/4/461.pdf. [123] Liu,H.andKawachi,K.ANumericalStudyofInsectFlight.JournalofComputa-tionalPhysics146(1998):124.Doi:10.1006/jcph.1998.6019. [124] Liu,T.,Rhew,R.,Kuykendoll,K.andJones,S.AvianWingGeometryandKinematics.AIAAJournal44(2006):954. [125] Lock,R.J.,Burgess,S.C.,Vaidyanathan,R.andLoveless,J.DevelopmentofaBiologicallyInspiredMulti-ModalWingModelforAerial-aquaticRoboticVehiclesthroughEmpiricalandNumericalModelingoftheCommonGuillemot,UriaAalge.JournalofBioinspirationandBiomimetics5(2010).DOI:10.1088/1748-3182/5/4/046001. [126] Loudini,M.andHall,E.AdvancesinRobotManipulators:TimoshenkoBeamTheoryBasedDynamicModelingofLightweightFlexibleLinkRoboticManipulators.(2010). [127] Love,M.,Wieselmann,P.,Zink,P.andYoungren,H.BodyFreedomFlutterofHighAspectRatioWings.AIAAStructures,StructuralDynamicsandMaterialsConference(2005). [128] Love,R.Experimentally-BasedFlappingFlexibleWingModel.(2010).Http://www.adaptivestructure.com/. [129] Love,R.andLind,R.IdenticationofAeroservoelasticModelsfromExperimentalFlapping-WingDeections.AIAAAtmosphericFlightMechanicsConference(2009). [130] .Experimentally-BasedAeroservoelasticSystemIdenticationandFeedforwardControlofFlexibleFlappingWings.AIAAAtmosphericFlightMechanicsConference(2010).AIAA-2010-750. [131] Love,R.,Schwartz,E.,andArroyo.A.SolarRay:AnAutonomousSolar-PoweredBiomimeticFlapping-WingUnderwaterVehicle.FloridaConferenceonRecentAdvancesinRobotics(2010). 191

PAGE 192

[132] Love,R.,Wu,P.,Lind,R.andIfju,P.Time-FrequencyAnalysisofAeroelasticDeformationsofFlappingWings.Proceedingsofthe47thAIAAAerospaceSciencesMeeting(2009). [133] Love,R.D.AnalysisofAeroelasticFlapping-WingSignalsforMicroAir-Vehicles.Masters'Thesis,UniversityofFlorida(2009). [134] Lucia,DavidJ.TheSensorCraftcongurations:Anon-linearAeroServoElasticchallengeforaviation.CollectionofTechnicalPapers-AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamicsandMaterialsConference3(2005):1768. [135] Manzo,J.AnalysisandDesignofAHyper-EllipticalCamberedSpanMorphingAircraftWing.Masters'Thesis,CornellUniversity(2006). [136] March,A.I.,Bradley,C.W.andGarcia,E.AerodynamicPropertiesofAvianFlightasaFunctionofWingShape.Proceedingsofthe2005ASMEInternationalMechanicalEngineeringCongressandExposition(2005). [137] Marmier,P.andWerely,N.MorphingWingsofaSmallScaleUAVUsingInatableActuatorsforSweepControl.Proceedingsofthe44thAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference(2003). [138] Mason,W.H.,Robertshaw,H.andInman,D.J.RecentExperimentsinAerospaceandDesignEngineeringEducation.Proceedingsofthe42ndAIAAAerospaceSciencesMeetingandExhibit(2004). [139] Maybeck,P.S.StochasticModels,EstimationandControl.AcademicPress,1979. [140] McGowan,A.R.,Wilkie,W.K.,Moses,R.W.,Lake,R.C.,Pinkerton-Florance,J.L.,Weiseman,C.D.,Reaves,M.C.,Taleghani,B.K.,Mirick,P.H.andWilbur,M.L.AeroservoelasticandstructuraldynamicsresearchonsmartstructuresconductedatNASALangleyResearchCenter.vol.3326.SPIE,1998,188.Doi:10.1117/12.310634. [141] McGowan,A.R.,HortaL.G.,Bryant,R.G.,CoxD.E.,Siochi,E.J.,Padula,S.L.,Washburn,A.E.,andHolloway,N.M.RecentResultsfromNASA'sMorphingProject.Proceedingsofthe9thAnnualInternationalSymposiumonStructuresandMaterials(2002).SPIEPaperNumber4698-11. [142] Mengesha,T.E.,Barraja,M.,Vallance,R.R.andMittal,R.ParametricStructuralModelingofInsectWings.JournalofBioinspirationandBiomimicry4(2009).DOI:10.1088/1748-3182/4/3/036004. [143] Miao,J.M.andHo,M.H.EffectofFlexureonAerodynamicPropulsiveEfciencyofFlappingFlexibleAirfoil.JournalofFluidsandStructures22(2006):401. 192

PAGE 193

[144] Misiti,M.,Misiti,Y.,Oppenheim,G.,andPoggi,J.WaveletToolboxForUsewithMatlabUsersGuide.(1997). [145] Moored,K.W.andBart-Smith,H.OptimizationofaTensegrityWingforBiomimeticApplications.SmartStructuresandMaterials(2006):3272.Doi:10.1117/12.658930. [146] .InvestigationofClusteredActuationinTensegrityStructures.Interna-tionalJournalofSolidsandStructures46(2009):3272. [147] Moored,K.W.,Hester,J.M.,Chang,W.,Smith,W.andBart-Smith,H.InvestigatingtheThrustProductionofaMyliobatoid-InspiredOscillatingWing.AdvancesinScienceandTechnology58(2008):25.Http://www.bartsmithlabs.com/muri/publications.html. [148] Moses,R.W.,Henderson,D.A.,Galea,S.C.,Manokaran,D.S.,Zimcik,D.G.,Wickramasinghe,V.,Pitt,D.M.,Pototzky,A.S.andGamble,M.A.ControllingBuffetLoadsbyRudderandPiezo-Actuation.ForumonAeroelasticityandStructuralDynamics(2005). [149] Mountcastle,AndrewM.andDaniel,ThomasL.Aerodynamicandfunctionalconsequencesofwingcompliance.ExperimentsinFluids46(2009).5:873.Doi:10.1007/s00348-008-0607-0. [150] Mueller,T.J.FixedandFlappingWingAerodynamicsforMicroAirVehicleApplications.(2002). [151] Mueller,T.J.,Ifju,P.G.,Kellogg,J.C.andShkarayev,S.V.IntroductiontotheDesignofFixed-WingMicroAirVehicles:IncludingThreeCaseStudies.(2007). [152] Mukhopadhyay,V.HistoricalPerspectiveonAnalysisandControlofAeroelasticResponses.JournalofGuidance,ControlandDynamics26(2003):673. [153] Munteanu,S.,Rajadas,J.,Nam,C.,andChattopadhyay,A.Reduced-order-modelapproachforaeroelasticanalysisinvolvingaerodynamicandstructuralnonlinearities.AIAAJournal43(2005).3:560.Aeroelasticanalysis;Gaussianpulseresponse;Reducedordermodel;Structuralnonlinearities;. [154] Murrow,H.N.andEkstrom,C.V.DronesforAerodynamicandStructuralTesting(DAST)-AStatusReport.AIAAJournalofAircraft16(1979):521. [155] Neal,D.A.Design,DevelopmentandAnalysisofaMorphingAircraftModelforWindTunnelExperimentation.Master'sThesis,VirginiaPolytechnicInstituteandStateUniversity(2006). [156] Newsom,J.R.DesigningActiveControlLawsinaComputationalAeroelasticityEnvironment.Ph.D.Dissertation(2002). 193

PAGE 194

[157] Ngo,L.V.,Seidel,H.,Kupke,W.andSchmid,U.SimulationandExperimentalResultsofaHot-FilmAnemometerArrayonaFlexibleSubstrate.ConferenceonMicro-Nano-TechnologiesforAerospaceApplications(2004):142. [158] Nicolai,L.M.andCarichner,G.E.FundamentalsofAircraftandAirshipDesign:VolumeIAircraftDesign.(2010). [159] Niksch,A.,Strganac,T.W.,Valasek,J.,andCarlson,L.A.SixDegreeofFreedomDynamicalModelofaMorphingAircraft.ProceedingsoftheAIAAAtmosphericFlightMechanicsConference(2009). [160] Nogarede,B.,Rouchon,J.,Henaux,C.andDuhayon,E.ElectroactiveMaterials:FromPiezomotorstoElectroactiveMorphing.IECONProceedings(2006):4437. [161] Noll,T.E.Aeroservoelasticity.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference(1990).NASA-TM-102620. [162] Noll,Thomas,Blair,Maxwell,andCerra,John.ADAM,AnAeroservoelasticAnalysisMethodForAnalogorDigitalSystems.JournalofAircraft23(1986).11:852. [163] Noor,A.K.StructuresTechnologyforFutureAerospaceSystems.(2000). [164] Norberg,U.M.L.,Brooke,A.P.andTrewhella,W.J.SoaringandNon-SoaringBatsoftheFamilyPteropodidae(FlyingFoxes,PteropusSPP.):WingMorphologyandFlightPerformance.TheJournalofExperimentalBiology203(2000):651. [165] Norris,A.G.,Palazotto,A.N.andCobb,R.G.StructuralDynamicCharacterizationofanInsectWing.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference(2010). [166] Nunes,L.C.S,Castello,D.A.,C.F.Matt,anddosSantos,P.A.M.ParameterEstimationUsingDigitalImageCorrelationandInverseProblems.MechanicsofSolidsinBrazil(2007). [167] Oberkampf,W.L.,Rutherford,B.M.,Diegert,K.V.,DeLand,S.M.andAlvin,K.F.ErrorandUncertaintyinModelingandSimulation.ReliabilityEngineeringandSystemSafety75(2002):333. [168] Oh,J.T.,Park,H.C.andHwang,W.ActiveShapeControlofaDouble-PlateStructuresUsingPiezoceramicsandSMAWires.SmartMaterialsandStructures10(2001):1100.Stacks.iop.org/SMS/10/1100. [169] Palmisano,J.,Ramamurti,R.,Liu,K.J.,Cohen,J.,Mengesha,T.,Naciri,J.,Sandberg,W.C.,Geder,J.andRatna,B.Bio-MechanismsofSwimmingandFlying:DesignDevelopmentandTestingofFlappingFinswithActivelyControlledCurvatureforanUnmannedUnderwaterVehicle.(2007):283. 194

PAGE 195

[170] Patel,S.C.,Koh,B.S.,Junkins,J.L.,Majji,M.andRediniotis,O.K.MorphingWing:ADemonstrationofAeroServoElasticDistributedSensingandControl.TiiMS(2005). [171] PeloubetJr.,R.P.YF16Active-Control-System/StrucutralDynamicsInteractionInstability.(1975). [172] Penning,KevinB.,Zink,P.Scott,Wei,Paul,DeLaGarza,AntonioP.,Love,MichaelH.,andMartinez,Juan.GLAanduttersuppressionforaSensorCraftclassconceptusingsystemidentication.CollectionofTechnicalPapers-AIAAAppliedAerodynamicsConference(2008). [173] Peters,D.Two-dimensionalIncompressibleUnsteadyAirfoilTheory-AnOverview.JournalofFluidsandStructures24(2008):295. [174] Pitt,D.M.andGoodman,C.E.FAMUSS-ANewAeroservoelasticModelingTool.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference(1992). [175] Ploemen,I.H.J.HybridModelingforMechanicalSystems:MethodologiesandApplications.JournalofDynamicSystems,Measurement,andControl121(1999):270. [176] Popov,A.V.,Botez,R.,Mamou,M.,Grigorie,A.andMebarki,Y.RealTimeMorphingWingOptimizationValidationUsingWind-TunnelTests.JournalofAircraft47(2010):1346.Doi:10.2514/1.47431. [177] Pornsin-Sirirak,T.N.,Ho,,C.,Tai,Y.andKeennon,M.Microbat:APalm-SizedElectricallyPoweredOrnithopter.ProceedingsofNASA/JPLWorkshoponBiomorphicRobotics(2001). [178] Putnam,T.W.andRobinson,M.R.ClosingtheDesignLooponHiMAT(HighlyManeuverableAircraftTechnology.NASADryden(1984). [179] Rakotomamonjy,T.,Ouladsine,M.,andMoing,T.L.Modelizationandkinematicsoptimizationforaapping-wingmicroairvehicle.JournalofAircraft44(2007).1:217.Doi:10.2514/1.22960. [180] Ramamurti,R.,Palmisano,J.,Ratna,B.,Geder,J.andSandberg,W.C.ComputationsofFlappingFlowPropulsionforUnmannedUnderwaterVehicleDesign.AIAAJournal48(2010).Doi:10.2514/1.43389. [181] Ramrkahyani,D.S.,Bharti,S.,Lesieutre,G.A.andFrecker,M.AircraftStructuralMorphingusingTendonActuatedCompliantCellularTrusses.JournalofAircraft42(2005):1615. 195

PAGE 196

[182] Ramrkahyani,D.S.,Frecker,M.,Lesieutre,G.A.,andBharti,S.AircraftStructuralMorphingUsingTendonActuatedCompliantCellularTrusses.AIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference(2004). [183] Rangarao,K.V.andMallik,R.DigitalSignalProcessing:APractitioner'sApproach.(2005). [184] Raymer,D.P.AircraftDesign:AConceptualApproach.(2006). [185] Reich,G.andSanders,B.IntroductiontoMorphingAircraftResearch.JournalofAircraft44(2007):1059.Doi:10.2514/1.28287. [186] RobinettIII,R.D.,Eisler,G.R.,Dohrmann,C.,Parker,G.G.,Wilson,D.G.,Feddema,J.andStokes,D.FlexibleRobotDynamicsandControls.(2001). [187] Roger,KennethL.,Hodges,GaroldE.,andFelt,Larry.ActiveFlutterSuppression-AFlightTestDemonstration.JournalofAircraft12(1975).6:551. [188] Rosenberger,L.J.PectoralFinLocomotioninBatoidFishes:UndulationVersusOscillation.TheJournalofExperimentalBiology204(2001):379. [189] Roth.AircraftSizingwithMorphingasanIndependentVariable:Motivation,Strategies,andInvestigations.AIAAsAircraftTechnology,Integration,andOperations2002Technical(2002). [190] Rozhdestvensky,K.V.andRyzhov,V.A.AerohydrodynamicsofFlapping-WingPropulsors.ProgressinAerospaceSciences39(2003):585. [191] Sallstrom,E.andUkeiley,L.Three-DimensionalAveragedFlowAroundFlappingWings.Proceedingsofthe38thAIAAFluidDynamicsConferenceandExhibit(2008). [192] Sane,S.P.TheAerodynamicsofInsectFlight.JournalofExperimentalBiology206(2003):4191.Doi:10.1242/jeb.00663. [193] Sane,S.P.andDickinson,M.H.TheControlofFlightForcebyaFlappingWing:LiftandDragProduction.TheJournalofExperimentalBiology204(2001):2607. [194] .TheAerodynamicEffectsofWingRotationandaRevisedQuasi-SteadyModelofFlappingFlight.JournalofExperimentalBiology205(2002):1087. [195] Sarma,G.R.TransferFunctionAnalysisoftheConstantVoltageAnemometer.ReviewofScienticInstruments69(1998):2385. 196

PAGE 197

[196] Scheuren,W.J.,Goodman,G.A.,Caldwell,K.A.andWegman,A.K.JointStrikeFighterPrognosticsandHealthManagement.ProceedingsoftheInternationalPoweredLiftConference(1998):38.1.7. [197] Schlesinger,S.TerminologyforModelCredibility.Simulation(1979):103. [198] Schneiter,G.R.ModelDesignationofMilitaryAerospaceVehicles.(1998). [199] Schreier,H.W.,Braasch,J.R.,andSutton,M.A.SystematicErrorsinDigitalImageCorrelationcausedbyIntensityInterpolation.OpticalEngineering39(2000):2915.Doi:10.1117/1.1314593. [200] Schuster,D.M.,Liu,D.D.,andHuttsell,L.J.ComputationalAeroelasticity:Success,Progress,Challenge.JournalofAircraft40(2003):843. [201] Schwarz,B.J.andRichardson,M.H.ExperimentalModalAnalysis.ProceedingsofCSIReliabilityWeek(1999). [202] Seigler,T.M.DynamicsandControlofMorphingAircraft.Ph.DDissertation,VirginiaPolytechnicInstituteandStateUniversity(2005). [203] Seigler,T.M.,Neal,D.A.andInman,D.J.DynamicModelingofLarge-scaleMorphingAircraft.ProceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference5(2006):3668. [204] Seigler,T.M.,Bae,J.,Neal,D.A.andInman,D.J.ModelingandFlightControlofLarge-ScaleMorphingAircraft.JournalofAircraft44(2007):1077.10.2514/1.21439. [205] .ModelingandFlightControlofLarge-ScaleMorphingAircraft.JournalofAircraft44(2007):1077.Doi:10.2514/1.21439. [206] Senjanovic,I.,Tomasevic,S.andMalenica,S.InvestigationofShipHydroelasticity.OceanicEngineering35(2008):523. [207] Shang,J.S.ComputationalFluidDynamicsApplicationtoAerospaceScience.AeronauticalJournal113(2009):619. [208] Shoele,K.andZhu,Q.NumericalSimulationofaPectoralFinDuringLabriformSwimming.TheJournalofExperimentalBiology213(2010):2038.Doi:10.1242/jeb.040162. [209] Shortelle,K.AerodynamicControlofMicroAirWeapons:MonthlyProgressReportMPR-0149-04.SystemDynamicsInternational,Inc.,ContractNo.FA8651-10-C-0149(2010). [210] Shyy,W.,Chimakurthi,S.K.,Trizila,P.,Kang,C.K.,Cesnik,C.E.S.,Aono,H.andLiu,H.RecentProgressinFlappingWingAerodynamicsandAeroelasticity.ProgressinAerospaceSciences46(2010):284. 197

PAGE 198

[211] Shyy,W.,Lian,Y.,Tang,J.,Viieru,D.,andLiu,H.AerodynamicsofLowReynoldsNumberFlyers.(2008). [212] Shyy,W.,Trizila,P.,Chang-kwon,K.,andAono,H.CanTipVorticesEnhanceLiftofaFlappingWing?AIAAJournal47(2009):289. [213] Simpson,J.O.,Bryant,R.G.,Cano,R.J.,Gates,T.S.,Hinkley,J.A.,Rogowski,R.S.,Wise,S.A.,andWhitley,K.S.InnovativeMaterialsforAircraftMorphing.IndustrialandCommercialApplicationsofSmartStructuresTechnologies3326(1998).3326-20. [214] Singh,B.DynamicsandAeroelasticityofHoverCapableFlappingWings-ExperimentsandAnalysis.Ph.DDissertation(2006).Http://hdl.handle.net/1903/6663. [215] Singh,B.andChopra,I.Insect-basedhover-capableappingwingsformicroairvehicles:Experimentsandanalysis.AIAAJournal46(2008).9:2115.Doi:10.2514/1.28192. [216] Sitz,J.J.AeroelasticAnalysisofAJoined-WingSensorcraft.Ph.D.Dissertation(2004). [217] Skillen,M.D.andCrossley,W.A.ModelingandOptimizationforMorphingWingConceptGeneration.NASA/CR-2008-214903(2007). [218] .MorphingWingWeightPredictorsandTheirApplicationinaTemplate-BasedMorphingAircraftSizingEnvironmentIIPartII:MorphingAircraftSizingViaMulti-levelOptimization.NASA/CR-2007-214860(2007). [219] Smith,M.J.C.SimulatingFlappingWingsUsinganAerodynamicPanelMethod.PhDThesis(1995). [220] Soa,A.Y.N,Tan,K.T.,Meguid,S.A.,andYeo,W.K.ShapeMorphingofAircraftWing:StatusandChallenges.MaterialsandDesign31(2010):1284.Doi:10.1016/j.matdes.2009.09.011. [221] Spedding,G.R.,Rosen,M.andHendenstrom,A.AFamilyofVortexWakesGeneratedbyaThrustNightingaleinFreeFlightinaWindTunnelOveritsEntireNaturalRangeofFlightSpeeds.JournalofExperimentalBiology206(2003):2313. [222] Sriram,P.,Hanagud,S.,andCraig,J.ModeShapeMeasurementusingaScanningLaserDopplerVibrometer.JournalofAnalyticalandExperimentalModalAnalysis7(1992):169. [223] Stanbridge,A.B.andEwins,D.J.ModalTestingUsingaScanningLaserDopplerVibrometer.MechanicalSystemsandSignalProcessing13(1999):255.Doi:10.1006/mssp.1998.1209. 198

PAGE 199

[224] Stanford,B.K.AeroelasticAnalysisandOptimizationofMembraneMicroAirVehicleWings.DoctoralDissertation,UniversityofFlorida(2007). [225] Stanford,B.K.andIfju,P.TheValidityRangeofLowFidelityStructuralMembraneModels.ExperimentalMechanics48(2008):697.Doi:10.1007/s11340-008-9152-2. [226] Strganac,T.W.,Lassiter,J.,Cole,S.,Silva,W.,Kurdila,A.,Reichenbach,E.,Cesnik,C.andLind,R.Aeroelasticity:State-of-the-ArtPractices.AIAAShortCourseNotes(2006). [227] Stubbs,M.D.KinematicDesignandAnalysisofaMorphingVehicle.Master'sThesis,VirginiaPolytechnicInstituteandStateUniversity(2003). [228] Su,W.andCesnik,C.E.S.DynamicResponseofHighlyFlexibleFlyingWings.AIAAJournal49(2011):324.Doi:10.2514/1.J050496. [229] Sun,C.T.MechanicsofAircraftStructures.(2006). [230] Sutton,M.,Chen,M.,Peters,W.,Chao,Y.,andMcNeill,S.ApplicationofanOptimizedDigitalImageCorrelationMethodtoPlanarDeformationAnalysis.ImageandVisionComputing3(1986):143.Doi:10.1016/0262-8856(86)90057-0. [231] Sutton,M.A.,Wolters,W.J.,Peters,W.H.,Ranson,W.F.,andMcNeil,S.R.DeterminationofDisplacementsUsingAnImprovedDigitalCorrelationMethod.ImageandVisionComputing1(1983):133.Doi:10.1016/0262-8856(83)90064-1. [232] Tai,H.ShapeSensingaMorphedWingwithanOpticalFiberBraggGrating.ProceedingsofSmartStructuresandMaterials(2005).Doi:10.1117/12.592764. [233] Takahashi,H.,Matsumoto,K.,andShimoyama,I.AirPressureSensorforanInsectWing.IEEEInternationalConferenceonMicroElectroMechanicalSystems(2009).Doi:10.1109/MEMSYS.2009.4805510. [234] Tangorra,J.L.,Hunter,I.W.,Madden,P.G.A.,Lauder,G.V.,Dong,H.,Bozkurttas,M.,Davidson,S.N.andMittal,R.TheDevelopmentofaBiologicallyInspiredPropulsorforUnmannedUnderwaterVehicles.IEEEJournalofOceanicEngi-neering32(2007):533. [235] Taylor,G.K.andThomas,A.L.R.AnimalFlightDynamicsII.LongitudinalStabilityinFlappingFlight.JournalofTheoreticalBiology214(2002):351.Doi:10.1006/jtbi.2001.2470. [236] Taylor,G.K.,Nudds,R.L.andThomas,A.L.R.FlyingandSwimmingAnimalsCruiseataStrouhalNumberTunedforPowerEfciency.Nature425(2003):707. 199

PAGE 200

[237] Taylor,N.V.,Jones,D.P.,Gaitonde,A.L.andAllen,C.B.ModelingtheBenchmarkActiveControlsWingthroughLinearandComputationalAeroelasticAnalyses.JournalofAircraft44(2007):1383.10.2514/1.22959. [238] Tennekes,H.TheSimpleScienceofFlight(FromInsectstoJumboJets).(1997). [239] Theodorsen,T.GeneralTheoryofAerodynamicInstabilityandtheMechanismofFlutter.NACATechnicalReports496(1934). [240] Tokhi,M.O.andAzad,A.K.M.FlexibleRobotManipulators-Modeling,SimulationandControl.Knovel,2008. [241] Tong,B.J.,Zhuang,B.J.,andCheng,J.Y.TheHydrodynamicAnalysisofFishPropulsionPerformanceanditsMorphologicalAdaptation.Sadhana18(1993):719. [242] Toomey,J.andEldredge,J.NumericalandExperimentalInvestigationoftheRoleofFlexibilityinFlappingWingFlight.Proceedingsofthe36thAIAAFluidDynamicsConferenceandExhibit(2006). [243] Trame,L.W.,Williams,L.E.,andYurkovich,R.N.ActiveAeroelasticOscillationControlontheF/A-18Aircraft.AIAAPaper(1985):94. [244] Trease,B.andKota,S.DesignofAdaptiveandControllableCompliantSystemswithEmbeddedActuatorsandSensors.JournalofMechanicalDesign131(2009).Doi:10.1115/1.3149848. [245] Triantafyllou,M.S.,Triantafyllou,G.S.andYue,D.K.P.HydrodynamicsofFishlikeSwimming.AnnualReviewofFluidMechanics32(2000):33. [246] Trizila,P.andKang,C.ASurrogateModelApproachin2Dversus3DFlappingWingAerodynamicAnalysis.ProceedingsoftheAIAAISSMOMultidiscioplinaryAnalysisandOptimizationConference(2008).AIAA-2008-5914. [247] Valasek,J.,Tandale,M.andRong,J.AReinforcementLearningAdaptiveControlArchitectureforMorphing.JournalofAerospaceComputing,Information,andCommunication2(2005). [248] Vialatte,F.,Martin,C.,Dubois,R.,Haddad,J.,Quenet,B.,Gervais,R.,andDreyfus,G.AMachineLearningApproachtotheAnalysisofTime-FrequencyMapsanditsApplicationtoNeuralDynamics.NeuralNetworks20(2007):194.Doi:10.1016/j.neunet.2006.09.013. [249] Voracek,D.AdaptiveStructures.AIAAAerospaceAmerica44(2006):72. [250] Vos,R.,Barrett,R.,Debreuker,R.andTiso,P.MorphingWingFlightControlViaPostbuckledPrecompressedPiezoelectricActuators.JournalofAircraft44(2007):1060.Doi:10.2514/1.21292. 200

PAGE 201

[251] Vos,R.,Debreuker,R.,Barrett,R.andTiso,P.Post-buckledPrecompressedElements:ANewClassofControlActuatorsforMorphingWingUAVs.SmartMaterialsandStructures16(2007):919.Doi:10.1088/0964-1726/16/3/042. [252] Vos,R.,Gurdal,Z.andAbdalla,M.MechanismforWarp-ControlledTwistofaMorphingWing.JournalofAircraft47(2010):450.Doi:10.2514/1.39328. [253] Walker,J.A.andWestneat,M.W.LabriformPropulsioninFishes:KinematicsofFlappingAquaticFlightintheBirdWrasseGomphosusVarius(Labridae).TheJournalofExperimentalBiology200(1997):1549. [254] Walker,S.M.,Thomas,A.L.R.andTaylor,G.K.PhotogrammetricReconstructionofHigh-resolutionSurfaceTopographiesandDeformableWingKinematicsofThetheredLocustsandFree-yingHoveries.JournaloftheRoyalSociety6(2009):351.Doi:10.1098/rsif.2008.0245. [255] Wang,Z.J.VortexSheddingandFrequencySelectioninFlappingFlight.JournalofFluidMechanics410(2000):323.Doi:10.1017/S0022112099008071. [256] Wang,Z.J.,Birch,J.M.,andDickinson,M.H.UnsteadyForcesandFlowsinLowReynoldsNumberHoveringFlight:Two-DimensionalComputationsvsRoboticWingExperiments.JournalofExperimentalBiology207(2004):449.Doi:10.1242/jeb.00739. [257] Warrick,D.R.,Tobalske,B.W.,Powers,D.R.,andDickinson,M.H.TheAerodynamicsofHummingbirdFlight.ProceedingsoftheAIAAAerospaceSciencesMeeting(2007). [258] Weiss,J.,Chokani,N.andComte-Bellot,G.ComparativeMeasurementsinMa=2.54FlowusingConstant-TemperatureandConstant-VoltageAnemometry.AIAAJournal43(2005):1140. [259] Weiss,P.WingsofChange.ScienceNews164(2003):359,362,365.Http://www.jstor.org/stable/4018925. [260] Weisshaar,T.A.MorphingAircraftTechnology-NewShapesforAircraftDesign.MultifunctionalStructures/IntegrationofSensorsandAntennasMeetingProceed-ings(2006).RTO-MP-AVT-141. [261] Weisshaar,TerrenceA.andNam,Changho.Aeroservoelastictailoringforlateralcontrolenhancement.JournalofGuidance,Control,andDynamics13(1990).3:458. [262] Whitmer,C.E.,Vu,P.,Kelkar,A.G.,andChavez,F.R.ModelingandControlofaMorphingAirfoil.ProceedingsoftheASMEDynamicSystemsandControlDivision72(2003):829. [263] Whitmer,C.E.andKelkar,A.G.RobustControlofaMorphingAirfoilStructure.Proceedingsofthe2005AmericanControlConference(2005). 201

PAGE 202

[264] Whittier,W.B.KinematicAnalysisofTensegrityStructures.Master'sThe-sis,VirginiaPolytechnicInstituteandStateUniversity(2002). [265] Wickenheiser,A.M.andGarcia,E.AerodynamicModelingofMorphingWingsUsinganExtendedLifting-LineAnalysis.JournalofAircraft44(2007). [266] Wieseman,C.MethodologyforUsingSteadyExperiemtnalAerodynamicDatatoImproveSteadyandUnsteadyAerodynamicAnalysis.Masters'Thesis,GeorgeWashingtonUniversity(1989). [267] Wilkie,W.K.,High,J.W.,Fox,R.L.,Hellbaum,R.F.,JalinkJr.,A.,Little,B.D.,Bryant,G.R.andMirick,P.H.Low-CostPiezocompositeActuatorforStructuralControlApplications.SPIE7thAnnualInternationalSymposiumonSmartStructuresandMaterials(2000).DOI:10.1117/12.388175. [268] Willmott,A.P.andEllington,C.P.MeasuringtheAngleofAttackofBeatingInsectWings:RobustThree-DimensionalReconstructionfromTwo-DimensionalImages.JournalofExperimentalBiology200(1997):2693. [269] .TheMechanicsofFlightintheHawkmothManducaSextaI.KinematicsofHoveringandForwardFlight.JournalofExperimentalBiology200(1997):2705. [270] .TheMechanicsofFlightintheHawkmothManducaSextaII.AerodynamicConsequencesofKinematicandMorphologicalVariation.JournalofExperimentalBiology200(1997):2723. [271] Wilson,J.R.MorphingUAVsChangetheShapeofWarfare.AerospaceAmerica(2004):23. [272] Wootton,R.J.FunctionalMorphologyofInsectWings.AnnualReviewofEntomology37(1992):113. [273] Wootton,R.J.FunctionalMorphologyofInsectWings.AnnualReviewofEntomology37(1999):113. [274] Wright,J.R.andCooper,J.E.IntroductiontoAircraftAeroelasticityandLoads.(2008). [275] Wu,P.ExperimentalCharacterization,Design,AnalysisandOptimizationofFlexibleFlappingWingsforMicroAirVehicles.Ph.D.Dissertation,UniversityofFlorida(2010). [276] Wu,P.andIfju,P.ExperimentalMethodologyforFlappingWingStructureOptimizationinHoveringFlightofMicroAirVehicles.Proceedingsofthe51stAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,andMaterialsConference(2010). 202

PAGE 203

[277] Wu,P.andStanford,B.StructuralDeformationMeasurementsofAnisotropicFlexibleFlappingWingsforMicroAirVehicles.ProceedingsoftheAIAAStruc-tures,StructuralDynamicsandMaterialsConference(2008). [278] .PassiveBendingandTwistingMotionDuringtheFlappingStrokeofaMicroElasticWingforThurstProduction.ProceedingsoftheAIAAAerospaceSciencesMeeting(2009).AIAA-2009-879. [279] Wu,P.,Stanford,B.K.,Sallstrom,E.,Ukeiley,L.,Love,R.,Ifju,P.andLind,R.AnExperimentalStudyonFlappingWingAeroelasticityinThrustProduction.Pro-ceedingsoftheAIAAStructures,StructuralDynamicsandMaterialsConference(2009). [280] Yan,J.,Wood,R.J.,Avadhanula,S.,Sitti,M.,andFearing,R.S.TowardsFlappingWingControlforaMicromechanicalFlyingInsect.ProceedingsoftheIEEEInternationalConferenceonRoboticsandAutomation4(2001):3901.Doi:10.1109/ROBOT.2001.933225. [281] Yang,S.,Qiu,J.andHan,X.KinematicsModelingandExperimentsofPectoralOscillationPropulsionRoboticFish.JournalofBionicEngineering6(2009):174. [282] Yang,T.,Wei,M.andZhao,H.NumericalStudyofFlexibleFlappingWingPropulsion.AIAAJournal48(2010):2909.Doi:10.2514/1.J050492. [283] Zdunich,P.,MacMaster,M.,Loewen,D.,DeLaurier,J.,Kornbluh,R.,Low,T.,Stanford,S.,Bilyk,D.andHoleman,D.DevelopmentandTestingoftheMentorFlapping-WingMicroAirVehicle.JournalofAircraft44(2007):1701.Doi:10.2514/1.28463. [284] Zbikowski,R.Sensor-richFeedbackControl:ANewParadigmforFlightControlInspiredbyInsectAgility.IEEEInstrumentationandMeasurementMagazine7(2004):19. [285] Zhao,Z.andRen,G.MultibodyDynamicApproachofFlightDynamicsandNonlinearAeroelasticityofFlexibleAircraft.AIAAJournal49(2011):41.Doi:10.2514/1.45334. 203

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BIOGRAPHICALSKETCH RobertLovewasborninMemphis,Tennesseein1984.Acuriousandimaginativechild,herstknewhewantedtoworkinaerospaceashewatchedagroupoftheAirForceF-16Thunderbirdsperformtheirbombburststunt.Hisyouthwasfullofadventuresallovertheworldashetraveledthroughover26countrieswhilelivinginRedlands,California,Daharan,SaudiaArabiaandJacksonville,Florida.HeattendedhighschoolatEpiscopalHighSchoolandwasapartofateamthatwasanationalnalistintheBotballroboticscompetition.DuringcollegeatAuburnUniversity,hedidresearchonsuperalloyjoiningandbiologically-inspiredanddecontamination-environment-capableroboticsintheMaterialsEngineeringLabs,andightcontrolofawing-in-ground-effectvehicleheconstructedintheAdaptiveAerostructuresLab.Healsoco-ledateamthatdesignedalowaspectratioyingwingfortheAmericanInstituteofAerospaceandAeronauticsDesign,Build,Flycompetition.HegraduatedmagnacumlaudeinMayof2007withbachelor'sdegreesinbothaerospaceengineeringandmaterialsengineering.AftermovingtotheUniversityofFlorida,hereceivedhismaster'sdegreeinaerospaceengineeringinMayof2009anddoctorateinAugustof2011.Duringgraduateschool,hehasworkedonanautonomous,biomimetic,apping-wingunderwatervehicle,anopen-source3Dprinter,andredesigninganornithoptertocaptureonboardvideo.RobertperformedthisresearchattheUniversityofFloridawhilestudyingunderDr.RickLindintheFlightControlLab.Hisresearchinterestsspanaeroservoelasticity,systemidenticationandcontrol,compositestructures,robotics,adaptivematerials,computervision,signalprocessing,modalanalysis,biomechanicsandnanostructures.Hisdoctoralresearchhasfocusedontheanalysis,modelingandcontroldesignofmorphingandappingwings.Heshareshisexcitementandinsightswithothersathttp://www.adaptivestructure.com. 204