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Analysis of Aeroelastic Flapping-Wing Signals for Micro Air-Vehicles

Permanent Link: http://ufdc.ufl.edu/UFE0024479/00001

Material Information

Title: Analysis of Aeroelastic Flapping-Wing Signals for Micro Air-Vehicles
Physical Description: 1 online resource (86 p.)
Language: english
Creator: Love, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aerial, aeroelastic, aeroservoelastic, aerospace, analysis, correlation, deflection, deformation, dic, digital, doppler, dynamics, engineer, flapping, fourier, ground, gvt, image, laser, ldv, mav, modal, mode, model, ornithopter, processing, signal, testing, uav, unmanned, vehicle, vibration, vibrometer, wavelet, wing
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Flapping-wing micro air-vehicles are being investigated for their potential to provide enhanced aerodynamic efficiency, maneuverability and gust tolerance. The flight dynamics of flapping-wing micro air-vehicles result from a complicated relationship between aerodynamics and structural dynamics. This relationship has both frequency-domain aspects and time-domain aspects that are each critical. As such, analyzing data from a flapping wing requires techniques that can process information related to both of these domains. This thesis introduces wavelet analysis as a tool to determine the frequency content of time-varying signals from a flapping wing testbed. Wavelet maps present a time-frequency domain representation that relates both time-domain and frequency-domain aspects. Data obtained with digital image correlation, including deflections and deformations while flapping and displacements under shaker excitation, is analyzed using wavelet processing from a set of wings with different structural dynamics and different flapping parameters. The resulting wavelet maps demonstrate the variations in energy content and temporal distribution associated with these signals. This thesis also examines wings of interest with a laser doppler vibrometer to obtain operating deflection shapes and their corresponding frequencies. These shapes and frequencies are compared to the results obtained using digital image correlation. The dynamics of a flapping-wing vehicle are inherently aeroservoelastic since the interaction of aerodynamics and structural dynamics are critical to performance and will be altered by any control effectors. Signals formed from deflections of a flapping wing using digital image correlation may provide a basis for an aeroservoelastic model of the flapping wing. This thesis generates a model of the aeroservoelastic dynamics for a wing as a function of the flapping frequency and flapping amplitude. These models are polynomial functions of each parameter and capture the nonlinear behavior. Most importantly, the resulting models are a basis from which to compute the wing deflections in response to any control command.
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 Love.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024479:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024479/00001

Material Information

Title: Analysis of Aeroelastic Flapping-Wing Signals for Micro Air-Vehicles
Physical Description: 1 online resource (86 p.)
Language: english
Creator: Love, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: aerial, aeroelastic, aeroservoelastic, aerospace, analysis, correlation, deflection, deformation, dic, digital, doppler, dynamics, engineer, flapping, fourier, ground, gvt, image, laser, ldv, mav, modal, mode, model, ornithopter, processing, signal, testing, uav, unmanned, vehicle, vibration, vibrometer, wavelet, wing
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Flapping-wing micro air-vehicles are being investigated for their potential to provide enhanced aerodynamic efficiency, maneuverability and gust tolerance. The flight dynamics of flapping-wing micro air-vehicles result from a complicated relationship between aerodynamics and structural dynamics. This relationship has both frequency-domain aspects and time-domain aspects that are each critical. As such, analyzing data from a flapping wing requires techniques that can process information related to both of these domains. This thesis introduces wavelet analysis as a tool to determine the frequency content of time-varying signals from a flapping wing testbed. Wavelet maps present a time-frequency domain representation that relates both time-domain and frequency-domain aspects. Data obtained with digital image correlation, including deflections and deformations while flapping and displacements under shaker excitation, is analyzed using wavelet processing from a set of wings with different structural dynamics and different flapping parameters. The resulting wavelet maps demonstrate the variations in energy content and temporal distribution associated with these signals. This thesis also examines wings of interest with a laser doppler vibrometer to obtain operating deflection shapes and their corresponding frequencies. These shapes and frequencies are compared to the results obtained using digital image correlation. The dynamics of a flapping-wing vehicle are inherently aeroservoelastic since the interaction of aerodynamics and structural dynamics are critical to performance and will be altered by any control effectors. Signals formed from deflections of a flapping wing using digital image correlation may provide a basis for an aeroservoelastic model of the flapping wing. This thesis generates a model of the aeroservoelastic dynamics for a wing as a function of the flapping frequency and flapping amplitude. These models are polynomial functions of each parameter and capture the nonlinear behavior. Most importantly, the resulting models are a basis from which to compute the wing deflections in response to any control command.
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 Love.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024479:00001


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ThankstotheAirForceOceofScienticResearchandtheUniversityofFlorida,whomadethisworkpossiblebysponsoringmeasapartofamulti-universityresearchinitiativeunderFA9550-07-1-0547andanAlumniFellowship.IwouldliketothankDr.RickLindformentoringmethroughouttheprocessofgraduateschool,andforprovidingfocusandmotivationwhereverandwhenevernecessary.IwouldalsothankDr.PeterIfjuandDr.LarryUkeileyforbeingonmycommitteeandfortheirwillingnesstoshareideasandresourcesthroughouttheprocess.SpecialappreciationisextendedtoPinWuandJohnSaxonattheUniversityofFloridaandBretStanfordofWright-PattersonAirForceBasefortheirassistanceinobtainingexperimentaldataandtoRobertoAlbertanioftheUniversityofFlorida-REEFforprovidingaccesstoexperimentalfacilities.ThewillingnessofPinWuandBretStanfordtoallowmetostandontheirshouldersprovidedanexampleofrealcollaborationandamodelforwhoIneededtobecomeinordertobeasuccessfulgraduatestudent.ThanksalsotomycolleaguesSankethBhat,JoeKehoe,BrianRoberts,BaronJohnson,DanielGrant,DavidEaton,RyanHurley,SeanRegisford,MujahidAbdulrahim,andDongTranintheFlightControlLabandErikSallstromattheUF-REEFfortheirwillingnesstoshareandsharpenideasaswellastoparticipateinthejoysandburdensassociatedwithgraduateschool.Thankstomyteacherswhoinvestedtheirtimeinsharingtheirknowledgeandthemanyotherfriendswhohavedirectlyorindirectlyimpactedmylife.ThanksalsotoDad,Mom,Shelley,andLaura;Iloveyoumorethanwordscanexpress.Nomanisanislandandwithoutallthesupportoftheseindividualsthisworkwouldnothavebeenpossible. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1Introduction ...................................... 13 1.1Motivation .................................... 13 1.2Contributions .................................. 14 2Background ...................................... 16 2.1ExperimentalTechniques ............................ 16 2.1.1DigitalImageCorrelation ........................ 16 2.1.2LaserDopplerVibrometry ....................... 17 2.1.3DigitalImageCorrelationofStructureUnderShakerExcitation .. 19 2.2SignalProcessing ................................ 19 2.2.1FrequencyDomainTechniques ..................... 19 2.2.2Time-FrequencyTechniques ...................... 20 2.3Reduced-OrderModelling ........................... 21 3Methodology ..................................... 24 3.1WingDesign ................................... 24 3.2MechanismDesign ............................... 26 3.3DigitalImageCorrelation ........................... 27 3.4LaserDopplerVibrometry ........................... 30 3.4.1Procedure ................................ 30 3.4.2FrequenciesforOperatingDeectionShapes ............. 32 3.4.3OperatingDeectionShapes ...................... 32 3.5DigitalImageCorrelationofStructureUnderShakerExcitation ...... 33 3.6SignalFormation ................................ 34 3.6.1TimeandFrequencyTechniques .................... 34 3.6.2Time-FrequencyTechniques ...................... 34 3.7Reduced-OrderModelling ........................... 34 3.7.1Identication ............................... 35 3.7.2Interpretation .............................. 36 5

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......................................... 37 4.1DigitalImageCorrelation ........................... 37 4.1.1Deections ................................ 37 4.1.1.1OverviewofSteadyStateSignalIdenticationandAnalysis 37 4.1.1.2ExtractingFeatures ...................... 39 4.1.1.3ComparingFlapping-WingParameters ........... 44 4.1.2Deformations .............................. 54 4.1.2.1ExtractingFeatures ...................... 55 4.1.2.2ComparingFlapping-WingParameters ........... 59 4.2LaserDopplerVibrometry ........................... 59 4.2.1Validation ................................ 59 4.2.2FrequenciesofOperatingDeectionShapes .............. 63 4.2.3OperatingDeectionShapes ...................... 65 4.2.3.1VariationsinShape ...................... 65 4.2.3.2VariationsinStructure .................... 65 4.2.3.3VariationsinMembraneMaterial .............. 66 4.2.3.4VariationsinStiness .................... 67 4.2.3.5ShapesatMaximumThrust ................. 67 4.3DigitalImageCorrelationofStructureUnderShakerExcitation ...... 68 4.4Reduced-OrderModelling ........................... 73 5ConclusionsandFutureWork ............................ 79 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 86 6

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Table page 3-1WingCharacteristics ................................. 25 4-1ModalFrequenciesforValidationTest ....................... 63 4-2WingModalFrequencies,(bc=burstchirpexcitation,s=sweepexcitation) ... 64 4-3DynamicParameters ................................. 74 7

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Figure page 1-1OverviewofExperimentalAnalysisMethodsPotentiallyUsefultoDeriveaReducedOrderModelForFWMAVs ............................. 15 2-110HzSinusoidTimeHistory,FastFourierTransformandWaveletAnalysis(Frequencyvs.TimeandMagnitudevs.TimeViews) ..................... 21 2-2ExampleMorletMotherWavelet .......................... 22 3-1WingsUsedAluminumBeamforValidation .................... 24 3-2SingleDegreeofFreedomFlappingMechanism,DesignedandBuiltbyWuforFlappingWingExperimentation ........................... 26 3-3DICExperimentalSetup ............................... 27 3-4ConstructionofFictionalRigidWing:ReferenceWing,FictionalRigidWingandFlexibleWing .................................. 29 3-5LDVExperimentalSetup .............................. 31 4-1SignalIdenticationandAnalysis .......................... 38 4-2Verticalvs.LateralDeectionSignals ........................ 40 4-3PeriodicNon-SinusoidalFeatures .......................... 41 4-4TemporalNatureofFlappingDeections ...................... 42 4-5Time-VaryingFeaturesinDeectionSignals .................... 43 4-6DeectionsResultingfromVariationsinStructuralDynamics .......... 44 4-7DeectionVelocitiesResultingfromVariationsinStructuralDynamics ..... 47 4-8DeectionVelocitiesResultingfromVariationsinStructuralDynamics ..... 48 4-9DeectionsResultingfromVariationsinFlappingFrequency ........... 49 4-10DeectionVelocitiesResultingfromVariationsinFlappingFrequency ...... 50 4-11DeectionVelocitiesResultingfromVariationsinFlappingFrequency ...... 51 4-12DeectionsResultingfromVariationsinFlappingAmplitude ........... 52 4-13DeectionVelocitiesResultingfromVariationsinFlappingAmplitude ...... 53 4-14DeectionVelocitiesResultingfromVariationsinFlappingAmplitude ...... 54 4-15DeectionsResultingfromVariationsinChordLocation ............. 55 8

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.............. 56 4-17ExtractingFeaturesinDeformationSignals .................... 57 4-18Verticalvs.LateralDeformationSignals ...................... 58 4-19DeformationsResultingfromVariationsinFlappingFrequency .......... 60 4-20DeformationsResultingfromVariationsinFlappingAmplitude ......... 61 4-21DeformationsResultingfromVariationsinChordLocation ............ 62 4-22AverageFrequencyResponseFunctionandFirstSixObservedOperatingDeectionShapesforAluminumBeam ............................. 63 4-23FrequencyResponseforVariationsinShape .................... 66 4-24FrequencyResponseforVariationsinStructure .................. 67 4-25FrequencyResponseforVariationsinMembraneMaterial ............. 68 4-26FrequencyResponseforVariationsinStiness ................... 69 4-27ExaminationofFrequencyResponseinRelationtoThrustProduction ..... 70 4-28SignalsfromDICofRectangularWingUnderShakerExcitation ......... 71 4-29OperatingDeectionShapesfromDICofRectangularWingUnderShakerExcitation 71 4-30SignalsfromDICofMembraneWingUnderShakerExcitation .......... 72 4-31OperatingDeectionShapesfromDICofMembraneWingUnderShakerExcitation ............................................. 73 4-32Wing-2AverageWingTipVerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) ...... 75 4-33Wing-2WingTipVertical-SignalStandardDeviationoftheErroratVariousFlappingFrequenciesandAmplitudes ........................ 75 4-34Wing-2AverageRootChord,LeadingEdge,VerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) .......................................... 76 4-35Wing-2AverageRootChord,TrailingEdge,VerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) .......................................... 76 4-36Wing-2AverageMid-Chord,LeadingEdge,Z-SignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) ... 77 9

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... 77 4-38Wing-2AverageWing-TipDeection,VerticalSignalCoecientsfor2ndOrderPolynomialAppoximationwithRespecttoFlappingFrequency ......... 78 4-39Wing-2AverageWing-TipDeection,VerticalSignalCoecientsfor2ndOrderPolynomialAppoximationwithRespecttoFlappingAmplitude ......... 78 10

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Flapping-wingmicroair-vehiclesarebeinginvestigatedfortheirpotentialtoprovideenhancedaerodynamiceciency,maneuverabilityandgusttolerance.Theightdynamicsofapping-wingmicroair-vehiclesresultfromacomplicatedrelationshipbetweenaerodynamicsandstructuraldynamics.Thisrelationshiphasbothfrequency-domainaspectsandtime-domainaspectsthatareeachcritical.Assuch,analyzingdatafromaappingwingrequirestechniquesthatcanprocessinformationrelatedtobothofthesedomains. Thisthesisintroduceswaveletanalysisasatooltodeterminethefrequencycontentoftime-varyingsignalsfromaappingwingtestbed.Waveletmapspresentatime-frequencydomainrepresentationthatrelatesbothtime-domainandfrequency-domainaspects.Dataobtainedwithdigitalimagecorrelation,includingdeectionsanddeformationswhileappinganddisplacementsundershakerexcitation,isanalyzedusingwaveletprocessingfromasetofwingswithdierentstructuraldynamicsanddierentappingparameters.Theresultingwaveletmapsdemonstratethevariationsinenergycontentandtemporaldistributionassociatedwiththesesignals.Thisthesisalsoexamineswingsofinterestwithalaserdopplervibrometertoobtainoperatingdeectionshapesandtheircorrespondingfrequencies.Theseshapesandfrequenciesarecomparedtotheresultsobtainedusingdigitalimagecorrelation. 11

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1 )andmaneuverability( 1 ; 2 )aswellasenhancedgusttolerance( 2 ).Suchappingisecientforforcegenerationofbothliftandthrust;however,theappingmotionisexceedinglycomplex( 3 ; 4 ).Thestructureisrequiredtohavesucientexibilitysuchthattime-varyingaerodynamicloadingsonthestructuredonotconictwithabilityofthewingtoproduceliftandthrust.Theresultingdynamicsareinherentlyaeroservoelasticsincetheaerodynamicsandstructuraldynamicsaretightlycoupledalongwithanycontroleectors.Assuch,theightdynamicsaredependentupontherelativedeectionsacrossthewingsoccuringthroughouttheappingcycle. Techniquesfordataanalysisthatcanextractinformationaboutdeectionsaresomewhatlessmaturethantheabilitytogeneratethatdata.Themodalpropertiesoftheaeroservoelasticdynamicswouldprovidereduced-ordermodels;however,thenonlinear,time-varyingpropertiesobservedduringeachappingcyclewouldviolateassumptionsofmodalanalysis( 5 ).Modelsoftheappingdynamicsmaybeobtainedusingparticleimagevelocimetry( 6 ),particleowvisualization( 7 )anddigitalimagecorrelation( 8 ; 9 );however,thesetechniquesgeneraterepresentationsofthetime-averageddynamicsacrossmultiplecycles. Waveletanalysisisabletogenerateatime-frequencyrepresentationofdatathatcapturesbothtime-domaincharacteristicsandfrequency-domaincharacteristics( 10 ).Suchanalysisreliesonlocalizedcorrelationtoknownwaveformswithoutassumptionsonlinearityortime-invariantproperties.Thesetime-frequencymapshavebeeneectivelyusedtoanalyzelimitcycleoscillations( 11 ),nonlinearnormalmodes( 12 ),neural 13

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13 )andnonlinearoscillators( 14 ).Foraappingwing,bothtimeandfrequencycharacteristicsareimportant.Theabilitytoobtaintimedependentfeaturesareimportantsinceanalysisofthequasi-steadydynamicsareinsucienttocapturebothofthesetypesofcharacteristics( 15 ; 16 ).Frequencydependentfeaturessuchasresonancecanenhanceeciencywhilerequiringminimalenergyinputandthereforearealsoimportanttounderstand( 17 ). Issuesrelatedtobothdesignandcontrolarefundamentaltothematurationofapping-wingvehicles.Inadditiontoobtainingreducedordermodelsthatwilldescribethewingbehavior,thetime-varyingpropertiesofawaveformassociatedwithresonancemustbeknownforlow-energyight( 18 ; 19 ).Schemesforornithoptersoftenachievecontrolbyvaryingappingfrequencyandassociatedwingdeections( 20 ; 21 ).Ineachcase,missionperformancedependsonunderstandingthetime-varyingdeectionsasafunctionoffrequency-varyingapping. 1-1 Thisthesisdemonstrateswaveletanalysisasabenecialandsometimesnecessarytechniquetoextractinformationaboutappingwings.Specically,time-frequencymapsareobtainedtorepresentdatafromasetofwingdesignsandappingconditions.Featuresinthesemapsareshowntoindicatetime-varyingaspectsofthedeectionsthatarediculttoextractusingtraditionaltechniquesofeithertime-domainanalysisorfrequency-domainanalysis.Variationsareclearlyshownthatrelatetodierencesinwingdesignanddierencesinbothappingfrequencyandappingamplitude.Also,variationsinthetime-frequencycharacteristicsarefoundacrossthewinginboththespan-wiseandchord-wisedirections.Forcompleteness,thethesisanalyzessignalsgeneratedbydeections,deformations(andassociatedvelocities)andoperatingdeectionshapesofappingwings. 14

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OverviewofExperimentalAnalysisMethodsPotentiallyUsefultoDeriveaReducedOrderModelForFWMAVs ExcludesNumerical/ComputationalMethodsIncludingFiniteElement,ComputationalFluidDynamic,ComputationalStructuralDynamicAnalyses Lastly,toprovideabriefexampleofhowthesesignalsmaybeusedtodevelopareduced-ordermodel,thisthesisintroducesamodelingschemetorepresenttheaeroservoelasticdynamicsofthepreviously-studiedwings.Asetofcoecientsforpolynomialsarecomputedtominimizethenormbetweenameasuredresponseandtheassociatedestimatedresponse.Thesemodelsaregeneratedatarangeofvaluesforthecontroleectorsofappingamplitudeandappingfrequency;consequently,theresultingmodelsareessentiallyabasisthatestimatesthedeectionatpointsalongthewinginresponsetoanyvariationsincontroleectors. 15

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2.1.1DigitalImageCorrelation 22 )todeterminedeections,deformations,andstrainsofastructure.DICiswelldescribedandestablishedtechniqueusinginexperimentalmechanics( 23 ; 24 ).DICexaminesmultipleimagesforrelationshipsbetweenregionsinthoseimagesandthendescribesthatrelationship.Typically,spatialandtemporalrelationshipsofstructuresaredeterminedwithDIC.Arandomspecklingpatternisappliedtothestructure.Twoormorecamerasarecalibratedandthelocationofthestaticstructureisdeterminedwithstereo-triangulationtechniques.Thecamerasthenusetemporaltrackingtoattempttocorrelatearegionofspecklesbetweensubsequentframes,specifyingacross-correlationcoecientsuchastheoneseeninEquation 2{1 wherethein-planedisplacementeld(u;v)isspeciedwithf(x;y)beingthegraylevelatcoordinatex;yfortherstimageandg(^x;^y)thegraylevelvalueatcoordinate(^x;^y)forthesubsequentimage.Thefandgaretheaveragegrayscalevaluesfortheregionand(u;v)arethedisplacementcomponentsforthesubsetcentersinthe(x;y)directions.( 25 ) @xdx+@u @ydy^y=y+v+@v @xdx+@v @ydy(2{1) Acalibrationpoint(region)isoftenspeciedtoinitiatethecorrelationanalysis.ThispointmustbeabletobetrackedacrossframesfortheDICsystemtodescribedeectionswithinreasonableerrorbounds.TheDICsystemusesthecorrelationprocesstosamplethedisplacementofmanypointsonthestructuresimultaneouslyinthreedimensions(whereapointrepresentsaregionofspeckles).Asignalmaybeformedbytrackingthe 16

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26 ; 27 )whentheDICsystemisabletocorrelatesubsequentregionsofspeckles. Forappingwingsystems,bothdeectionsanddeformationsareimportant.Deectionsexpressthemovementoftheentirewing.DeformationshavebeenobtainedfromDICdataforappingwingsandaremayoermoreinsightintotheappingsystemsincethedeformationsremoveanyinuenceduetovariationsinthemechanism( 8 ; 9 ). 28 ).TheLDVsystemusingasinglebeamlasertoassessthemodalcharacteristicsofasystemwasdescribedin1992( 29 ).TheLDVsystemshinesalaserontothestructureasitisvibratingundersomeexcitationandmeasuresthescatteringofthelightreectedbackbythestructure.ThephaseshiftofthereectedlightwavecausedbytheDopplereectisproportionaltotheobject'svelocityandthebeamwavelengthandthereforemaybeusedtogeneratethefrequencyresponseofthestructure( 30 ).LDVhasbeenusedonstructuresaslargeasbridges( 31 )tostructuresassmallascarbonnanotubes( 32 ). Modalanalysisiscommonlyusedtoassessthestructuraldynamicsofasystemandisgenerallyarststepinpredictingandworkingtowardcontrollingvibrationsoranalyzingthestructuraldynamicsofasystem.Inaddition,modalanalysismaybeusedtomeasureastructure'svibrationpropertiesinordertocomparethemwithatheoreticalorniteelementmodel,toproduceamathematicalmodelofacomponentwhenitisdesiredtointegrateintoapre-existingstructure,topredicttheeectsofchangesintheoriginalstructure,ortodeterminethedynamicforcesormaterialpropertiesinastructure( 33 ). 17

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34 ). Traditionalexperimentalmethodsformodalanalysismayusecontactornon-contactmethods( 35 ).Contactmethodsuseaninputexcitationgeneratedwithashakerorhammertoapplyaforcetothestructure.Variousoutputs(e.g.deectionsorvelocities)aremeasuredatmultiplepointsonthestructure.Straingagesoraccelerometersplacedatvariouslocationsonthestructureareoftenusedforthispurpose.Fornon-contactmeasurements,laserdopplervibrometry(LDV)isoftenusedformeasurementwhileashakerisstillusedtoexcitethestructure.Excitationsignalsvarywidely,butmayincludesinedwells,sweeps,andvariousformsofchirpswhicheachhavedierentadvantagesbasedontheparticularsituationinvolved.Anon-contactmethodsuchasLDVisanobviouschoiceforperformingmodalanalysisonaFWMAVwingduetothelowweightofthewing.Astrictlypropermodalanalysisofthestructuraldynamicswouldbeperformedinvacuum,howeverperformingmodalanalysisisairiscommon. Traditionalmodalanalysisreliesonfourmainassumptions:linearity,timeinvariance,reciprocity,andobservability( 36 ).Experimentaltechniquesfordoinglinearmodalanalysisarewelldevelopedbutexperimentaltechniquesformodalanalysisoftime-varyingandnonlinearsystemsarenotwelldeveloped( 5 )sincesuchsystemsviolatethesefundamentalassumptions.Itiscommontolinearizethesystemmodelbystatingthatstructuraldeformationsaresmallandassumingtimeinvarianceduetoperiodicityofthesystemtodealwithsystemnonlinearities.Theseassumptionsareinsucientforappingwingsduetolargedeectionsofthewingandperiodicitywhichmayormaynotbeevident,especiallyifthecontrolactuationthroughappingfrequencyorappingamplitudeisbeingchangedintime.Anotherpossibilitythathasbeenusedtoanalyzerotatingsystemswithsimilarchallengesistoanalyzeoperatingdeectionshapes( 37 ; 38 ).Anoperatingdeectionshapeisdenedasthedeectionofastructureataparticular 18

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39 ).Theseshapesareaccessibleforappingwingsundershakerexcitation,andanoperatingdeectionshapeforappingcouldbedenedasthedeectionshapeforagivenappingfrequencywheretheappingmechanismprovidestheexcitation.Althoughtechniquesthatgobeyondtraditionalmodalanalysisarerequiredtoanalyzeaappingwing,LDVtestingresultinginoperatingdeectionshapesandtheircorrespondingfrequenciesmaybeusedasarststeptounderstandthebasicstructuraldynamicsofthewings.Operatingdeectionshapesexpressthemodesofthesystemundersomeloadingcondition.Thereforewemaydeneanoperatingdeectionshapeforawinginstaticairandwhileapping. 40 ),aswellastoestimatestructuralparametersofasimplebeam( 25 ).LikeLDVorothermodalanalysistechniques,DICisusedasaninversemethodtoestimateboundaryconditionsandcharacterizationofmaterialpropertiesforuseinnumericalmodels( 41 ; 42 ).TheuseofDICtoobtainoperatingdeectionshapesforappingwingsandgenerallyonstructuresthesizeofthewingsinthisstudyhasnotyetbeeninvestigated.DICcouldbeveryhelpfulonthisscale,sinceitobtainsnon-contactmeasurementsofmanypoints(regions)simultaneously.Inaddition,DICmaybeabletoobtainoperatingdeectionshapesandtheircorrespondingfrequencieswithoutanypriorknowledgewhencombinedwiththeabilityofwaveletanalysistolocalizetime-frequencycomponents. 2.2.1FrequencyDomainTechniques 19

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2{2 inthefrequencydomain.ThediscreteversionoftheFourierTransformisshowninEquation 2{3 TimeinformationislostwhentheFourierTransformisusedtotransformthesignalintothefrequencydomainunlessthesignalisstationary.Toobtaintimeandfrequencycontentsimultaneously,adierentapproachisrequired( 43 ). 43 ).Wavelettransformsallowtheuseoflongtimeintervalstoobtainpreciselow-frequencyinformationandshorterregionstoobtainprecisehigh-frequencyinformation( 43 ).Wavelettransformstaketheformofeitheradiscretewavelettransform(DWT)oracontinuouswavelettransform(CWT).TheirusemayprovidefurtherinsightintodynamicalsystemsthanFouriertransformalone( 10 ).Wavelettransformsrelateaninputsignaltoabasisfunctiondenedasthemotherwavelet,similartohowtheFFTrelatestheinputsignaltoasuperpositionofcosinetermswhenittransformsasignalfromthetimedomaintothefrequencydomain.However,waveletsareofniteandvariablelength,sowaveletanalysiscanidentifynearlyinstantaneousfrequencychangesinthesignalwhiletheFFTcannotbecauseitisrelatedtoaninnitetimesignal.Thispropertyofwaveletsallowsthewavelettransformtosimultaneouslydisplaythethreerelevantdimensions(time,frequency,andmagnitude)asseeninFigure 2-1 fora10Hzsinusoid.Theadditionaltimeinformationcanbeparticularlyusefulfortheanalysisofnon-linearandtime-varyingsystems( 11 ).Thewaveletsignalsmayalsobeviewedonthetimevs.magnitudeaxisifdesired.Although 20

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Figure2-1. 10HzSinusoidTimeHistory,FastFourierTransformandWaveletAnalysis(Frequencyvs.TimeandMagnitudevs.TimeViews) TheMorletwaveletasseeninFigure 2-2 iscommonlyusedwhendealingwithdynamicsystems.Inparticular,nonlinearitiesandtime-varyingstructuraldynamicshavebeeninvestigatedusingtheMorletwaveform( 12 ),( 11 ).Thebasicequationforthemotherwavelet,theMorletwaveletandthefrequencycorrespondingtothemagnitudeandscaleareseeninEquations 2{4 2{5 and 2{6 respectively( 11 ). 21

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ExampleMorletMotherWavelet maybedescribedasacombinationofcomponentsofthatbasis.Thisapproachisusedinlinearcontroltheoryandstructuraldynamicsthroughmodalanalysis.Inthecaseofappingwings,especiallyonthescaleofFWMAVs,anewapproachmustbefoundduetothenonlinearandtime-varying(althoughoftenperiodic)natureofmostappingwingsystems. Modelsofaeroservoelasticityareoftengeneratedusinghigh-delitycomputationalapproaches.Severalstudieshaveintroducedcontrolmodelingtodoublet-latticecodesthatarealreadyusedforutterprediction( 44 { 47 ).Theseapproachesgeneratefrequency-domainmodels;however,theyrequireatheoreticalmodelofthesystemphysics. Identicationofaeroservoelasticmodelsfromightdatahasalsobeeninvestigated.Anapproachtoestimatemodelsandassociateduncertaintywasformulatedusingarobustmini-maxschemethatevaluatedwavelet-basedmodalparameters( 48 ).Ablock-orientedapproachwasformulatedthatconsiderednonlinearitiesinthedynamics( 49 ).Computationalmodelsforappinghaveaccountedforaeroelasticcouplingwithanunsteadypanelmethodcombinedwithniteelementmethod( 50 )andcomputationaluiddynamics( 51 )andconcludedthatchangingexibilitycouldenhancetheaeroelasticcharacteristicsofthewingandultimatelyightperformance. Experimentalresultshavealreadyindicatedthebasicnatureofaeroservoelasticbehaviorforasetofwings( 52 ).Thisstudydemonstratedthevariationsinthetime-frequency 22

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3-1 ,aredesignedfortheinvestigationofapping.TheassociateddimensionsandweightsforthesewingsarelistedinTable 3-1 .Eachwinghasanaspectratioof7.65andshapeassociatedwithaZimmermanplanformformedbytwoellipsesthatintersectattheirquarter-chordpoint.Variationstothewingsareintroducedbydierencesinthematerialandstructuraldesign. Figure3-1. WingsUsedAluminumBeamforValidation MovingToptoBottom,thenLefttoRighta)Wing-1(RadialBatten,Capran),b)Wing-2(ParallelBatten,Capran),c)Wing-3(Aluminum),d)Wing-4(L1B1,Capran),e)Wing-5(L1B2,Capran),f)Wing-6(L2B1,Capran),g)Wing-7(SingleBatten,Latex),h)Wing-8(RadialBatten,Latex),i)Wing-9(ParallelBatten,Latex),j)Wing-10(Rectangle,Aluminum),k)Wing-11(Aluminum) ThealuminumbeamisconstructedforvalidationoftheLDVsystemandconsistsofacutaluminumsheetwithahole3.45mmindiameter.Wings3,10,and11aremadeofsolidaluminumsheetthatisspraypaintedwhiteandthenspeckled. Theconstructionofallcarbonberwingsusea3-layer12kbidirectionalcarbon-berroottriangledimensionedas6mm12mmwithaninnerlayerofunidirectionalcarbon-berbattens.Thistriangleensuressucientstinessattherootchordofthewing.Theleadingedgesofthecarbonberwingsstartatawidthof1.2mmandtaperto0.8mmatthetip.BattensforWing-1andWing-2aresinglestripshavingwidthof 24

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WingCharacteristics WingSpan(mm)Chord(mm)Thick(mm)Weight(g) Al-Beam137.518.00.372.25Wing-175.926.3n/a0.14Wing-277.126.5n/a0.12Wing-376.526.30.320.26Wing-475.025.6n/a0.09Wing-575.725.4n/a0.10Wing-675.526.0n/a0.10Wing-L2B275.126.2n/a0.11Wing-L3B159.025.8n/a0.12Wing-L3B267.025.5n/a0.13Wing-781.127.9n/a0.24Wing-881.327.0n/a0.30Wing-980.627.7n/a0.30Wing-1069.017.70.280.27Wing-1180.525.50.280.28 0.8mmmadefromunidirectionalpre-pregcarbonber.Forwings4-6thenumberofcarbonberstripsformingthestructureofthewingsisvaried.Wing-4hasasinglestripforboththeleadingedgeandbattens,formingaveryexiblewing.Wing-5maintainsasinglestripfortheleadingedge,butusesadoublestripforthebattens.Wing-6hasadoublestripfortheleadingedge,butsinglestripsforthebattens.Wing-L2B2,L3B1,andL3B2arenotshowninFigure 3-1 sincetheirstructurallayoutwasthesameasWings-4,5,and6.ThenumberfollowingtheLrepresentsanincreaseinthenumberofcarbonberstripsalongtheleadingedgeandthenumberfollowingtheBrepresentsthenumberofcarbonberstripsintheparallelbattens.Forwings7-9,thewidthofthecarbonberstripsfortheleadingedgeandbattensare1.2mm. Thecarbon-berwingsarelaiduponaatplateandcuredfor2hoursat250oF.Afterthecarbonberstructuresarecured,themembranesareaxedwithsprayglue.Wings1,2,4,5,6,L2B2,L3B1,andL3B2haveCapranmembraneswhilewings7,8,and9haveLatexmembranes. 25

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9 ),seeninFigure 3-2 .ThelinearappingmechanismisdrivenbyaMaxon15WbrushlessDCmotorEC16andpoweredbyastandardpowersupply.Themechanismincludesa57/13reductionratioplanetarygearhead,256countsperturnencoder,andanEPOS24controller.Theapperutilizesaslider-crankmechanismtotransformtherotationalmotionofthemotortolinearmotionofthereciprocatorwhichthenactuatestheappingmotionthroughalinkage.Theamplitudeisadjustedbyvaryingtheo-centerdistanceofthecrankmodule.TheappingfrequencyisadjustedbyvaryingthevelocityofthemotorwithLabviewsoftware.Flappingamplitudesrangingfromapproximately+/-0to+/-60degreesatappingfrequenciesbetween0and45Hzareattainabledependingonthewingweight.Wingsareattachedtothemechanismwithcyanoacrylateglue.Themotionproducedbythemechanismhasatmosta2percentbiaserrorcomparedtoanidealsinusoid. Figure3-2. SingleDegreeofFreedomFlappingMechanism,DesignedandBuiltbyWuforFlappingWingExperimentation 26

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3-3 .Attopperformance,thetwoPhantomv7camerasformingtheDICsystemcaneachstore2800picturesof800600pixelsat4800frames-per-second(fps).Accdsensorbooststhesignal-to-noiseratioandenablesshutterspeedsof1/500,000s.TwoSigma28-300mmf/3.5-5.6lensesareusedinthetests.Thehigh-speedcamerasarecalibratedwithadotgridwithpredeterminedspacingandCorrelatedSolutionsVicSnapcommercialsoftwarebetweenanyexperimentalrunswhenthecamerasormechanismaremoved.Theappropriategridisselectedtollmostoftheactiveimageframe.Themeasurementsareinitiatedbyobtainingareferencepictureofthewingatthecentralmid-planelocationoftheappingcycle.Flappingisthenactivatedandsubsequentdeectionsareobtainedrelativetothisreferencelocation.Exposuretimesareadjustedwithanychangesincaptureframeratebutaregenerallyaround100mstoprovidesucientlightinginthecapturedframes.Toensureadequatesampling,theframerateissettocapture50imagesperappingcycleatanyvalueofappingfrequency;consequently,theframeratevariedbetween500and1500framespersecondforalldatapresentedinthisthesis. Figure3-3. DICExperimentalSetup 27

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FortheDICanalysis,thevemainvariablesinthisexperimentarewingtype,appingamplitude,appingfrequency,winglocation,anddisplacementcomponent.Experimentalrunsarereferencedwiththese5variables.Threetypesofwings(Wings1-3),3appingamplitudes(+/-10Degrees,+/-17Degrees,+/-35Degrees),4appingfrequencyproles(10Hz,20Hz,30Hz,Dynamicfrom0-30Hzin1second)andasignalforeachof5winglocations(RootLeadingEdge(RLE),RootTrailingEdge(RTE),MidwingLeadingEdge(MLE),MidwingTrailingEdge(MTE)andWingtip(WT))areexamined.Foreachofthesewinglocationsthedisplacementsignalsinthree-dimensionalspacearecaptured.ThisthesisfocusesononlytheverticalandlateraldisplacementsintheZandYdirectionshowninFigure 3-4 sincetheirmagnitudeswerelargest.Thesewingstationsareselectedsincethelocationofthetrailingedgecomparedtotheleadingedgeisanimportantfactorwhenrelyingonasimplelinearactuatorwithpassivewingdeectionsatthetrailingedgetoproducethrust.Measurementsfromthewingrootandtipcompleteabasicoutlineofthewing.Forsteady-stateapping,thesystemiscommandedtoapforafewsecondswiththeLabviewuserinterfaceandthenamanualtriggerstartsdatacollectionwiththeVICSnapsoftware.Forthedynamicsweep,thetriggerfortheDICsystemispressed,followedimmediatelybytheLabviewcommandtoramptheappingfrom0-30Hz.Whiletheinitialtimingisunknown,synchronizationisnotrequiredsincethisthesisisconcernedwithtrendsindicatedbythewaveletanalysistechnique. OncethedataisobtainedwiththeVICSnapsoftware,thedataisanalyzedwithCorrelatedSolutionsVIC3DsoftwaretodetermineboththeinitialfulleldX,Y,Zcoordinatesofthereferenceframe,aswellasthedisplacementsofthewingineachdirectionforeachframecaptured.TheVIC3Dtracksthedeectionsofacalibrationpoint 28

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ConstructionofFictionalRigidWing:ReferenceWing,FictionalRigidWingandFlexibleWing andpointswithinaspeciedareaofinterestthroughallframesobtained.ThedatawasimportedintoMATLABandthereferencewingistranslatedtoazeropositionbymovingthemaximumYcoordinateandmaximumXcoordinateofthereferenceframetozero.ThisadjustmentformsarighthandedcoordinatesystemwiththeYaxisrunningfromtheroottotipalongtheleadingedge,theXaxisrunningfromtheleadingedgetotrailingedgeattherootchord,andtheresultantzaxisupfromthetopofthewing.AsignalisformedbytrackingthedisplacementofagivenpointrepresentingaregionofspecklesintheX,Y,orZdirectionatagivenlocationacrossframes.Theresultingsignalsinthethesishaveerrorsindeectionlessthan1mm( 53 )despitesignicantoutofplanedisplacement. ThesignalsareanalyzedwiththeDiscreteFourierTransformwithsamplingsetequivalenttoframerateoftheDICsystem.ThecontinuoustimewavelettransformisusedwiththeMorletwaveletandthenumberofscalessetgreaterthan200toanalyzethesamesignals.Velocitysignalsareobtainedandanalyzedforsomecasesusingacentraldierencingalgorithmtoapproximatetheslopeofthetimehistory.Deformationsarealsoobtainedusingtheprocesspreviouslydescribed( 8 ; 9 )wheretherigid-bodymotionsforaappingwingaredeterminedbyisolatingthreepointsclosetotheleading-edgerootofthewing.Thissetofpointsmaybeassumedtodescribetherigid-bodymotionofthewing. 29

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9 ; 8 ). 3.4.1Procedure ThewingstructureisattachedtoamountthatwasscrewedintoaLingDynamicSystemsV201/3-PA25Eelectrodynamicshakertofacilitateaconsistentexcitation.Thesetupisplacedonavibrationisolatedlabtableexposedtoambientairinalargeroom.AloadcellisattachedtothestingerbetweentheshakerandthestructuretomeasuretheinputoftheshakerforincorporationinthePolytechsoftware.WhilethePolytechsoftwarecancommandmanytypesofinputexcitationswiththeshaker,fortheseexperimentsonlytheburstchirp,periodicchirp,sweep,periodicsweep,andsinglefrequencysinusoidareconsidered.DICisalsoperformedinconjunctionwiththeshakersetuptoexamineDIC'sabilitytoidentifymodalinformation.ApictureoftheLDVsetupisseeninFigure 3-5 PolytechsoftwareisusedtoobtaintheoutputdatafromtheLDVandtocalculatethefrequencyresponseofthestructureforallwings.Tostart,thelaserisautofocusedonasinglepointonthewingandatleast5calibrationpointsarespecied.ThecalibrationisautomaticallyperformedbythePolytechsoftware.Agridofpointstomeasurethe 30

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LDVExperimentalSetup Note:UnusedLoadCell,SignalFeedbackFromAcquisitionSystem responseofthewingisspecied,eachofwhichwasanalyzedwiththeLDV.Aminimumof50pointsweresampledforeachwing,withresultsforWings4,5,6,L2B2,L3B1andL3B2sampled300pointspertest. Burstchirpexcitationisusuallyfoundtoprovidethefrequencyresponsespectrumwiththehighestsignaltonoiseratio,soitisoftenselectedastheexcitationmethod.Ingeneral,burstchirpexcitationproducesacleanfrequencyresponsespectrumandreasonablyaccuratemodalfrequencies.However,ifestimationofmodalparametersusingcurvettingisrequired,suchasdamping,itisnotadvisabletouseburstchirpexcitationbecausethemeasurementscontaindistortion.Inthiscaseparameterestimationisnotconsidered,soburstchirpexcitationisprimarilyused.TheLDVobtainsdatathroughthecourseofaboutthreeburstchirpsforeachpointonthegrid,averagingtogivethefrequencyresponseatthatpoint.Tocomparewiththeburstchirpresults,sweeporperiodicsweepexcitationisoftenused.Undersweepexcitationgenerallyveormoresamplesaretakenforeachpoint.ThePolytechsoftwareautomaticallyresamplespointsiftheerrorsarenotwithinacceptablebounds.Forallexcitationmethods,frequencieswerevariedfrom0-1000Hzwithsweepsover1s.ThePolytechsoftwarethenusesanaveragingalgorithmtogenerateanaveragefrequencyresponsespectrumfortheentirewing. 31

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32

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Toobtaintheoperatingdeectionshape,theshakerissettoperformasweepexcitationataknownrate.TheDICsamplingrateissetto1300fpstoavoidaliasing,especiallyatfrequenciesbelow130Hz.ObtainingthesignalofthedeectionsisstraightforwardusingthesameDICanalysis.Itisnon-trivialistoidentifytheprecisemomentintimewhenthestructureisrespondingtotheshakerexcitationatagivenfrequency.Sincetheexcitationisatime-varyingsignalandwaveletanalysisisabletoisolatetheenergylocatedatagivenfrequencyandtime,itispossibletousewaveletanalysisofthedeectionsignaltoidentifyboththetimeandfrequencywherethedeectionofaparticularpointrepresentstheoperatingdeectionshapespecictoadesiredfrequency.Thedeectionofallpointsatthattimeareplotted,producingtheoperatingdeectionshapeofthewing.Itisanticipatedthattheresponsewillbesubstantialenoughtoidentifyanoperatingdeectionshape,yetitisdesirabletomitigatethereductioninaccuracyduetolaginthesystem.Thereforetheleadingedgemid-pointwasusedtoidentifythetimeforeachoperatingdeectionshapeobtainedwiththeDICsystem.Theapproachassumesthatallpointsarebeingexcitedatthatfrequencyatthegiventimeofthesinglepointandthereforedoesnotaccountforlaginthestructurebetweenpoints.Iftheideaisdevelopedfurther,thepointintimewherethedesiredfrequencyisexcitedcouldbeidentiedusingthewaveletanalysisforallpointsandthentherealoperatingdeectionshapecouldbeplottedusingthetimespecicdeectionofeachpoint. 33

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3.6.1TimeandFrequencyTechniques 34

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3{1 ThecoecientsinEquation 3{1 willvarywiththeappingfrequency.Assuch,amth-orderdependencyon!isintroducedasinEquation 3{2 AnadditionaldependencyisintroducedtothecoecientsinEquation 3{2 torepresentapth-orderdependencyonappingamplitudeasinEquation 3{3 Theset,S,isintroducedtodescribeallthecoecientsinthemodelasinEquation 3{4 Amodelisidentiedbychoosingcoecients,S,suchthatthesimulatedresponse,y,matchesameasuredresponse, 3{5 minSk Themodelisthentransformedfromatime-domainrepresentationintoaphase-domainrepresentation.Thistransformationisnecessarybecausethemodel,asformulatedinEquation 3{1 ,isonlyvalidforasinglewaveformthatlastsfromt=[0;T].Suchamodelwouldgenerateaninnite-sizedeectionastimeincreasedwhichisobviouslynotrealistic.Anappropriatetransformationcanbenotedthattheappingisperiodic;consequently, 35

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3{6 Thetimeisthencomputedast=1 3{1 isthustransformedintotherepresentationinEquation 3{7 whichisvalidforanyphaseof2[0;360]. Finally,themodelofawingisformulatedbysimplycombiningthemodelsassociatedwithindividualpoints.Amatrixisthusderivedwithdimensionsdeterminedbythenumberofmeasurementstakenalongthechord-wiseandspan-wisedirections. Synthesisofightcontrollerscanbeperformedusingthismodelifthebasishassucientrichness.Theconceptnotesthattheresponseatanyfrequency,!,andamplitude,,canbederivedfromtheparameterizationacrossthesetoftestedconditions.Themth-orderpolynomialin!fromEquation 3{2 andthepth-orderpolynomialinfromEquation 3{3 providethisderivationtoanyvalueofthecontroleectors.Assuch,theaeroservoelasticresponsecanbecomputedfromthisbasis. 36

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4.1.1Deections 4-1 todemonstratethediversityofinformationresultingfromtime-domainrepresentation,frequency-domainrepresentation,andtime-frequencydomainrepresentation. Trendsseenareconsistentwithwhatwouldbeexpectedforaappingwing.Forthetimehistory,asexpected,themagnitudesincreasegoingfromtheroottotipandthemagnitudeofverticaldeection(appingupanddown)ismuchlargerthanlateraldeection(pointonwingmovestowardstheroot).Thepeakfrequencyofthelateraldeectionisapproximatelydoublethefrequencyofverticaldeectionsincethetimesignaloscillatesbetween0andamaximumnegativevalueintime.Thisfrequencydoublingresultsfromthepointonthewingmovinginthenegativelateraldirectiononboththe 37

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SignalIdenticationandAnalysis SignalAnalysisforWing-2at+/-17degand20HzFlappingFrequency,a)FullLength,TimeResolved3DDeectionPlotwithRespecttoReferenceWingandBoxBoundingMaximumDisplaments,b)FullTimeHistoryofVertical()andLateral(|)Deections,c)DiscreteFourierTransformofVertical()andLateral(|)Deections,d)WaveletRepresentationofVerticalDeection,e)WaveletRepresentationofLateralDeection,f)3CycleZoomforTimeHistoryofVertical()andLateral(|)DeectionsforRootChordLeadingEdge(Row1),RootChordTrailingEdge(Row2),Mid-SpanLeadingEdge(Row3),Mid-SpanTrailingEdge(Row4),andWingTip(Row5) upstrokeanddownstroke.Whenlookingatthemagniedtimehistory,itisalsopossibletonoteavariationinthepeakmagnitudesintimeforthemid-winglateraldeection,withanapparentlagatthetrailingedgeofthemid-wingindicatedbyasharperpeakinboththeverticalandlateraldeectionsignals. Thefrequency-domaindataclearlyidentiestheprimarypeaksforlateraldeectionstobeequaltotheappingfrequencyof20Hz,whiletheverticaldeectionshaveastrongpeakattherootattheappingfrequencyof20Hz,graduallyreducingasthesamplesmovetowardthewingtip.TheFourierTransformoflateraldeectionsdemonstratesthatthereisapeakatboth20Hzand40Hzforthemidplane.Incontrasttothemidplane, 38

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4-2 focusesonthedierencebetweentheverticalandlateralsignalsanalyzedwiththewavelettransform.Thewaveletplotfortheverticalsignalhasalmostalltheenergycenteredatthemainappingfrequencyof20Hz,whilethewaveletplotoflateralsignalonthefarrightdemonstratesanaccumulationofenergyat40Hz.ThebottomthreeplotsinFigure 4-2 areforthesamesignals,butadierentsubsectionofthetimehistory.Therelationshipbetweentheenergyat40Hztotheenergyat20Hzisseentochangeasthefrequencyandmagnitudeofthepeaksshiftslightlyintime. 4-3 ,whichshows7subsequentappingcyclesoverlayedontopofeachother.Variationsseenathigherfrequencyandappingamplitudesuchasthoseat35degand30HzareduetoissuesobtainingcorrelationwiththeDICsystem,notalackofperiodicityintheactualappingcycle.Whilethissubstantialperiodicitymaysuggestthatwaveletanalysisisnotrequired,variationswithintheappingcycleandvariationscausedbychangingactuationorresonancerequiretime-frequencyanalysistobefullyunderstood. 39

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Verticalvs.LateralDeectionSignals a)DisplacementTimeHistoryofVerticalSignal(Dot-Dash)andLateralSignal(Solid),b)WaveletVerticalSignalandc)WaveletLateralSignalforWing-3at+/-10degFlappingAmplitudeand20HzFlappingFrequencyforTimeInterval0.35-0.45sec(TopRow)andTimeInterval0.095-0.195sec(BottomRow) Thetemporalnatureoftheappingdeectionshavingperiodicnon-sinusoidalfeaturescorrelateswithtime-varyingfrequencycontent.Suchafeatureisnotedbyexaminingthemid-spanstationalongthetrailingedgeofWing-3whileappingat30Hzwithanamplitudeof+/-10deg.TheresultingdeectionsareshowninFigure 4-4 alongwithfrequency-domainplotsandwaveletplotsfor0.1s. Thedeectionsinthelateraldirectionhaveespeciallynotabletime-varyingcomponents.Thetime-domaindeectionsclearlyshowabeatingphenonemonofpeakswithperiodicmagnitudes.Thefrequenciesofthesepeaksareshowninthefrequency-domainplot;however,thetemporalnatureofthebeatingisonlyseeninthewaveletmaps.Thepeaksofpositivedeectionshowalargevalueat0.22sfollowedbyasmallervalueat0.23swithanotherlargevalueat0.25sinthetime-domainplots;correspondingly,thewaveletmapshowssimilarchangesinpeakmagnitudeatthesetime 40

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PeriodicNon-SinusoidalFeatures WingtipVerticalDisplacementsofWing-2at10,17,and35degFlappingAmplitudeand10,20,and30HzFlappingFrequency values.Thenatureoftheappingisnotedbytheverticaldisplacementwhichshowsthelargedeectionat0.22scorrespondstothewingcrossingthecenterlocationduringtheupstrokewhilethesmalldeectionat0.23scorrespondstothewingcrossingthecenterlocationduringthedownstroke.Thisasymmetryisnotedinboththetime-domainresponsesandtheassociatedtime-frequencymapsbutnotinthefrequency-domainplots. The90Hzcomponentoftheresponseisanotherfeatureforwhichthewaveletmapprovidesadditionalinsightascomparedtoonlyafrequency-domainplot.Inthiscase,thewaveletmapsinFigure 4-4 indicateatemporalvariationinmagnitudeforthis90Hzenergy.Atemporalrelationshipisclearbetweenthebeatingofthe60Hzcontributionandthe90Hzcontribution.

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TemporalNatureofFlappingDeections ResponseofWing-3at10degFlappingAmplitudeand30HzFlappingFrequency:a)Time-DomainRepresentationinVerticalDirection()andLateralDirection(|),b)WaveletRepresentationofLateralDirection,c)WaveletRepresentationofVerticalDirection,d)Frequency-DomainRepresentationinVerticalDirection(|)andLateralDirection(),e)WaveletCorrelationofLateralDirection,f)WaveletCorrelationofVerticalDirection Asetofdataisgeneratedthatconsidersthedeectioncharacteristicsinresponsetodierentapping;specically,thedeectionsinFigure 4-5 reectresponsetoasine-dwellappingat30Hzandsine-sweepappingfrom0-30Hzin0.5s.ThisdatacorrespondstotheWing-3wingatanamplitudeof+/-10deg. Thetransientnatureofthesweepclearlyaectstheresponseasnotedbythedramaticdierenceintheresponsesat0.5sbutsimilarityintheresponsesat1.0s.Essentially,theincreasetoappingfrequencystopsat0.5sbutthedeectiontakesanother0.2stosettletothenalsteady-statevalue.Thistemporalnatureofthedecaycannotbenotedinthefrequency-domainplotsbutisevidentinthewaveletmaps.ComparisonswithresultsfromLDVtestsseeninTable 4-2 donotpredictthislarge 42

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Time-VaryingFeaturesinDeectionSignals ResponseofWing-3at10degFlappingAmplitudeforDwellat30HzFlappingFrequency(top)andSweepbetween0-30HzFlappingFrequency(bottom)forVerticalDeection()andLateralDeection(|):a)Time-DomainRepresentations,b)Frequency-DomainRepresentations,c)WaveletRepresentationofVerticalDeection,d)WaveletRepresentationofLateralDeection concentrationaroundwhentheappingnears20Hz,howeverthewaveletanalysisdemonstratesthatthelateralsignalhassignicantenergypresentnear40Hzduetothefrequencydoubling.Thisenergymayhavesomerelationshiptotherstbendingoperatingdeectionshapeseenat41.9HzfromtheLDV.ThisrelationshipbetweenaresonantresponseseenwiththeDICandLDVresultswarrantsfurtherinvestigationinthefuture. Additionally,theresponselosessomeofitssinusoidalnatureduringthesweep.Thespreadofenergyacrossmultiplefrequenciesisindicativeofaperiodicsignalthathassomelevelofnon-sinusoidalcharacteristics.Thedecayinmagnitudeafterthesweepfrom0.5sto0.7sisaccompaniedbyadecayinthespreadofenergyacrossfrequenciesand,consequently,adecreaseinnon-sinusoidalcomponentstotheresponse.Thesinusoidal 43

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4-6 resultfrommotioninthespan-wiselateraldirectionwhileappingat20Hzwithanamplitudeof+/-10deg. Figure4-6. DeectionsResultingfromVariationsinStructuralDynamics ResponseofWing-3(top)andWing-1(bottom)at10degFlappingAmplitudeand20HzFlappingFrequencyforLateralDeectionatMid-SpanTrailing-EdgePoint()andWingTip(|):a)Time-DomainRepresentations,b)Frequency-DomainRepresentations,c)WaveletRepresentationofMid-SpanTrailing-EdgePointforFrequenciesabove20Hz,d)WaveletRepresentationofWingTipforFrequenciesabove20Hz

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Thetemporaldistributionofenergybetweenthelocationsoneachwingisalsoevidentinthewaveletmaps.TheWing-3wingactuallyshowssomeinvertedbehaviorinthatthenegativedeectionatthemid-spanlocationshowstheextraenergywhereasthepositivedeectionatthewing-toplocationshowsthisbehavior;inconstrast,theWing-1wingshowsthatthepositivedeectionisassociatedwiththeextraenergyatboththemid-spanlocationandwing-tiplocation. 4-7 demonstratesanexampleofhowknowingthetimeandfrequencycontentsimultaneouslyfromthewaveletrepresentationprovidesinsightintotimehistoryofthevelocitysignal.ForthemembraneWing-1,thetimehistoryshowspeaksthataresharplypointedasthewingapproachesmaximumvelocityinboththepositiveandnegativedirections.Itisworthnotingthatthedownstrokeachievesahighervelocityasexpected.Afterthepeakvelocity,thevelocityofthewingdeclinesquickly,levelsoandthendropsinanearlysinusoidalmanner.Thisincreaseinvelocityisthesnapthathasbeenobservedinexiblewings( 1 ).TheFourierTransformoftheverticalsignalprovidesevidencethatthereisenergyat20,40and60Hz,butisunabletodemonstratehowthehigherfrequencycontentshapesthetimehistory.Whenobservingthewaveletmaps,thebulkof 45

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ByexaminingFigure 4-7 forWing-3,itisnotedthatthepeakvaluesofthevelocity,whilemostlysinusoidal,seemtohavebeenclippedbyhighfrequencyenergy.Thesnapthathasbeenobservedinthehighlyexiblemembranewingisnotpresentforthiswing.Thewaveletplotsdemonstrateclearlythatthehighfrequencycontenthasnosignicantpeaksaectingthewingresponse.Similarpatternsareobservedinboththewingmidtrailingedgeandthewingtipfortheverticalsignals. ThesituationisquitedierentforthelateralsignalsasshowninFigure 4-8 .Fromthetimehistory,themembranewingstillmovesinaperiodicmannerwhilethealuminumwinghassignicantlydierentpeakvaluesevenifonlyeveryotherpeakisexamined.ThepeaksremainwelldenedforallsignalsintheFourierTransformforWing-1,whiletheyareabitmorediculttoobtainforWing-3.However,thewaveletrepresentationsdemonstratethatthesignalforWing-1ismuchmoreperiodicthanWing-3whenindividualfrequenciesareisolated.ThefrequencyspeciccontentshowninthewaveletplotsindicatesaclearvariancewithrespecttotimeforWing-3.WhilethereasonforthisphenomenaisnotcertainandmaybeduetoeectsfromthemechanismandrelativelylargerweightofWing-3,theeectisclearlyseenwiththewaveletrepresentation. 4-9 alongwiththeirfrequency-domainrepresentationsandwaveletmapsforboththemid-spanlocationandwing-tiplocation. Thewaveformassociatedwiththedeectionisclearlydierentsuchthatthesinusoidalnatureismorepronouncedinthetime-domainmeasurementsforthefaster 46

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DeectionVelocitiesResultingfromVariationsinStructuralDynamics ComparisonofVerticalVelocitiesforWing-1(TopRow)vs.Wing-3(BottomRow)at20HzFlappingFrequencyand+/-10degFlappingAmplitudeusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) apping.Thefrequency-domainrepresentationindicatessomedenitivepeaksatthecorrectfrequenciesforthefasterappingbutislessinstructivefortheslowerapping.Thewaveletmapsareabletoaccountforthenon-sinusoidalnatureoftheresponseandthusprovideanaccuratedescriptionofthefrequencycontentateachinstantintimeforeachresponse. Also,thevariationsacrossthewingareagainnotedusingthewaveletmaps.Thetime-domaindeectionsandassociatedwaveletmapsarenearlyidenticalatthemid-spanlocationandwing-tiplocationfortheslowerappingbutshowdistinctdierencesforthefasterapping.Thetemporalnatureofthefrequencycontentisclearlyshowninthewaveletplotsbythefamiliartripletofpeaksforpositivedeectionfollowedbyasingle 47

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DeectionVelocitiesResultingfromVariationsinStructuralDynamics ComparisonofLateralVelocitiesforWing-1(TopRow)vs.Wing-3(BottomRow)at20HzFlappingFrequencyand+/-10degFlappingAmplitudeusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) peakfornegativedeection.Therelativemagnitudeoftheenergyassociatedwiththesepeaksisquitedierentattheselocations.Eventheshapeofthenegativepeakissharperatthewing-tiplocationtoindicatethedeectionismoresinusoidalherethanatthemid-spanlocationwithitsroundedpeakinthewaveletmapforthefasterapping. 48

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DeectionsResultingfromVariationsinFlappingFrequency ResponseofWing-2at10degFlappingAmplitudeand10HzFlappingFrequency(top)and30HzFlappingFrequency(bottom)forLateralDeectionatMid-SpanTrailing-EdgePoint()andWingTip(|):a)Time-DomainRepresentations,b)Frequency-DomainRepresentations,c)WaveletRepresentationofMid-SpanTrailing-EdgePointforFrequenciesabove10Hz,d)WaveletRepresentationofWingTipforFrequenciesabove10Hz 4-12 alongwiththeirfrequency-domainrepresentationsandwaveletmapsforboththemid-spanlocationandwing-tiplocation. 49

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DeectionVelocitiesResultingfromVariationsinFlappingFrequency ComparisonofVerticalVelocitiesfor10Hz(TopRow)vs.30Hz(BottomRow)FlappingFrequencyforBPCWingat+/-10degFlappingAmplitudeusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) ThedierenceintheresponsesisquiteevidentinFigure 4-12 forboththetime-domainrepresentationsandthewaveletmaps;however,thefrequency-domainplotsdonotshowmuchdierenceexceptforsmallchangesinpeakamplitude.Inparticular,thewaveletmapsindicatetheresponseismoresinusoidalforthehigheramplitudeofappingascomparedtotheloweramplitudeofapping.Suchcomparisonisderivedbynotingtherelativelynarrowbandsforpeaksinthelowerrowofwaveletmapscomparedtothewidebandsforpeaksintheupperrowofwaveletmaps. Also,thevariationbetweentheupstrokeanddownstrokeisquiteevidentinthetime-domainrepresentationsandwaveletmapsbutcannotbedeterminedinthefrequency-domainrepresentations.TheupperrowofwaveletmapsinFigure 4-12 show 50

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DeectionVelocitiesResultingfromVariationsinFlappingFrequency ComparisonofLateralVelocitiesfor10Hz(TopRow)vs.30Hz(BottomRow)FlappingFrequencyforBPCWingat+/-10degFlappingAmplitudeusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) atripletofpeaksduringtheupstrokebutonlyasinglepeakduringthedownstroketoindicateatemporally-localizedvariationinthefrequencycontent. 51

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DeectionsResultingfromVariationsinFlappingAmplitude ResponseofWing-2at20HzFlappingFrequencyand10degFlappingAmplitude(top)and35degFlappingAmplitude(bottom)forLateralDeectionatMid-SpanTrailing-EdgePoint()andWingTip(|):a)Time-DomainRepresentations,b)Frequency-DomainRepresentations,c)WaveletRepresentationofMid-SpanTrailing-EdgePointforFrequenciesabove10Hz,d)WaveletRepresentationofWingTipforFrequenciesabove10Hz VariationsinChordLocation 4-15 fromamid-spanlocationontheleadingedgeandtrailingedgeofWing-2whileappingat30Hzwithanamplitudeof10deg. Clearlythedeectionsattheselocationsvarytoindicatesometorsionaldynamicisresultingfromtheapping.Inparticular,thewaveletmapsindicatethetrailing-edgedeectionisdominatedbyasinglefrequencywhereastheleading-edgedeectionhasadistincttemporalvariationtoitsfrequencycontent.

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DeectionVelocitiesResultingfromVariationsinFlappingAmplitude ComparisonofVerticalVelocitiesfor+/-10deg(TopRow)vs.+/-35deg(BottomRow)FlappingAmplitudeforWing-2at20HzFlappingFrequencyusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) Figure 4-16 demonstratesdierencesinthehighfrequencycontentofWing-2at+/-35degappingampltudefor10Hzapping.Theplotusesthewaveletrepresentationtoexplorethehigherfrequencycontentasitislocatedintime.Thelagsinhighfrequencycontributionstotheoverallsignalsareevidentinboththeverticalandlateralsignals.WhiletheFourieranalysisdemonstratesthepresenceofenergyatthesehigherfrequencies,itgivesnoinformationaboutthetemporalnatureofthatcontent.Noconclusionsaremadeastothecauseofthishighfrequencycontent;however,Figure 4-16 demonstrateshowwaveletanalysismaygivecluesregardingwheretolookforhigherfrequencyphenomenathatmaycontributetotheoveralleeciencyofaappingwing. 53

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DeectionVelocitiesResultingfromVariationsinFlappingAmplitude ComparisonofLateralVelocitiesfor+/-10deg(TopRow)vs.+/-35deg(BottomRow)FlappingAmplitudeforWing-2at20HzFlappingFrequencyusinga)TimeHistoryofMid-WingTrailingEdge(|)andWingTip(),b)FourierTransformofMid-WingTrailingEdge(|)andWingTip(),c)WaveletRepresentationofMid-WingTrailingEdge(AllFrequencies),d)WaveletRepresentationofWingTip(AllFrequencies) 9 ).Oncesignalshavebeenformed,theyareanalyzedwith 54

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DeectionsResultingfromVariationsinChordLocation ResponseofWing-2at10degFlappingAmplitudeandChirpbetween0-30HzFlappingFrequency:a)WaveletRepresentationatMid-SpanLeading-EdgePointforFrequenciesabove18Hzandb)WaveletRepresentationofMid-SpanTrailing-EdgePointforFrequenciesabove18Hz Thehighfrequencycontentispresentinthedeections;however,thelargemagnitudeoftherigid-bodymotionshidetheinformation.Inmanycases,thedistinctionbetweentheupstrokeanddownstrokecannotbedeterminedwhenlookingatdeformationsignalsbecauseoftheamountofhighfrequencyinformation.Itisworthnotingthatdeformationsignals,especiallyforlowerappingamplitudes,lowerfrequencies,andlateralsignalshavemagnitudesthatareneartheexpectederrormagnitudesandarethereforealsopronetoexperimentalnoise.However,theknowndynamicsaregenerallyvisible,especiallyathigherappingfrequenciesandappingamplitudes.Thereforewhilethiseectshouldbekeptinmindthedeformationsignalsshouldprovideusefulinformation. 4-17 .Theverticaldeformationsindicatethattime-varianthighfrequencycomponentsarepresentforboththemid-spanleadingedgeandwingtip,whichrepresents 55

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DeectionsResultingfromVariationsinSpanLocation DisplacementSignalAnalysisforWing-2at+/-35degFlappingAmplitudeand10HzFlappingofthea)MidTrailingEdgevs.b)WingTip,TimeHistoryVerticalDeection()andLateralDeection(|)(1stRow),DiscreteFourierTransformVerticalDeection()andLateralDeection(|)(2ndRow),WaveletRepresentationofVerticalSignalforFrequenciesabove35Hz(3rdRow),WaveletRepresentationofLateralSignalforFrequenciesabove35Hz(4thRow) thegeneraltrendwhenmovingacrossthespanofthewings.Thehighfrequencycontentoftheoveralldeectionsignalismuchmorevisiblewhenanalyzingthedeformationsignal.Thetime-frequencylocationsofthesehighfrequencies,aswellastheactualshapeofthesweepfrom0-30Hzarestillseenclearlyinthewaveletplot,whilethetimehistoryandfrequencyrepresentationmakeanydenitiveconclusionsdicult. 4-18 .Whiletheverticaldeformationsarestillfairlywelldenedwithmostenergycenteredaround20Hz,thelateraldeformationsignalismorediculttointerpret.Time-varyingmagnitudeoffrequencyspeciccomponentsareseenclearlyinthewaveletplotswithlargermagnitudespikesfortheverticaldeformationsat20Hzfortimesfrom0.05-0.15s

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ExtractingFeaturesinDeformationSignals WingMid-SpanLeadingEdge(TopRow)andWingTipVerticalDeformations(BottomRow),Wing-2,+/-10degFlappingAmplitude,0-30HzSweepFlappingFrequencya)TimeHistory,b)FourierTransform,c)WaveletRepresentation thanfor0.15-0.5s,despiteappingataconstantratethroughoutthesamplingperiod.Asimilarconcentrationisseenforthelateraldeformationsfrom0.2-0.3s.Analysisofthedeformationmeasurementsmaketheseinsightspossible,whenthelargerigidwingmotionsmaskthesedynamicsinthedeectionsignalsasseenbycomparingFigure 4-18 57

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4-12 .Whilethisisamoredramaticexampleoftime-varyingmagnitudesofthemostdominantfrequenciesinthewaveletmaps,time-varyingmagnitudesareoftenseenforthedeformationsignalsatallappingfrequenciesandappingamplitudes. Figure4-18. Verticalvs.LateralDeformationSignals WingTipVerticalDeformations(TopRow)andWingTipLateralDeformations(BottomRow),Wing-2,+/-10degFlappingAmplitude,20HzFlappingFrequencya)TimeHistory,b)FourierTransform,c)WaveletRepresentation 58

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4-19 .Relativelyhighermagnitudehighfrequencyenergyisseenfor10Hzapping,whilefor30Hzapping,theenergyismoreconcentratedat30Hzasexpectedfromtrendsfordeectionswithrespecttoappingfrequency.TheFourieranalysisandtimerepresentationsgiveareasonableanalysisat30Hz,butanalysisat10Hzisgreatlyfacilitatedbythewaveletrepresentation. 4-20 demonstratesthedierencebetweendeformationsignalsfordierentappingamplitudes.Relativelyhighermagnitudehigh-frequencyenergyisagainpresentataround60Hzandthefrequencytailextendingonthedownsideoftheupstrokeisevidentforbothcases,where+/-10degappingshowsthephenomenamoreclearly.TheFourieranalysisandtimerepresentationsgiveareasonablycompleteanalysisat+/-35deg,butanalysisat+/-10degisfacilitatedbythewaveletrepresentation. 4-21 .Bothsignalshavesignicantlylargetimeandfrequencyvaryingcomponentsthatareclearlyrevealedbythewaveletrepresentations.Whilethedistinctionsaremuchmoredramaticfor10Hzand+/-10deg,oncetheamplitudeisincreasedto+/-35deg,thewaveletrepresentationsshowverylittledierenceandmuchsmallertime-varyingcomponents. 4.2.1Validation 3-1 ,Young'sModulus,E=6:89E10,density, 59

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DeformationsResultingfromVariationsinFlappingFrequency WingTipVerticalDeformations10HzFlappingFrequency(TopRow)and30HzFlappingFrequency(BottomRow),Wing-2,+/-10degFlappingAmplitude,a)TimeHistory,b)FourierTransform,c)WaveletRepresentation 54 )usingEquation 4{1 whereiisaconstant,Listhelengthofthebeam,EisYoung'sModulus,misthedensityperunitareaandI=(beamwidth)3(beamthickness)=12isthecrosssectionalinertiaisalso 60

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DeformationsResultingfromVariationsinFlappingAmplitude WingTipVerticalDeformations+/-10degFlappingAmplitude(TopRow)and+/-35degFlappingAmplitude(BottomRow),Wing-2,20HzFlappingFrequencya)TimeHistory,b)FourierTransform,c)WaveletRepresentation performedtocomparewiththevalidationtest.Errorsaregenerallylessthan+8percentiftheexperimentalresultisassumedtobethetruth.TheLDVtestisexpectedtobeclosertorealitysincethetestisperformedinair,whilethestructuralmodespredictedbyFEAandbasicbeamtheoryassumethebeamisinavacuum.ShapesfrombothLDV 61

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DeformationsResultingfromVariationsinChordLocation VerticalDeformationsfortheMid-SpanLeadingEdge(TopRow)andMid-SpanTrailingEdge(BottomRow),Wing-2,+/-10degFlappingAmplitude,10HzFlappingFrequencya)TimeHistory,b)FourierTransform,c)WaveletRepresentation andFEAmatchcloselyfortherstbending,secondbendingandrsttorsionwithslightdierencesobservedoneachsideofthebeam,aswouldbeexpectedduetomanufacturingimperfectionsfortheLDVtest,asymmetricmeshingfortheFEA,andslightlydierent 62

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4-1 andFigure 4-22 m1=2;i=1:875;4:694;7:855(4{1) Table4-1. ModalFrequenciesforValidationTest Al-BeamMode12345678 LDV62.563.1382393430443n/an/aFEA65.665.640641347047411291156SBT63.0n/a397n/an/an/a1113n/a Figure4-22. AverageFrequencyResponseFunctionandFirstSixObservedOperatingDeectionShapesforAluminumBeam 4-2 Ingeneraleitherburstchirporsweepexcitationarecapableofidentifyingthedesiredfrequenciesforthesewings,showingdierencesontheorderofafewHertzwhichcorrespondstotheaccuracyavailablefromthedataacquisitionsystem.ThestinessofthewingisrelativelyindicatedbythefrequenciesobtainedwiththeLDVsystemwhereahigherfrequencyisindicativeofastierwing.ComparingWings1-3indicatesthatforanidenticalshape,acarbonberandmembranewingmaybemadesignicantlystierthananaluminumwing.Stierbattensalsoraisethefrequencyofmembraneshapesasseen 63

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Wing-165163303350n/an/aWing-261150280380n/an/aWing-342180275432n/an/aWing-4(L1B1)bc235097123152198Wing-4(L1B1)s234997121152199Wing-5(L1B2)bc224597132250364Wing-5(L1B2)s224499132256369Wing-6(L2B1)bc42n/a124233248n/aWing-6(L2B1)s4284126204249418Wing-L2B2bc4151133167261366Wing-L2B2s415284204249418Wing-L3B1bc59104138256303675Wing-L3B1s59n/a139255302676Wing-L3B2bc67n/a135515n/an/aWing-7n/an/an/an/an/an/aWing-8214261120n/an/aWing-9214264n/an/an/aWing-1039239286n/an/an/aWing-1134161246430n/an/a Table4-2. WingModalFrequencies,(bc=burstchirpexcitation,s=sweepexcitation) bycomparingtherelativefrequenciesofWing-5toWing-6.TheeectisseentobefairlylinearwhencomparingtherstfrequencyofWing-4,Wing-6andWing-L3B1whichaddasinglelayerofcarbonbertotheleadingedgeforeachwing. Inadditiontothewingsexaminedindetail,derivativesofWing-4notedasWing-L2B2,Wing-L3B1andWing-L3B2aretested.Thesewingsarestienoughthattheconstantenergyinputdoesnotexcitethewingenoughtorevealhigherfrequencymodes.HowevertrendsseenpreviouslyinWings-4,5and6(alinearincreaseinmode1frequencywiththeadditionofaleadingedgebatten,andlittlechangeinmode1frequenciesduetoadditionofparallelbattenswithshifthigherseenforhigherfrequencymodes)continuewiththesethreederivatives.ThemodalfrequenciesisolatedarelistedinTable 4-2 .Theadditionofanadditionalcarbonberstripfortheparallelbattenshaslittleeectonthebendingoperatingdeectionshapes,butappearstosubstantiallyincreasethefrequenciesofthemembranedependentoperatingdeectionshapesfoundathigherfrequencies. 64

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4-23 .Whiletherstbendingshapeappearsforthemostpartasexpected,theboundaryconditioninthetopleftofbothwingscausestheshapestobeslightlyasymmetricfromexpected.Wing-11showstheexpecteduniformrstbendingshapefollowedbyasecondbendingonthetopandbottomasexpected.Wing-10showsthebasicrstandsecondbendingaswellasthersttorsionalmodeasexpected,althoughtheboundaryconditiondoesagainslightlyaltertheseshapes. 4-24 below.ForthealuminumWing-3,theshapesappearverysimilartowhatwouldbeexpectedforarst,second,third,andfourthbendingmodeofacantileverbeam.However,bothmembranewingspresentsubstantiallydierentshapes.Therstandsecondbendingshapesarefairlysimilartoeachother,wherethefrequenciesareslightlyhigherfortheradialbattenarrangement,suggestingthiswinghasaslightlygreaterstiness.Thetwohigherfrequencyshapesrevealtheunderlyingbattenstructure, 65

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FrequencyResponseforVariationsinShape AverageFrequencyResponseFunction(Row1)andFirstThreeObservedOperatingDeectionShapesfora)Wing-11(Column1),b)Wing-10(Column2) asbothshapesforWing-2havenodallinesfollowingthepatternoftheparallelbattensandbothshapesforWing-3havenodallinesfollowingthedirectionoftheradialbattens. 4-25 .Alltheseshapesareattainedwithburstchirpexcitation.ThesweepexcitationoerednoadditionalinformationfortheLatexwings.Itisimportanttonotethatitisonlythesamplingarea,notthewingplanformthatisslightlydierent.TheLatexmembranewingsstillappeartopresentamuchlessdenedoperatingdeectionshape,althoughfurtherworkincludingnersamplingandamoreconsistentboundaryconditionmayhavecontributedtothesignicantlybetterresultfortheCapranwings.BycomparingtheburstchirpresultsseenhereforWing-1andWing-2tothesweepresults 66

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FrequencyResponseforVariationsinStructure AverageFrequencyResponseFunction(Row1)andFirstFourObservedOperatingDeectionShapesfora)Wing-3(Column1),b)Wing-2(Column2),andc)Wing-1(Column3) inFigure 4-24 itmaybeconcludedthatbothformsofexcitationmaybeusedtogivesatisfactoryoperatingdeectionshapes. 4-26 .Theadditionofacarbonberstriptotheleadingedgegenerallydoesnotaltertheoperatingdeectionshapessubstantiallyalthoughtheadditionofacarbonberstriptotheparallelbattensdoessomewhataltertheshapes. 55 ).Forthreewings,theoperatingdeectionshapesaregraphedatthefrequencyofexcitationidenticaltowherethepeakaveragethrustandpeakthrustareobservedasseeninFigure 4-27 .Alltheseshapesareattainedwithsweepexcitation.Whilethepeakthrustdoesfallin 67

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FrequencyResponseforVariationsinMembraneMaterial AverageFrequencyResponseFunction(Row1)andFirstSixObservedOperatingDeectionShapesfora)Wing-9(Column1),b)Wing-8(Column2),c)Wing-2(Column3),andWing-1(Column4) betweentherstandsecondshapeforallthreewings,theworkpresentedisnotenoughtostateanydenitiverelationshipbetweentheoperatingdeectionshapeandthemaximumthrustproduced. 68

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FrequencyResponseforVariationsinStiness AverageFrequencyResponseFunction(Row1)andFirstSixObservedOperatingDeectionShapesfora)Wing-4(Column1),b)Wing-5(Column2),andc)Wing-6(Column3) FourierTransform.Ifaspikeisseeninthetimehistorythewaveletmapmaybeusedtoconrmthatthestructureisbeingexcitedatthattimeatoneofthesuspectedresonantfrequencieswiththewaveletmap.Excitedmodesinthewaveletmapgenerallyhaveaspikeofenergyatthersttimetheyareexcitedbythesweep,withenergythendyingoutastimepasses.ThedeectionfromtheDICisdisplayedattherstmomentintimewhenthestructureexhibitsexcitation. TheoperatingdeectionshapesobtainedusingDICarecomparedwiththeoperatingdeectionshapesobtainedwiththeLDVforWing-10bycomparingFigure 4-28 andFigure 4-23 .TheshapesareverysimilartothoseobservedfromtheLDVsystem,althoughoflowermagnitude,whichisreasonablesincetheshakerissweepingthroughthefrequencyrangequicklyanddoesnotinputmuchenergyatanygivenfrequency.DICanalysiswouldbenetfromaslowersweep,althoughabalanceneedstobedrawnbetween 69

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ExaminationofFrequencyResponseinRelationtoThrustProduction AverageFrequencyResponseFunction(Row1)andOperatingDeectionShapesforPeakAverageThrust(Row2)andPeakThrust(Row3)fora)Wing-4(Column1),b)Wing-5(Column2),andc)Wing-6(Column3) sweeprate,camerasamplingrate,andthetotalmemoryavailabletothecamera.Itshouldbenotedthattheslantedpartwherepartofthebeamismissingislostwhentheareaofinterestisspeciedduringpost-processing.Inthefuture,thislosscouldbemitigatedsuchthatonlytheboundaryconditionatbottomrightismissing. ThesameprocedureisusedforWing-9withresultsshowninFigures 4-30 and 4-31 .WhenexaminingtheresultsforWing-9,peakpickingothefrequencyspectrumsuggeststhat21,45,62and101Hzshouldbeinvestigatedaspotentialsourcesforoperatingdeectionshapes.ThesecorrespondroughlytothosefrequenciesfoundbytheLDVsystemof21,42and64Hz,withtheexceptionofthe101Hzpeakwhichisnotseen.Thetimehistoryindicatesthat0.4869and1.743salsoshouldbeinvestigated.Thewaveletplotsconrmthattheenergyat20Hzat0.4869sisduetotherstbendingmode,whichisvisualized.Thetime0.5446scorrespondswiththefrequencycloserto61Hz,andformsanoperatingdeectionshapeunlikeanythatareobservedwiththeLDVsystem.Zoominginonthe100Hzregioninthewaveletplotrevealsasmallamountofenergythathasbeen 70

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SignalsfromDICofRectangularWingUnderShakerExcitation a)VerticalDisplacementsTimeHistoryofWing-10UnderSweepExcitation,b)FourierTransformc)WaveletPlotofPointUsedtoIdentifyTime-FrequencySpecicOperatingDeectionShapes Figure4-29. OperatingDeectionShapesfromDICofRectangularWingUnderShakerExcitation a)OperatingDeectionShapefromDICforWing-10forFirstBending,b)OperatingDeectionShapefromDICforSecondBending,c)OperatingDeectionShapefromDICforFirstTorsion excitedstartingatatimeof0.5677s.PlottingtheresultingoperatingdeectionshaperevealsashapeverysimilartothemembranemodesobservedfromtheLDVforother 71

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ThesuccessoftheexampleforWing-9demonstratesthepowerofobtainingoperatingdeectionshapeswiththeDICsystemwhencombinedwithsignalprocessingtechniqueslikethetimehistory,FouriertransformandWavelettransform.Theexamplesuggeststhatthistechniquecouldbeusedtoidentifyoperatingdeectionshapesofmembranewingsonthisscale,evenperhapsforwingsthattheLDVsystemstrugglestoanalyze.TheLDVisunabletodetermineanyclearoperatingdeectionshapesforthiscase,whiletheDICmethodshowsthreemodesquiteclearly.WhileknowledgefromtheLDVsystemmayhelprenethefrequencylocationofanoperatingdeectionshape,theDICprocessshowsexcellentpotentialforfuturedevelopmentasatooltoobtainoperatingdeectionshapesofsystemswhichnormallyhavebeenquitediculttoanalyze. Figure4-30. SignalsfromDICofMembraneWingUnderShakerExcitation a)VerticalDisplacementsTimeHistoryofWing-9UnderSweepExcitation,b)FourierTransformc)WaveletPlotofPointUsedtoIdentifyTime-FrequencySpecicOperatingDeectionShapes IntheseexamplesanalyzingtheDICdatafromwingsundershakerexcitation,weseehowasynthesisofinformationfromthetimehistory,theFouriertransformandWavelettransformallcontributetoidentifyinganoperatingdeectionshape.Itisevidentthat 72

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OperatingDeectionShapesfromDICofMembraneWingUnderShakerExcitation a)OperatingDeectionShapefromDICforWing-9forFirstBending,20Hz,0.4869s,b)OperatingDeectionShapefromDIC,61Hz,0.5446s,c)OperatingDeectionShapefromDICIndicativeofaMembraneMode,100Hz,0.5677s 4-3 .TheseparametersarecommonlyusedascontroleectorsincontrolstrategiesforFWMAVs. 73

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DynamicParameters ControlEectorSymbol FlappingFrequency!FlappingAmplitude 4-3 ,apolynomialtisformedforagivenappingamplitudeandappingfrequency.Figure 4-32 showstheaveragesignalfromthe7segmentsinadditiontothe5thorderpolynomialtforthewingtip.ThestandarddeviationoftheerrorforthepolynomialtisshowninFigure 4-33 toprovideanestimateoftheaccuracyofthepolynomialt.Whilegreateraccuracywouldbedesirable,thismodelispresentedasanexampleofhowthesignalsformedpreviouslycouldbeusedtoformareduced-ordermodel. Figures 4-34 4-35 4-36 ,and 4-37 demonstratethepolynomialtsfortheleadingedgeandtrailingedgerootchordandtheleadingedgeandtrailingedgemid-chordrespectively. Sincetheexperimentalspaceisdiscretizedinto3appingfrequenciesand3appingamplitudes,secondorderpolynomialsareselectedforpolynomialtsacrossappingfrequencyandappingamplitude.Figure 4-38 showsthepolynomialcoecientsforthesecondorderpolynomialapproximationwithrespecttoappingfrequency.Figure 4-39 demonstratesthecoecientsforthesuccessivesecondordertacrosstheappingamplitude. 74

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Wing-2AverageWingTipVerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) Figure4-33. Wing-2WingTipVertical-SignalStandardDeviationoftheErroratVariousFlappingFrequenciesandAmplitudes SingleLine=10Hz,X-Line=20Hz,*-Line=30Hz;SolidLine=10deg;DashedLine=17deg;Dash-DottedLine=35deg 75

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Wing-2AverageRootChord,LeadingEdge,VerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) Figure4-35. Wing-2AverageRootChord,TrailingEdge,VerticalSignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) 76

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Wing-2AverageMid-Chord,LeadingEdge,Z-SignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) Figure4-37. Wing-2AverageMid-Chord,TrailingEdge,Z-SignalatVariousFlappingFrequenciesandAmplitudeswith5thOrderPolynomialApproximation(DashedLine) 77

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Wing-2AverageWing-TipDeection,VerticalSignalCoecientsfor2ndOrderPolynomialAppoximationwithRespecttoFlappingFrequency LefttoRight:6thOrderCoecientto0thOrderCoecient,SolidLine=10deg;DashedLine=17deg;Dash-DottedLine=35deg Figure4-39. Wing-2AverageWing-TipDeection,VerticalSignalCoecientsfor2ndOrderPolynomialAppoximationwithRespecttoFlappingAmplitude LefttoRight:6thOrderCoecientto0thOrderCoecient,SolidLine=10Hz;DashedLine=20Hz;Dash-DottedLine=30Hz 78

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Themainhypothesesdefendedbythisthesisare: 1.DICisafeasiblemeanstoanalyzedeectionsanddeformationsofwingswhileapping(Rearmed( 8 ; 9 )).Resultssummarizingthedeections,deformationsandoperatingdeectionshapeswithDICareconsidered.Theperiodicitynormallyevidentforappingwingdeectionshasbeenobserved.WhileanalyzingatimehistoryorDiscreteFourierTransformofthesedeectionsmaybeusefultoobserveandmodelthemainperiodicfeatures,therearecircumstanceswherefurtheranalysismayberequired. 2.Waveletanalysisisausefulandsometimesnecessarymeanstoanalyzedeectionsanddeformationsofappingwings(New).Waveletanalysisisproventobeausefultooltounderstandperiodic,non-sinusoidalfeaturesaswellasnon-periodictime-varyingfeaturesobservedindeectionsanddeformationsofappingwings.Sincethedeectionshapehasarelationshiptothethrustthewingisabletoproduce,understandinghowthehigherfrequencycontentaectstheshapeofthetimehistorysignalisessentialtodesigninganyoptimalwing.Waveletanalysisthereforeprovidesdesignerswithaveryusefultooltounderstandanddesignthesewings.Inaddition,waveletanalysisisshowntohelpunderstandresonantbehaviorandcaseswhencontroleectorsareutilized. 3.LDVisafeasiblemeanstoanalyzeoperatingdeectionshapesofbiologicallyinspiredmembranewingsusedforapping.Theseoperatingdeectionshapesprovideinsightintothestructuraldynamicsofaappingwingsystem(Rearmed( 56 )). 4.DICcombinedwithwaveletanalysisisafeasiblemeanstoanalyzeoperatingdeectionshapesatknownfrequencies(Rearmed( 40 )).DICiscombinedwithwaveletanalysistoobtainoperatingdeectionshapesofabiologically-inspiredmembraneappingwingundershakerexcitationwithoutpriorknowledgeofthecorrespondingfrequency(New).ThefeasiblityofDICtoobtainoperatingdeectionshapeshasbeenvalidatedbycomparisontotheresultshownfromanLDVsystem(New). 79

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ThesynthesisofDIC,LDVandWaveletAnalysisprovidesahopethatsoonwewillbeabletoperformreduced-ordermodelingofaappingwingsystemwithsomelevelofdelity.Whilesignicanthurdlesremaintosystematicreduced-orderaeroservoelasticmodelingandcontroldesignofamissioncapableappingwingmicroair-vehicle,techniquessuchasDIC,LDV,andWaveletAnalysisformafoundationtobuildon. 80

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RobertLovewasborninMemphis,Tennesseein1984.Acuriousandimaginativechild,herstknewhewantedtoworkinaerospaceashewatchedagroupoftheAirForceF-16Thunderbirdsperformtheirbombburststunt.Hisyouthwasfullofadventuresallovertheworldashetraveledthroughover22countrieswhilelivinginRedlands,California,Daharan,SaudiaArabiaandJacksonville,Florida.HeattendedhighschoolatEpiscopalHighSchoolandwasapartofateamthatwasanationalnalistintheKIPRRoboticscompetition.DuringcollegeatAuburnUniversityhedidresearchonsuperalloyjoiningandbiologically-inspiredanddecontamination-environment-capableroboticsintheMaterialsEngineeringLabs,aswellasightcontrolofwing-in-ground-eectvehiclesintheAdaptiveAerostructuresLab.Healsoco-ledateamthatdesignedalowaspectratioyingwingfortheAIAADesignBuildFlycompetition.HegraduatedmagnacumlaudeinMay2007withB.S.degreesinbothaerospaceengineeringandmaterialsEngineering.Robertisasecond-yeargraduatestudentattheUniversityofFloridastudyingunderDr.RickLindasamemberoftheFlightControlLab.Heisinterestedinmodalandimageanalysis,signalprocessing,andadaptivestructures.Heisactivelyinvolvedinmultiplesocialnetworkingsites,usingweb-basedtechnologiestokeepabreastofcurrenttechnology.Heshareshisinsightswithothersatoneofmanywebsitesathttp://ornithtopters.wordpress.com.Despitehiswideranginginterests,hismainresearchfocusesontheanalysisandcontroldesignofappingwingmicroair-vehiclesthesizeofhummingbirds. 86