A Model for Roll Stall and the Inherent Stability Modes of Low Aspect Ratio Wings at Low Reynolds Numbers

MISSING IMAGE

Material Information

Title:
A Model for Roll Stall and the Inherent Stability Modes of Low Aspect Ratio Wings at Low Reynolds Numbers
Physical Description:
1 online resource (211 p.)
Language:
english
Creator:
Shields, Matthew C
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Aerospace Engineering, Mechanical and Aerospace Engineering
Committee Chair:
MOHSENI,KAMRAN
Committee Co-Chair:
UKEILEY,LAWRENCE S
Committee Members:
LIND JR,RICHARD C
EMMEL,THOMAS C

Subjects

Subjects / Keywords:
aerodynamics -- lowaspectratiowings -- lowreynoldsnumber -- rollstall -- stabilityandcontrol -- windtunneltesting
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:
The development of Micro Aerial Vehicles has been hindered by the poor understanding of the aerodynamic loading and stability and control properties of the low Reynolds number regime in which the inherent low aspect ratio (LAR) wings operate. This thesis experimentally evaluates the static and damping aerodynamic stability derivatives to provide a complete aerodynamic model for canonical flat plate wings of aspect ratios near unity at Reynolds numbers under $1\times10^5$. This permits the complete functionality of the aerodynamic forces and moments to be expressed and the equations of motion to solved, thereby identifying the inherent stability properties of the wing. This provides a basis for characterizing the stability of full vehicles. The influence of the tip vortices during sideslip perturbations is found to induce a loading condition referred to as roll stall, a significant roll moment created by the spanwise induced velocity asymmetry related to the displacement of the vortex cores relative to the wing. Roll stall is manifested by a linearly increasing roll moment with low to moderate angles of attack and a subsequent stall event similar to a lift polar; this behavior is not experienced by conventional (high aspect ratio) wings. The resulting large magnitude of the roll stability derivative, $C_{l,\beta}$ and lack of roll damping, $C_{l,p}$, create significant modal responses of the lateral state variables; a linear model used to evaluate these modes is shown to accurately reflect the solution obtained by numerically integrating the nonlinear equations. An unstable Dutch roll mode dominates the behavior of the wing for small perturbations from equilibrium, and in the presence of angle of attack oscillations a previously unconsidered coupled mode, referred to as roll resonance, is seen develop and drive the bank angle $\phi$ away from equilibrium. Roll resonance requires a linear time variant (LTV) model to capture the behavior of the bank angle, which is attributed to the variation in the $C_{l,\beta}$ derivative. These are purely aerodynamic modes which are demonstrated to be inherently present in LAR wings. To compare the impact of the roll stability derivative at high and low aspect ratios, a model for roll stall is developed which represents the tip vortices as infinite line vortices and estimates their influence on the surface pressure distribution of the wing; results for the roll moment coefficient are favorably compared with experimental data, and are used to compute $C_{l,\beta}$. By estimating the induced spanwise lift acting on a rolling wing, the roll damping derivative may also be computed and, along with the roll stability derivative, used to populate a simplified stability matrix for LAR wings. Solving for the eigenvalues of this system of equations at aspect ratios ranging from the near-unity values applicable to MAVs to high aspect ratio configurations reveals fundamentally different stability regimes. At cruise conditions, aspect ratios below 3.3 do not experience significant roll damping and the large magnitudes of roll stall instigate the divergent Dutch roll mode described by an unstable, complex eigenvalue. At higher aspect ratios above $AR = 4.6$, the eigenvalues cross into the left side of the complex plane and the lateral mode becomes stable, causing the wing to behave in a conventional, high aspect ratio manner. The disparity in lateral stability regimes between high and low aspect ratios at this Reynolds number suggests a potential explanation for why MAVs are prone to lateral instabilities, as their wings are inherently affected by unique flow physics which are not experienced by more conventional aircraft with a longer span.
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 Matthew C Shields.
Thesis:
Thesis (Ph.D.)--University of Florida, 2014.
Local:
Adviser: MOHSENI,KAMRAN.
Local:
Co-adviser: UKEILEY,LAWRENCE S.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2014
System ID:
UFE0045928:00001


This item is only available as the following downloads:


Full Text

PAGE 1

AMODELFORROLLSTALLANDTHEINHERENTSTABILITYMODESOFLOWASPECTRATIOWINGSATLOWREYNOLDSNUMBERSByMATTSHIELDSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2014

PAGE 2

c2014MattShields 2

PAGE 3

ToKim,thelightattheendofthewindtunnel 3

PAGE 4

ACKNOWLEDGMENTS Asiscustomaryatthispointofreectionattheendofathesis,Ihaverealizedthatanumberofinuentialpeoplehavemadeanimpactonmeandmyabilitytoaccomplishthistask.Whileafewshortwordsatthebeginningofathesisseemaninadequatewaytoproperlythankthem,Iwillatleastmaketheattempt.Ihavealwaysappreciatedthatmyparentshavesupportedmydecisions,buthaveensuredthatI,myself,amcondentthatIhavemadethecorrectchoice-likely,thisphilosophyhasnotbeentestedasrigorouslyasduringmyPhDsaga.IhopethattheyknowhowinstrumentaltheyhavebeenformetogettowhereIamtoday.IamalsoenormouslygratefultomyGranpdaCliffforbeinganinspirationalrolemodelforourfamilyandgivingmeanunderstandingoftheimportanceofhardworkanddedication.Toconcludethefamilysegment,Iappreciatemybrother'spersistentlackofinterestinmyresearchandourextensiveconversationsaboutanyalternatetopicwecouldthinkof-manytimes,thisprovidedanecessarybreakfromthework.Itisstrangetothinkthat,overthecourseofthenextphaseofmylife,Kamranwillbeconspicuouslyabsent.Ihavelearnedsomuchfromhimasamemberofhisgroup,andIamproudtosaythatIconsiderhimamentorandafriend.WhenIstartedworkingforKamranasanundergraduateMAVmonkey,IthinkitissafetosayIhadnoideawhatIwasgettinginfor;asitturnsout,myexperiencesinhisgrouphaveunquestionablyshapedwhoIamtoday,andIamabetterpersonforit.Thankyou,Kamran,andhopefullywecanworktogethersometimeinthefuture;IlookforwardtobaskingintheEyeofSauronagain.Finally,toKim,theamazingpersonwithwhomIhavesharedmystresses,frustrations,failures,(few)successesand,nowmypost-graduatelife:theexcitementofnishingmythesisistrivialincomparisonwiththerewardofnallybeingwithyou.Thoseyearsofcross-countryphonecallsandmonthlyrendezvousseemlikeadistantmemory,butIwillneverforgethowtheworkweputintostayingtogetherwasthefuelfor 4

PAGE 5

nishingmyPhD.WearetogetheratlastandIamsogratefultobeapartofyourlife.Youarethemostimportantpersoninmyworldandalwayswillbe.Thankyou.Tonish,afewwisewordstakenfromasmallsamplingofthetunesIplayedincessantlythroughoutthecourseofthisthesis: Doyouseethewaythattreebends?Doesitinspire?Reachingouttocatchthesun'sraysAlessontobeappliedArewegettingsomethingoutofthis,All-encompassingtrip?Vedder,McCready Fearlesslytheidiotfacedthecrowd,smilingMercilessthemagistrateturned`round,frowning.Andwho'sthefoolwhowearsthecrown?GodowninyourownwayAndeverydayistherightdayAndasyouriseabovethefearlinesinthefrown,YoulookdownHearthesoundofthefacesinthecrowd.Gilmore,Waters,Mason,Wright Nothin'lefttodobutsmile,smile,smile.Garcia,Hunter 5

PAGE 6

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 9 LISTOFFIGURES ..................................... 10 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTION ................................... 16 1.1BackgroundandMotivation:MicroAerialVehicleDevelopment ...... 16 1.2AircraftStabilityandControl:AHistoricalPerspective ........... 18 1.3StabilityandControlofMicroAerialVehicles ................ 20 1.4EffectsofLowReynoldsNumber ....................... 22 1.5EffectsofLowAspectRatio .......................... 28 1.6ThesisScopeandContributions ....................... 33 2APPARATUSANDEXPERIMENTALTECHNIQUES ............... 35 2.1WindTunnel ................................... 35 2.1.1TunnelCharacteristics ......................... 35 2.1.2ModelPositioningSystem ....................... 36 2.2DataAcquisition ................................ 38 2.2.1MicroLoadingTechnology(MLT)ForceBalance ........... 38 2.2.2DataAcquisitionHardware ....................... 39 2.3TestModels ................................... 41 2.3.1FlatPlateModels ............................ 41 2.3.2WindTunnelBlockage ......................... 42 2.3.2.1Solidblockage ........................ 42 2.3.2.2Wakeblockage ........................ 43 2.3.2.3Streamlinecurvatuve .................... 44 2.4DataReductionTechniques .......................... 45 2.4.1StatisticalAnalysisTools ........................ 45 2.4.2BalanceCalibrationUsingConstantLoads .............. 47 2.4.3StaticAerodynamicLoads ....................... 51 2.4.4ComparisonwithPublishedResults .................. 51 2.4.5TheForcedOscillationTechnique ................... 53 2.4.6DynamicDataAcquisitionandPostprocessing ............ 56 2.4.6.1Synchronizationofmotionanddataacquisition ...... 57 2.4.6.2Datapostprocessing ..................... 58 2.4.7TestingProcedure ........................... 60 2.4.8TestCase:KnownSignalwithAddedNoise ............. 60 6

PAGE 7

2.4.8.1Procedure .......................... 61 2.4.8.2Testfunctionresults ..................... 62 3EXPERIMENTALRESULTS ............................. 66 3.1SmokewireVisualizationofFlatPlateWingsinSideslip .......... 66 3.1.1RectangularWings ........................... 66 3.1.2TaperedWings ............................. 67 3.2StaticLongitudinalLoading .......................... 67 3.2.1VaryingAspectRatioRectangularPlates ............... 67 3.2.2VaryingTaperRatiosforaFlatPlatewithAR=1 .......... 73 3.2.3EffectsofSidesliponTaperedWings ................. 74 3.3StaticLateralLoading ............................. 77 3.3.1RollMomentforRectangularWings ................. 78 3.3.2RollMomentforTaperedWings .................... 80 3.3.3RollMomentwithZeroSideslip .................... 82 3.3.4SideForceandYawMoment ..................... 84 3.4StabilityDerivativeEstimates ......................... 86 3.5MeasuredDampingDerivatives ........................ 88 3.6EffectsofWingletsonTaperedWings .................... 92 3.6.1LongitudinalLoads ........................... 92 3.6.2LateralLoadsandRollStall ...................... 98 3.6.3EffectonCl, .............................. 102 3.7AerodynamicLoadingforthePIPERMAV .................. 102 3.8SummaryofExperimentalResults ...................... 106 4STABILITYCHARACTERISTICSOFLOWASPECTRATIOWINGS ...... 109 4.1LongitudinalInstability ............................. 110 4.2NonlinearEquationsofMotionandAerodynamicDependencies ..... 112 4.3LinearizedModelsofLowAspectRatioModes ............... 115 4.4PurelyLateralMotion ............................. 118 4.5EffectsofAngleofAttackPerturbations:TheRollResonanceMode ... 121 4.6ImpactofInitialPhaseShiftsonRollResonance .............. 125 4.7LinearTimeVariantModelforRollResonance ............... 126 4.8AttenuationofRollResonanceMode ..................... 128 4.9EffectsofDampingDerivativesontheWingResponse ........... 129 4.10DiscussionofResults ............................. 130 5ROLLSTALLMODELINGANDCOMPARISONWITHHIGHASPECTRATIOS 140 5.1StabilityFormulation .............................. 140 5.2AModelforRollStall .............................. 144 5.2.1DenitionofTermsandCoordinateSystem ............. 145 5.2.2EffectiveAngleofTrailingVortices .................. 146 5.2.3InducedPressureonWingSurface .................. 148 5.2.4ComputationofRollMoment ..................... 151 7

PAGE 8

5.2.5ValidationwithExperimentalData ................... 151 5.3RollDampingatLowAspectRatios ..................... 154 5.3.1HighAspectRatioModel ........................ 154 5.3.2EffectsofAspectRatioonLiftCurveSlope ............. 156 5.3.3TheNatureofRollDamping ...................... 156 5.3.4DerivationoftheDampTimeParameter ............... 158 5.4StabilityAnalysisatVaryingAspectRatio .................. 161 5.5FutureResearchDirections .......................... 165 5.5.1ImplementationofRollStallasaLateralControlMechanism .... 165 5.5.2PassiveandActiveStrategiesforIncreasedStability ........ 167 6SUMMARYANDCONCLUSIONS ......................... 169 APPENDIX ADERIVATIONOFTHEAIRCRAFTEQUATIONSOFMOTION .......... 177 BMLTBALANCEOPERATION ............................ 185 CALTERNATEREPRESENTATIONOFDAMPINGDERIVATIVES ........ 189 REFERENCES ....................................... 203 BIOGRAPHICALSKETCH ................................ 211 8

PAGE 9

LISTOFTABLES Table page 2-1Freestreamcenterlineturbulenceintensitiesforincreasingowvelocities ... 35 2-2MLTbalancemaximumloadsanderrors(providedbyMMT) ........... 39 2-3Testmodeldimensions ............................... 41 2-4MLTbalanceanalysisfora10gload ........................ 49 2-5Recordedloadsforaatplateat=10 ..................... 52 2-6Recordedloadsforaatplateat=20 ..................... 52 2-7Inputparameterstotestcase ............................ 61 2-8Resultsformeasuredtimedelay .......................... 65 3-1ValuesofCl,forvariousplanformgeometriesandtrimangles ......... 86 3-2ValuesofCn,forvariousplanformgeometriesandtrimangles ......... 88 3-3Flatplatemodeldimensions. ............................ 92 3-4Lateralstabilityderivativesduetoforvaryingcongurations. ......... 107 3-5PIPERMAVgeometricproperties. ......................... 108 4-1Parametersoflateralstabilitymodesatequilibriumanglesof0=5and15. 121 4-2Normalizedrootmeansquaredeviations ..................... 128 4-3Effectsofscaledstabilityderivativesoneigenvaluesat0=5. ......... 129 5-1Comparisonofstabilityparametersoffull44systemwith33approximation. 142 5-2ComparisonofCl,valuesobtainedfromexperimentaldataandmodel. .... 153 9

PAGE 10

LISTOFFIGURES Figure page 1-1ExamplesofMicroAerialVehicles(MAVs) ..................... 17 2-1Velocityproles .................................... 37 2-2ModelPositioningSystemmountedunderthetestsection ............ 38 2-3MicroLoadingTechnology(MLT)forcebalance .................. 39 2-4SchematicofDAQandmotioncontrolhardware ................. 40 2-5RawvoltageoutputofMLTwithandwithoutltering ............... 47 2-6Convergenceofsamplemean ........................... 50 2-7Systemvalidation:comparisonwithresultsfromMuellerandPelletier ..... 53 2-8Encodermeasurementsforapitchoscillation ................... 58 2-9Frequencyresponseandrequiredlow-passltercharacteristics ......... 59 2-10TestsignalandwhitenoisegeneratedwithSNR=0.01 ............. 62 2-11Frequencycontentofarticiallygeneratedband-limitednoise .......... 63 2-12Plotofnoisysignalandlteredsignal ....................... 63 3-1SmokewirevisualizationforaatplatewingwithAR=1 .............. 68 3-2SmokewirevisualizationforaatplatewingwithAR=1and=0.25 ...... 69 3-3AerodynamicloadsonrectangularatplateplanformswithvaryingAR ..... 70 3-4AerodynamicloadsonrectangularatplateplanformswithvaryingAR ..... 71 3-5AerodynamicloadsonrectangularatplateplanformswithvaryingAR ..... 72 3-6Aerodynamicloadsonatplateplanformswithvaryingtaper .......... 74 3-7Liftcoefcientofrectangularandtaperedatplatesinsideslip ......... 75 3-8Dragcoefcientofrectangularandtaperedatplatesinsideslip ........ 76 3-9Quarterchordpitchingmomentcoefcient ..................... 77 3-10Rollmomentcoefcientforrectangularatplatewingsinsideslip ........ 79 3-11Loadingasymmetriescreatedbythepresenceoftheupstreamtipvortex ... 80 3-12RollmomentcoefcientfortaperedatplatewingswithAR=1insideslip .... 82 10

PAGE 11

3-13Rollmomentcoefcient(Cl)witherrorbars .................... 83 3-14Sideforce(CSF)andyawmoment(Cn)coefcientsforatplatewings ..... 87 3-15Dampingderivativesduetorollrate,@=@p ..................... 89 3-16Dampingderivativesduetopitchrate,@=@q .................... 90 3-17Dampingderivativesduetoyawrate,@=@r .................... 91 3-18Liftcoefcientvsangleofattackforataperedplate(=0.75) .......... 93 3-19Dragcoefcientvsangleofattackforataperedplate(=0.75) ......... 94 3-20Quarterchordpitchingmomentcoefcientvsangleofattack .......... 95 3-21Liftcoefcientvsangleofattackforataperedplate(=0.25) .......... 96 3-22Dragcoefcientvsangleofattackforataperedplate(=0.25) ......... 97 3-23Quarterchordpitchingmomentcoefcientvsangleofattack .......... 98 3-24Lateralloadcoefcientsfor=0.5 ......................... 100 3-25Lateralloadcoefcientsfor=0.75 ........................ 101 3-26LongitudinalloadsforthePIPERMAVinsideslipatRe=7.5104 ........ 103 3-27LateralloadsforPIPERMAVmodelsinsideslipatRe=7.5104 ........ 104 4-1Illustrationofstaticlongitudinalstability ...................... 111 4-2Illustrationofpitchinstability ............................. 112 4-3Comparisonoflinearandnonlinearresponseat0=5and0=0 ...... 119 4-4Comparisonoflinearandnonlinearresponseat0=15and0=0 ..... 120 4-5Effectsofdivergentlateralmodeinitialconditionsat0=15and0=0 ... 122 4-6Comparisonoflinearandnonlinearsolutionswith(t)=3sin(!t) ....... 132 4-7Diagramofrollresonancemotion .......................... 133 4-8Variationsinnonlineartimehistoriesof~xlatforvaryingphases .......... 134 4-9Approximationofdivergentresponseusingalineartimevariant(LTV)model 135 4-10Effectsofdampingderivativesonthenonlinearlateralresponse ........ 136 4-11Effectsofdampingderivativesonthenonlinearrollresonance ......... 137 4-12Effectsofdampingderivativesonthenonlinearlateralresponse ........ 138 11

PAGE 12

4-13Effectsofdampingderivativesonthenonlinearrollresonance ......... 139 5-1Comparisonofnonlinearandlinearmodels .................... 143 5-2Denitionoftrailingvortexmodelcoordinatesystem ............... 146 5-3Variablesforcomputingthepressuredistributionfromthetrailingvortices ... 148 5-4Rollmomentmodel ................................. 152 5-5Spanwise-asymmetricinducedlocalangleofattack ............... 155 5-6Comparisonoftheliftcurveslopeandaspectratioofatplatewings ...... 157 5-7SectionalliftforceL(y)generatedbyrotatingwing ................ 158 5-8NondimensionalparameterTfforincreasingangleofattack ........... 161 5-9Complexeigenvaluesforhighandlowaspectratiowings ............ 163 5-10Realandcomplexeigenvaluemagnitudes ..................... 164 B-1SystemrepresentationoftheMLTbalance .................... 185 C-1Normalforcedampinginpitch@CZ @qforanAR=1wingatRe=7.5104 ... 190 C-2Pitchmomentdampinginpitch@Cm @qforanAR=1wingatRe=7.5104 ... 191 C-3Rollmomentdampinginpitch@Cl @qforanAR=1wingatRe=7.5104 .... 192 C-4Rollmomentdampinginyaw)]TJ /F6 7.97 Tf 6.67 -4.58 Td[(@Cl @rforanAR=1wingatRe=7.5104 .... 193 C-5Rollmomentdampinginroll@Cl @pforanAR=1wingatRe=7.5104 ..... 194 C-6Normalforcedampinginroll@CZ @pforanAR=1wingatRe=7.5104 .... 195 C-7Normalforcedampinginroll@Cm @pforanAR=1wingatRe=7.5104 .... 196 C-8Normalforcedampinginpitch@CZ @qforanAR=2wingatRe=7.5104 ... 197 C-9Pitchmomentdampinginpitch@Cm @qforanAR=2wingatRe=7.5104 ... 198 C-10Rollmomentdampinginpitch@Cl @qforanAR=2wingatRe=7.5104 .... 199 C-11Rollmomentdampinginyaw)]TJ /F6 7.97 Tf 6.67 -4.57 Td[(@Cl @rforanAR=2wingatRe=7.5104 .... 200 C-12Normalforcedampinginroll@CZ @pforanAR=2wingatRe=7.5104 .... 201 12

PAGE 13

C-13Normalforcedampinginroll@Cm @pforanAR=2wingatRe=7.5104 .... 202 13

PAGE 14

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyAMODELFORROLLSTALLANDTHEINHERENTSTABILITYMODESOFLOWASPECTRATIOWINGSATLOWREYNOLDSNUMBERSByMattShieldsMay2014Chair:KamranMohseniMajor:AerospaceEngineeringThedevelopmentofMicroAerialVehicleshasbeenhinderedbythepoorunderstandingoftheaerodynamicloadingandstabilityandcontrolpropertiesofthelowReynoldsnumberregimeinwhichtheinherentlowaspectratio(LAR)wingsoperate.ThisthesisexperimentallyevaluatesthestaticanddampingaerodynamicstabilityderivativestoprovideacompleteaerodynamicmodelforcanonicalatplatewingsofaspectratiosnearunityatReynoldsnumbersunder1105.Thispermitsthecompletefunctionalityoftheaerodynamicforcesandmomentstobeexpressedandtheequationsofmotiontosolved,therebyidentifyingtheinherentstabilitypropertiesofthewing.Thisprovidesabasisforcharacterizingthestabilityoffullvehicles.Theinuenceofthetipvorticesduringsideslipperturbationsisfoundtoinducealoadingconditionreferredtoasrollstall,asignicantrollmomentcreatedbythespanwiseinducedvelocityasymmetryrelatedtothedisplacementofthevortexcoresrelativetothewing.Rollstallismanifestedbyalinearlyincreasingrollmomentwithlowtomoderateanglesofattackandasubsequentstalleventsimilartoaliftpolar;thisbehaviorisnotexperiencedbyconventional(highaspectratio)wings.Theresultinglargemagnitudeoftherollstabilityderivative,Cl,andlackofrolldamping,Cl,p,createsignicantmodalresponsesofthelateralstatevariables;alinearmodelusedtoevaluatethesemodesisshowntoaccuratelyreectthesolutionobtainedbynumericallyintegratingthenonlinearequations.AnunstableDutchrollmodedominates 14

PAGE 15

thebehaviorofthewingforsmallperturbationsfromequilibrium,andinthepresenceofangleofattackoscillationsapreviouslyunconsideredcoupledmode,referredtoasrollresonance,isseendevelopanddrivethebankangleawayfromequilibrium.Rollresonancerequiresalineartimevariant(LTV)modeltocapturethebehaviorofthebankangle,whichisattributedtothevariationintheCl,derivative.ThesearepurelyaerodynamicmodeswhicharedemonstratedtobeinherentlypresentinLARwings.Tocomparetheimpactoftherollstabilityderivativeathighandlowaspectratios,amodelforrollstallisdevelopedwhichrepresentsthetipvorticesasinnitelinevorticesandestimatestheirinuenceonthesurfacepressuredistributionofthewing;resultsfortherollmomentcoefcientarefavorablycomparedwithexperimentaldata,andareusedtocomputeCl,.Byestimatingtheinducedspanwiseliftactingonarollingwing,therolldampingderivativemayalsobecomputedand,alongwiththerollstabilityderivative,usedtopopulateasimpliedstabilitymatrixforLARwings.Solvingfortheeigenvaluesofthissystemofequationsataspectratiosrangingfromthenear-unityvaluesapplicabletoMAVstohighaspectratiocongurationsrevealsfundamentallydifferentstabilityregimes.Atcruiseconditions,aspectratiosbelow3.3donotexperiencesignicantrolldampingandthelargemagnitudesofrollstallinstigatethedivergentDutchrollmodedescribedbyanunstable,complexeigenvalue.AthigheraspectratiosaboveAR=4.6,theeigenvaluescrossintotheleftsideofthecomplexplaneandthelateralmodebecomesstable,causingthewingtobehaveinaconventional,highaspectratiomanner.ThedisparityinlateralstabilityregimesbetweenhighandlowaspectratiosatthisReynoldsnumbersuggestsapotentialexplanationforwhyMAVsarepronetolateralinstabilities,astheirwingsareinherentlyaffectedbyuniqueowphysicswhicharenotexperiencedbymoreconventionalaircraftwithalongerspan. 15

PAGE 16

CHAPTER1INTRODUCTION 1.1BackgroundandMotivation:MicroAerialVehicleDevelopmentTheincreasedinterestintheeldoflowReynoldsnumberaerodynamicscanessentiallybeattributedtotheDARPAinitiativeofthemid-1990'stodeveloplightweight,inexpensive,maneuverablevehiclesformilitaryapplications[ 1 ].ThisledtotherstdenitionofaMicroAerialVehicleasanaircraftsmallerthan6in.(15cm)inalldimensionsandweighinglessthan5oz.(150g).Withthesedesignparametersinmind,AerovironmentwasabletodeveloptheBlackWidowMAVwhichewforthersttime(briey)in1996andwassuccessivelyiterateduponforseveralyearsuntilanalcongurationwasachievedandownonAugust10,2000[ 2 ].TheBlackWidow,showninFig. 1-1A wasfabricatedmainlyfromsolidfoamwitha6in.span,5.4in.centerlinechordandatotalmassof56.5g.InthepaperpublishedbyAerovironment,itisinterestingtonotethatwhilesignicantdataareprovidedfortheefciencyofthepropulsionsystemandthemassbudgetsofthevehicle,noaerodynamicdataaregivendespitetheauthorsuseofliftinglinetheoryandboundarylayerapproximationstomodelthevehicledrag[ 2 ].FollowingthesuccessoftheBlackWidow,anumberofinstitutionsbegandevelopingtheirownMAVsforamultitudeofmissions.Somenotableexamplesincludethe4.5in.wingspanMAVandtheexiblewingvehiclesdevelopedbytheUniversityofFlorida[ 3 4 ];thedualpropellerNRLMITEairvehicledevelopedforautopilottesting[ 5 6 ];anornithopterwithaclosedloopcontrolsystemcapableofautonomousightusingtelemetryacquisition[ 7 ];theTYTO-30biplanecongurationsdesignedtoincreasethemaximumattainablelift[ 8 ];theDelyMicro,aappingwingornithopterweighingonly3gandcapableofcarryingacamera[ 9 ];andthePrototypeInteractivePERformance(PIPER)MAVdevelopedbytheauthorandassociateswhileafliatedwiththeUniversityofColorado[ 10 ](showninFig. 1-1B ).Whilethesevehicleswere 16

PAGE 17

A BFigure1-1. ExamplesofMicroAerialVehicles(MAVs).A)Aerovironment'snaldesignoftheBlackWidowMAV,circa2000[ 2 ],B)PIPERMAVdevelopedatCU;photocourtesyofMattShields,2009. developedwithdifferentgoalsinmindrangingfromentriesintheInternationalMicroAerialVehiclecompetitiontodevelopingaplatformforsensornetworking,acommonthemeisthelackofaccurateaerodynamicmodeling.WhilelargerscaleUAVscanbedesignedusingapproximationsofinviscidaerodynamicssuchasvortexpanelmethodsutilizedinopensourcecodes,thesetechniquesfailtocapturetheuniqueaerodynamicbehaviorassociatedwiththelowaspectratio(LAR)wingscommontoMAVs.ThegoalofthisthesisistoconductasystematicexperimentalinvestigationintotheinherentstabilitypropertieswhicharisefromtheaerodynamicregimeofLARwings.Previousworkshavenotcomprehensivelystudiedeitherlateralloads(rollmoment,yawmoment,sideforce)orviscousdampingofthesewings,bothofwhicharerequiredtodevelopacompleteanalyticalmodel.TheexperienceofighttestingMAVshasrepeatedlyindicatedinstabilitieswhichresultfromlateralgustperturbations;thisresearchwilldemonstratethat,unlikeconventionalaircraftinwhichlateralloadingismostlydependentupongeometricfeaturessuchasverticaltails,LARwingsexperiencesignicantlateralloadspurelyduetotheaerodynamicightregimeassociatedwiththe 17

PAGE 18

shortwingspan.Furthermore,theseforcesandmomentsareshowntoinducedivergentstabilitymodesforsimplerectangularatplatewingsbereftofadditionalgeometricfeatures.ThisrepresentsanewperspectiveinMAVdesign,astheinherentdynamicsofthepurewingmustrstbeconsideredinthecourseofvehicledevelopmentandthenincorporatedintocontrollawsspecicallytailoredforLARiers. 1.2AircraftStabilityandControl:AHistoricalPerspectiveTherstconsiderationofthestabilitycharacteristicsofayingvehiclewasconductedbyF.W.Lanchesterin1908,inwhichherstdescribedtheconceptsofliftanddrag,postulatedamodelforthetrailingvorticesofanaircraft,andrstintroducedtheconceptofstabilitymodesbydescribing`Phugoid'oscillations[ 11 ].Lanchester'stheorieswerelargelybasedonobservationsfromhisownighttestinginwhichvaryingthecenterofgravityoftheaircraftcausedsmalllongitudinaloscillations;hethenformedaquantitativemodelbasedpredominantlyontherotationalvelocityofthecenterofgravityaboutaspeciedorigin.Althoughhisresultswerenotwellreceivedatthetimeofpublication,Lanchester'sworkformedabasisfortheconsiderationofaerodynamicstability.Amorerigorousformulation,andonemoresimilartotherigidbodyequationsofmotioncurrentlyusedtodescribeaircraftdynamics,wasformulatedbyBryanin1911[ 12 ].Bryanrecognizedtheseparatelateralandlongitudinalcomponentsoftheforcesandmomentsonanaircraftanddevelopedamathematicalmodelfortheassociatedoscillations.Healsointroducedtheconceptofstabilityderivatives,acommoncomponentofstabilityanalysisinmoderntimes.Thesearedenedasthepartialderivativeofaloadcomponentwithrespecttoaowcomponent;thatis,theeffecttheowconditionshaveonaspecicloadparameter.SimilartoLanchester,Bryan'smodelwasinherentlybasedupontheassumptionthattheproportionalitiesoftheaerodynamicforceswereknowntoscalewiththesquareoftheaircraft'svelocity,andthatanyrotationalmotionswouldresultinashiftingofthecenterofpressureand 18

PAGE 19

asubsequentmomentbeingimpartedtotheaircraft.ApointtonoteaboutBryan'smodelwasthatratederivativeswereforthemostpartneglected;heacknowledgedtheirexistencebutdidnothaveasatisfactorymodelforthem.Thiswouldbeimproveduponinthefollowingyears.Bryan'smajorgoalwastodenebasicgeometricandightparameterswhichwouldrenderayingaircraftstable.Thiswasgreatlyexpandeduponinthe20thcenturyastheavailabletoolsforaerodynamicanalysisbecamemoreadvanced.Currently,detailedtheoryofaircraftstabilitycanbefoundinanyightdynamicstextbook[ 13 14 ].Thegoalsoftheseearlymodelsweretoprovidesimplestabilityconsiderationsforaircraftbeforetheavailabilityofcomputationaltoolswhichcouldsimplysolvethenonlinearequationsofmotion.Vehicleswithhighaspectratiowingsyingatsteady,levelconditionsarenicelymodeledbythelinearizedanddecoupledequationsofmotion(derivedfullyinAppendix A );however,theincreasinglymaneuverableaircraftdevelopedinthesecondhalfofthetwentiethcenturybegantoexhibitightcharacteristicsnotwellrepresentedbythesimpliedequations.AwellrenownedexampleofthiswasthecontributionofrollratetolongitudinalinstabilitieswasrstdiscussedbyW.H.Phillipsin1948[ 15 ].Heconsideredtheproblemofghteraircraftwithshortwingspansandthemajorityofthefuelbeingstoredinthefuselage,whichresultedinsmallmomentsofinertiaabouttherollaxis(Ixx).Thehighrollratescommandedbypilotscausethenonlineartermsintheequationsofmotiontobecomesignicant,andtheinertialpropertiesoftheaircraftcoupletherollrelatedtermswiththepitchandyawaxes.ThistypeofanalysisissimilartotheconsiderationswhichmuchbeaddressedforLARwings,inwhichnonlineartermscannotnecessarilybeignoredinthederivationofthemodel.AnumberofadditionalstudieswereaddressedthroughwindtunneltestingbytheNationalAdvisoryCommitteeforAeronautics(NACA)and,later,bytheNationalAeronauticsandSpaceAdministration(NASA)inthesecondhalfofthe 19

PAGE 20

20thcentury.Thegoalofthesetestsweretoinvestigatetherapidlyexpandingightenvelopesandnonlinearregimesexperiencedbyhighperformanceaircraft;windtunnelexperimentationwastypicallyfocusedonthepitchingcharacteristicsofpurewingsandairfoils[ 16 18 ]aswellasstaticanddynamicstabilitycharacteristicsofscaledaircraft[ 19 22 ]andre-entryorsuborbitalvehicles[ 23 25 ].SignicantcouplingofstabilityaxeswasoftenuncoveredathighanglesofattackorMachnumbersandinsomecaseswassatisfactorytoexplaininstabilitiesorcrashesexperiencedduringighttesting;moreinformationaboutcouplinginaircraftbydynamicscanbefoundin(forexample)thereviewarticlebyDay[ 26 ].Theseinstabilitiesrarelyoccurredatequilibriumight,however,andcouldoftenbepreventedorrecoveredfrombyanexperiencedpilotorsimplecontrolsystem.Moreconventionalaircraft,suchasjettransports,weredevelopedwithbasicliftinglineandpotentialowmodelswhichwereeffectivefortheightregimeunderconsideration.UntiltheconceptofMAVswassuggestedinthe1990's(whichalsorequiredsignicantadvancesinmicroelectronics),littleattentionwaspaidtolowaspectratio,lowReynoldsnumberaerodynamicsorthepotentialfortheuniqueowbehaviortoinstigatenewdynamicresponses.Whenthesevehicleswereinitiallybeingdesignedandown,however,itquicklybecameapparentthatconventionalaerodynamicswouldnotsufcetopredictthecharacteristicsofMAVs,andthatmoreextensivetechniqueswouldberequiredtofullydescribetheirstabilityandcontrolproperties. 1.3StabilityandControlofMicroAerialVehiclesTheequationofmotionformulationusedforconventionalaircraftdoesnotnecessarilyprovideanaccuraterepresentationoftheightdynamicsofMAVs;theassumptionsmadeinsimplifyingthenonlinearequationscannotalwaysbeappliedtotheLARwingsofinterestforMAVight.Asaresult,verylittleanalyticalworkhasbeendonetoconsiderthestabilitypropertiesofthesevehicles;instead,thedevelopmentofreliableightplatformstypicallyhasdependeduponiterativeprocedures 20

PAGE 21

andtheintuitionandexpertiseofthedesigners[ 4 ].Typically,attemptstoinvestigateaerodynamicderivativesandstabilitymodesofMAVshasbeenlimitedbyalackofavailablewindtunneldata;forexample,someworkhasbeendonebytheUniversityofFloridainconjunctionwithNASA'sBasicAerodynamicsResearchTunneltodevelopasimulationmodelandassessthecontrolcharacteristicsofanaeroelasticxedwingMAV[ 3 4 ].Unfortunately,thewindtunneldatawaslimitedtosmallsideslipanglesandsorollstallwasnotconsidered(althoughthelargemagnitudeofCl,wascommentedupon[ 3 ]);furthermore,nodynamicderivativeswerecomputedsoacompletestabilityanalysiscouldnotbeundertaken.Thesimulationmodeldevelopedwasanonlineardynamicinversionmodelwhichdidnotrequiretheuseofstabilityderivativesandwasnotexpectedtoprovideaparticularlygoodrepresentationofthevehicle'sdynamics;inaddition,thelateraldirectionalforcedatawerelimitedandsomewhatunreliable,theaerodynamicmodelwasdevelopedinanadhocmanner,andnononlineartermswereconsideredintheequationsofmotion[ 4 ].Theinvestigationprovidedabasisforfuturedevelopmentofautonomouscontrol.AmorecompletefunctionalmodelforthesametypeofMAVwasdevelopedbyAlbertani,etal[ 27 ].AseriesofwindtunneltestswereperformedtomeasureCL,CDandCmfunctionalrelationsforeachcoefcientwereobtainedusingalinearregressionanalysis.Thiswasjustiableastherangeofanglestesteddidnotdisplaysignicantnonlinearities;however,theauthorsalsodidnotconsiderthecrosscoupledeffectsonthefunctionals,whichcanbesignicantduetothepresenceofrollstallonLARwings.Itisalsonotclearhowthedynamiceffectsofpitchrateweremeasuredduringthewindtunneltesting.AmorecomprehensivefunctionalmodelwasdevelopedbyBabcock,etal,forsimilarexiblewingstestedwithanewlydesignedpitch-plungepositioningsystemwhichallowedmeasurementsofpitching,plungingandrotation(acombinationofplungeandpitch)[ 28 ].Thefunctionalmodelwasagainlimitedtothelinearrangeofliftandonlydependenceon,_,_andqwasconsideredfor 21

PAGE 22

CL,CDandCm.TheinuenceofthemeasuredcoefcientsonthestabilitymodesofaMAVwerenotdiscussed.AdditionalworkattheUniversityofFloridahasusedparameteridenticationbasedtechniquestotstabilitycoefcientstopolynomialexpressionsfortheaerodynamicloadsandestimatethevehicleresponse.Garcia,etal,usedighttestdatatodevelopapproximatemodelsoftheMAV'sightdynamicsforthepurposeofusingamorphingwingforrollcontrol[ 29 ].An`adhoc'approximationofthelateraldirectionalmodelwasattempted;however,asignicantnumberofassumptionsweremadetoreachthenalformofthemodel.DatafromadifferentMAVwasusedtoestimatetheDutchrollandspiralmodes;furthermore,loadsduetosideslipandyawwereconsideredtobesmallduetogeometricargumentswhichdonotaccountforrollstall.Asaresult,theproposedcontrollerwasonlycapableofmoderateimprovementsintheobservedightoscillations.AcommonthemeoftheaforementionedstudiesonthestabilityandcontrolpropertiesofMAVsistheapplicationofsystemidenticationtechniquestodeterminethefunctionalrelationshipsbetweentheaircraftresponsevariablestotheaerodynamicloads.Whilethismethodisusefulatcapturingtheeffectsofnonlinearrelationshipsbetweenthesysteminputsandoutputs,itinherentlyassumesthattheformofthemodelisavailable[ 30 ];thatis,therelevantstatevariablesareknown.ThisisavalidtechniquebutcannotnecessarilybeappliedtoLARwingsduetotheunknownaerodynamicmodelassociatedwithlowReynoldsnumberowphenomena.Anin-depthunderstandingoftherelevantoweffectsisimportanttounderstandthenatureofloadingdependenciesforLARwings,whichcanbeusedtodevelopafullaerodynamicmodel. 1.4EffectsofLowReynoldsNumberForxedwingMAVs,Reynoldsnumbersofinterestaretypicallybetween104and105;investigationsintotheeffectsoftheassociatedowconditionsonaerodynamicperformancehavebeenunderwaylongbeforetheDARPAMAVinitiative.Perhapsthe 22

PAGE 23

earlieststudyofaerodynamiccoefcientsatReynoldsnumbersontheorderof1105wasconductedbySchmitzinthe1930'sforthepurposeofstudyingmodelairplanes[ 31 ].Heutilizedwindtunneltestingtoinvestigatethinatplates,thincamberedplatesandathickcamberedairfoilbymeasuringtheliftanddrag.BytestingthegeometriesatReynoldsnumbersbetween2104and2105hewasabletoidentifyatransitionregionnearRe=1105inwhichadramaticincreaseinliftanddecreaseindragareobserved.Thisimprovementinperformancecanbeexplainedasthetransitiontoturbulenceoftheowaroundthewing.Schmitz'sworkwasnotusedforanypracticalapplications;itwouldhavebeenimpossibletodevelopaviableMAVwithouttheasyetunheardoflithiumbatteries,micro-electronicsandhighdensityfoam/carbonbermaterialsusedincurrentMAVs.However,theresultsofSchmitz'stestsservedtoidentifythechallengesfuturedesignerswouldencounterwiththeaerodynamicregimethatwouldcometodeneMAVs.InvestigationsoftheowphenomenaassociatedwithlowReynoldsnumberbegansoonafterSchmitz'sworkasthesignicanteffectsoftheseconditionsonaerodynamicperformancebecameapparent.Oneoftherstdirectionsthisresearchfocusedonwastheobservedlaminarseparationoftheboundarylayerowundercertainexperimentalconditions[ 32 35 ].WhilethefreestreamReynoldsnumberwasslightlyhigherthantheMAVregionofinterest(intheorderof3)]TJ /F8 11.955 Tf 12.65 0 Td[(5105),owseparationneartheleadingedgeunderthepresenceofastrongadversepressuregradientwasobservedusinghotwireanemometry,surfacepressuremeasurementsandsurfaceowvisualization.Itwasclearthatundercertainowconditions,includingfreestreamReynoldsnumberandturbulenceintensity,theturbulentstressesintheseparatedowentrainedsufcientexternaluidtoincreasethepressureofthedetachedshearlayer.Alargeenoughpressureincreasecouldcausetheowtoreattachtotheairfoil[ 36 ].Theseearlystudiesrecognizedtheeffectofthisseparation,andtheassociatedpocketofreversedow,onthepressuredistributiononthesurfaceoftheairfoil;namelyanincreasedsurface 23

PAGE 24

pressureinthevicinityoftheseparatedregion.AlsoobservedwasatendencyforthebubbletodisappearatsmallReynoldsnumbers(astheowneverreattachesafterseparation)andaboveacriticalReynoldsnumber(astheowisentirelyturbulentinitially)[ 33 ].Mueller'sgroupatNotreDameextensivelyinvestigatedthelaminarseparationbehaviorofamultitudeoftestmodelswiththeadditionalbenetofimprovedmeasurementtechniquesanduniqueowvisualizationmethods.MuellertypicallytestedatReynoldsnumbermorerelevanttoMAVs(althoughhisresearchstartedwellbeforetheDARPAinitiativeoftheearly1990s).Hewasoneofthersttousesmokewirevisualizationtocaptureimagesoftheseparationbubbleandcomparedtheresultswithempiricaltheoriesofseparationbubblelengths[ 36 ].Inthesamestudy,Muelleralsotookhighspeedmovieswith16mmlmanddetectedunsteadybehavioroftheseparationbubble,whichappearedtogrowandshrinkinlength(measuredasapercentofthechord)overaperiodof0.6seconds[ 36 ].Itisinterestingtonotethatthisbehavioroccursatalowangleofattackof4withnodynamicmaneuveringoftheairfoil;thelowReynoldsnumberbehavior(Re=1.5105fortheaforementionedtest)inducedunsteadyowonthetopsurfaceoftheairfoilduetothepresenceofthebubble.AdditionalworkbyMuellerwasusedtobetterdenethetransitionlocationandassociatedboundarylayerthicknessonWortmannFX63-137andEppler387airfoils[ 37 38 ].Mueller'sextensiveworkinthearealeadshimtoconcludethatavailableempiricalmodelsfortheseparationoftheshearlayerappearquestionable,andthatthesignicanteffectsofReynoldsnumberandfreestreamdisturbancesmustbeaccountedforinlowReynoldsnumberloadpredictions[ 37 39 ].Morerecentstudieshaveindicatedthat,whilethefreestreamReynoldsnumberdoesaffecttheshapeofthebubble,theresultingimpactonliftanddragarenotnecessarilyassignicantasthatcausedbyvariationsinangleofattack;however,theliftcyclesaresubstantiallyinuencedbyasinusoidallyvaryingfreestreamwhichsimulatesagustyenvironment,creatinghysteresisinliftandin 24

PAGE 25

someinstancesinitiatingaself-excitationinexiblemembranewings[ 40 42 ].Finally,improvedexperimentalcapabilities(i.e.,time-resolvedParticleImageVelocimetry(PIV))nowmakeitpossibletoresolvethequasi-periodicdevelopmentofvorticalstructureswithinthebubblewhichcontributetotheburstingoftheLSBwhentheyreachacriticalstate;thisimprovedunderstandingoftheformationandevolutionoftheLSBmayfacilitateactiveowcontroltechniquesforimprovedaerodynamicefciencies[ 43 ].AsimilarphenomenontotheLSBwhichhasalsobeeninvestigatedindetailistheleadingedgevortex(LEV).LambourneandBryer[ 44 ]andEarnshaw[ 45 ]weresomeofthersttoexperimentallyobserveandclassifytheformationoftheLEVonslenderdeltawings.TheyfoundthattheLEVformswhenthesharpleadingedgecreatesaseparatedshearlayerwhichsubsequentlyrollsupandformsavorticalstructurealongtheleadingedge.Earnshawalsofoundthatthisvortexcouldbeseparatedintothreeregions:aninnervortexcorewithaxialvelocitiestwicethatofthefreestream;aviscoussub-coremanifestedbyhighpressureandvelocitygradients;andanouterregionwherethetraceofthevortexsheetcanbemarginallyobserved[ 45 ].AsimultaneoustheoreticalstudybyHallwasperformedwhichattemptedtomodeltheowinsidethevortexcoreusingasymptoticmatching[ 46 ].WhiletheseearlyapproachesprovidedsomegoodqualitativeunderstandingofthedevelopmentoftheLEV,theresearchwasmostlyconstrainedtoslenderdeltawingswithmoreofafocusonsupersonicow.Asaresult,thereremainssomedoubtastowhetherastrong,coherentLEVformsonaMAV-typeplanformthewayitdoesforaslenderwingathighincidenceangles;forexample,asvisualizedbyPayne,etal[ 47 ].SomeresultsfornonslenderdeltawingsatmoderateReynoldsnumbersdosuggest,however,thatatleastaweakLEVwillformalthoughittendstobesusceptibletoearlyburstingduetothestrongadversepressuregradient[ 48 49 ].Thisvortexbreakdownismanifestedbyasuddendivergenceofthestreamtubesinthecorewhichexpandintoeitherabubbleorspiralform[ 50 ]. 25

PAGE 26

Thediscoverythatsteadystateaerodynamicsdidnotaccuratelypredicttheliftforcecreatedbyappingwingsledtotherealizationthatthevortexliftcreatedbytheleadingedgevortex(LEV)ispredominantlyresponsiblefortheightofinsectsandbirds[ 51 53 ].Thisstructureformsduringthedownstrokeofaappingwing(orforwardmotionofatranslatingwing)whentheowseparatesoverthesharpleadingedgeandrollsuptoformacoherentvortex[ 54 ].Thelowpressurecoreofthevortexlocallyaugmentstheliftattheleadingedgeandthuscontributestotheforcegeneratedbythewing(onecanalsoconsidertheeffectsoftheLEVincontributingtothenetcirculationofthewing,inwhichcasetheLEVaugmentstheliftbyuseofpotentialowtheory)[ 52 ].Ultimately,atsomelocationinthewing'sappingcycleoraccelerationprole,theboundvorticitybecomestoogreatandtheunstablevortexshedsfromtheleadingedge,carryingwithitthebenecialaspectsofincreasedliftonthewing.SincetheLEVwasrecognizedasacriticalcontributortoappingwingaerodynamics,aconsiderableamountofworkhasbeenconductedtounderstandthenatureofthisowphenomenaandtocorrelatecomputationalmodelswithexperimentalresults.FlowvisualizationandPIVhavebeenusedextensivelytocharacterizethedynamicsoftheLEVforawidevarietyofkinematicsrangingfromthree-dimensionalrotationtoclap-and-ingmechanisms;comparisonsbetweenlocalmeasurementsoftheowphysicsattheleadingedgewiththegloballoadmeasurementsofthewinghaveshownhowtheforcesataninstantdependonthetimehistoryofthevortexinteractions[ 52 54 58 ].Directnumericalsimulationshavebeenabletoreproducetheseresultswithreasonableaccuracyandprovidedetaileddepictionsofthevorticityeldsaroundappingwingsatanyinstantinthemotioncycle[ 57 59 62 ].Reducedordermodelshavebeendeveloped(andcontinuetoberened)whichreplicateboththeexperimentalandsimulationresultsatafractionofthecomputationaltimeandwhichmaysoonbeusableaseitherdesigntoolsorbaselinedynamicalmodelsforcontrolsystems[ 63 64 ].Inessence,asubstantialdegreeofinformationabouttheowphysicsoftheLEVand 26

PAGE 27

itseffectsonwingloadinghasbeenrevealedinrecentyears,andcontinuedprogressintheeldmaysoonmakeitpossibletousethisknowledgeforthedesignofappingwingMAVs.AdditionalinvestigationoftheeffectsofvortexsheddinginthewakeofinclinedatplateswasconductedbyLamandLeung[ 65 ].Theyexperimentallymeasuredvelocityspectrainthewakeofnominally2D(AR33)airfoilsatReynoldsnumbersbetween5103and2.5104.Theydetectedasymmetricvortexsheddinginthewakewhichwasattributedtotheinteractionbetweentheleadingandtrailingedgevortices.TheLEVwasobservedtoformandthenconvectalongaportionofthewingsurfacebeforedetachingatthetrailingedge.Thisisanextremelyinterestingresultasitindicatesthetime-dependentnatureoftheowoverastationaryMAVwing.AstheLEVgrowsandsheds,thetimeaveragedforcemayremainthesamealthoughinstantaneousuctuationscanbeattributedtothedevelopmentofthevorticalstructures.TheinuenceoftheLEVontheaerodynamicloadingofalowReynoldsnumberwingismostlyassociatedwiththelowpressureintheinnercore.Thisinturncreatesalocalliftaugmentationasthedecreasedpressurecontributestothepressuredifferentialbetweenthetopandbottomsurfacesofthewing.Itshouldbenotedthatthiscanalsocontributetoasymmetricmomentsaboutthecenterofgravityofthewing.Inaddition,ifvortexbreakdownoccurs,theresultinglossofthelowpressurecorecanthencauseareductioninlocalliftandamoredisturbedowelddownstreamoftheburst.AftertheDARPAinitiativeintheearly1990s,moreattentionwasfocusedontheglobalaerodynamicpropertiesoflowReynoldsnumberairfoilsforthepurposeofcategorizinglift,dragandpitchingmomentofpotentialMAVdesigns.TheimportanceofhavingaccurateloadcellsforMAV-scalemeasurementsbecameapparent.Laitonetestedaseriesofairfoilswithaspectratiosof6andwasabletoaccuratelymeasuredragusingaforcebalancesensitiveto0.01g,twoordersofmagnitudebetterthanthebalanceusedbyMuellerinhisearlierwork[ 66 ].Heconcludedthatasmallamountof 27

PAGE 28

camber(5%)improvedtheliftperformanceofthewingbyincreasingthestallangleofattack,whereasdecreasingtheReynoldsnumberbelow50,000greatlydegradedtheperformanceofthewing.Healsonotedthenonlinearnatureoftheliftcurveslopeandthedeparturefromthepredictedslopeof2predictedbypotentialowtheory.Seligconductedextensivewindtunneltestingtocategorizethelift,dragandpitchingmomentofavarietyofairfoilsunderdifferentowconditionsbetweenReynoldsnumbersof6104-3105[ 67 ],butdidnotlookcloselyattheassociatedowphenomena.PelletierandMueller[ 68 ]andTorresandMueller[ 69 ]alsoconductedcanonicalinvestigationsofatplategeometries;however,theirresearchwasmorefocusedontheinuenceofsmallaspectratiosandwillbediscussedinthefollowingsection.Theextensiveinvestigationsofthelaminarseparationbubble(LSB)andhaveprovidedasignicantamountofknowledgeabouttheowconditionswhichcausethebubbletoform,themechanismsofturbulententrainmentandowreattachment,thenatureoftherecirculatingowinsidethebubble,theexpectedlocationsofseparationandreattachment,andthepressuredistributionaffectedbythebubble.Similarly,thedevelopmentoftheLEVhasbeenexperimentallystudiedandisreasonablywellunderstood.Whiletheaforementionedstudiesyieldanexcellentunderstandingofthesephenomena,therehasbeenminimalworkdonetoquantifyandmodeltheeffectsonaerodynamicloadingandstability.ThepresenceoftheLSBhasbeendetectedonthePIPERMAVseeninFig. 1-1 [ 70 ]. 1.5EffectsofLowAspectRatioThepreviouslymentionedowphenomenaweretypicallyinvestigatedfornominally2Dorhighaspectratiomodels(withtheexceptionofthedeltawingmodelsintheLEVstudies,whicharenotfeasibleMAVplanforms).Lowaspectratio(LAR)wingsfaceadditionalcomplicationsduetotheirsmallwingspans;thismuchhasbeenapparentsincetheearlydaysofwindtunneltestingasLARgeometriesweretestedtoinvestigatetheirefciencyascontrolsurfaces[ 71 72 ].Whilethesetestswereconductedathigher 28

PAGE 29

ReynoldsnumbersthanwouldberelevantforaMAV(Re>3105),itbecameclearthatthesmalleraspectratiosexperiencedanincreasedvalueofCLmaxandstall.Zimmermanreportedonthisbutdidnotcloselyinvestigatethenatureofthisbehavior[ 71 ];Winterwasabletomeasureincreasednegativepressurealongthesidesofthewingwhichheattributedtocrossowfromthetipvortices[ 72 ].Theformationofthetipvortexhasbeenstudiedindetailsincetheearlydays(seeforexampleMcCormick,etal[ 73 ])anditiswellunderstoodthatthisstructureformsduetothepressuredifferentialbetweenthesuctionsideandthepressuresideofthewing[ 13 ];however,itwasnotuntilrecentlythatthesignicanceoftheseeffectsforMAVswasconsidered.PelletierandMueller[ 68 ]providedoneoftherstthoroughstudieswhichexplicitlyinvestigatedlowaspectratio,lowReynoldsnumberwingswiththegoalofprovidinginformationforMAVdesigners.Thewingsweresemi-innitemodelsmountedverticallyinthewindtunnel(allowingthetipvortextopropagateoveroneofthewingtips)andweretestedatReynoldsnumbersbetween6104and2105.Thestudywasnotspecicallyfocusedonaspectratio,althoughseveralweretested;otherpointsofinterestincludedtheeffectsoffreestreamturbulence,trailingedgegeometryandwingcamber.Still,theresultsforlowReynoldsnumberLARwingsmatchedthosefoundnearly70yearspreviously:areducedaspectratioledtoanincreasedstallangleandliftcoefcient.TorresandMueller[ 69 ]extendedonthisworktomorecloselyinvestigatethebehaviorofLARwings.TheydemonstratedthattheaspectratioofthewingwasthemostimportantparameterinMAVaerodynamics,surpassingplanformgeometryandReynoldsnumber.Thehighstallangleswereattributedtothetipvortexcrossowpreventingowseparationinangleofattackregionswhichwouldtypicallyhavestalledforahigheraspectratiowing.TorresandMuelleralsoattemptedtomatchexperimentaldatatoexistinganalyticalmodelsforlift,dragandpitchingmoment.Whiletheyfoundgoodagreementforsomelimitedcases,itwasclearthattheseequationswereonlyvalid 29

PAGE 30

foralowrangeofanglesofattackinwhichtheeffectsoftipvorticeshadnotyetbeguntodominate.ThesignicanceoftheresultsfromTorresandMuellerledtofurtherstudyoftipvorticesforLARatplateandMAV-typewingswithanincreasingfocusonowvisualization,PIV,andsimulations.Viieru,etal,conductedexperimentalandnumericalinvestigationscategorizingthewakestructureofthetipvorticesandtheresultanteffectontheloadingofaMAVwing[ 74 ].Bycomparingtestcaseswithandwithoutendplates(or,winglets)mountedonthewingtipsitwaspossibletoevaluatethedirecteffectofthetipvortex.ShieldsandMohseniinvestigatedtheeffectsofsideslipontheaerodynamicloadingofvariousatplategeometriesatReynoldsnumbersbetween5104and1105[ 75 ].Itwasdeterminedthatatincreasedsideslipanglesupto=35,thedownstreamtipvortexhadlittleeffectonthewingasitwasmostlyconvecteddownstream;however,theupstreamvortexpropagatedacrossasignicantportionofthewingandwasabletokeeptheowattached(asvisualizedbysurfacetuftmeasurements).Asaresult,evenatincreasedsideslipangles,thestallangleandvalueofCL,maxremainedhigh.Itwasalsofoundthattheincreasedsideslipangleimpactedtheslopeofthepitchingmomentand,thus,thestabilityderivativeCm,.Thisderivativeistypicallynotconsideredforlateralstabilitymodes;however,thevariedvalueatincreasedsideslipanglesindicatesthecouplingbetweenlateralandlongitudinalmodesforLARwingsatlowReynoldsnumbers.SomeinterestingcomputationalworkwasconductedbyTairaandColoniusinvestigatingtheinteractionbetweenthetipvortexandtheLEVfortranslatingLARwings[ 76 ].ItshouldbenotedthattheReynoldsnumberstestedinthisstudywerefarlowerthanthoseofxedwingMAVs(typicallyRe=300);however,theirresultswereobservedtoagreewiththoseofFreymuth,etal,whotestedatRe=5200[ 77 ].Inessence,whilethequantiableeffectsoftheowphenomenawillnotbeidenticalin 30

PAGE 31

thedifferentReynoldsnumberregimes,itappearsthatfromaqualitativebasissomeofthetrendsareremarkablysimilar.InbothcasesitwasobservedthatforLARwings(AR1),thetipvortexandtheLEVinteractedstrongly.Foranimpulsivelystartedwing,theLEVwasseentoformduetotheseparatedshearlayerattheleadingedge,feedingvorticityintothestructureandenhancingtheliftoftheplate.Forhigheraspectratioplates(AR>2),theLEVshedsaperiodically;whentheaspectratioisunity,thedownwardsvelocityfromthetipvortexkeepsthevortexsheetformedattheleadingedgeattachedtothesurfaceofthewing.ThisbehaviorallowsthehighstallanglesassociatedwithLARwings.Itwasalsoobservedthatathighanglesofattack,thetipvorticesinteractedstronglywiththeLEVandthetrailingedgevortexinthewake,contributingtoanaperiodicunsteadystate.Thisbehaviorhasalsobeenstudiedforpitching/plungingatplatewings,whichindicatesthatthenatureoftheplanformgeometry(suchastheleadingedgesharpness)andthekinematicsofthemotionaffectthesizeandstrengthofthetipvortexand,thus,theinteractionwiththeLEV[ 78 ];thisindicatesthatthedynamicmotionofthemodelwillbesignicantinassessingtheaerodynamiceffectsofthetipvortices.TairaandColoniusalsoinvestigatedtaperedleadingedgegeometriesandfoundthatthesheddingoftheLEVitselfcanbeattributedtothereleaseofbuiltupvorticityinthestructure.Fortherectangularleadingedgepreviouslydiscussed,theonlymechanismtodischargethisvorticitywasthroughshedding;forataperedorellipticleadingedge,thevorticitygeneratedattheleadingedgefedintotheformationofthetipvortexandconvectsdownstreamwiththerolledupvortex[ 79 ].ThisisasignicantresultforMAVaerodynamics,asitindicatestheeffectsofleadingedgegeometryontheuidbehavioronthesurfaceoftheentirewing;theseeffectsareinnowayconnedtothevicinityoftheleadingedge.JianandKe-QinconsideredalowaspectratioMAV-typewingataReynoldsnumberof1104atanglesofattackupto45[ 80 ].Theydetectedsignicantasymmetry 31

PAGE 32

intheoweldabovethewingabove=11.Notonlydothetipvorticesappeartohavedifferentstrengthsandsizes,theywandererraticallyinalateraldirectionastheyinteractwithadditionalvorticalstructuresonthesuctionsideofthewing.ThisisexpectedtoaffectthelateralloadingofMAVsastheyawmomentvarieswithtime.ThisstudyrepresentedoneoftherstindicationsthatatlowReynoldsnumbers,thelateralloadingofLARwingscouldbeaffectedbythetrailingvortices.Whilethevastmajorityoftheliteraturestillpertainstoliftanddragmeasurements,laterinvestigationssuchasthoseconductedbyGresham,etal,showthatthetrailingvorticescanalsoexciterolloscillationsinLARwings[ 81 ].Furthermore,thefrequencyoftheserolloscillationswerecorrelatedwithapitchmotionprescribedtothewingatahighincidenceangle(>20),suggestingapotentialcross-couplingofthelateralandlongitudinalstabilityaxes[ 82 ].ThesepreviousstudiesprovidedabasisforanewdirectionofinvestigationbyShieldsandMohseni;inordertobetterunderstandtheimpactofthetipvorticesonlateralloadsofLARwings,aseriesofexperimentsincorporatingsmokewirevisualizationandforcebalancemeasurementswereconducted[ 83 ].Whenexaminingthelateraldirectionalloads(sideforce,yawmomentandrollmoment)ofsideslippingatplatewingsataReynoldsnumberof7.5104theauthorsdiscoveredthattherollmomentincreasedlinearlytosignicantmagnitudesbeforestallinginamannersimilartoaliftpolarnear=20.Thisbehavior,describedas`rollstall',wasattributedtotheasymmetricdistributionofthetipvorticescreatedbytheincidentsideslipangle;thevortexemanatingfromtheupstreamwingtippropagatedoverthesurfaceofthewingwhereasthedownstreamvortexwasconvectedawayfromthewingsurfacebythefreestreamow.Thisresultsinanasymmetricspanwisecirculation(andthuslift)distribution;furthermore,thelowpressurecoreoftheupstreamvortexcontributesalocalliftaugmentationwhichalsocontributestotherollmoment.Thisowasymmetrycausesthemagnitudeoftherollstabilityderivativetobecomesignicant;typicalresultsindicatedthatjCl,j0.15,greaterthantherangeof)]TJ /F8 11.955 Tf 9.3 0 Td[(0.1Cl,0consideredto 32

PAGE 33

constitute`goodhandlingqualities'foranaircraft[ 13 ].TheeffectsoftherollstabilityderivativeonthegustsensitivityofsmallaircrafthavebeenconsideredbyPisanoandLawrence,whodesignedaGustInsensitiveAircraft(GIA)whichwassymmetricaboutthex)]TJ /F3 11.955 Tf 12.34 0 Td[(yplane;asaresult,thevalueofCl,wasnominallyzeroandtheaircraftshowedanimprovedresponsetolateralgustperturbations[ 84 ].Itshouldbenotedthattheaircraftinquestionhadawingspanofapproximately3feetandahigheraspectratiothanaMAV,andthuswaslesssusceptibletotherollstalleventdescribedinthiswork.TheinuenceofthetipvortexasymmetryontherollmomentofLARwingsinsidesliphasnotbeenpreviouslydiscussedintheliterature,anditwillbeshowntohavesignicantimplicationsfromastabilityandcontrolperspective.Thesenewlydiscoveredowphysicsformthebasisforthisthesis. 1.6ThesisScopeandContributionsThecentralmessageofthisdissertationisthattheloadingconditionscreatedbyrollstallinducestabilitymodeswhichareuniquelyinherenttolowaspectratiowings.Unlikeconventional,highaspectratiowingswhicharepredominantlyconsideredtobeliftcreatingdevices,theshortwingspansandsmallmomentsofinertiaassociatedwiththelowaspectratiocasepermitsthesemodestodevelop.Thisthesiswilldescribeandcategorizedthesemodes,includingtheinherentlycross-coupleddependenceonangleofattackandsideslip.Furthermore,amodelwillbedevelopedwhichdemonstrateshowtheinducedpressurefromthetrailingvorticescreateseffectsofrollstallforLARwings;thismodelwillbeusedtocomparethestabilityparametersofhighandlowaspectratiowings,andtodenitivelyshowthatthelatterexistinafundamentallydifferentstabilityregime.Theresultsobtainedforcanonical(atplate)wingsmaythenpotentiallybeextendedtoprovideabasisforthestabilityanalysisoffullMAVs.ThespeciccontributionstotheeldoflowReynoldsnumberaerodynamicsandcontrolareasfollows: 33

PAGE 34

Design,fabricateandvalidateaModelPositioningSystem(MPS)capableoftestingfourdegreesoffreedominthewindtunnelasnoresultsexistintheliteratureforlateral(ie,sideslip)perturbations. Usingbothsmokewirevisualizationandstaticloadmeasurements,providetherstdescriptionofrollstallforLARwings. ConducttherstcanonicalinvestigationofaerodynamicdampingderivativesofLARwingsusingaforcedoscillationtechnique,permittingadynamicmodelforthewingstabilitymatrixtobedeveloped. Numericallyintegratethenonlinearequationsofmotionandcomparetheresultswiththezero-inputresponseofthelineartimeinvariant(LTI)model;characterizethedivergentDutchrollresponseofLARwings. Deneanddescribetherollresonancemode,thecrosscoupledmodewhichdevelopswhenlateralperturbationsoccurinthepresenceoflongitudinaloscillations.Considertheimpactoftherollstabilityderivativeusingalineartimevariant(LTV)model. Deriveamodelforrollstallbycalculatingtheinuenceofthetrailingvorticesonthesurfacepressuredistributionofrectangularwings. Modifythepre-existingmodelforrolldampingtoestimateatwhichaspectratiosCl,pbecomessignicant Obtainasimplied(33)stabilitymatrixforthelateraldynamicsofLARwingswhichrequiresonlytherollstabilityandrolldampingderivatives;usingthedevelopedmodels,computeandcomparethestabilityparametersforhighandlowaspectratiowings. AnalyzetheshiftinstabilityregimesfromhightolowaspectratiosatlowReynoldsnumbers.DiscussthepotentialimplicationsforfutureMicroAerialVehicledevelopment. 34

PAGE 35

CHAPTER2APPARATUSANDEXPERIMENTALTECHNIQUESThissectiondescribesthehardwarecurrentlyinplaceusedforexperiments,equipmentwhichmustbeobtainedandusedforfuturetesting,anddiscussesthedatareductionandanalysistechniquesused. 2.1WindTunnelAlltestsconductedforthisresearchwereconductedusingthePrototunnelattheUniversityofColoradoWindandGustCharacterizationLaboratoryandatthePERClabattheUniversityofFlorida.Thislowturbulence,opencircuit,closedjetwindtunnelhasa14in.14in.testsectionwithan8.5:1contractionratio.Analuminumhoneycombscreenwith0.25in.aperturesisusedasaowstraightenerwithanemeshscreenimmediatelydownstreamwhichhelpstoreducethefreestreamturbulenceintensityinthetunnelto0.2%intherangeofvelocitiestested(20ft/s).Thetestsectionis5ftlongwiththetestmodelsmounted4ftdownstreamofthecontractionconeexit. 2.1.1TunnelCharacteristicsSomeworkhasbeendonecharacterizingtheowenvironmentinthePrototunnel;muchofthishasbeendocumentedinpreviouspublicationsfromthegroup[ 70 75 ].StandardwindtunneltechniquessummarizedbyPope[ 85 ]wereincluded,suchastheuseofthehoneycombscreenandnemesh,improvedthefreestreamcenterlineturbulenceintensityfromaninitialvalueofover4%totheorderof0.2%;nalresultsforincreasingfreestreamvelocitiesaredisplayedinTable 2-1 Table2-1. Freestreamcenterlineturbulenceintensitiesforincreasingowvelocities Freestreamvelocity(ft/s)Turbulenceintensity 21.40.0030232.00.0024242.70.0017052.30.0022363.70.00227 35

PAGE 36

Abasicinvestigationofboundarylayerthicknessandcenterlineowqualitywasalsoconductedbyperforminghotwiretraversesacrossthex-andy-directionsofthetestsection(ie,acrosssectionofthetunnel)atthedownstreamlocationwheremodelsaremountedonthepositioningsystem.TestswereconductedatfreestreamvelocitiesofU0=17and55ft=s,correspondingtotheminimumandmaximumrangesofvelocitiesatwhichthetunnelcanmaintainsteadyow.Datawascollectedat156stationsatthedesiredcrosssection,formingagridofdatapointsconsistingof13rowsand12columns.Datawassampledatfs=8192Hzforadurationof10sateachlocation;10separaterecordswerecollectedandthenensembleaveragedtoobtainthenalvelocitymeasurement.TheresultsareshowninFig. 2-1 .Boundarylayereffectsarevisiblenearthetopofthetestsection;theprobecouldonlybemounted1in.fromthesidewallsand0.5in.fromthebottom,whereasitcouldbeplacedwithin0.25in.ofthetop.Slightasymmetriesarenoticeable,particularlytheslightincreaseinvelocitywithinaninchofthelefttunnelwall.Thisisnotaboundarylayereffect,butisattributedtotemperaturedriftinthehotwirestowardstheendofthedataacquisitionprocess.TheowinthecenterofthetestsectionissatisfactoryforMAVtesting,withspatialvariationsontheorderof5%ofthedesiredvelocity.Careisalsotakenduringexperimentstoensurethatthemodeldoesnotextendintotheboundarylayerofthetunnel,northepotentiallyincreasedvelocityregionnearthesidewalls.Futureworkwillbedonetoreassessthevelocityprolewithanautomateddataacquisitionsystemtoreducethetemperaturedriftandtobetterdenethevelocityprole;however,forthegloballoadmeasurementsconductedinthisinvestigationtheowqualityisseentoproduceaccurateresults. 2.1.2ModelPositioningSystemTheobjectiveofthisresearchistomeasureaerodynamiceffectsatawidevarietyofightangles.ManylowReynoldsnumbertunnelsutilizeanexternalforcebalancetotestatplategeometriesmountedtoendplatesnearthewallsofthetestsection,enablingcomparisonwith2Dresults.Thishasthedisadvantageofeffectivelyblocking 36

PAGE 37

A BFigure2-1. VelocityprolesofPrototunneltestsectionatthedownstreamlocationoftheMPS.A)U0=17ft=s,B)U0=55ft=s. thetipvorticesfrompropagatingoverthewingtips,whichisthemainphenomenabeinginvestigated.TorresandMuellerusedaninternalforcebalancemountedtothetrailingedgeofthetestmodel,allowingfulldevelopmentofthetipvortices[ 69 ].Sometunnelsemployadynamicpositioningsystemtoinducepitchingandplungingmotionofatestmodel[ 86 ].ThePrototunnelisoutttedwithaModelPositioningSystem(MPS)whichwasdesignedandbuiltbytheauthorstoincreasetheavailabledegreesoffreedomwhichcanbetestedwhilemaintainingtheeffectsoftipvortices.TheMPS(showninFig. 2-2 )ismountedunderthetestsectionandutilizesaseriesofthreeDanaherMotionNemasteppermotorsforroll,pitchandyawactuationandaHaydonKerk57000seriesnon-captivelinearactuatortocontrolthemodelinaverticalplunge.ThesemotorscanrotatetheMPS180inroll,40inpitch,and180inyaw,inadditiontobeingabletoplungeatotaldistanceof14.ItshouldbenotedthattheserangesaretheallowabledisplacementsoftheMPSitselfanddonotnecessarilyreectthemaximumrangethatcanbetestedforaspecicmodel.Testmodelsathigheranglescanpotentiallybreachtheboundarylayerofthetunnelorproduceforceswhichoverloadthebalance;thussometestcasesarenottakentotheabsolutemaximumof 37

PAGE 38

A BFigure2-2. ModelPositioningSystemmountedunderthetestsection.A)Schematicwithblockarrowsindicatingavailabledegreesoffreedomthatcanbeactuatedbyeachmotor(gurenottoscale),B)FabricatedMPS;photocourtesyofMattShields,2009. correspondingMPSaxis.Modelscanbeaccuratelypositionedtowithin0.1inangleofattackand0.5insideslip.Multipleaxescanbecontrolledsimultaneouslytoinducemaneuveringight(ie,combiningrollandpitchtomimicabankedturn)byrunningLabviewsoftwareandNationalInstrumentshardware(aPXIMotionController,aUMIUniversalMotionInterface,andfourPacicScienticP70530controllers). 2.2DataAcquisition 2.2.1MicroLoadingTechnology(MLT)ForceBalanceAllmeasurementswereconductedusingtheMicroLoadingTechnology(MLT)forcebalance,custombuiltbyModernMachineandTool(MMT)Co.inNorfolk,VAandshowninFig. 2.2.1 .ThissixcomponentinternalbalancewasdesignedtomeasureMAV-scaleloadswithahighaccuracy;thespecicationsprovidedbyMMTforeachchannelcanbefoundinTable 2-2 .The24straingaugeswereconguredas6fullWheatstone 38

PAGE 39

bridges;adetaileddescriptionoftheiroperationisprovidedinAppendix B .AsignicantamountofcalibrationtestingwassubsequentlyconductedonthebalancetovalidateitsperformanceinthespecicconnesofthePrototunnelexperiments(andwillbediscussedinSection 2.4 ). Figure2-3. MicroLoadingTechnology(MLT)forcebalance.PhotocourtesyofMattShields,2009. Table2-2. MLTbalancemaximumloadsanderrors(providedbyMMT) ChannelMaxloadMaxerrorTypicalMAVload[%offullscaleload][=30,=20,Re=105] Normalforce3lbs0.13%0.39lbsAxialforce2lbs0.19%0.07lbsPitchingmoment5inlb0.12%0.13inlbRollingmoment3inlb0.18%0.12inlbYawingmoment3inlb0.20%0.06inlbSideforce2lbs0.14%0.04inlb 2.2.2DataAcquisitionHardwareTheDAQsystemusedinthisinvestigationiscomprisedprimarilyofNationalInstrumentshardware.Twoseparatedatapathsarerequiredfortheforcebalanceandhotwiredata,respectively.Fortheformercase,aSCXI-15208-channeluniversalstraingaugemoduleismountedinsideaSCXI-1000chassiswhichcaneasilysamplethe6analogchannelsoftheMLTbalanceinterfacedthroughaSCXI-1314mountingblock.TheSCXI-1520enablessimultaneoussamplingonallchannelsandfeaturesa4-polelow-passButterworthlterwithavailablecutofffrequenciesof10Hz,100Hz,1kHzand10kHz.AnimportantfeatureoftheSCXI-1520toconsideristheminimumvoltagerange 39

PAGE 40

of10mV;duetotheexpectedoutputoftheMLTbalanceontheorderof100V,theVofthemoduleisactuallytoolargefortheMLTbalance,whichcancreatesomequantizationerrorsforlowforceoutputs(specicallyaxialforce)accordingtotherelationeq=V 2MwhereVisthevoltagerangeandMistheresolutionoftheADCconverter.Thiswillbediscussedfurtherinthenextsection.ThelteredsignalispassedintoaPXI-6229MultifunctionDAQ,whichmitigatesmuchoftheaforementionedquantizationerrorwitha16-bitresolutionADCconverter.HotwiredataisacquiredwithDantecsingleandcrossprobesinterfacedwithavariableresistanceDantecMiniCTA54T30whichhasabandwidthof5-10kHzandisoptimizedforusewithwireprobes.ThesignaliscollectedbyaNIBNC-2120shieldedconnectorblockandsenttothesamePXI-6229DAQusedforstraingaugemeasurements.TheseparatedatapathsandthepreviouslydiscussedmotioncontrolhardwarearedepictedintheschematicinFig. 2.2.2 Figure2-4. SchematicofDAQandmotioncontrolhardware 40

PAGE 41

2.3TestModels 2.3.1FlatPlateModelsFlatplate(0%camber)modelsweremanufacturedforthisinvestigationinamannercommontootherlowaspectratiowindtunnelstudies.Theplatesweremachinedwitha5:1ellipticallyroundedleadingedgeandhadthicknesstorootchordratiosbetween2.4%and5.4%.ThesmalldimensionsofthePrototunneltestsectionrequirethatmaximumspanofamodelmustbelessthan10in.toensurethatitdoesnotintrudeuponthetunnelwallboundarylayer.Themodelsweremachinedfromacrylicinsteadofthemoretraditionalaluminumtoreducetheweightthatthebalanceneedstosupport.ThedimensionsofthemodelsarelistedinTable 2-3 Table2-3. Testmodeldimensions ARcroot,cm.ctip,cm.b,cm.Diagram Rectangularplanforms0.7516.916.912.71 115.215.215.21 1.510.210.215.21 37.67.622.91 Taperedplanforms115.211.415.20.75 115.27.615.20.5 115.23.815.20.25 Wingletswerealsomachinedtoinvestigatetheeffectsofreducingthetipvortexowonthetopsurfaceofthetaperedwingsandwereusedinvarioustestcases.Theseadditionswere1.5in.highandthesamelengthasthetipchord(1.5in.,3in.,and4.5in.).Theleadingedgewasroundedononesidewitha5:2ellipticalgeometryandwasatontheotherside,allowingittobemountedushtothewing.Theywereattachedtothe 41

PAGE 42

winginthreecongurations:entirelyabovethewing,centeredonthewing,andentirelybelowthewing,asshowninTable 2-3 ,inordertodeterminewhichgeometryhadthegreatesteffectontheaerodynamics.ModelsweremountedtotheMLTbalanceattheirthreequarterchordusingacylindricalmountattachedushwiththebottomsurfaceofthemodelalongthecenterline.Althoughthemountwastaperedtoreducedrag,itwasinitiallyaconcernthatits0.5in.diameterwouldhaveasignicantimpactontheloading;however,trialsrunwiththemountonthetopsurfaceandbottomsurfaceofthewingaswellasresultswiththemountdragsubtractedoutproducedidenticaldata.Hence,interferencefromthemountwasdeemedtobenegligible. 2.3.2WindTunnelBlockageThepresenceofthemodelsinthetestsectioncauseblockageeffectswhichcanalterthemeasuredaerodynamicloads.Forthesimpliedgeometriestestedinthisexperiment(forexample,withnopropellersattachedtothemodels),themostsignicanteffectsarecreatedbysolidblockage,wakeblockage,andstreamlinecurvature[ 69 85 ].Inordertocorrectfortheseadverseeffects,thewingisrepresentedbyavortexsystemandamethodofimagesisusedtorepresenttheclosedwalleffectsontheowwithinthetestsection.Theassociatedcorrectionstothemeasuredloadswillbediscussedseparatelyinthefollowingsections. 2.3.2.1SolidblockageTheprojectedfrontalareaofthemodelinthetestsectionreducestheareatheairowsthrough;duetoBernoulli'stheorem,thiswillincreasethevelocityoftheowoverthewing.Itshouldbenotedthatthevelocityincrementissmallerthanwhatwouldbecalculatedfromadirectmassconservationstandpointasthestreamlinesnearthemodelarelessaffectedthanthestreamlinesnearthetunnelwalls[ 85 ].Still,itisnecessarytodeterminetheincreased(average)velocityinthevicinityofthemodelandtoapplythiscorrectiontothemeasuredloads.Thebodyisrepresentedbyasource-sinkpairandthewallsaremodeledwithaninnitedistributionofimages.Thevelocity 42

PAGE 43

incrementSBisdescribedbyEq. 2 : SB=U0 U0,u=K11(wingvolume) C3=2,(2)inwhichK1isthebodyshapefactorbasedonthethicknesstochordratio,1isafactorbasedontheratioofthegeometricspanofthemodeltothewidthofthetestsection,Cisthecrosssectionalareaofthetestsection,andthesubscriptureferstotheuncorrectedquantities.K1and1areconservativelyestimatedfromdatapresentedbyRaeandPope[ 85 ];thevaluesarebothnominallyconstantforanymodeltestedandarenominallyequalto0.9. 2.3.2.2WakeblockageThewakebehindawingwillhaveadecreasedvelocityrelativetothefreestreamowduetothedragcreatedbythebody.Asthevolumeofairenteringthetestsectionremainsconstantduringatest,theowvelocityoutsideofthewakemustincreaseduetocontinuity.Thiscreatesavelocityandpressuregradientnearthemodel.Asaresult,acorrectionmustbeappliedtothevelocityandthedragcoefcient,whichisgivenbyEq. 2 : WB=U0 U0,u=Ssin 4CCD,u,(2)inwhichSisthewingarea,istheangleofattack,andCD,uistheuncorrecteddragcoefcient.AnexplicitcorrectiontoCDisalsoimplemented,andisgivenby: CD,WB=K11(wingvolume) C3=2CD,u,(2)Itshouldbenotedthattheseequationsarespecicallydesignedforstreamline(attached)ow;whileseparationoccursattheleadingedgeofaatplateathighanglesofattack,thethreedimensionalowfromthetipvorticeshelpstokeeptheowattachedathigheranglesofattack[ 69 75 ].Thus,correctionsforfullyseparatedowsonbluffbodies(see,forinstance,Maskell'spaper[ 87 ])arelessapplicablethanthestreamlinedequations.Thecorrectionsfrom 2 and 2 arecombinedandapplied 43

PAGE 44

totheexpectedvalueoffreestreamvelocityusedinthenondimensionalizationofthemeasuredloads;thecorrectioninEq. 2 isapplieddirectlytothedragcoefcient. 2.3.2.3StreamlinecurvatuveAstheclosedtestsectionofthePrototunnelpreventsthenaturalcurvatureofthestreamlineswhichwouldoccurforowaroundamodelinaninnitelywidetestsection,thewingarticiallyappearstohaveanincreasedcamberthanitstruevalue.Thisresultsinincreasedangleofattack,liftandpitchingmomentaboutthequarterchord.Therelevantcorrectionsarecomputedfromtheequivalentimagevortexsystemtodeterminetheeffectiveangleofattackandthedependentlongitudinalloads. SC=2(Ssin=C)CL,(2)inwhich2isafactorrepresentingtheincreaseofboundary-inducedupwashbehindthewing(estimatedtobe0.1usingdatafromPope[ 85 ])andistheboundarycorrectionfactor(nominally0.4forthewingspansusedintheclosed,rectangulartestsection).TheangleofattackincrementgivenbyEq. 2 isusedtocorrecttheliftandpitchingmomentcoefcients: CL,SC=)]TJ /F8 11.955 Tf 9.29 0 Td[(SCaCM,SC=)]TJ /F8 11.955 Tf 9.29 0 Td[(0.25CL,SC,(2)whereaistheliftcurveslope.Insummary,thecorrectionstothemeasuredloadsaretypicallynegligibleasthecorrectioncoefcientsaredependentuponthethicknesstochordratioofthethin,atplates.Typicallythemagnitudeofthecorrectionsarefoundtobelessthan1%ofthe 44

PAGE 45

measuredloads.Thesetofcorrectionsforallmeasureddataare: CL=CL,u(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2(SB+WB))+CL,SC,CD=CD,u(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2(SB+WB))+CD,WB,CM=CM,u(1)]TJ /F8 11.955 Tf 11.95 0 Td[(2(SB+WB))+CM,SC,CSF=CSF,u(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2(SB+WB)),Cl=Cl,u(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2(SB+WB)),Cn=Cn,u(1)]TJ /F8 11.955 Tf 11.96 0 Td[(2(SB+WB)).(2)Finally,anumberofstudieshavebeenconductedtoassesstheimpactofoscillatingairfoilsorwingsonunsteadyaerodynamicloading;typically,thesestudiesareconcernedwithcompressibleowsandmodelingreectedacousticdisturbancesoffofthetunnelwalls[ 18 88 89 ].Thisisnotaconcernforthefullyincompressibleconditionsofthisinvestigation.Somepreviouslypublishedresultshaveindicatedthatthemotionofthewingmaystillintroduceliftaugmentationorphaselead/lagforsmallertestsections(H=c<3,whereHistheheightofthetunnel)[ 90 ].Thedynamictestsinthisinvestigationwereconductedwithwingsofc=4in.,andalengthratioofH=c=3.5;possiblyasaresultofthis,noliftvariationwasobservedbetweenpostprocesseddynamicdataandcorrespondingstaticmeasurements.Asaresult,nocorrectionsforunsteadywallinterferencewereconsidered;asthemeasureddampingderivativesarefoundtobeatleastanorderofmagnitudesmallerthanthestaticderivatives(discussedinChapter 3.5 ),anyminorerrorsintroducedbythisassumptionwillnotsignicantlyimpactthedynamicmodelsofthewingaerodynamics. 2.4DataReductionTechniques 2.4.1StatisticalAnalysisToolsTocharacterizethesignalobtainedfromthebalanceaseriesofcommonstatisticalanalysismethods.Whenasignalisusedtoestimateaphysicalloaditisnecessarytocharacterizethebiasandrandomerrorsassociatedwiththemeasurementofadataset 45

PAGE 46

x.Thesearegiven,respectively,by b^E[^])]TJ /F7 11.955 Tf 11.96 0 Td[((2) ^q E[(^)]TJ /F1 11.955 Tf 11.95 0 Td[(E[^])2](2)whereistheparameterbeingestimated,^istheestimateoftheparameter,andEistheexpectationoperator.Eqs. 2 and 2 canbenormalizedbytogivetherelativebiasandrandomerrorsbandr,respectively.Thecondenceintervalsforthesamplemeanaregivenbythe`2sigma'convergencevalueasalldatawereaveragedoverN=31records.Thus,theestimateforthemeanofadatasetxisgivenby x=x+2s p N(2)wherexisthesamplemean,Nisthenumberofsamplesandsistheunbiasedestimateofthestandarddeviation: s2=1 N)]TJ /F8 11.955 Tf 11.96 0 Td[(1NXk=1(xk)]TJ /F8 11.955 Tf 12.14 0 Td[(x)2(2)AnexaminationoftherawdatasampledbytheunloadedMLTbalanceat100Hzwasconductedtoinvestigatethenecessityofananaloglter.Thepureoutputofthebalancewasseentoexhibitsignicantdriftwhenconnectedtoaoating5Vexcitationpowersupply,asseeninFig. 2-5A .Thiswascorrectedbygroundingthenegativeterminalofthesupply,resultingintheconstantmeansignalseeninFig. 2-5B .Itshouldalsobenotedthattherangeoftheoutputssignicantlychangedbetweenthegroundedandungroundedcase,greatlyreducingtheassociatedquantizationerror.However,itisclearfromFig. 2-5B thatalow-frequencyperiodicnoisecomponentispresentinthedata.TheSCXI-1520moduleisequippedwithalowpassButterworthlterwhichcanbesetat10Hz,100Hz,1kHz,or10kHz.Applyinga1kHzltertothedataproducesthesteadyoutputseeninFig. 2-5C ;itshouldbenotedthatusingthe10kHzlterdoesnotaffecttheperiodicnoise.Thus,itisclearthatahighfrequencynoisesourcebetween 46

PAGE 47

1kHzand10kHzisaliaseddowntoalowfrequencycontaminationinthesignal;thisappearstobeentirelyremovedwiththeuseofa1kHzlter. A B CFigure2-5. RawvoltageoutputofMLTwithandwithoutltering.A)Balanceoutputwithungrounded5Vexcitation,B)Balanceoutputwithgrounded5Vexcitation,C)Balanceoutputwithgrounded5Vexcitationand1kHzlowpassButterworthlter. 2.4.2BalanceCalibrationUsingConstantLoadsPriortoconductingwindtunneltestingwiththeMLTbalance,itwasnecessarytoperformaseriesofstaticloadingmeasurementstodeterminethebehaviorofthebalanceunderidealconditions(ie,nodynamiceffectsfromwind).Thiswasachievedbyloadingthebalancewithaseriesofknownweightsandcomparingthemeasuredloadswiththeexpectedvalues.Thisinvolvestakingtwoseparatesetsofdata;aninitial`zeroreferencedataset'withnoloadsonthebalanceandasecondsetwiththeweightmountedonthecorrectaxis;subtractingthetwovaluesprovidestherelative 47

PAGE 48

forceandmomentcausedbytheweight.ThemeasuredvoltagesforeachloadarethenconvertedtophysicalloadsusinganiterativeprocedurespeciedbythemanufacturersanddescribedintheAIAAStrainGaugeStandard[ 91 ].Usingnormalforce(NF)asanexample: NF=NFNFSC)]TJ /F8 11.955 Tf 11.95 0 Td[((interactionsonNF)(2)whereNFisthenormalforceinpounds,NFistheMLToutputinV=5VandNFSCisthenormalforcesensitivityconstantthatconvertsvoltagetopounds(providedbyMMT).ThenalterminEq. 2 accountsfortheinteractionsbetweenthechannels;forexample,aloadonthenormalforcewillnaturallyalsoaffectthepitchingmomentoutput.MMTprovidesa276interactionmatrixwhichismultipliedbytheoutputofallsixchannels;thesummationofall27interactivecomponentsforeachchannelprovidesthenalterminEq. 2 .Thisequationmustbesolvediterativelyuntilthesolutionconverges.Thedesiredconvergencetolerancewassetto10)]TJ /F5 7.97 Tf 6.58 0 Td[(8;hence,allmeasurementshavethesamedegreeofprecision.Thenumberofiterationsistypicallybetween8and10.Resultsarepresentedhereforstaticloadingofa10g(0.022lb)weighthungfromthebalancewhilemountedontheMPSinthetunnelwiththewindandmotoroff;asthesensitivitycoefcientsprovidedbythecompanyareinunitsofpoundsandinch-pounds,resultsarepresentedinthesameunits.Itshouldbenotedthattheorientationofthebalancefortheseriesoftestswassuchthattheappliedforcesandmomentswerenegativeforeachaxis;thisgeometryisselectedsothatwind-ontestswillproduceapositiveliftforce.Thesamplemeanwithcondenceintervals,relativebiasandrandomerror,andthestationarityofthedatawith95%condencelevelswerecomputedusingEqs. 2 2 andthereversearrangementtestasdescribedbyBendatandPiersol[ 92 ].Foreachrecord,10ksamplesweretakenatasamplingrateof10kHz;astherearenodynamicsinthecurrentstudy,therewasnodangerofthefastsamplingrateeliminatingslowertimescalesfromthedata.TheresultsarepresentedinTable 2-4 48

PAGE 49

Table2-4. MLTbalanceanalysisfora10gloadfornormalforce(NF),axialforce(AF),pitchingmoment(PM),rollingmoment(RM),yawingmoment(YM)andsideforce(SF). ChannelSamplemeanCondenceintervalbrStationary?[lb,in-lb][lb,in-lb] NF-0.021182.24e)]TJ /F8 11.955 Tf 11.95 0 Td[(53.8%.0006%YAF0.02005.10e)]TJ /F8 11.955 Tf 11.95 0 Td[(59.0%.0004%YPM-.02683.85e)]TJ /F8 11.955 Tf 11.95 0 Td[(52.6%.0006%YRM-0.00469.92e)]TJ /F8 11.955 Tf 11.95 0 Td[(54.0%.004%YYM-0.02602.52e)]TJ /F8 11.955 Tf 11.95 0 Td[(50.79%.0004%YSF-0.02204.02e)]TJ /F8 11.955 Tf 11.95 0 Td[(50.24%.0005%Y Asignicantamountofimportantinformationcanbeobtainedfromtheresults.First,therelativebiaserroronmostchannelsisfoundtobelessthan5%ofthefullscaleload.AxialforceisslightlyhigherduetodamagesustainedbythebalanceduringthemovefromtheUniversityofColorado;asmallcrackdevelopedinontheaxialforcebeamsandcausedthezerostochangesignicantly.ThebalancewasregagedbyMMTbutanewsensitivityconstantwasnotcomputed;thus,thecalibrationisoffbyaround9%forthesmallestloadsmeasured,althoughtheerrorisreducedforhigherloads.Thishasbeendeemedacceptableforthecurrentuseofthebalance,althoughfuturecalibrationcanbeconductedtoadjustthesensitivityconstantofEq. 2 .Itshouldbenotedthatthe2condenceintervalandbiaserrorremainlowenoughthatthechannelstillproducesconsistentresults.Thecondenceintervalsonallchannelsareontheorderof10)]TJ /F5 7.97 Tf 6.59 0 Td[(5lborin)]TJ /F3 11.955 Tf 11.96 0 Td[(lb,againindicatingtheprecisionoftheresults.Aninterestingpointtonoterelatestothestationarityofthedata;asexpected,theloadedaxeswerefoundtobestationarytoa95%condencelevelusingthereversearrangementtest(thecolumninTable 2-4 referstoboththedataandzeroreferenceset).However,theunloadedaxeswerenotalwaysfoundtobestationary;anexampleofthisisillustratedinFig. 2-6 inwhichtherunningaveragesofthesamplemeanandautocorrelationareshownforthedatasetobtainedwiththenormalforceaxisloaded.InFig. 2-6A themeanofthenormalforceisseentoconvergetotheoverallsample 49

PAGE 50

meanwithin2000samples;however,thesideforcegraduallyincreasesanddoesnotreachthenetsamplemeanuntilnearly10,000samples.ThisindicatesaveryslightdegreeofdriftinthestraingaugesusedintheMLTbalance.Whenanaxisisloaded,thedominanteffectonthechannelistheforceormomentcreatedandanypotentialdriftisundetectable;however,anunloadedchannelisallowedtowanderaboutitsnaturalzero.Thisisbelievedtobeathermaleffect;asthebalanceispoweredbyaconstant5Vexcitationthereisacurrentowingthroughthe44AWGwireseveniftheyareunloaded.Overtime,thiswillcausethewirestoheatupandaltertheoverallresistanceintheWheatstonebridgecircuit.Thisbecomesnoticeableifthebalanceisinuseformanyhourswithoutbeingturnedoff,astheerroronallchannelsbeginstoincrease;however,theresultsshowthatwhilethereversearrangementtestshowsthedatatobenonstationary,within100samplesallofthecasesshowninFig. 2-6 remainwithin1%boundsofthesamplemeanandautocorrelationfunction.Thissatisesthedenitionofweaklystationarydata,andtheminorvariationsseeninFig. 2-6B havenodiscernibleeffectonthecalculatedloads.Thusitisimportanttoexaminetheactualdatafromthebalanceinadditiontosimplyrelyingonatestsuchasthereversearrangementtest. A BFigure2-6. Convergenceofsamplemeanforloaded(normalforce,NF)andunloaded(sideforce,SF)axes.Atestweightof50gwasapplieddirectlytotheNFaxis.A)Normalforce(loaded),B)Sideforce(unloaded). 50

PAGE 51

2.4.3StaticAerodynamicLoadsHavinginvestigatedtheperformanceoftheMLTbalanceusingpurestaticloads,thenextstepwasmountingaatplatewingontheMPSinthePrototunneltestsectionandmeasuringtheaerodynamicloads.Typicallythisisdonetomeasurelift,dragandpitchingmomentcoefcientsforthewing.Again,10ksampleswereobtainedatsamplingfrequenciesof1kHzand10kHztodetermineanyvariationintheresultsduetosamplingfrequency.TheresultsaredisplayedinTables 2-5 and 2-6 .Itshouldbenotedthatunlikethestaticloadingcase,anestimateofthebiaserrorcannotbeobtainedasthereisnoknownloadtocompareto.Theresultsshowminimalchangeinsamplemeanestimatesduetotheincreasedsamplingfrequencyfornormalforce,axialforceandpitchingmoment;however,rollingmoment,yawingmoment,andsideforce(for=20)changesignicantlyasthesamplingfrequencychanges.Whilethedataarestillconsideredstationarywith95%condencebythereversearrangementtest,thechangeinthemeanvalueindicatesthatsomedynamicsarepresentwhichareaffectingtheresultsforthelateraldirectionalloads.Itshouldalsobenotedthatwhencomparingthetwoanglesofattack,thecondenceintervalsforthemeanincreasebyanorderofmagnitude,indicatingthatthevarianceinthedatahasincreased.Therandomerrorremainslowduetothehighnumberofsamplesperrecord. 2.4.4ComparisonwithPublishedResultsWhiletheresultsinTables 2-5 and 2-6 showsomeunsteadybehavior,itshouldbenotedthatthemeanvaluesarebelievedtobeaccurate.Thiscanbeconrmedbycomparinglift,dragandpitchingmomentcoefcientswithclassicalresultspublishedbyMuellerandPelletierforasimilartestcase[ 68 ].Athinatplatewingwithanellipticallyroundedleadingedgeandaspectratioof3wasmountedinthetunnelandrunthroughanangleofattacksweepataReynoldsnumberof8104.Themoderatelyhighaspectratiowaschosentomitigatetheeffectsoftipvorticesasmuchaspossible,asthisisoneofthefeaturesbeinginvestigated.TheresultsshowninFig. 2-7 showgood 51

PAGE 52

Table2-5. Recordedloadsforaatplateat=10withc=4in.atRe=8104;statisticsaveragedoverN=31records ChannelSamplemeanCondenceintervalrStationary?[lb,in-lb][lb,in-lb] fs=1kHzNormalforce(NF)0.05933.24e)]TJ /F8 11.955 Tf 11.96 0 Td[(43.80e-5YAxialforce(AF)0.00754.89e)]TJ /F8 11.955 Tf 11.96 0 Td[(52.78e-5YPitchingmoment(PM)0.21801.13e)]TJ /F8 11.955 Tf 11.96 0 Td[(31.05e-4YRollingmoment(RM)0.00111.81e)]TJ /F8 11.955 Tf 11.96 0 Td[(45.96e-5NYawingmoment(YM)0.00202.98e)]TJ /F8 11.955 Tf 11.96 0 Td[(47.56e-5YSideforce(SF)0.00166.0e)]TJ /F8 11.955 Tf 11.95 0 Td[(52.31e-5Yfs=10kHzNormalforce(NF)0.05572.60e)]TJ /F8 11.955 Tf 11.96 0 Td[(45.92e-5YAxialforce(AF)0.00817.33e)]TJ /F8 11.955 Tf 11.96 0 Td[(53.79e-5YPitchingmoment(PM)0.20431.18e)]TJ /F8 11.955 Tf 11.96 0 Td[(31.69e-4YRollingmoment(RM)-2.89e-48.78e)]TJ /F8 11.955 Tf 11.96 0 Td[(58.64e-5YYawingmoment(YM)0.003585.73e)]TJ /F8 11.955 Tf 11.96 0 Td[(51.13e-4YSideforce(SF)0.00132.98e)]TJ /F8 11.955 Tf 11.96 0 Td[(53.28e-5Y Table2-6. Recordedloadsforaatplateat=20withc=4in.atRe=8104;statisticsaveragedoverN=31records ChannelSamplemeanCondenceintervalrStationary?[lb,in-lb][lb,in-lb] fs=1kHzNormalforce(NF)0.15482.72e)]TJ /F8 11.955 Tf 11.96 0 Td[(36.44e-5NAxialforce(AF)0.01303.41e)]TJ /F8 11.955 Tf 11.96 0 Td[(44.66e-5NPitchingmoment(PM)0.50887.33e)]TJ /F8 11.955 Tf 11.96 0 Td[(31.89e-4NRollingmoment(RM)0.00111.29e)]TJ /F8 11.955 Tf 11.96 0 Td[(31.09e-4YYawingmoment(YM)0.00367.63e)]TJ /F8 11.955 Tf 11.96 0 Td[(41.38e-4YSideforce(SF)0.00213.03e)]TJ /F8 11.955 Tf 11.96 0 Td[(34.11e-5Yfs=10kHzNormalforce(NF)0.16061.44e)]TJ /F8 11.955 Tf 11.96 0 Td[(36.47e-5NAxialforce(AF)0.01321.34e)]TJ /F8 11.955 Tf 11.96 0 Td[(44.51e-5YPitchingmoment(PM)0.52513.68e)]TJ /F8 11.955 Tf 11.96 0 Td[(31.81e-4NRollingmoment(RM)-0.003713.79e)]TJ /F8 11.955 Tf 11.96 0 Td[(41.01e-4YYawingmoment(YM)0.003284.96e)]TJ /F8 11.955 Tf 11.96 0 Td[(41.44e-4YSideforce(SF)0.00101.68e)]TJ /F8 11.955 Tf 11.96 0 Td[(44.36e-5Y agreementwiththeseresultsandwerefoundtobeconsistentformultipletrials.Furtherinformationaboutthetestingprotocolisdescribedin[ 75 ].Theaccuracyofthevalidationresultsindicatethatminorissuesintheexperimentalsetupup,suchasslightlyow 52

PAGE 53

asymmetriesinthetestsectionandbiaserrorintheaxialforcechannelofthebalance,donotadverselyaffectthenalmeasurements. A B CFigure2-7. Systemvalidation:comparisonwithresultsfromMuellerandPelletier[ 68 ]forarectangularatplateplanformwithAR=3atRe=8104.A)Liftcoefcientvs.angleofattack,B)Dragcoefcientvsangleofattack,C)Quarterchordpitchingmomentcoefcientvsangleofattack. 2.4.5TheForcedOscillationTechniqueThemethodsdescribedtothispointaresufcienttoobtaintheaerodynamicloadingofawinginthetestsectionwithaxedposition,andthuspermitestimationofthestaticaerodynamicderivatives(CL,,Cl,,etc.);however,acompleteanalysisofthestabilitycharacteristicsofLARwingsalsorequiresthedampingderivativeswhichrepresenttheviscousdampingcreatedbythewing.ExperimentalestimationofaerodynamicdampingderivativesofNASAaircrafthavebeenconductedthroughoutthelatterhalfofthe20thcentury;whiletheproceduresbecamemorerenedovertheyearsitstill 53

PAGE 54

remainsachallengingandinfrequentlyusedtechnique[ 93 ].TheexperimentaldifcultiesaremagniedfortheinherentlysmallerloadsandassociatedmeasurementresolutionissuesforMAVs;however,thePrototunnelfacilitiesweredesignedtoaddressthisveryproblem.ThissectionwillreviewtheforcedoscillationtechniqueusedfordampingderivativeestimationwithspecicreferencetotheMPSandMAVconsiderations;theseresultsaresummarizedinanadditionalpublicationbytheauthor[ 94 ].Itisinstructivetodiscussthetheorybehindtheaerodynamicsignicanceofdampingderivatives.Ratederivativesdependonthetranslationalandangularvelocitiesoftheaircraftandcreaterestorativeloadingswhichopposethedirectionofthemotion;theyarethereforephysicallyidenticaltoadashpotdamperinadynamicalsystemanalysis.Fromaphysicalstandpoint,dampingrepresentsadelaybetweentheaerodynamicloadingandthemotionofthewing/aircraft;therefore,inthecontextofLARwings,onemustconsiderhowthemotionofthemodelaffectstheformationofthethree-dimensional,viscousowphenomena(LEV,tipvortices,rollstall,etc)andwherethedelaysmayoccurphysically.Astheaerodynamicdampinginuencesthesysteminasimilarmannertoadashpot,itislogicaltotestforthisparameterinamannerreminiscentofharmonicoscillators;thisprovidesthebasisfortheforcedoscillationtechnique.Todescribethetechnique,consideraforced,damped,periodicoscillationinyawasanillustrativecase.Themotionisthengovernedbythesecondorderlineardifferentialequation[ 95 ]: I +C_ +k =N0cos(!t),(2)inwhichIistheyawmomentofinertia,Cisthe(constant)dampingterm,kistherestorativespringterm,N0istheamplitudeoftheforcingfunctionand!istheangularvelocityofthemotion.Itshouldalsobenotedherethatthisisasingledegreeoffreedom(DOF)equation,whichimposestheconstraintontheprescribedmotionthatallotherdegreesoffreedommustbexedortheequationisnolongervalid.Furthermore, 54

PAGE 55

thisimpliesthattherotationalmotionmustoccuraboutthecenterofgravity(cg)ofthemodel;otherwise,atranslationalvelocity(andcorrespondingforce)componentwillexist.ThislimitstheavailablemodelswhichcanbetestedontheMPStohavingachordoflessthan4in.Thesolutionofthisequationcanbeexpressedintheform (t)= 0cos(!t)]TJ /F7 11.955 Tf 12.35 0 Td[(),inwhichrepresentsthephaselagbetweentheforcingtermandthemotionofthebody.Theposition,velocityandaccelerationof canbeexpandedusingtrigonometricidentities: (t)= 0cos(!t)]TJ /F7 11.955 Tf 11.95 0 Td[()= 0(cos(!t)cos+sin(!t)sin),_ (t)=)]TJ /F7 11.955 Tf 9.3 0 Td[( 0!sin(!t)]TJ /F7 11.955 Tf 11.96 0 Td[()=)]TJ /F7 11.955 Tf 9.3 0 Td[( 0!(sin(!t)cos)]TJ /F8 11.955 Tf 11.95 0 Td[(cos(!t)sin), (t)=)]TJ /F7 11.955 Tf 9.3 0 Td[( 0!2cos(!t)]TJ /F7 11.955 Tf 11.96 0 Td[()=)]TJ /F7 11.955 Tf 9.3 0 Td[( 0!2(cos(!t)cos+sin(!t)sin).(2)SubstitutingthetermsofEq. 2 intoEq. 2 andgroupingcos(!t)andsin(!t)termsyieldsthefollowingtwoexpressions: )]TJ /F3 11.955 Tf 11.96 0 Td[(I 0!2cos+C 0!sin+k 0cos=N0, (2) )]TJ /F3 11.955 Tf 9.3 0 Td[(I 0!2sin)]TJ /F3 11.955 Tf 11.96 0 Td[(C 0!cos+k 0sin=0. (2) Finally,thetwoequationscanbesolvedalgebraicallyforthedampingtermCandaninertial/springtermk)]TJ /F3 11.955 Tf 11.96 0 Td[(I!2: k)]TJ /F3 11.955 Tf 11.96 0 Td[(I!2=N0cos 0, (2) C=N0sin 0. (2) WhileEq. 2 canbeappliedtoanydynamicalsystem,thecoefcientshavespecicsignicanceforanaircraftastheyrepresentthestabilityderivativeswhich 55

PAGE 56

arisewhenlinearizingtheaircraftequationsofmotion.Theconventionalyawmomentderivativeformulation[ 95 ]ofI,CandkaregiveninEq. 2 I=N )]TJ /F3 11.955 Tf 11.96 0 Td[(IzC=N_ )]TJ /F3 11.955 Tf 11.96 0 Td[(N_k=)]TJ /F3 11.955 Tf 9.3 0 Td[(N.(2)ThetermsinEq. 2 includebothderivativesduetotheyawangle andthesideslipangle;thisisduetothekinematicallycoupledrotation.Asaresult,varyingtheyawanglewillalsoalterthesideslipangleandthestabilityderivativesduetobothperturbationswillbepresent;asthesideslipangleisthenegativeoftheyawangle(forsmallperturbations)thesignsareoppositeforbothterms.Itisimportanttonotethatthephysicalnatureofthesederivativesaredifferent(momentsduetoyawratearetypicallyattributedtotheverticaltailswhereasmomentsduetotranslationalaccelerationsareduetothepropagationdelayofowstructuresmovingalongthewing);hence,bothmustbeconsideredindependently.Asapureyawmotionwillinherentlyalterboththeyawandsideslipangles,thisresultsincoupleddampingderivativeswhicharetypicallyconsideredtobelinkedinstabilityandcontrolanalyses.Insteadofindependentlymeasuringtheyawandsideslipderivatives,thecombinationofthetwoisusedandtypicallyyieldsgoodapproximationsforshort-periodtypemodes[ 93 ].Derivativesdueto_and aretypicallyconsiderednegligible. 2.4.6DynamicDataAcquisitionandPostprocessingItisclearthatthecrucialaspectofexperimentallydeterminingthedampingderivativesofLARwingsisaccuratelymeasuringthephaselagbetweenthemodeldisplacementandtheaerodynamicloading.Specically,thisinvolvesrecordingthetemporaldelaybetweenthepositionanddataandthencomputingthecorrespondingphaselag(=)]TJ /F7 11.955 Tf 9.3 0 Td[(!tlag).Thereareseveralissuestoaddress;specically,synchronizationofDAQandmotion,balanceresolution,signal-to-noise(SNR)ratioandtheassociated 56

PAGE 57

ltering,andthereliability/repeatabilityofthedata.Thesewillbeaddressedinthissection,andatestcasewillbeaddressedtoindicatetheeffectivenessofthealgorithm. 2.4.6.1SynchronizationofmotionanddataacquisitionThetimedelaybetweenthemotionandloadingisindicativeoftheaerodynamicdampingcausedbytheow;somepublishedresultsindicatethatexpectedtimedelaysformoreconventionalaircraft(withverticaltails)areintheorderof0.01s[ 20 ].WhilethismaynotnecessarilyapplytoLARwings,itisafairinitialestimateforthedesiredresolutionoftheDAQsystem.ThisresolutioncaneasilybeachievedwiththeuseoftheNationalInstrumentshardwareusedtocontroltheMPS[ 83 ];whenthemotorencoderreachesacertainposition,abreakpointtriggercanbesentfromthemotioncontrollertotheDAQhardwaretoinitializedataacquisition(withexpectedlinedelaysintheorderofnanoseconds).Fortheexperimentalsetupcurrentlyinplace,acosinetrajectoryisimpartedtothewing;attheinitial(andmaximum)angulardeection,thepositionofthemotorandencoderaresettozero;thus,alloftheactualmeasurementsarenegative.Itisdesirabletostartthedataacquisitionatapointinthemotionwherethemotorvelocitiesareslower(toreduceanyuncertaintyintheposition);thus,thebreakpointshouldbesentclosetothestartofthemotion.Currentlythebreakpointoccurswhentheencoderpositionreaches-1counts;thatis,atthebeginningoftherststep.Fig. 2-8 showsthemeasuredanddesiredmotorposition.Hence,itispossibletocorrelatetheexactpositionofthemotorandthebeginningofthedatasamplingsequence.Theamplitudeofthemotionissettobeaconstantvalueof1foralldegreesoffreedomtested;publishedexperimentalresultshaveindicatedthatrigid,lowaspectratiowingsdonotexperiencesignicantlydifferentowphysicsforamplitudesunder5[ 82 ]. 57

PAGE 58

Figure2-8. Encodermeasurementsforapitchoscillationatf=1Hzandamplitudeof0=1;motormotionisseentocloselytrackthedesiredtrajectoryoftheaxis.Thebreakpointlocationwhichinitiatesdataacquisitionismarkedatthebeginningoftherststep. 2.4.6.2DatapostprocessingTheissuesofsignal-to-noiseratio(SNR),digitallteringandbalanceresolutionareallinherentlylinked(andtoalesserextent,theavailablerangeofmotionoftheMPSplaysanadditionalrole).Theproblemcanbestatedas: Forcedoscillationtestingistraditionallyconductedforsmallperturbationssoastoremainwithinthelinearregime(max<5)suchthatEq. 2 remainsvalid.Attheselowdisplacements,theexpectedaerodynamicloadsaretypicallyquitesmall(althoughtheyarewithinthestaticrangethatcanbeaccuratelymeasuredwiththeMLTbalancewithlessthan5%biaserror). Forthelowexpectedmagnitudes,thevibrationalnoiseimpartedbytheMPSmotionwilltypicallybeonthesameorderofmagnitude(orgreaterthan)theactualsignal(ie,SNR0.1). Inordertoaccuratelycomputethetimelag,acleansignalmustbeused;therefore,itisnecessarytolterthesignaltoeliminateanynoise.However,digitalltershavetheirowndynamicswhichmustbeconsideredtoensurethattheydonotcorrupttheactualdata. 58

PAGE 59

Fig. 2-9 showsthefrequencycontentofthenormalforce(NF)channeloftheMLTbalanceforaharmonicoscillation(f=2Hz);itisclearthatafrequencycomponentexistsatthesamefrequencyasthemotionofthewing,indicatingthatthereisanaerodynamicloadbeinggeneratedduetotheperiodicmotion(tareloadsarealsocomputedandtheFastFourierTransform(FFT)showsnocomponentatf=2Hz,indicatingthatthisisnotaninertialload).Clearly,though,thereissignicantnoiseathigherfrequencieswhichmuchbereduced. A BFigure2-9. Frequencyresponseandrequiredlow-passltercharacteristicsforpitchmotionatf=1Hz;noattenuationispresentatthecarrierfrequencyandallcontentisremovedabovetheminimumnoisethresholdof10Hz.A)Frequencycontentofnormalforcechannel,B)Theoreticalandactuallterresponseforfc=3Hz. Alow-passltercanbeappliedtoeliminatehigherfrequencynoise;however,itisnotsufcienttosimplyapplyalterandtoassumethatthedataisextractedcorrectly.Theprimaryissuetoconsideristhatanylterimpartsitsowndynamicstothesystem,specicallyaphaselag.Severaltechniquescouldbeapplied;forexample,alinearphaselter(suchasaBessellter)couldbeimplementedwhichwouldallowtheexactdelaycreatedbyltertobecalculated.However,Bessellters(andotherlinearlters)donotexhibitstronggainpropertiesandthusdonoteliminatesufcientnoisetoprovideacleansignalforanalysis.Thetechniqueusedintheexperimentisa`zero-phase'lter.Thismethodltersthesignalintheforwarddirectionandthenreversesthedataarrayandperformsthesamelteringoperation,eliminatingthephaselagentirely. 59

PAGE 60

Furthermore,itwillbeseentobeadvantageousinthelteringprocessthatthevibrational/resonance/electronic/etcnoiseareconsistentlyatleastthreetimeshigherthanthecarrierfrequenciesoftheMLTsignal.AvailableratesoftheMPSkeepthemotionfrequenciesatorbelow3Hz(forthemostpart);nosignicantnoiseispresentbelow10Hz.Evenwiththezero-phaselteringtechnique,lowfrequencynoisecomponentscancausesignicantdistortionofthesignal;thisisfortunatelyseentonotbeanissuewiththeMLTdata.Thiswillbedemonstratedwiththeuseofatestfunction. 2.4.7TestingProcedureThestepstakentocollectdataatagivenincidenceangle(0,0)aredescribedbelow: 1. Loadindataandtaresets;40periodsperrecord. 2. ComputeFFTofdata;detectanycomponentsofdataattheinputfrequency. 3. Setfcattypically2Hzgreaterthanthecarrierfrequency;performzero-phaseltering. 4. Computezerocrossingsofltereddataandmeasuredmotorposition. 5. Eliminatetherstandlastperiodsduetoendeffectsoflter;subtractthezerocrossingsofpositionanddatatodeterminethemeasureddelays. 6. Averagethedelays;computecondencebounds. 7. Computephaselagfromtimedelayandcomputestatic/dampingderivatives. 8. Repeatfortenrecordstodeterminerepeatabilityofestimate.Thisprocedureenablescondenceboundstobecomputedonthedelayestimatesofeachrecordandalsoontherepeatabilityofthedata. 2.4.8TestCase:KnownSignalwithAddedNoiseToconrmthatthealgorithmworksproperly,atestcaseusingaknownsinusoidalfunctionsimilartotheexpecteddatafromtheexperimentwasused.Band-limitedGaussiannoiseandaknowntimedelayoft=0.01swereaddedtothedata;theresultantsignalwasthenlteredusingsingle-directionalandzero-phaseltering 60

PAGE 61

methodsforcomparison.Thedesiredresultisanaccuratemeasurementofthetimedelay. 2.4.8.1ProcedureThefollowingtwofunctionsweredenedtorepresentthemotionofthewing(w)andthe(assumedtobe)delayedaerodynamicloading(w,lag): w=Asin(2ft),w,lag=Asin(2ft)]TJ /F7 11.955 Tf 11.95 0 Td[(),(2)whereAistheamplitudeofthemotion,fisthefrequencyofthemotion,tisatimearrayandisthephaselagbetweenthemotionandthedata(denedas=)]TJ /F8 11.955 Tf 9.3 0 Td[(2ft).ThephysicalparametersusedforthetestcasearegiveninTable 2-7 ;itshouldbenotedthatthemagnitudeofwisdenedtobesmalltorepresentsomeoftheweakersignalsencounteredintheexperiment. Table2-7. Inputparameterstotestcase f[Hz]T=1/f[s]fs[Hz]NA[V]t[s]SNRfc[Hz] 2.50.4100040310)]TJ /F5 7.97 Tf 6.59 0 Td[(60.010.014 Eq. 2 clearlydepictstwosinusoidalfunctionswithaconstantphaselag;tomimictheexpectedexperimentalresults,GaussiannoisewiththeSNRgiveninTable 2-7 mustbeaddedtothedata.ThisiscreatedusingMatlab'srandn.mfunction(normallydistributedrandomnoise)withavariancedenedbySNR=2signal=2noise.ThetestSNRisdenedtobe0.01,anorderofmagnitudeweakerthanthequalityoftypicalexperimentaldata.ThecleansignalandthegeneratednoisearrayareseeninFig. 2-10 :Animportantaspectofthegeneratednoiseismatchingthefrequencycomponentspresentintheactualdata;asseeninFig. 2-9 ,minimalnoiseispresentforf<10Hz.Thus,thegeneratednoiseisrunthroughahigh-passlterwiththecutofffrequencyfcsetat10Hz.Thebandwidth-limitednoiseisplottedinFig 2-11 .Theresultsofusing 61

PAGE 62

A BFigure2-10. TestsignalandwhitenoisegeneratedwithSNR=0.01.A)Sinusoidalinputsignal,w,B)Generatedwhitenoise. band-limitedandfullbandwidth(`whitenoise')willbeinvestigated.ItcanbeseeninFig. 2-11 thatsomelowfrequencynoisestillremainsinthebandwidthlimitedsignal;thisactuallyrepresentsascenariowithworsenoisecontaminationatthecarrierfrequencyofthemotionthanseeninexperiment.Thus,satisfactoryvalidationofthealgorithmusingthisarticiallygeneratedsignalindicatesthatitwillbesuccessfullyabletoremovethevibrationalnoiseoftheexperimentaldata.Thecombinedcleansignalandband-limitednoisearethenzero-phaselteredto(ideally)recovertheoriginal,cleansignal.Thecutofffrequencyofthelow-passzero-phaselterissetat4Hz.Thezerosofthelteredsignalarethencomputed(ifthezero-crossingdoesnotoccurexactlyatameasureddatapoint,alinearinterpolationisusedtondtheapproximatezerobetweenthepositive/negativevalues).Sinceagivenperiodhastwozerocrossings,thenumberofmeasuredtimedelaysisequalto2N;theseresultsareaveragedandthe95%condenceboundsarecomputed. 2.4.8.2TestfunctionresultsFig. 2-12 showstheresultsforthegeneratednoisysignalandthesubsequentlylteredresult.Itcanbeseenthatthemagnitudeofthenoiseissignicantlygreaterthan 62

PAGE 63

Figure2-11. Frequencycontentofarticiallygeneratedband-limitednoisedesignedtoreproduceexperimentalconditionswithnovibrationalnoiseatf<10Hz;somenoiseisstillpresentatlowerfrequencyduetorolloffoflterbuttheeffectsareminimal. theamplitudeofthesignal;however,themagnitudeofthelteredresultisclosetothedesiredresult. A BFigure2-12. Plotofnoisysignalandlteredsignal;zero-phaselteringeliminatesthenoiseanddoesnotintroduceanydistortionofthesignalwiththeexceptionofsomeedgeeffectsneart=0andsmallvariationinamplitudeofsomepeaks.A)Delayedsignalw,lagcontaminatedbyband-limitednoise,B)Filteredsignalanddesiredw,lag. 63

PAGE 64

Fig. 2-12 doesnoteasilyshowthemeasuredtimedelays;thebestmethodistabulatingtheresultsfor5consecutivetrialstodeterminetheaccuracy(biaserror),condenceboundsandrepeatabilityoftheresult.TheseresultsareshowninTable 2-8 forazero-phaselterwithbandwidthlimitednoise;azero-phaselterwithwhitenoise;andaunidirectionallterwithbandwidthlimitednoise.Inallcasesa4thorderButterworthlterwithacutofffrequencyoffc=5Hzisused.Condenceboundsfortheindividualcasesareshowninthecorrespondingcellsofthetable.Itisclearfromtheresultspresentedthatthezero-phaselterprovidesthemostaccurateresults.Whenlow-frequencynoiseispresentsurroundingthecarriersignal(the`whitenoise'case),signicantdistortionispresent.Althoughtheaveragedresultofmultipletrialsisquiteclosetotheactualresultoftlag=0.01s,thestandarddeviationisquitesignicant.Furthermore,thecondenceboundsontheresultsfortheindividualtrialsarelarge(sometimesintheorderof50%ofthemeanvalue),indicatingthattheestimatesofthezero-crossingsarewidelydistributed.Useoftheunidirectionallterimpartsasignicantdelayofitsowntothemeasureddata;thisdelayisconstantfortheband-limitednoiseandvariesforthewhitenoiseasexpected.Itwouldbeentirelypossibleintheactualexperimenttocomputethelterdelayandextractthephysicaltimedelays;however,theuseofthezero-phaselterisseentobemorethansatisfactoryinachievingaccurateresultsfromthenoisydata.ItshouldbenotedthatthistestcasewasconductedforanextremelylowSNRof0.1,whichinspirescondencethatifanydatasignalispresentduetothemotionofthewing,meaningfulandaccurateresultscanbeextractedevenwithahighnoisecontent.Thisisafortunateeffectofthewingoscillationfrequencybeingfarenoughremovedfromthevibrationalnoiseassociatedwiththesystemthatthelteringdoesnotintroduceanydistortionintotheresults. 64

PAGE 65

Table2-8. Resultsformeasuredtimedelayusingzero-phaseandunidirectionallteringtechniques. Trial:FilterNoise12345tlag,avgtlag[%oftlag] Zero-phaseBand-limited0.010.010.010.010.010.0108.7e)]TJ /F8 11.955 Tf 11.95 0 Td[(67.3e)]TJ /F8 11.955 Tf 11.96 0 Td[(69.5e)]TJ /F8 11.955 Tf 11.96 0 Td[(68.5e)]TJ /F8 11.955 Tf 11.95 0 Td[(68.9e)]TJ /F8 11.955 Tf 11.96 0 Td[(6Zero-phaseWhite0.0110.00820.01690.00910.00750.010537.80.00420.0040.00440.00360.0039UnidirectionalaBand-limited-0.0785-0.0785-0.0785-0.0785-0.0785-0.078506.0e)]TJ /F8 11.955 Tf 11.95 0 Td[(56.0e)]TJ /F8 11.955 Tf 11.96 0 Td[(55.8e)]TJ /F8 11.955 Tf 11.96 0 Td[(56.4e)]TJ /F8 11.955 Tf 11.95 0 Td[(56.0e)]TJ /F8 11.955 Tf 11.96 0 Td[(5UnidirectionalWhite-0.0297-0.0349-0.0293-0.0288-0.0344-0.031429.70.00430.00460.00420.0040.0047 aRequiredlow-passltertobereducedtofc=3Hztoeliminatenoise 65

PAGE 66

CHAPTER3EXPERIMENTALRESULTSTheuncertaintiessurroundingthenatureofloadingonLARwingspreventstheformulationofanaerodynamicmodelbasedpurelyonconventional(liftingline)theory.Toaccuratelydeterminetheforceandmomentdependenciesofthesewings,anextensivewindtunneltestingprogramwasconductedtocharacterizetheloadingofLARwingsforawiderangeofowconditions.Knowledgeofhowthelongitudinalandlateralloadsareaffectedbystaticanddynamicdisplacementsfromequilibriumpermitstheformulationofanaerodynamicmodelwhichcanbeusedtonumericallysolvethenonlinearequationsofmotionortoobtainstabilityderivativesatagivenequilibriumconditionforuseinalinearapproximation.Acriticalelementofthistestingistheinvestigationofloadsatincreasingsideslipangles,whichhasnotpreviouslybeenconductedforLARwings.Toinvestigatethis,angleofattacksweepswereperformedatvaryingsideslipanglestodeterminetheeffectsonthelongitudinalandlateralstaticloads;aforcedoscillationmethodwasthenusedtoestimateaerodynamicdampingderivativesataseriesofincidenceangles(0and0).ThistypeoftestinghasnotpreviouslybeenconductedforLARwingsatlowReynoldsnumbers.Furthermore,smokewirevisualizationwasconductedtoqualitativelydeterminetheowbehavioratvariousincidenceanglesandtousethisknowledgetointerprettheforcebalancemeasurements.Thevisualizationresultsarediscussedrstandarethenusedtoexplainthemeasuredforcesandmoments. 3.1SmokewireVisualizationofFlatPlateWingsinSideslip 3.1.1RectangularWingsInordertodeterminethebehavioroftheowaroundawinginsideslip,smokewirevisualizationwasconductedforallmodelsusedintheinvestigation.APhantomv210highspeedcamerawasmountedabovethetestsectionandusedtocapturetheimagesofthethreedimensionalowaroundthewingtips.Thesmokewirewassituated 66

PAGE 67

immediatelybelowtheleadingedgeinordertofocusonthestructureofthetipvorticesinsteadoftheleadingedgeseparation;forthetaperedwingcases,itwasplacedjustbelowtheupstreamwingtipvertex.ResultsareshowninFigs. 3-1 and 3-2 .Themostimportantpointtonotefromthesmokewireresultsisthebehaviorofthetipvorticesatincreasingsideslipangles.AsseeninFig. 3-1B ,forexample,thetipvortexontheupstream(left)wingtippropagatesoverthesurfaceofthewinginacoherentvorticalshape.Thedownstream(right)tipvortexisconvectedawayfromthewingbythefreestreamow;thiseffectismorepronouncedatasideslipangleof=)]TJ /F8 11.955 Tf 9.3 0 Td[(20asseeninFig. 3-1C .Whentheangleofattackisincreasedfrom=10to30,thepropagationoverthewingissomewhatsimilarforeachsideslipcasealthoughthestructureofthevortexisnowdisruptedbytheincreasingadversepressuregradient.Spanwiseowfromlefttorightstillexistsalthoughnocoherentvorticalstructureisvisible. 3.1.2TaperedWingsThetaperedwinginsideslipshowninFig. 3-2 alsoexhibitsthetipvortexasymmetryandthedisruptedstructureathigheranglesofattack.BycomparingFigs. 3-1C and 3-2C itisclearthatthetaperedgeometryleaveslittlesurfaceareadownstreamofthewingtiptobeaffectedbytheattachedtipvortex,andwillthereforebelessaffectedbythetipvortexasymmetrythantherectangularwings. 3.2StaticLongitudinalLoading 3.2.1VaryingAspectRatioRectangularPlatesAsabaselinesetoftests,atplateswithaspectratiosof0.75,1,1.5and3weretestedatvelocitiesbetween5.2m/sand18.7m/s,correspondingtofreestreamchord-basedReynoldsnumbersof5104,8104and1105.Figures 3-3 3-5 displaytheresultsofthetests.ItshouldbenotedthatnodataispresentforAR=3inFig 3-5 astheshorterchordwouldrequireahighervelocitythanthePrototunnelcanattaintoachieveaReynoldsnumberof1105.10ksampleswerecollectedateachangleof 67

PAGE 68

A B C D E FFigure3-1. SmokewirevisualizationforaatplatewingwithAR=1at=10and30andincreasingsideslipanglesforRe=7.5104.A)=10,=0,B)=10,=)]TJ /F8 11.955 Tf 9.3 0 Td[(10,C)=10,=)]TJ /F8 11.955 Tf 9.3 0 Td[(20,D)=30,=0,E)=30,=)]TJ /F8 11.955 Tf 9.3 0 Td[(10,F)=30,=)]TJ /F8 11.955 Tf 9.3 0 Td[(20. attackatasamplingrateof10kHztomitigaterandomerror.TheresultsofthesetestsdisplaythesamebehaviorassurmisedbyTorresandMueller[ 69 ]anddepictedintheowvisualizationresultsinSection 3.1 ;adecreasedaspectratioallowsthecrossowproducedbythetipvorticestopropagatefurtheralongthewingandcreateanonlinearincreaseinliftathighanglesofattack.Anotherpointofinterestinthedataisthedragpolarinthestallregime,whichalsodisplaysnonintuitiveresultswhichwerenotdiscussedbyTorresandMueller.Althoughnotalltestcasescouldbetakencompletelytostallbehaviorduetogeometricconstraintswithinthetunnel,thosethatdidstall(asdeterminedbythelocationofCLmax)recordedadecreaseindragasangleofattackwasincreased.ThiscanbestbeseeninFig. 3-5B ,whereboththeAR=1andAR=1.5casesstallandthenexperience 68

PAGE 69

A B C D E FFigure3-2. SmokewirevisualizationforaatplatewingwithAR=1and=0.25at=10and30andincreasingsideslipanglesforRe=7.5104.A)=10,=0,B)=10,=)]TJ /F8 11.955 Tf 9.3 0 Td[(10,C)=10,=)]TJ /F8 11.955 Tf 9.3 0 Td[(20,D)=30,=0,E)=30,=)]TJ /F8 11.955 Tf 9.3 0 Td[(10,F)=30,=)]TJ /F8 11.955 Tf 9.3 0 Td[(20. adecitindragforseveraldatapoints;intheAR=1casethedragcoefcientdoesnotrecovertoitsmaximumvalueof1.61measuredatanangleofattackof43.ThisdiffersconsiderablyfromaconventionaldragpolarforhigheraspectratiowingsathigherReynoldsnumbers,whichwillexhibitacontinuedincreaseindragafterstall.Itissurmisedthatthisbehavior,similartotheliftpolar,canbeattributedtotheeffectsoftipvortices.Asthevorticesgrowinsizeandstrengthathigheranglesofattackandconsumeagreatersectionofthewing,thestallangleisincreasedtoarelativelyhighangleofattackaround45.AsseeninSection 3.1 ,atthispointthecrossowduetotipvorticesdissipatescausingtheattachedowoverthewingtoseparateandinducestall.Thereducedimpactofthetipvorticeslowerstheinducedangleofattackand,thus,theinduceddragsothatthemajorityofthedragonthewingisproledrag.Itis 69

PAGE 70

A B CFigure3-3. AerodynamicloadsonrectangularatplateplanformswithvaryingARatRe=5104.A)Liftcoefcientvs.angleofattack,B)Dragcoefcientvsangleofattack,C)Quarterchordpitchingmomentcoefcientvsangleofattack. thisreductionininduceddragthatcausesthedragpolartodecreaseafterthewingstalls.Astheangleofattackcontinuestoincrease,theresultingincreaseindragisduepredominantlytoproledrag.Analcontributiontothereductionindragisthelossofthelowpressurecoreassociatedwiththetipvortex,whichwasreportedonbyTairaandColonius[ 76 ].ThistheoryofdependenceofdragonthepresenceoftipvorticesalsohelpstoexplainwhytheAR=1caseexperiencesalargerdecreaseinCDafterstallthanAR=1.5.Thesmalleraspectratioreliesonnonlinearliftfromthetipvorticestoreachits 70

PAGE 71

A B CFigure3-4. AerodynamicloadsonrectangularatplateplanformswithvaryingARatRe=8104.A)Liftcoefcientvs.angleofattack,B)Dragcoefcientvsangleofattack,C)Quarterchordpitchingmomentcoefcientvsangleofattack. highstallangle,andwhentheowdoesdetacharound43thereisagreaterreductionininduceddragthanwhenthelargeraspectratiostallsat25.Itshouldalsobenotedthatalthoughatthishighangleofattacktheincidencegeometryofthewingbeginstoresemblethatofabluffbody,nounsteadyeffectsontheloadingcausedbyasymmetricvortexsheddingweremeasured.TheStrouhalnumberofvortexsheddingbasedonthefreestreamvelocityandprojectedfrontalheightofthemodelisexpectedtobe0.2fortherangeofReynoldsnumbersinthisinvestigation[ 65 96 ];thiscorrespondstoanexpectedsheddingfrequencyintheorderof30Hz.Asthemeasuredexperimentaldata 71

PAGE 72

isthemeanvalueofaonesecondsampletime,theseunsteadyeffectswereaveragedouttoobtainthemeanloads. A B CFigure3-5. AerodynamicloadsonrectangularatplateplanformswithvaryingARatRe=1105.A)Liftcoefcientvs.angleofattack,B)Dragcoefcientvsangleofattack,C)Quarterchordpitchingmomentcoefcientvsangleofattack. ItisclearfromFigs. 3-3 3-5 thataspectratioisthemostimportantfactorinmeasuringtheeffectsoftipvortices.Althoughfreestreamvelocity(andthusReynoldsnumber)doesplayaroleintheloading,onlysmallvariationsarepresentforagivenaspectratiobetweenReynoldsnumbersof5104and1105.Asthegeometriceffectsofthewingaresoprevalentintheresults,thenextlogicalstepisexaminingataperedmodeltoexaminehowfurtherchangestothewingshapeaffecttheaerodynamics.As 72

PAGE 73

theresultsfromdifferentReynoldsnumberswerenotsignicantlydifferent,onlydataatRe=1105willbepresentedfortheremainingdata. 3.2.2VaryingTaperRatiosforaFlatPlatewithAR=1Asetoffourplateswithtaperratiosof=0.25,0.5,0.75and1werefabricatedfortesting,withthetaperratiodenedas=ctip=croot.Forthistest,theaspectratiowasheldconstantat1asthepreviousdataindicatedthatthiswouldallowthebeststallcharacterization.Allmodelswerefabricatedwitharootchordof6in.sothatthefreestreamtunnelvelocity(and,thus,theturbulenceintensity)wouldbeconstantforeachtest.Modelsweretestedatafreestreamvelocityof38.1ft/s(11.6m/s),correspondingtoaReynoldsnumberof1x105.Themeasuredloadsnormalizedbythedynamicpressure,planformareaandrootchordofthewingareshowninFig. 3-6 .AsthemodelstestedwereofAR=1,eachwassuccessfullytakentostall.Fig. 3-6A showsthatthestallangleforthedifferenttaperratiomodelswasmostlyunchangedandremainedaround42foreachwing.Thebiggestdifferencebetweenthemodelsisthegreaterliftcoefcientoftherectangularplatesomewhatunsurprisinggiventhemuchlargerliftingsurfaceofthe=1caseandtheincreasedimpactofthereversedowoverthetaperedtrailingedgeseenintheowvisualization.Themaximumliftcoefcientsofthecases=0.75and=0.5arecorrespondinglylower;however,the=0.25modelhasahigherCLmaxthantheothertaperedmodelsdespiteitssmallerliftingsurface.Atanglesofattackgreaterthan20thetipvortexbecomesincreasinglyprevalentandbeginstointeractwiththeleadingedgevortex,althoughthecrossoweffectsarelessthanthosefeltbytherectangularplate.Still,theinteractionallowstheowtoremainmostlyattachedathigheranglesofattack.Thismuchwasconrmedusingowvisualization.FromFig. 3-6A ,itisclearthattheinteractiveeffectsoftheLEVandthetipvorticesatlowtaperratiosandhighsideslipanglesresultinabenecialincreaseinCLmax. 73

PAGE 74

A B CFigure3-6. Aerodynamicloadsonatplateplanformswithvaryingtaper;AR=1,Re=1105.A)Liftcoefcientvs.angleofattack,B)Dragcoefcientvsangleofattack,C)Quarterchordpitchingmomentcoefcientvsangleofattack. 3.2.3EffectsofSidesliponTaperedWingsItisclearfromthetestresultsoftheprevioussectionsthatLARierscanbegreatlyaffectedbycrossowoverthesurfaceofthewing,particularlyathighanglesofattack.Tothispoint,onlythezerosideslipcasehasbeenconsidered;whilethisisausefulstudy,theultimategoalofthisresearchistodevelopabetterunderstandingoflowReynoldsnumberaerodynamicstofacilitateMAVdesign.Theseaircrafthavebeenobservedtobehighlysusceptibletolateralperturbationssuchasgusts;thus,itisalsoimportanttoinvestigatehowalateralcomponentoffreestreamvelocityaffectstheloadingonthemodels.TheMPSwasdesignedtocombinedegreesoffreedomsuch 74

PAGE 75

assideslipandpitch,allowingtheseeffectstobeaccuratelytested.Forthesetests,themodelwasrotatedtosideslipanglesof=10,20and35andwasthenpitchedthroughthesamerangeofanglesofattacktestedat=0.Thetunnelvelocitywaskeptat11.6m/stosimulateaconditioninwhichthewingislaterallyperturbedfromitsequilibriumightvelocityandnowcontinuesmovingforwardatanewlyinducedsideslipangle.Inessence,=0isconsideredtobea`cruise'ightcondition.ResultsforthedifferenttaperratiosareshowninFigs. 3-7 3-9 A B C DFigure3-7. LiftcoefcientofrectangularandtaperedatplatesinsideslipatAR=1andRe=1105.A)=1,B)=0.75,C)=0.5,D)=0.25. Theresultsofthetestsatvarioussideslipanglesshowsthat,whiletheaerodynamicperformancedegradessomewhatasthewingyaws,thisbehaviorismostlyconnedtohighangleofattackregions(>20).ItcanbeseeninFig. 3-7 thatincreasingthe 75

PAGE 76

A B C DFigure3-8. DragcoefcientofrectangularandtaperedatplatesinsideslipatAR=1andRe=1105.A)=1,B)=0.75,C)=0.5,D)=0.25. sideslipanglecausesadecreaseinliftcurveslopeathigheranglesofattack,whichisattributedtoincreasinginteractionsbetweentheasymmetrictipvorticesandtheLEV.Itisalsonoticeablethattheincreasedspanwiseowfromthetipvortexpreventssharpstallsorincreasesinnonlinearliftathighervaluesof.ThemostsignicanteffectofincreasingsideslipangleisvariationinpitchingmomentasseeninFig. 3-9 .When=35thezeroslopeofCMaround=0skewstowardsthenegative,indicatingthatnotrimconditionexistswheretheaerodynamicsumoflongitudinalmomentsiszero.Atanglesofattackabove20,theslopeismoregentlethanatzerosideslipwhichrepresentsadecreaseinthederivativeCM,,orareducedabilitytorecoverfrompitchperturbations.Still,thevariationsinthegradientsarereasonablysmallandtheresultsof 76

PAGE 77

A B C DFigure3-9. QuarterchordpitchingmomentcoefcientofrectangularandtaperedatplatesinsideslipatAR=1andRe=1105.A)=1,B)=0.75,C)=0.5,D)=0.25. Fig. 3-9 indicatethatthepitchmoment,liketheliftanddrag,donotvarysignicantlyatnormalequilibriumconditionsfor<35. 3.3StaticLateralLoadingLateralloadswereinvestigatedforatplatewingswithAR=0.75,1,1.5and3.Themostimportantloadsmeasuredinthisstudyweretherollingandyawingmomentsofthemodelstested(sideforceisinherentlyrelatedtotheyawmoment,andistypicallyofnegligiblemagnitude,soisnotdiscussedingreatdetailhere).Bothvaluesarecomputedaboutthecenterofmassofthemodel.Thetestingwasconductedbetweenanglesofattackof)]TJ /F8 11.955 Tf 9.3 0 Td[(24and42atsideslipanglesof=0,10,20and35.Itshould 77

PAGE 78

benotedthattheAR=3casewasnottestedat=35becausethelongerspanofthismodelinfringesupontheboundarylayerofthetestsectionatthishighsideslipangle.TheresultswillbepresentedindividuallyinFigs. 3-10 3-14 3.3.1RollMomentforRectangularWingsTherollmomentsmeasuredforrectangularatplatewingsatincreasingsideslipanglesaredisplayedinFig. 3-10 .Oneoftherstpointstonoteishowtheshapeoftheplotsforaspectratiosupto1.5looksqualitativelysimilartoaliftpolar,manifestedbyalinearslopeatlowanglesofattackleadingtoadistinctstallregion.Thisbehavior(hereafterreferredtoas`rollstall')isnotawellknownphenomena;theauthorsarenotawareofanysimilarexperimentalresultsforLARwingsatlowReynoldsnumbers.Thephysicalexplanationforthisresultisattributedtothepresenceofthetipvortex,particularlyontheupstreamwingtipwhenthewingyaws.ItisclearfromFig. 3-1 thatthecoreoftheupstreamtipvortexislocatedabovethesurfaceofthesideslippingwingwhilethedownstreamtipvortexhaslittleeffectontheowabovethetestmodel;thisconditionisuniquetoLARwings.TheinducedvelocitiescreatedbytheupstreamvortexatachordwisestationonthewingcanbecomputedusingtheBiot-SavartlawandaqualitativeexampleisshowninFig. 3-11 .Thisschematicshowshowanupwashisactuallycreatednearthewingtipwhiletherestofthewingexperiencesadownwash,resultinginanincreasedeffectiveangleofattacknearthetipanddecreasedincidenceangleselsewhere.Theproportionalspanwisevariationinsectionalliftcreatesapositiverollingmoment.Inadditiontotheeffectsofthetipvortexasymmetryonthespanwiseloadingofthewing,atincreasingsideslipanglesandanglesofattacktheseparatedshearlayerattheupstreamedgewillexperiencegreatervelocitygradientsandacorrespondinglygreaterstrengthoftherolled-upvortex[ 97 ].Thelowpressurecoreactstolocallyaugmenttheliftontheupstream(left)wingtip;asthereisnocorrespondingdownstreamtipvortexabovethesurfaceofthewingtobalancethisforce,anadditionalcomponentofpositive 78

PAGE 79

rollmomentisinduced.Athigheranglesofattack,thestrongeradversepressuregradientdisruptsthecoherentnatureofthetipvortex,asseeninFig. 3-1F .Thiscanbeconsideredsimilartovortexbreakdownindeltawings[ 47 50 ].Thereducedstrengthofthetipvortexdecreasesitsimpactontheinducedangleofattackandthelocalpressureaugmentation,correspondingtotherollstallseeninFig. 3-10 A B C DFigure3-10. RollmomentcoefcientforrectangularatplatewingsinsideslipatRe=7.5104.A)AR=0.75,B)AR=1,C)AR=1.5,D)AR=3. WhiletheresultsshowninFigs. 3-10A 3-10C arequalitativelysimilar,thehigheraspectratiocaseinFig. 3-10D displaysamorenonlinearrollmomentslope.ThisisunliketheliftperformanceofLARwings,inwhichhigheraspectratiosexhibitamorelinearslopeandaspectratiosnearunityarenoticeablynonlinear[ 68 69 75 ].Atlowanglesofattack,thehigheraspectratiowingexperiencesamoreconventional 79

PAGE 80

Figure3-11. Loadingasymmetriescreatedbythepresenceoftheupstreamtipvortexabovealowaspectratiowing.Theinducedvelocitesduetothevortexcreateanupwashandanincreaseinliftnearthewingtip;furthermore,thelowpressurecoreofthevortexalsocreatesitsownliftvectorandassociatedrollmoment. aerodynamicloading(ie,dominatedbytheboundcirculationofthewing)whichideallyshouldnotcreateasignicantrollmomentatanysideslipangleforathinwing[ 13 14 ].Atlowtheslopeoftherollmomentpolarisnonzerobutsmallforanysideslipcaseexcept=0andisthenobservedtoatleasttripleinmagnitudeabove=10.Atthesehigheranglesofattack,theowseparationismorepronouncedandtheinuenceofthetipvortexismoreprevalent;asaresult,therollmomentbehavesmoresimilarlytotheloweraspectratiocases.Itshouldbenotedthatwingswithaspectratiosbetween2and3havetraditionallybeenconsideredanintermediateaspectratio[ 98 ];tipeffectshavemoreimpactthantheydoforaconventionalaircraftwing(AR3)butthereissignicantlylessinteractionbetweenthetipvorticesandtheLEVthanthereisforaLARwingthatismoretypicalofMAVplanforms(AR1).TheAR=3casethusdisplayssomesimilaritiestoaconventionalwing(asseenwiththesmallmagnitudeoftherollmomentsfor<10)aswellasaloweraspectratiowing(manifestedbythesignicantincreaseinClfor>10). 3.3.2RollMomentforTaperedWingsToinvestigatetheeffectsofleadingedgegeometry,aseriesoftaperedwingsweretestedatthesameanglesofattackandsideslipastherectangularwings.Astheaspect 80

PAGE 81

ratioofalltaperedwings(basedonthemidchordspan)isequaltounity,theAR=1resultsfromFig. 3-10B areincludedtoprovideabaselinereference.TheresultsareshowninFig. 3-12 .Inseveralways,theresultsaresimilartothoseoftherectangularplanforms.Itisclearthatincreasingthesideslipangleresultsinagreatervalueoftherollingmoment;whilethiscanbemainlyattributedtothepresenceofthetipvortex,itisalsoimportanttonotethatforlowervaluesofthetaperratioanincreasedamountofspanwiseowisfedintothedevelopingtipvortexbytheLEV[ 76 ].Thisvorticity-sinkmechanismhelpstostrengthenthetipvortexandimprovesitsabilitytowithstandanadversepressuregradient.ThistrendisnoticeableinFig. 3-12 ;for=1and0.75,rollstalloccursbetween20and30inbothcasesbutthepost-stallgradientismuchmoregentleforthetaperedwing,indicatingthatthedisruptionoftheupstreamtipvortexislessdrasticthanforrectangularwings.Decreasingthetaperratioto0.5preventsrollstallfromoccurring,althoughtheslopedoesdecreasearound=20,asseeninFig. 3-12C .Thissuggeststhatthe=0.5caseismoreeffectiveatfeedingvorticityfromtheLEVintothetipvortexathigheranglesofattackthanthehighertaperratiocases,resultinginamorecoherenttipvortexthatcanwithstandbreakdownbetterthanonarectangularwing.Whenthetaperratioisdecreasedto0.25(ie,smallestwingtips)themagnitudeoftherollmomentissignicantlydecreasedafterremainingrelativelyconstantoverthepreviousplots.Thiscanbeattributedtothesmallwingtiponlyallowingalimitedchordwisedevelopmentofthetipvortex.Thegeometryofthe=0.25winghastheleastsurfaceareadownstreamofthewingtipandthustheaforementionedeffectsofthetipvortex'sinducedvelocityareminimized.Inaddition,thewingdoesappeartoexperiencerollstallaround=20,unlikethe=0.5case;however,thereisasubstantialrecoveryzonewheretherollmomentincreasestoavaluegreaterthanthestallvalue.ThismayrepresenttheangleofattackwheretheformationoftheLEVweakensanddoesnotcontributemuchtothestrengthofthetipvortex;thenominallyconstantvalueofroll 81

PAGE 82

momentseenathigheranglesispurelyduetotherollupoftheshearlayeroverthewingtip. A B C DFigure3-12. RollmomentcoefcientfortaperedatplatewingswithAR=1insideslipatRe=7.5104.A)=1,B)=0.75,C)=0.5,D)=0.25. 3.3.3RollMomentwithZeroSideslipInthecourseoftheexperimentitwasobservedthattherollmomentofawingwithzerosideslipwasnotnecessarilyzero,aswouldbeexpectedforawingwithidenticaltipvortices;thisisslightlynoticeableinFig. 3-10 andismoreprominentforthetaperedwingcasesinFig. 3-12 .Inaddition,thewingswiththeaspectratioofunitywerehighlysusceptibletodeviationsinrollmomentduetoslightperturbationsinsideslipaboutthebaselineconditionofzero.Toillustratethis,thesquarewingwithAR=1wassweptthroughtheusualrangeofanglesofattackatsideslipanglesincrementallyaboveand 82

PAGE 83

belowzero;theresultsfortherollmomentareshowninFig. 3-13 .Atanapproximatelyzeroangleofsideslip,neithertipvortexshouldbecompletelysweptawayfromthewingtocreatethelateralasymmetryexperiencedbythesamewinginhighsideslip,andthusnorollmomentshouldbeproduced.FromthedatainFig. 3-13 itisclearthatarollmomentiscreatedwithnegligiblesideslipperturbations,andthatthegradientofthiscurveisdependentonthesignofthesideslipangle.ItisalsonoticeablethattheslopesofClvsarenotperfectlysymmetricfor!0)]TJ /F1 11.955 Tf 10.41 2.96 Td[(and0+;thiswouldimplythatslightlydifferentstabilityderivativeswouldneedtobecomputedforpositiveornegativesideslipperturbationsabout=0.However,whenerrorbars(correspondingtotheminimumresolvablerollmomentwhichyieldslessthan5%biaserror)areadded,itcanbeseeninFig. 3-13 thatthesmallvariationsinthegradientofClnear=0arewellwithintheboundsoftheMLTbalance'sresolution.Hence,theseasymmetriescanbeattributedtoexperimentaluncertaintyandtheassumptionofsymmetryabout=0remainsvalid.Thereisasmalloffsetinrollmomentatzerosideslip(Cl)]TJ /F8 11.955 Tf 23.81 0 Td[(0.025)butasthestabilityderivativesarecomputedfromtheslopeoftheClvscurve,thiswillnotaffectthecomputationoftherollstabilityderivative. Figure3-13. Rollmomentcoefcient(Cl)witherrorbarsforaAR=1atplatewingatnear-zerosideslipanglesatRe=7.5104. 83

PAGE 84

Whilethenonzerorollmomentat=0canbepartiallyattributedtoexperimentaluncertaintiesinsideslipangle,imperfectionsinleadingedgegeometryandadevelopingasymmetryofthetipvorticesalsoleadtoanunbalancedoweldonthesuctionsurfaceofthewing.Theinuenceofthetipvorticescancauseasignicantvariationinlaminarseparationonairfoils,movingthepointofdetachmentasfarbackasx=c=60%forAR=2atanglesofattackaslowas=3[ 99 ];attheloweraspectratiosinvestigatedhere,thisinuenceisevenmorepronounced.Inaddition,Gresham,etal,observedthatLARwingsexperiencenonzerotrimbankanglesatzerosideslipduetotheunevendevelopmentoftipvortices,occasionallyrequiringanglesupto=15toachievezerorollmomentatlowanglesofattack.Thiswasattributedtotheroundedleadingedgeoftheatplatemodelstested,asthereisnotauniformseparationpointontheleadingedge;abeveledleadingedgewasusedtobringthetrimangleclosertozero[ 81 ].Thedisparitybetweenthedevelopmentoftheleadingedgevorticesontheleftandrightsidesofthewing,whichinturninuencesthestrengthoftherespectivetipvortices,greatlyimpactstherollmomentoftaperedwingsatzerosideslipasseeninFig. 3-12 .Itshouldbenotedthatthedifferenttaperratiosexhibitbothpositive(=0.5)andnegative(=0.75,0.25)slopes,indicatingthattheresultsarecausedbythespecicmodelconditionsandnotanypotentialowinhomogeneityinthetestsection.Thiswasconrmedbyrunningasecondseriesoftestswiththemodelmountedupsidedown,inwhichtheslopesweresubsequentlyreversed.ThisresultindicatesthecomplexityofthecompetingeffectsoftheoweldassociatedwithLARwings,particularlyatanaspectratioofunityinwhichthereissignicantinteractionbetweentheLEVandthetipvortices. 3.3.4SideForceandYawMomentTheresultsoftheprevioussectionsshowthesignicanceoftherollmomentatincreasinganglesofattackandsideslip.Theremaininglateralloadstobeconsideredarethesideforceandyawmoment,whichweretypicallyfoundtobeanorderof 84

PAGE 85

magnitudesmallerthantheotheraerodynamicloadsonthewings.AselectionofresultsforwingsofAR=3andAR=1with=1and0.25aredisplayedinFig. 3-14 forillustrativepurposes.Sideforceismeasuredinthebody-centeredcoordinatesystem,andthusisdifferentfromthecross-windforce.Whilethemagnitudesoftheforceandmomentscoefcientsaresmallatallanglestested,itisalsoclearthatthetrendsexhibitedbytheloadsaresimilarforbothmodels.Itcanbeseenthatincreasingthesideslipangleresultsingreatersideforceandyawmomentcoefcients,althoughfortheAR=1casetheformertendstowardszeroatanglesofattackabove=15.Thesideforceisabyproductoftheinduceddragcreatedbytheattachedupstreamtipvortex,whichhasacomponentactinginthe+ybdirectionofthewing.ThereductioninmagnitudeathigheranglesofattackcorrespondstotherollstallregionseeninFig. 3-10 ;asthetipvortexisdisruptedbytheadversepressuregradient,thereducedstrengthofthevortexdiminishesitscontributiontotheinduceddragandthusthesideforce.Theonlycaseinwhichthisreductiondoesnotoccurisfor=0,wherethesideforcecreationismoreaffectedbytheasymmetricseparationalongtheleadingedge.Asimilartrendisobservedtoalesserextentforthe=0.25caseinFig. 3-14E wheretheimpactofthetipvortexanditsassociatedinduceddragarelimitedbythereducedwinggeometrydownstreamofthewingtip,asseeninFig. 3-2F .Thedatafortheyawmomentshowstheinuenceofbothaspectratioandplanformgeometry.TheAR=3caseinFig. 3-14B showsscattereddatawithcoefcientsontheorderofCn0.001andnomeaningfuldifferencesbetweenthedifferentyawangles.ThemagnitudeofthedatafortheAR=1caseissomewhatlarger(Cn0.01)andisseentoincreaseslightlywithsideslipangle;thistrendcontinuesforthetaperedwingalthoughtheyawmomentisseentoremainnearlyconstantoverallanglesofattackforagivensideslipangle.Again,thedifferencebetweentherectangularandtaperedcasecanbeattributedtotheeffectsoftheupstreamtipvortex,whichisreducedforthecaseof=0.25.Itisalsointerestingtonotethatwhilethesideforceispredominantly 85

PAGE 86

positivetheyawmomentisnearlyalwaysnegative.Bothareattributedtothecomponentoftheinduceddragcreatedbytheupstreamtipvortex;asthevortexextendsfurtherinboardalongthewingnearthetrailingedge(asseeninFig. 3-1 ),agreaternetsideforceexistsonthedownstreamportionofthewingandcreatesanegativeyawmoment.Whilethemagnitudeofthesideforceandyawmomentindicatesthattheassociatedstabilityderivativesarelikelynegligible,itisstillinstructivetoconsidertheowphysicscausingtheresults. 3.4StabilityDerivativeEstimatesThedatainFigs. 3-10 3-14 canbeusedtoestimatethestabilityderivativesCl,,Cn,andCSF,bycollectingthedataatagivenangleofattackoverarangeofsideslipangles.AMAVtypicallyiesatanangleofattackof10[ 10 ];thederivativeswerecomputedat=5,10,15and20asarealisticrangeofightangles(resultsfor0werenominallyzero).TheresultsforCl,andCn,arepresentedinTables 3-1 3-2 afterconvertingtheanglestoradians;CSF,hadmagnitudessimilartoCn,butisnotincludedhereforbrevity.Yawmomentismeasuredreportedaboutthequarter-chord. Table3-1. ValuesofCl,forvariousplanformgeometriesandtrimangles Planform=5=10=15=20 AR=0.75-0.062-0.106-0.161-0.233=1AR=1-0.101-0.167-0.238-0.313=1AR=1.5-0.053-0.086-0.123-0.177=1AR=3-0.019-0.048-0.105-0.179=1AR=1-0.047-0.086-0.150-0.173=0.75AR=1-0.057-0.088-0.105-0.128=0.5AR=1-0.036-0.055-0.067-0.051=0.25 86

PAGE 87

A B C D E F Figure 3-14.Sideforce(C SF )andyawmoment(C n )coefcientsforatplatewingsin sideslipatRe=7.5 10 4 .A) C SF for AR =3, =1 ,B) C n for AR =3, =1, C) C SF for AR =1, =1 ,D) C n for AR =1, =1,E) C SF for AR =1, =0.25 ,F) C n for AR =1, =0.25. 87

PAGE 88

Table3-2. ValuesofCn,forvariousplanformgeometriesandtrimangles Planform=5=10=15=20 AR=0.750.1250.1130.0990.093=1AR=10.0380.0310.0220.028=1AR=1.50.0370.0320.0240.021=1AR=30.0140.00380.00180.0027=1AR=10.0430.0350.0270.033=0.75AR=10.0740.0680.0680.065=0.5AR=10.2140.2100.2070.211=0.25 ItcanbeseenfromtheresultsprovidedthatthemagnitudeofCl,istypicallyhigherthanCn,,althoughitshouldbenotedthatthereareseveralcasesinwhichthemagnitudesoftherespectivederivativesaremuchcloser.Specically,forthetaperratio=0.25inwhichtheimpactoftheupstreamtipvortexislimitedbythegeometryofthetaperedwing,theyawstiffnessisactuallygreaterthantherollstabilityderivative.Thus,whilerollstabilityappearstobetheparametermostinuencedbysideslipthisisnotauniversalresultforallconceivableMAVplanforms.ThisindicatesthechallengesinherenttoMAVdesignastheuniqueowphenomenaassociatedwithLARwingsatlowReynoldsnumberscannotbeeasilypredictedevenwithgeometricallysimilarplanforms.Itshouldstillbenotedthatthemostsignicantimpactofrollstallfromavehiclecontrolperspectiveisinrollasthisaxiswillalmostuniversallyhavethesmallestmomentsofinertia[ 3 ],andthuswillbesubjectedtothehighestangularaccelerationsforagivenmagnitudeoftheappliedmoment. 3.5MeasuredDampingDerivativesTheresultspresentedinSection 2.4.6 indicatethatitispossibletoextractmeaningfulandreliableestimatesoftheaerodynamicdampingderivatives.Alarge 88

PAGE 89

testmatrixwasdevisedanddampingderivativeswerecomputedforrotationalvelocities(@=@p,@=@q,@=@r);derivativesduetotranslationalaccelerations,@=@_and@=@_,werefoundtobenegligible.ResultsforoscillationsinagivendegreeoffreedomareplottedinFigs. 3-15 3-17 fortheAR=1wingtestedataReynoldsnumberof7.5104;eachgurecontainstheresultsforanyloadwhichdemonstratednon-negligibledampingtosimplifythepresentationofthedata.Resultsarepresentedinbodyaxes(normalforceandaxialforce)insteadofstabilityaxes(liftanddrag)toavoidconfusionbetweentheliftforceandrollmoment,whicharebothoftenrepresentedbythevariableL.Analternaterepresentation,showingeachindividualderivativeonaseparateplotforeachoscillationfrequency,isprovidedforthereader'sreferenceinAppendix C Figure3-15. Dampingderivativesduetorollrate,@=@p,foranAR=1wingatRe=7.5104oscillatedatfrequenciesoff=0.5,1,1.5,2,2.5,3Hzandincidenceanglesof=0(),10(),20(4),30(5),40(),50(D)and=0( ),10( ),20( ),35( ). TheresultsdisplayedinFigs. 3-15 3-17 representtherstexperimentalmeasurementsoftheaerodynamicdampingderivativesofLARwingsatlowReynoldsnumbers.The 89

PAGE 90

Figure3-16. Dampingderivativesduetopitchrate,@=@q,foranAR=1wingatRe=7.5104oscillatedatfrequenciesoff=0.5,1,1.5,2,2.5,3Hzandincidenceanglesof=0(),10(),20(4),30(5),40(),50(D)and=0( ),10( ),20( ),35( ). largestmagnitudesareseentobethedampingofnormalforceinpitch(CZ,q)whicharenominallyanorderofmagnitudelargerthananyotherderivativemeasured;however,thetrendspresentforallderivativesareconstant.Thelargestdampingisnoticeableatthelowestreducedfrequenciestestedandthemagnitudeofthederivativesconvergetowardsasmallbutnitevalueatthehigherfrequencies.Allaerodynamicderivativesmeasuredwerenegative,implyingpositive(stable)damping.Somecross-coupledderivativeswereobserved,specicallydampinginrollmomentwithpitch(Cl,q)anddampinginnormalforce/pitchmomentwithrollrate(CZ,pandCm,p)atincreasedsideslipangles,althoughthemagnitudesweretypicallysmall.Additionally,therolldampingderivativeCl,pwasconspicuouslyabsentwhennosideslipanglewaspresent;thissurprisingresultisattributedtothesmallwingspansoftheLARwing 90

PAGE 91

Figure3-17. Dampingderivativesduetoyawrate,@=@r,foranAR=1wingatRe=7.5104oscillatedatfrequenciesoff=0.5,1,1.5,2,2.5,3Hzandincidenceanglesof=0(),10(),20(4),30(5),40(),50(D)and=0( ),10( ),20( ),35( ). experiencingonlyminorvariationsininducedvelocityduetotherollrate.Thelackofrolldampingisaplausibleexplanationforthe`jittery'natureofMAVight.Flighttestingexperiencehasshownthat,evenatcruiseconditions,MAVscanexperienceperiodic,small-amplituderolloscillationswhichcanadverselyaffectmissionprolessuchasimaging[ 100 101 ];thisisreferredtoas`jitter'herein.Analsignicantresultisthat,forphysicalfrequenciesbetween1
PAGE 92

3.6EffectsofWingletsonTaperedWingsAlthoughthefocusofthisthesisisprimarilyrelatedtotheaerodynamicandstabilitypropertiesofpureLARwings,additionaltestswereconductedwithwingletsaddedtothetaperedplanformsof=0.25,0.5and0.75todeterminetheimpactofdisruptingthetipvortexformationonthepreviouslypresentedstaticloads;theaddedinertiaofthewingletspreventsthemeasurementofdampingderivativesduetomotorconstraints.Wingletswiththesamethicknessandleadingedgegeometryasthewingwerefabricatedtohavethesamelengthasthetipchordofthetaperedwing.Thewingletscouldbemountedabove,below,orcenteredontheplaneofthewingtodeterminewhichcongurationhadthemostsignicanteffectontheloading.ThedimensionsofthetaperedmodelsandwingletsareincludedinTable 3-3 Table3-3. Flatplatemodeldimensions. ARcroot,cm.ctip,cm.b,cm.t,cm.Wingletdim.,cm.Diagram 115.211.415.20.40.7511.43.80.4 115.27.615.20.40.57.63.80.4 115.23.815.20.40.253.83.80.4 3.6.1LongitudinalLoadsSomecomputationalworkhasbeendonebyViieru,etal[ 74 102 ]toinvestigatetheeffectsofwingletadditionstoaMAVwingatloweranglesofattack(6and15),withresultsindicatinganincreaseofliftanddragaswellasthelift-to-dragratio.Itisdesirabletotestthesecongurationsathigheranglesofattacktocapturethemoredominanteffectsofthetipvortices.Inaddition,testswererunforthesametaperratiosandsideslipanglesusedtoproduceFigs. 3-7 3-9 .Wingletsweretestedinthreecongurations(mountedabovethewing,centeredonthewing,andmountedbelowthewing)todeterminewhichsetuphadthemosteffectonthecrossow.Again,testswererunatRe=1105. 92

PAGE 93

A B C DFigure3-18. Liftcoefcientvsangleofattackforataperedplate(=0.75)atvarioussideslipanglesandwingletcongurationswithAR=1andRe=1105.A)=0,B)=10,C)=20,D)=35. Selectedresultsfromtaperratiosof=0.75and=0.25areshowninFigs. 3-18 3-23 .Theseplotshelptoillustratethesignicanceofthetipvorticesonaerodynamicperformance.ThegeneralresultsfromViieru,etal,areconrmed;boththeliftcoefcientandthedragcoefcientincreaseatloweranglesofattackandatallsideslipangles[ 74 102 ].Thisistobeexpectedasthewingletshelptominimizethereductionininducedangleofattackandthusgeneratemorelift.Theincreaseindragasthemodelyawsisobviousasthewingletspresentagreaterimpedancetotheowandcausetheproledragtoincreasesignicantly.Bothcoefcientsareatamaximumwiththewingletsbelowthewingandaminimumwithabovewingconguration.An 93

PAGE 94

A B C DFigure3-19. Dragcoefcientvsangleofattackforataperedplate(=0.75)atvarioussideslipanglesandwingletcongurationswithAR=1andRe=1105.A)=0,B)=10,C)=20,D)=35. interestingpointtonoteinFig. 3-18D isthattheliftcoefcientisnonzeroatanangleofattackof=0forthecenteredandbelowwingcongurations.Atthishighlateralperturbation,withsomesurfaceareaofthewingletbelowthewing,ahighenoughblockageiscreatedthatthepressurebelowthewingisgreatlyincreasedandcausesanincreaseofliftevenatsmallanglesofattack.WhileitseemsthatthiscouldpotentiallyprovideabenetinMAVdesign,thecorrespondingdragpolarindicatesthatinthissamerangeofangleofattackthedragcoefcientincreasesfrom0.05toalmost0.4,negatingtheperformancebenetinlift.Inaddition,thesamenonzeroCMderivativeat 94

PAGE 95

A B C D Figure 3-20.Quarterchordpitchingmomentcoefcientvsangleofattackforatapered plate(=0.75)atvarioussideslipanglesandwingletcongurationswithAR =1andRe= 1 10 5 .A) f =0 ,B) f =10 ,C) f =20 ,D) f =35 lowanglesofattackasseeninFig. 3-9 indicatesthatthelocationoftheaerodynamic centerisnotconstantforanyoftheanglestested. PerhapsthemostsignicantresultsshowninFig. 3-18 isthedrasticreductionin nonlineareffectsforthecenteredandbelowwingcongurations.Onlytheabovewing geometrydisplaystherecoveryinliftcoefcientafterstallattributedtothecontributionof crossowfromthetipvortices.ConsideringthedragpolarsinFig. 3-19,onlythecase withtheabovewingcongurationdemonstratesanysignicantnonlineareffects.This isadirectresultofminimizingtheeffectsoftipvorticesastheirdevelopmentisimpeded bythepresenceofthewingletsbelowthewingandthustheycannotcontributetothe 95

PAGE 96

CLrecovery.ItisclearfromFig. 3-18 thatattemptingtoblockthecrossowbyusinganabovewingcongurationislesseffectiveinreducingtheeffectsofthetipvortexasnonlinearliftingeffectsremainprevalentat=35.Itisbelievedthattheeffectsfromthewingletsaresignicantlyreducedwithanincreaseinsideslipangleasthefreestreamowsimplyconvectsthevortexdownstream.Whilethelinearrelationshipbetweenangleofattackandloadingisbenecialforsimplifyingaerodynamicanalysis,itmustalsobenotedthatthedecreaseinstallandCLmaxaswellasthesignicantincreaseinCDindicateadeteriorationinightconditionswiththereducedcrossow. A B C DFigure3-21. Liftcoefcientvsangleofattackforataperedplate(=0.25)atvarioussideslipanglesandwingletcongurationswithAR=1andRe=1105.A)=0,B)=10,C)=20,D)=35. 96

PAGE 97

A B C DFigure3-22. Dragcoefcientvsangleofattackforataperedplate(=0.25)atvarioussideslipanglesandwingletcongurationswithAR=1andRe=1105.A)=0,B)=10,C)=20,D)=35. Forthelowertaperratioof=0.25,therstobservationfromFigs. 3-21 3-23 isthethatthewingletcongurationhassignicantlylessinuencethanthe=0.75caseastheresultsforthethreegeometriesareverysimilar.Thisistobeexpectedasthesmallerwingletsurfaceareahaslesseffectontheloading;however,itisclearthatthenonlineareffectsareessentiallyeliminatedatallsideslipanglesasseeninFig. 3-21 .Astheangleofattackincreasesandthewingbeginstostall,eventhesmallerwingletgeometryissufcienttopreventtheliftincreaseathighanglesofattack.Theresultsforallsideslipconditionsshowthatliftcurveslopesarenearlyidenticaltothenowingletscaseuntilapproximately=25wheretheeffectsoftipvortices,ifnottamperedwith, 97

PAGE 98

A B C DFigure3-23. Quarterchordpitchingmomentcoefcientvsangleofattackforataperedplate(=0.25)atvarioussideslipanglesandwingletcongurationswithAR=1andRe=1105.A)=0,B)=10,C)=20,D)=35. allowthewingtoproduceanincreasedstallandCLmax.Interestingly,thecaseswiththewingletsactuallyproducelowerdragatthesehighanglesofattackastheinduceddrag,whichissignicantforthemoredominanttipvortices,isreduced.Fig. 3-22D indicatesthatthewingletsdohavetheeffectofincreasingthedragcoefcientatlowanglesofattackforthesamereasonaspreviouslystated. 3.6.2LateralLoadsandRollStallThesamebatteryoflateralloadtestswerethenconductedataReynoldsnumberof7.5104sothattheinuenceofthewingletsonthelateralloadingcouldbeassessed. 98

PAGE 99

AsampleofresultsareshowninFig. 3-24 and 3-25 fortaperratiosof0.5and0.75,respectively.Therstpointtonotefromtheresultsisthattheyawmomentandsideforceareconsistentlynegativeandpositive(respectively).Thesignoftheloadsisconsistentwhetherthewingletsareaboveorbelowthewing,andcanbeattributedtotheincidentowuponthewinglets.Whenthewingissetatanegativesideslipangle,anormalowincidentuponthewingletsexists;thiscreatesaforceinthe+ydirectionalongthewingcorrespondingtothepositivesideforce.Itcanbeassumedthatthisowisreasonablyuniformalongthechordofthewingandtheforceissymmetricaboutthehalfchord;thus,theyawmomentaboutthequarterchordofthewingisnonzeroandnegative.Therollmomentismoresignicantfromacontrolperspective;whilethemagnitudesaretypicallysimilartotheyawmoment,itisimportanttorecognizethatthemomentsofinertiaabouttherollaxis(Ixx)aremuchsmallerthanabouttheotheraxes[ 3 4 ].ThismakestherollmomentmoresignicantforMAVsthanforconventionaltransportaircraft(inwhichIxxismorecomparablewiththeotherinertiacomponentsduetothefuelstorageinthewings[ 14 ]).Thus,therollmomentwillhaveagreatereffectontheresponseofaMAVthantheyawmoment.ComparisonofthedatainFigs. 3-24A and 3-24B ,aswellasFigs. 3-25A and 3-25B ,indicatesthattheshapeofthecurvesareantisymmetric.Thisistobeexpectedasthegeometryisessentiallyreversed;thewingletsdowncongurationatanegativeangleofattackmirrorsthewingletsupcongurationatapositiveangleofattack.InthecontextofMAVcontrol,thewingletsdowncongurationisofthemostinterest.WhencomparedtoFig. 3-12 itisclearthatwhilearollstalleventstilloccurs,themagnitudeoftherollmomentisdrasticallydecreased.Thepresenceofthewingletsbelowthewingdisruptstheformationoftipvorticesandreducestheassociatedrollmoment.Above-wingwingletsarelesseffectiveatthisdisruption,andinfactincreasethemagnitudeoftherollmomentinsomecases. 99

PAGE 100

A B C D E FFigure3-24. Lateralloadcoefcientsfor=0.5atplatewingsinsideslipatRe=7.5104withvaryingwingletcongurations.A)ClforAR=1,=.5,wingletsdown,B)ClforAR=1,=.5,wingletsup,C)CnforAR=1,=.5,wingletsdown,D)CnforAR=1,=.5,wingletsup,E)CSFforAR=1,=.5,wingletsdown,F)CSFforAR=1,=.5,wingletsup. 100

PAGE 101

A B C D E FFigure3-25. Lateralloadcoefcientsfor=0.75atplatewingsinsideslipatRe=7.5104withvaryingwingletcongurations.A)ClforAR=1,=.75,wingletsdown,B)ClforAR=1,=.75,wingletsup,C)CnforAR=1,=.75,wingletsdown,D)CnforAR=1,=.75,wingletsup,E)CSFforAR=1,=.75,wingletsdown,F)CSFforAR=1,=.75,wingletsup. 101

PAGE 102

Figs. 3-24A and 3-25A showpromisingresultsinreducingthemagnitudeoftherollmomentandthussimplifyingMAVcontroltechniques.Whilethecharacteristicsoftherollpolararesignicantlydifferentbetweenthewingletsonandoffcongurations,itisinterestingtonotethatthelongitudinalloadsarenot[ 75 ].Thisisanimportantconsiderationinrollstallmitigation,asthemodicationofthewinggeometrycanbeappliedtoreducetherollmomentinsideslipperturbationswithoutadverselyaffectingtheliftingcharacteristicsofthewing. 3.6.3EffectonCl,Whilethecross-coupledstabilityderivativeCl,isseentobesignicantforthePIPERMAV(andthecanonicaltaperedwingtestcases),therollstabilityderivativeCl,istypicallyofmoreconcernforMAVcontrolasthemagnitudeisevenlarger.Inaddition,MAVsareknowntobeespeciallysensitivetolateralperturbationsduetotheirsmallmomentsofinertiainthelateralplane;thus,asimilarforcevariationmaybemoresignicantwhenappliedlaterallytoaLARwing.Computationofthisvalueinvolveschoosingatrimangleofattackandthendeterminingthevariationinrollmomentwithrespecttotherangeofsideslipanglestested()]TJ /F8 11.955 Tf 9.3 0 Td[(200).ItwasfoundthatCl,couldbelinearlyrepresenteduptoasideslipangleof=)]TJ /F8 11.955 Tf 9.3 0 Td[(15;largerperturbationsof=)]TJ /F8 11.955 Tf 9.29 0 Td[(20resultedinsomenonlinearities.ThevaluesofCl,fortaperedwingswithoutwingletsarelistedinTable 3-1 .Theresultsoftheprevioussectionsshowedthecontributionofthewingletstoreducingtherollmomentatpositiveanglesofattack;thecorrespondingvaluesofCl,atequilibriumanglesoftrim=10and20aregiveninTable 3-4 alongwithCn,andCSF,forcomparison. 3.7AerodynamicLoadingforthePIPERMAVToconrmthatrollstallisindeedarelevantphenomenonforfullMAVs,thePIPERwasmountedontheMPSatincreasingsideslipanglesandsweptbetweenanglesofattackof)]TJ /F8 11.955 Tf 9.3 0 Td[(1530ataReynoldsnumberof7.5104;thetestswererepeatedwiththetailsurfacesremovedtoassesstheimpactoftheverticaltailsontheloading.A 102

PAGE 103

selectionofresultsareshowninFigs. 3-26 and 3-27 ;onlythelongitudinalloadsfortheMAVwithtailsattachedareshownastheadditionalgeometrydidnotsignicantlyaffectthelift,dragorpitchingmoment.Allmomentsaredenedaboutthequarterrootchordofthemodel. A B CFigure3-26. LongitudinalloadsforthePIPERMAVinsideslipatRe=7.5104.A)Liftcoefcientvsangleofattack,B)Dragcoefcientvsangleofattack,C)Quarter-chordpitchingmomentvsangleofattack. Theplotsforthelongitudinalloads(CL,CD,CM)inFig. 3-26 typicallyshowlittlevariationforallsideslipanglestested;theexceptiontothisistheincreaseinCDforthehighersideslipanglesduetothegreaterproledrag.Thelargestvariationisobservedintherollmomentpolars,whichvarysignicantlybetweensideslipanglesforboththetailonandoffcongurations.Fig. 3-27A showsagraduallyincreasingrollmoment 103

PAGE 104

A B C D E FFigure3-27. LateralloadsforPIPERMAVmodelsinsideslipatRe=7.5104.A)ClfortheMAVwithtails,B)ClfortheMAVwithouttails,C)CnfortheMAVwithtails,D)CnfortheMAVwithouttails,E)CSFfortheMAVwithtails,F)CSFfortheMAVwithouttails. coefcientuptoanangleofattackof12,atwhichpointanapparentrollstalloccursforthefullPIPERmodel.ThetaillessMAV(Fig. 3-27B )createsalargerslopeleadingup 104

PAGE 105

toastallatthesameangleofattackbutareducedCl,max.Amoderatenonzeroslopeisnoticeableatasideslipangleofzero;thiscanbeattributedtoleadingedgeasymmetriescreatingaspanwisevaryingstagnationpoint[ 81 ].Thisaffectstheformationofboththetipvorticesandtheleadingedgeseparationbubbleandhasbeenfoundtocreatesmall,butnonzero,rollmomentswithzerosidesliporbankangles[ 81 ];theresultsfromthisstudyshowthattheasymmetrictipvorticescreatesmalllateralloadsforboththePIPERandtheatplatewingsatzerosideslipangle.Finally,itisinterestingtonotethatthepresenceofverticaltails(Fig. 3-27A ),evenat=0,resultsinenoughofaninteractionthattheslopeapproacheszerobutthenominalvalueofClisnegative.Thisindicatesthecomplex,interactivenatureoftheowaroundaMAVwingcreatesmanycompetingeffectswhichneedtobecarefullycharacterized.TheinferencetobedrawnfromFigs. 3-27A and 3-27B isthat,whilethetailsurfacescontributesignicantlytotherollmomentatlowanglesofattack,thespanwiseloadingcreatedbytheasymmetrictipvorticesbecomesdominantatanglesofattackabove=10.Althoughtheincidentowontheverticaltailsinherentlycreatesapositiverollmomentaboutthex-axis,thestalleventpresentatallsideslipanglesaround=12createsasignicantnonlinearityinlateralloading.Theliftpolar(Fig. 3-26A )showsareasonablylinearslopeinthesamerangeofangleofattack,whichsuggeststhatthismaybepotentiallinearizationpointfortheaerodynamicloading;however,whenconsideringtherollmoment,itisclearthatthelateralloadingexperiencessignicantnonlinearitiesatthesametrimcondition.Thus,thenatureoflateralandlongitudinalloadsforaMAVindicatesthatwhileonecomponentcanbeeasilymodeledwithalinearapproximation,thesamemodelisnotnecessarilyapplicabletoforcesonotheraxes.RollstallcreatesanadverseloadingconditionwhichrapidlychangesthelateralforcesexperiencedbyaLARwing;thus,furtherinvestigationintomitigationtechniquesisdesirable. 105

PAGE 106

3.8SummaryofExperimentalResultsAsignicantamountofstaticanddynamicdatahasbeenpresentedinthischapter;priortodiscussingtheramicationsoftheseresultsforthestabilitypropertiesofLARwings,abriefsummaryofthekeyndingsisprovided.Resultsthathavebeendiscussedbypreviousstudiesarenotlistedforbrevity(forexample,thedepartureoftheliftcurveslopefromthetheoretical2value[ 69 103 ]). Whilethelongitudinalloads(lift,drag,pitchingmoment)experiencedsomeunexpectedbehaviorathighanglesofattack,suchasadragdecitafterstall,attributedtotheinuenceoftipvortices,littlevariationwaspresentatincreasingsideslipangles.Assuch,longitudinalloadsarenominallyindependentofsideslipangleupto=35. Lateralloading(rollmoment,yawmoment,sideforce)werefoundtobesignicantlyinuencedbysideslipperturbations;rollstallwasdiscoveredandismanifestedbyalinearlyincreasingslopeofrollmomentagainstangleofattackleadingtoastalleventnear=20.Thisbehaviorisnotexperiencedbyhighaspectratiowingsandisattributedtoasymmetricspanwiseloadingcreatedbydisplacedtipvorticesduringsideslip.ThisisauniquefeatureofLARwings. Rollstallalsoinducesayawmomentwhich,liketherollmoment,exhibitsadependencyonbothangleofattackandsideslip.Thesameistrueforsideforcealthoughthemagnitudeistypicallysmall. Stabilityderivativescreatedbyrollstallaresignicantfromavehiclecontrolperspective;therollstabilityderivative,Cl,,islargerthanthedesiredvalueforgoodhandlingaircraft,andthecrosscoupledderivativesCl,andCn,existforsmallperturbationsfromequilibriumconditions(=0). AnexperimentalsurveyofdampingderivativesforLARwingshasbeenconductedforthersttime.Derivativesduetotranslationalrates(@=@_and@=@_)werefoundtobenegligible.Normalforcedampinginpitch(@Z=@q)experiencedthelargestmagnitude.Crosscoupledderivatives,suchas@Z=@pand@l=@q,wereobservedatincreasedsideslipangles.Finally,therolldampingderivative@l=@pwasnominallyzeroat=0,suggestingacauseofthejitterynatureofMAVight. Testingoftaperedplanformswithwingletsindicatesthatthemagnitudeofrollstallmaybereducedwithwingletsconguredbelowthewingastheformationofthetipvorticesaredisruptedalthoughthisdoesinduceapenaltyindrag. StaticaerodynamicloadsofthePIPERMAVindicatethatrollstallisstillprevalentforfullvehiclesinadditiontoatplatewings. 106

PAGE 107

Table3-4. Lateralstabilityderivativesduetoforvaryingcongurations. PIPERPIPER=0.25=0.25=0.5=0.5=0.75=0.75(notail)(withtail)(w/ldown)(w/lup)(w/ldown)(w/lup)(w/ldown)(w/lup) Cl,trim=10-0.198-0.3810.0007-0.144-0.039-0.288-0.048-0.28trim=20-0.212-0.27-0.017-0.0880.023-0.190.021-0.126Cn,trim=100.0060.3540.1760.1490.1620.2170.1620.287trim=200.0380.2170.1810.1320.1980.2030.2150.281CSF,trim=10-0.184-1.08-0.393-0.305-0.516-0.592-0.584-0.727trim=20-0.324-0.785-0.396-0.28-0.625-0.566-0.772-0.667 107

PAGE 108

Table3-5. PIPERMAVgeometricproperties. WingVerticaltailsurfaces b,cm.20.3Maxheight,cm.8.9c,cm.17.3Maxwidth,cm.7.6Dihedralangle,deg.8Incidenceangle,deg.8Surfacearea,cm.2290Surfacearea,cm.264.5Maxcamber,cm.1.1@.19cSideproleof Maxreex,cm.0.2@.82ctailgeometryInverseZimmerman Tailposition planform(topview)relativetowing 108

PAGE 109

CHAPTER4STABILITYCHARACTERISTICSOFLOWASPECTRATIOWINGSTheexperimentalresultsofthepreviouschapterprovideaclearindicationofthecomplexnatureofLARwingloading;therealizationthatthetipvortexasymmetryofrollstallinducessignicantrollandyawmomentsevenintheabsenceofverticaltailgeometryhasnotpreviouslybeenconsidered.Withoutaccountingfortheeffectsofrollstall,specicallythemagnitudesofthederivativesCl,,Cn,,Cl,andCn,,thestabilitycharacteristicsofLARierscannotbeaccuratelymodeled.Theinherentdynamicsofthesewingsmustbecorrectlyunderstoodtodevelopmorerigorousdesigntoolsandbetterightcontrolsystems;sizingtailsurfacesanddesigningautopilotswithoutcompensatingforrollstallwillresultinineffectivecontrolauthorityforthevehicle.Assuch,itisnecessarytomodeltheresponseofaLARwingtoperturbationsfromequilibriumandtocharacterizethemodalresponseinasimilarmannertothatofafullaircraft.ThischapterwilldemonstratethatLARwingsexperiencestabilitymodeswhichdevelopentirelyfromtheiruniqueaerodynamicregime,andparticularlyduetotheeffectsofrollstall.Anaccuratesolutionisobtainedbynumericallyintegratingthenonlinearequationsofmotion,andlinearmodelsaresubsequentlyusedtomatchtheresponsetobetterdescribetheobservedresponse.ItshouldbenotedherethattheseresultsdonotnecessarilycorrespondtoresponsesoffullMicroAerialVehicles;onlythepurewingtrajectoryisconsidered.Thepurposeistoclearlyindicatethatthesewingsinitiatetheirownharmonicresponsetoperturbationsfromequilibriumwithnocontributionfromtailsurfaces,wingtaper,activecontrolsurfaces,etc.ThecurrentimplementationofthesewingsinMAVdesignwithoutconsiderationofthesedynamicsrepresentsafundamentalmisunderstandingofthecomplexitiesofLARaerodynamicsandstability;animprovedknowledgeofhowLARwingsrespondtodisturbanceswillgreatlyfacilitatefutureMAVdesign. 109

PAGE 110

Rollstallpredominantlyaffectsthelateralloads-rollmoment,yawmomentandsideforce-withoutgreatlyimpactingthelongitudinalloadingofLARwings.Assuch,themostsignicantstabilityimplicationswillberelatedtotheresponseofthelateralvariables,,pandr,whichwillbediscussedindetailinthischapter.ExperiencefromighttestingMAVshasshownthatlongitudinalstabilityisstraightforwardtoachieveusingareexedairfoil(orhorizontaltail)andacorrectlylocatedcenterofgravity;however,alateralequilibriumismoredifculttomaintainasminorperturbationscancreateunrecoverablerolloryawmotions.Thepresenceofrollstallprovidesanexplanationforthisbehavior,andwillbequantitativelydescribedinthischapter. 4.1LongitudinalInstabilityPriortoconsideringthelateralresponseofLARwingstodisturbancesfromequilibrium,thenatureoflongitudinalstabilityshouldbediscussed.Symmetricwingsofanyaspectratioarepassivelyunstableinpitchasitisnotpossibletoobtainanegative(restoring)pitchstabilityderivativeandastaticallytrimmedwingwithouteitherahorizontaltailorcambered/reexedairfoilprovidingbalancedmoments[ 13 ].Thepitchstabilityofawingoraircraftisrelatedtotwoconditions:theabilitytotrim(orbalance)themomentsaboutthecenterofgravity,andarestoringpitchstabilityderivative@M=@whichreturnsthebodytoequilibriumafteranangleofattackperturbation.TheseconditionsaredescribedbyEqs. 4 and 4 ,respectively: XMcg,trim=Mwing+Mtail=0,(4) @Mwing @=@L @xnp c)]TJ /F3 11.955 Tf 13.15 8.23 Td[(xcg c<0.(4)ItcanbeseenfromEq. 4 that,forapositivevalueoftheliftcurveslope@L=@,thepitchstabilityderivativeisonlynegativeifthelocationoftheneutralpoint,xnp,isforwardofthecenterofgravity,xcg.Foraconventionalaircraftwithawingandhorizontaltailconguration,Eqs. 4 and 4 aresatisedasbothliftingsurfacesproduceoppositemomentsaboutthecenterofgravity;furthermore,asthecgisforwardofthe 110

PAGE 111

centerofliftonthewing,astablepitchstabilityderivativeiscapableofreturningtheaircrafttothisequilibriumconditionafteranangleofattackperturbation.ThismuchisillustratedinFig. 4.1 .ItshouldbenotedthatthesameresultisoftenachievedforMAVswiththeuseofacambered/reexedairfoilwherethereexedtrailingedgeprovidesthesamerestoringmomentastheliftfromthetail. Figure4-1. Illustrationofstaticlongitudinalstabilityforaconventionalwing/horizontaltailconguration;PM=0and@M=@<0asthecenterofgravityisforwardoftheneutralpoint. Forapuresymmetricwingwithnohorizontaltailorreex,itisnotpossibletomeetbothpitchstabilitycriteria.Ifthecenterofgravityisforwardoftheneutralpoint,asillustratedinFig. 4-2A ,thewingliftstillinducesanose-downpitchingmomentbutthereisnohorizontaltailtoprovidearestoringmoment.Ifthecenterofgravityislocatedattheneutralpoint,atrimconditioncanbeachievedasnomomentsareproducedaboutthecg(asseeninthestaticmomentplotssuchasinFig. 3-4C );however,thisalsomeansthatnorestoringmomentispresenttorecoverfromvariationsinangleofattack.ThisisportrayedinFig. 4-2B .Thus,solutionstotheequationsofmotionfortheatplatewingsinvestigatedinthisresearchwillbedominatedbythepitchinstabilityandwillnotaccuratelyportraythelateraldynamics;furthermore,simplyxingaconstantangleofattackwilleliminateanypotentialdependenciesonlongitudinalmotionwhichareexpectedtobesignicantduetothemagnitudeoftherollstallderivatives.Neitherofthesearedesirablescenarios.Itisofinterestinthisinvestigationtoanalyzeboththepurelylateralmodes(constant)createdbyrollstallandtoalsodeterminehowthecross-coupledloads 111

PAGE 112

A BFigure4-2. Illustrationofpitchinstabilityforasymmetricwingwithoutatail;eitherthewingcannotbetrimmedasthemomentsareunbalanced( 4-2A )ornorestoringmomentisprovidedtorecoverfromangleofattackperturbations( 4-2B ).A)xcg
PAGE 113

aregiveninEq. 4 and 4 : 8>>>>>>>>><>>>>>>>>>:XYZ9>>>>>>>>>=>>>>>>>>>;=W g8>>>>>>>>><>>>>>>>>>:_u_v_w9>>>>>>>>>=>>>>>>>>>;+W g26666666664qw)]TJ /F3 11.955 Tf 11.95 0 Td[(rvru)]TJ /F3 11.955 Tf 11.96 0 Td[(pwpv)]TJ /F3 11.955 Tf 11.95 0 Td[(qu37777777775,(4) 8>>>>>>>>><>>>>>>>>>:lmn9>>>>>>>>>=>>>>>>>>>;=8>>>>>>>>><>>>>>>>>>:Ixx_p)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixz_rIyy_qIzz_r)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixz_p9>>>>>>>>>=>>>>>>>>>;+26666666664qr(Izz)]TJ /F3 11.955 Tf 11.95 0 Td[(Iyy))]TJ /F3 11.955 Tf 11.95 0 Td[(pqIxzpr(Ixx)]TJ /F3 11.955 Tf 11.96 0 Td[(Izz)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(p2)]TJ /F3 11.955 Tf 11.95 0 Td[(r2Ixzpq(Iyy)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixx)+qrIxz37777777775.(4)whereWistheweightofthebody,u,vandwarethetranslationalcomponentsofvelocity,p,qandraretherotationalcomponentsofvelocity,X,YandZandl,mandnaretheforceandmomentcomponentsaboutthebodyaxes,Ixx,IyyandIzzarethemomentsofinertiaaboutthebodyaxes,andIxzistheproductofinertiaaboutthexandzaxes;duetosymmetryaboutthex)]TJ /F3 11.955 Tf 11.59 0 Td[(zplane,Iyz=0.ThisisavalidassumptionfortheLARwingsusedinthisinvestigationaswellasforconventionalaircraft.Dottedtermsindicateatimederivative.Thelinearizedsystemcanbewritteninaformsuitablefornumericalintegration: _u=X W=g)]TJ /F3 11.955 Tf 11.95 0 Td[(qw+rv)]TJ /F3 11.955 Tf 11.96 0 Td[(g(sin),_v=Y W=g)]TJ /F3 11.955 Tf 11.95 0 Td[(ru+pw+g(cos)(sin),_w=Z W=g)]TJ /F3 11.955 Tf 11.95 0 Td[(pv+qu+g(cos)(cos),_p=Ixx)]TJ /F3 11.955 Tf 13.15 8.08 Td[(I2xz Izz)]TJ /F5 7.97 Tf 6.59 0 Td[(1l+Ixz IzzN)]TJ /F3 11.955 Tf 11.96 0.01 Td[(qrI2xz Izz+Izz)]TJ /F3 11.955 Tf 11.96 0.01 Td[(Iyy)]TJ /F3 11.955 Tf 11.96 0 Td[(pq(Iyy)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixx)Ixz Izz)]TJ /F3 11.955 Tf 11.95 -0.01 Td[(Ixz,_q=1 Iyym)]TJ /F3 11.955 Tf 11.95 0 Td[(pr(Ixx)]TJ /F3 11.955 Tf 11.96 0 Td[(Izz)+)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(p2)]TJ /F3 11.955 Tf 11.95 0 Td[(r2Ixz,_r=Izz)]TJ /F3 11.955 Tf 13.18 8.09 Td[(I2xz Ixx)]TJ /F5 7.97 Tf 6.59 0 Td[(1n+Ixz IxxL)]TJ /F3 11.955 Tf 11.96 0 Td[(pqIyy)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixx)]TJ /F3 11.955 Tf 13.15 8.09 Td[(I2xz Izz)]TJ /F3 11.955 Tf 11.96 0 Td[(qr(Izz)]TJ /F3 11.955 Tf 11.95 0 Td[(Iyy)Ixz Ixx+Ixz.(4) 113

PAGE 114

inwhichthex,yandzweightvectorshavebeenexpressedintermsoftheEuleranglesand.ThetermsofEq. 4 areinuencedbytheaerodynamicanglesandratesofthewing;anaccurateknowledgeoftheloaddependenciesisrequiredforacorrectmodel.TheconventionalTaylorseriesexpansionoftheseloadsiswellknownforhighaspectratiowingsandtypicallyassumesthatitispossibletodecouplelateralandlongitudinalaxes[ 13 ].Duetothesignicantdifferencesintheassociatedowregime,determiningthedependenciesofLARwingsrequiresexperimentalevidencewhichconsiderspotentialcross-coupledloadingandaerodynamicdamping[ 75 ].TheexpansionsofX,Y,Zandmaboutazerosideslipanglearefoundtobenominallyidenticaltohighaspectratiocases;however,thepresenceofrollstallandtheassociatedanddependenciesalterthenatureofthelandn(rollandyawmoments)foranAR=1wing: l=@l @+@l @+@l @pp+@l @qq+@l @rr,n=@n @+@n @+@n @rr.(4)OfmostsignicantinterestinEq. 4 isthepresenceofthecross-coupledderivatives@l=@and@n=@,whichcouplestheevolutionofthelateralmomentstovariationsinangleofattack.Asthemodelstestedtoobtaintheseexperimentaldependenciesweresimpleatplate,rectangularwings,thesederivativesarecreatedpurelyduetotheaerodynamicloadingofLARwings.Whentheaspectratioisincreasedtothree,@n=@disappearsand@l=@issignicantlyreduced[ 83 ];thisindicatestheuniqueaerodynamicregimeofLARwings,andsuggeststhepossibilityofstabilitymodeswhicharespecictothistypeofvehicle.Analsimplicationtotheequationsofmotionisachievedbyconstrainingthepitchangletobeconstant;therefore,inadditiontothe_uand_wequationsofEq. 4 beingeliminatedfromtheequationsofmotionduetotheprescribed(t)trajectory,thepitchrateqisalsoequaltozero.Thus,allcrosscouplingduetolongitudinalvariationsisdue 114

PAGE 115

tothevaryingverticalvelocityasopposedtocoupledperturbationsofanglesofattackandpitch.Furthermore,foratplatewingsthecrossproductofinertiaIxzisnegligible.Thesesimplicationsreducetheequationsofmotiontoamoreunderstandableform: _u=U0cos((t)),_v=Y W=g)]TJ /F3 11.955 Tf 11.95 0 Td[(ru+pw+g(cos)(sin),_w=U0sin((t)),_p=l=Ixx,_q=0,_r=l=Izz.(4)Inessence,thebankandyawanglesaredirectreectionsoftherollandyawmoments,respectively;thefunctionalityofeachisgiveninEq. 4 .TheEulerangles,andaredenedintheusualway.Numericallyintegratingtheseequationsforaprescribedlongitudinaltrajectoryof(t)andagivenlateralperturbation(v,,p,r)yieldsthetimehistoriesofthelateralstatevariablesandcanbeusedtoidentifystabilitymodescreatedbytheinuenceofrollstallonthewingaerodynamics.Althoughtheintegrationiscarriedoutinthebodyaxesofthewing(u,v,w),resultswillbepresentedinthemoreconventionalstabilityaxeswhere =tan)]TJ /F5 7.97 Tf 6.59 0 Td[(1w u,=sin)]TJ /F5 7.97 Tf 6.59 0 Td[(1v U.(4) 4.3LinearizedModelsofLowAspectRatioModesAlinearizedmodelcanbedevelopedfromthenonlinearequationsofmotioninEqs. 4 and 4 byassumingthattheEulerangleperturbationsaresmall;thatthewingisanAR=1atplatewhereIxz=0andIxx=Iyy;thattheinitialvaluesofsideforceandroll,yawandpitchmomentsarezero;andthattheproductsofrates(qr,pr) 115

PAGE 116

arenegligible.Itshouldbenotedthatthepwterminthesideforceequationisretainedduetothepotentiallyhighrollratesassociatedwithlowaspectratiowings,althoughitistypicallyignoredforconventionalaircraft.Itislinearizedabouttheinitialverticalvelocitycomponentsuchthatpw=pw0.Thenallinearequationsofmotionbecome: X)]TJ /F3 11.955 Tf 13.15 8.09 Td[(W ggcos0=W(_u+qw0),Y+W ggcos0=W(_v+rU0)]TJ /F3 11.955 Tf 11.96 0 Td[(pw0),Z)]TJ /F3 11.955 Tf 13.15 8.09 Td[(W ggsin0=W(_w)]TJ /F3 11.955 Tf 11.95 0 Td[(qU0),l=Ixx_p,m=Iyy_q,n=Izz_r.(4)wherethetermsrepresenttheforceandmomentperturbationsfromequilibrium.ThesetermscanbeexpandedusingaTaylorseriesandalinearsystemfortheresponseofthewingcanbeconstructed.ThedependenciesofEq. 4 areincorporatedintoEq. 4 ;therollmomentduetopitchrate@l=@qisneglectedasqisconstrainedtobezero.TheensuinglinearizedsystemisdenedinEq. 4 : 116

PAGE 117

2666666666666666666666666410000001000000W gU00000001000000Ixx000000Izz377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:____p_r9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;=266666666666666666666666640)]TJ /F7 11.955 Tf 9.3 0 Td[(!200001000000YYWcos0W gw0)]TJ /F3 11.955 Tf 10.49 8.09 Td[(W gU00000100ll0lplr0nn000377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:_pr9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;.(4)wherethersttwoequations(and_)arearticiallyconstructedtoinduceasinusoidalvariationinangleofattackcorrespondingtoEq. 4 .StabilityderivativesarepresentedindimensionalformsothatthesolutiontoEq. 4 producestimehistoriesofthestatevariablesinphysicalunits.Notethattheseequationscaneasilybenondimensionalizedtoprovideamorefundamentalanalysistool;thisformulationwillbeprovidedanddiscussedinChapter 5 .Forthetimebeing,however,itmorestraightforwardtoconsidertheresponseofthewinginphysicalunits.Unliketheformulationofthelateralequationsofmotionforaconventionalaircraft,thepureLARwingdoesnotexperiencestabilityderivativedependenciesonYr,Yp,npandnr;also,thelpderivativeisoftennegligibleexceptathighvaluesofthesideslipangle.ThesimpliednonlinearequationsofmotionandthelinearizedsystemofequationspresentedinEqs. 4 and 4 ,respectively,aresolvedforthetimehistoriesofthelateralstatevariables: ~xlat=fprgT.(4)Theaccuracyofthelinearmodelinrepresentingthenonlinearequationsofmotionisassessedandthenatureofthewingresponseisdiscussedandquantied.Initially, 117

PAGE 118

theangleofattackisheldconstanttodeterminethepurelateralresponseinSection 4.4 ;thesinusoidaltrajectoryofEq. 4 isthenprescribedandtheensuingeffectsonthemodalresponseofthewingarediscussedinSections 4.5 4.7 4.4PurelyLateralMotionTheinitialconditionresponseofthesystemofequationsdescribedinEq. 4 isevaluatedforequilibriumanglesofattackof0=5and15andanarbitrarylateralperturbationof~xlat,0=f1000gT.Theresults,showninFigs. 4-3 and 4-4 ,demonstratethedevelopmentofaperiodic,divergentmode.Despitethelargemagnitudesoftheperturbations,whichwouldseemtoinvalidateseveralassumptionsmadeinthelinearizationprocess,thenonlinearequationsarematchednicelyforasmuchasthreesecondsofmotion.Assuch,theresultsofalinearanalysiscanaccuratelypredictthestabilitycharacteristicsofthewing.Theparametersofthemodeshapescomputedfromsolvingtheeigenvalueproblem,includingtheeigenvalues()andeigenvectors(~v),normalizedmagnitudeandphasingofthestatevariables,thedamping()andundampednaturalfrequenciesofperiodiclateralmodes(!n,lat),arelistedinTable 4-1 .Asseeninthesimulationresults,adivergentandoscillatorymodeexistswithsignicantparticipationbyallfourvariables;aheavilydampedmodeisalsopresentbutwillnotbediscussedindetailhere.Thelargestcontributionisduetotherollrate,whichisdrivenbylargemomentscreatedbyrollstallandcorrespondinglowmomentsofinertiaaboutthex-axis.TherelativemagnitudesandphasedelaysbetweenthevariablesapproximatelycorrespondwiththoseexpectedforDutchrollofaconventionalaircraft;furthermore(althoughnottabulated)themagnitudesoftheyawheadingangleareapproximatelyequaltothatofthesideslipangle[ 104 ].AlthoughthephasinganglesarenotidenticaltotypicalDutchrolls,likelyduetothesmallyawmoments,thisresponsewillbereferredtohereinasathedivergentDutchrollbehaviorasitisqualitativelysimilartoatypicalDutchroll.Theundampednaturalfrequenciesareall 118

PAGE 119

A B C DFigure4-3. Comparisonoflinearandnonlinearresponseat0=5and0=0;solidline:nonlinearsolution,dashedline:linearmodel.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. below!n,lat=10.2rad/s[f=1.6Hz],whichplacesthemwithintherangeoffrequenciesatwhichthedampingderivativesweremeasuredexperimentally.Inessence,theseresultsdemonstratehowtheloadingcreatedbyrollstall-predominantlythelargemagnitudeoftherollmomentduetosideslip,l-inducesalinear,butdivergent,Dutchroll-typemodeinherenttolowaspectratiowings.TheoscillatoryandunstableresultsofFigs. 4-3 and 4-4 ,computedforanarbitraryinitialcondition~xlat,0whichdoesnotspecicallyexciteanyofthemodeslistedinTable 4-1 ,indicatethatthismodedominatestheresponseofthewingevenforsmallsideslip 119

PAGE 120

A B C DFigure4-4. Comparisonoflinearandnonlinearresponseat0=15and0=0;solidline:nonlinearsolution,dashedline:linearmodel.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. perturbations.Thisisconrmedbyconsideringtheresponseofthestatevariablestoinitialconditionswhichproducethemode,computedastherealpartofthenormalizedcomplexeigenvectorfromTable 4-1 ;thisresponseisplottedinFig. 4-5 for0=15andisseentocloselyresembletheresponsetothearbitraryinitialconditionofFig. 4-4 .Althoughtheamplitudeofoscillationsincreases,resultinginalesseffectivelinearapproximation,thesimilarityoftheplotsindicatesthatthelateralresponseofLARwingsisdominatedbythedivergentharmonicbehaviorofthisunstableDutchrollmode. 120

PAGE 121

Table4-1. Parametersoflateralstabilitymodesatequilibriumanglesof0=5and15. ~vj~vj[]!n,lat[rad/s] 51.15.7i8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.3 0 Td[(.0004.054i.032.162i.961)]TJ /F8 11.955 Tf 9.3 0 Td[(.201.074i9>>>>>>=>>>>>>;8>>>>>><>>>>>>:.057.1721.2239>>>>>>=>>>>>>;8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.3 0 Td[(90)]TJ /F8 11.955 Tf 9.3 0 Td[(790)]TJ /F8 11.955 Tf 9.29 0 Td[(1609>>>>>>=>>>>>>;-.1925.82)]TJ /F8 11.955 Tf 9.3 0 Td[(1.651.69i8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.3 0 Td[(.022)]TJ /F8 11.955 Tf 9.3 0 Td[(.261.266i.881)]TJ /F8 11.955 Tf 9.3 0 Td[(.114.268i9>>>>>>=>>>>>>;8>>>>>><>>>>>>:.025.4231.3319>>>>>>=>>>>>>;8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.29 0 Td[(359)]TJ /F8 11.955 Tf 9.29 0 Td[(1340)]TJ /F8 11.955 Tf 9.29 0 Td[(1139>>>>>>=>>>>>>;.6992.37151.3510.1i8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.3 0 Td[(.002.032i.013.097i.993)]TJ /F8 11.955 Tf 9.3 0 Td[(.049.012i9>>>>>>=>>>>>>;8>>>>>><>>>>>>:.032.0991.0519>>>>>>=>>>>>>;8>>>>>><>>>>>>:)]TJ /F8 11.955 Tf 9.3 0 Td[(94)]TJ /F8 11.955 Tf 9.3 0 Td[(820)]TJ /F8 11.955 Tf 9.29 0 Td[(1679>>>>>>=>>>>>>;-.13310.2)]TJ /F8 11.955 Tf 9.3 0 Td[(2.339.73i8>>>>>><>>>>>>:.008.002i)]TJ /F8 11.955 Tf 9.3 0 Td[(.337.141i.922)]TJ /F8 11.955 Tf 9.3 0 Td[(.024.124i9>>>>>>=>>>>>>;8>>>>>><>>>>>>:.009.3961.1389>>>>>>=>>>>>>;8>>>>>><>>>>>>:16)]TJ /F8 11.955 Tf 9.29 0 Td[(1570)]TJ /F8 11.955 Tf 9.29 0 Td[(1009>>>>>>=>>>>>>;.9232.53 4.5EffectsofAngleofAttackPerturbations:TheRollResonanceModeConventionalformulationsoftheDutchrollmodeassumethatitisunaffectedbythelongitudinalmotionoftheaircraft;however,duetotheloadscreatedbyrollstallforLARwings,therollandyawmomentsexhibitadependencyontheangleofattack.Furthermore,thehighvaluesofpandrseeninFigs. 4-3 4-5 cancausethepwandrunonlineartermstobecomesignicantinthesideforceequation.Asinusoidalangleofattacktrajectoryof(t)=A0sin(!t)isprescribedinthelinearandnonlinearequationstoinvestigatetheeffectsontheresponseofthelateralvariables.TheamplitudeissettoA0=3;whilethisisanarbitraryvalue,resultsarefoundtobenominallyidenticalforA05.Largeramplitudeswillnaturallyaffecttheresponse 121

PAGE 122

A B C DFigure4-5. Effectsofdivergentlateralmodeinitialconditionsat0=15and0=0onthelinearandnonlinearmodels'response;solidline:nonlinearsolution,dashedline:linearmodel.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. ofthestatevariables;however,thepurposeofthisstudyistodeterminehowsmallperturbationsfromequilibriummayinitiateunstablemodes.Hence,asmallamplitudeoscillationismaintained.Thefrequency,!,andphase,,oftheangleofattackinputarevariedtoassesstheensuringeffectsonthewingresponse;selectedvaluesof!arebasedonthenaturalfrequenciesofthelateralmodesseeninTable 4-1 suchthat!=!n,lat=0.5,1and2.Thelinearequationsaretrimmedabout0=5and0=5;thenonzerotrimsideslipangleisutilizedsothatthecross-coupledrollstallderivativesY,l 122

PAGE 123

andnarealsononzero.TimehistoriesofthestatevariablesforthesetestparametersaredisplayedinFig. 4-6 .Theresultsofthesesimulationsindicatethatthepresenceoflongitudinalperturbationsdoesaffecttheresponseofallfourlateralvariables,althoughtheconstraintthattheamplitudeoftheangleofattackvariationsremainsmallcausesonlysubtleeffectsinmostcases.TheadditionalrestoringforcesandmomentsattributedtolandNincreasethefrequencyandthemagnitudeoftheresponserelativetothepurelateralcasesofFigs. 4-3 4-5 .Thelinearizedmodelprovidesareasonableapproximationforthetrajectoryofthewingforangleofattackvariationsatfrequenciesof!=!n,lat=0.5and2.Whilesomeovershootispresentathighersideslipdeectionswherethevaluesforthestabilityderivativescomputedatatrimconditionof0=5arenolongervalid,themainfeaturesoftheresponsearecorrectlycaptured.Whenthefrequencyof(t)isequaltothenaturalfrequencyofthelateralresponse,however,itisclearthatthelineartimeinvariantmodelofEq. 4 breaksdown.ThemostsignicantvariationisthatofthebankangleshowninFig. 4-6E ;theresultsofthenonlinearsimulationdonotoscillateaboutzerobutinsteaddrifttowardsaconsiderableamplitude()]TJ /F8 11.955 Tf 9.3 0 Td[(150withinthe3secondsofthesimulation).Thisbehaviorisonlyobservedwhen!iswithin50%ofthenaturalfrequenciesoftheoriginallateralmode,andsuggeststhattherelativephasingof(t)and(t)inthesecasescausesafundamentalshiftintheaerodynamicloadingconditions,andresultingstabilitycharacteristics,ofLARwings.ThisdescriptionoftheinteractiveeffectsofoscillationsinanglesofattackandsidesliponthebankanglestabilityofLARwingsrepresentsamodewhichhasnotpreviouslybeendescribed;furthermore,itisamodewhichisseentoarisefromsmallperturbationsfromanequilibriumightcondition.Astheresponseofishighlydependentuponthefrequenciesof(t)and(t)oscillationshavingsimilarvalues,thismodewillbereferredtohereinasthe`rollresonance'mode. 123

PAGE 124

AschematicdepictingthedevelopmentofthersttwocyclesofrollresonanceisshowninFig. 4-7 foraninitialperturbationwhichexcitesthedivergentlateralresponse(r0<0=s,p0>0=s);simultaneousviewsofthesideslipangle(withtrailingtipvorticesincluded),thebankanglefromadownstreamperspective,andtheangleofattackareprovided.Thelateraldisplacementofthewingisnotshown.RelativemagnitudesoftheangulardisplacementsandratesaretakenfromthesimulationresultsshowninFig. 4-6 .Theeffectsoftheangleofattackontherelativetipvortexstrengthareincluded,andsignicantlyinuencetherollresonance.Themaximaandminimaoftheangleofattacktrajectoryareseentocoincidewiththelargestpositiveandnegativevaluesofthesideslipangle,respectively,inFigs. 4-7B 4-7D 4-7F and 4-7H .Rollstallcausestherollmomenttoscalelinearlywiththeangleofattackforlowaspectratiowings[ 83 ];therefore,atthepositivesideslipdisplacements(Fig. 4-7B ),thestrengthofthetipvorticesincreaseswiththeangleofattackandimpartsagreatermagnituderollmomentthanwouldbegeneratedbyawingatconstantangleofattack.Similarly,duringnegativesideslipdisplacements(Fig. 4-7D )thelowerangleofattackreducesthestrengthofthetipvorticesandtheassociatedrestoringrollmoment.Asthewingcontinuestooscillateinsideslipduetoarestoringyawmoment(alsogeneratedbythetipvortexasymmetryofrollstall[ 83 ])thedisparityintipvortexstrengthandrollmomentmagnitudeperpetuatesandthebankangleofthewingisdriveninthenegativedirection;whileitstilloscillatesandmayreachalessnegativeangle(Fig. 4-7F ),itneverreturnstoawings-levelconguration.Itshouldbenotedthatiftheangleofattackdoesnotchange,thestrengthofthetipvorticesalsoremainsnominallyconstantandtherestoringrollmomentremainsinproportionwiththesideslipangle;thisresultsinthedivergentDutchrollmodeseeninFigs. 4-3 4-5 .Somepreviousexperimentalresultshavedemonstratedthat,athighanglesofattack(>20),aprescribedpitchmotioncanintroduceself-excitedrolloscillationsatthesamefrequency[ 82 ],althoughtheowphysicsaredifferentfromresonanceasthesideslipangleiszerointhepreviousstudy. 124

PAGE 125

Furthermore,rollresonancemayoccuratsmalllateralperturbationsfromcruise,asdiscussedabove,anddonotdependuponthenonlinear,highangleofattackregime. 4.6ImpactofInitialPhaseShiftsonRollResonanceTofurtherinvestigatehowthephasingbetweentheanglesofattackandsideslipeffectstherollresonancemode,thenonlinearequationsofEq. 4 areintegratedfortheinitialconditionscorrespondingtothedivergentDutchrollmodeandangleofattacktrajectorieswithphaseshiftsofphaseshiftsof=0,90and180.Thesemaybeconsideredtorepresentinitiallongitudinalconditionscorrespondingtopositiveornegativeverticalgustperturbationsatt=0s.ThestatevariablesresponseisshowninFig. 4-8 .Comparisonoftheresponsesof(t)and(t)revealsinherentdifferencesintheevolutionoftheseangles.Thesideslipangleoscillatesperiodicallyabout=0withunboundedamplitudes;thetimeconstantsofthemotiondonotvarysignicantlyfortherangeofphaseshiftsimpartedto(t),althoughtheamplitudesoftheoscillationsexperiencesomedifferencesatincreasingtime.Therollresonance,however,demonstratessignicantdependenceonthephaseshiftofthelongitudinalmotion.Regardlessofthephaseoftheangleofattackconditionsprescribedtothewing,theinitialdisplacementsarepositivewithnominallyidenticalmagnitudes.Bytheendoftherstperiodatt1s,however,therolldivergencebecomesdominantasthebankangleincreaseswithoutbound;additionally,thesignoftheresponseisseentodependuponthephaseofthe(t)trajectoryasphaseshiftsof=90and180leadtoapositivedivergencewhereas=0and)]TJ /F8 11.955 Tf 12.53 0 Td[(90createsanegativedivergence.Thus,whentheangleofattackincreasesimmediatelyaftert=0sif=90and180thedirectionofthebankangledivergenceispositive;aninitialdecreaseinangleofattackresultsinanegativedisplacement. 125

PAGE 126

4.7LinearTimeVariantModelforRollResonanceWhiletheintegrationofEq. 4 providesthefullnonlinearresponseofthewingtoasetofinitialperturbationconditions,amoresimpliedmodelwhichprovidesmoreinsightintothenatureoftherolldivergenceisofinterest.Duetotheangleofattackvariationswhichinstigatetherollresonancemode,atimeinvariantmodeldoesnotaccuratelycapturethebehaviorofthesystemasthestabilityderivativesofEq. 4 willevolvewiththeinstantaneousvalueof(t);thismuchisseeninFig. 4-6 .Furthermore,theinitialconditionresponsemethodcommonlyusedtosolvethelinearsysteminEq. 4 willnotpredictthedepartureinbankanglefromequilibriumseeninFig. 4-8B asthistechniqueinherentlyassumesaformofthesolutioneAtwhichmayoscillateabouttheorigin,orsmoothlygrowordecay,butnotboth.Theseconditionssuggesttheneedforatimevariantmodel,whichaccountsforvariationsinthestabilityderivativeswithangleofattack.Thestatevectormustbeevaluatedateverytimesteptocapturethedivergentnatureofthebankangle.Alineartimevariant(LTV)modelisgivenbyEq. 4 : 2666666666666664W gU0000010000Ixx0000Izz37777777777777758>>>>>>>>>>>>>><>>>>>>>>>>>>>>:___p_r9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;=2666666666666664YWcos0W gw0)]TJ /F3 11.955 Tf 10.49 8.09 Td[(W gU00010l((t))0lplrn00037777777777777758>>>>>>>>>>>>>><>>>>>>>>>>>>>>:pr9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;.(4)ThismodeldiffersfromEq. 4 byallowingtherollstabilityderivativeltovarywithangleofattackand,implicitly,time.Inaddition,astheangleofattackvariationsareconstrainedtobe3andarethereforetypicallysmallerthanthesideslipperturbations,thedirecteffectsofthecrosscoupledderivativesY,landnareignored.TheresultsoftheintegratedLTVsystemarecomparedwiththefullnonlinearresponseforaninputangleofattackfrequencyof!=!n,latandarecomparedwiththestatevariableresponsesofFigs. 4-6B 4-6E 4-6H and 4-6K .TheresultsaredisplayedinFig. 4-9 126

PAGE 127

UnlikethelineartimeinvariantmodelofEq. 4 ,theLTVmodelprovidesareasonableapproximationofthebehavioroftherollresonancemode.Fig. 4-9B depictstheresponseofthebankangleinwhichthedirectionofthedivergence,theamplitudeandfrequencyoftheoscillations,andthemagnitudeofthedeviationfromtheequilibriumangleof0=0arecapturedbytheintegrationofEq. 4 .Astheonlystabilityderivativeallowedtovarywithtimeisl,itisclearthattheeffectsofangleofattackperturbationsontherollstabilityderivativesignicantlyimpactstherollresponseofLARwings;effectsduetothecrosscoupledderivativeswouldbecomesignicantforlargeramplitudeoscillationsin(t).AlthoughthethedifferencesinrollratepbetweenthenonlinearandLTImodelsareslight,asseeninFig. 4-6H ,theseminordiscrepanciescausedbythevariationinlattheoscillatinganglesofattackwillcausethetotalrollmomenttoleaneitherpositiveornegative.CombinedwiththelowIxxmomentsofinertiaofLARwings,thisskewnessoftherollmomentcreatesthedivergentresponseofthebankangle.ItisinterestingtonotethatthisbehaviorcanstillbemodeledbyalinearsystemsuchasEq. 4 ,althoughthevalueoflmustbeupdatedforevensmallvariationsinangleofattack(3).Inordertocomparethedifferentlinearapproximationsusedinthispaper,anormalizedrootmeansquaredeviation(RMSD)iscomputedforthelinearandnonlinearmodelsusingEq. 4 : xRMSD=vuut 1 nnXi=1xNL(i))]TJ /F3 11.955 Tf 11.96 0 Td[(xlin(i) jxNLjmax!(4)wherexNLandxlinarethenonlinearandlinearresponses,respectively.ThevaluexRMSDisnormalizedbythemaximumvalueofthenonlinearsolutionoverthedomainofinterestsothattherelativemagnitudesofdifferentvariablesmaybecompared.Itwascomputedforthestatevariablesofallsimulationresultspresentedinthispaperfor0
PAGE 128

qualitativeresultsobtainedbyobservation.RollresonanceisseentoincreasetheRMSdeviationbyanorderofmagnitudewhentheangleofattackoscillationsoccuratthenaturalfrequencyofthelateralmode(!=!n,lat=1).TheimplementationoftheLTVmodel,listedinthebottomrowofTable 4-2 ,reducesthedeviationbetweenthelinearandnonlinearmodelsbyover50%fort<1sandbyaround75%overtheentire3seconddurationofthesimulation.Similarreductionsarepresentfortheotherstatevariables,againindicatingthebetterapproximationoftheLTVmodel. Table4-2. Normalizedrootmeansquaredeviationsbetweenlinearmodelsandnonlinearsolutions. xRMSD(t<1)xRMSD(t<3)00! !n,latFigureprpr 500 4-3 0.040.050.030.050.250.280.160.081500 4-4 0.050.050.080.070.150.160.200.051500 4-5 0.040.040.020.040.550.600.510.10550.5 4-6A 4-6D 4-6G 4-6J 0.110.090.120.130.360.400.370.38551 4-6B 4-6E 4-6H 4-6K 0.450.530.570.191.080.631.050.76552 4-6C 4-6F 4-6I 4-6L 0.340.370.290.150.380.580.390.29551 4-9 0.200.230.200.090.440.180.360.53 4.8AttenuationofRollResonanceModeAsboththedivergentDutchrollandrollresonancemodesarefoundtobeunstableevenforsmallperturbationsfromequilibriumight,itisdesirabletodetermineamechanismformitigatingtheseresponses.Arangeofscalingfactorsareappliedtoltodeterminetheeffectsontheensuingeigenvalues;theresultsaretabulatedinTable 4-3 .Itcanbeseenthatreducingtherollstabilityderivativebyafactorof1/11producespositiverealpartsoftheeigenvaluesandthusdrivesthesystemstable,attenuatingthedivergentDutchrollmode;alargerscalingfactorincreasesthemagnitudeofthenegativerealeigenvaluecomponentandthusimprovesthestabilitycharacteristics.Thereducedimpactoflontheinstantaneousloadingofthewingpreventstheevolutionofa 128

PAGE 129

skewedrollmomenthistoryandthusthebankangleisnolongeradverselyaffectedbycoupledangleofattackandsidelslipperturbations.Itshouldbenotedherethatsimilarresultsareachievedbyincreasingthenderivative,alsolistedinTable 4-3 ,effectivelystiffeningthedirectionalstabilityofthewingandpreventingthesideslipangle(andensuingrollmoment)fromgrowingtoolarge. Table4-3. Effectsofscaledstabilityderivativesoneigenvaluesat0=5. ScalingfactorScaledderivative 1l,n1.15.7i1/11l)]TJ /F8 11.955 Tf 9.3 0 Td[(.0014.3i1/20l)]TJ /F8 11.955 Tf 9.3 0 Td[(.144.2i6n)]TJ /F8 11.955 Tf 9.3 0 Td[(.00312.1i20n)]TJ /F8 11.955 Tf 9.3 0 Td[(.2121.9i 4.9EffectsofDampingDerivativesontheWingResponseTodeterminetheimpactofthemeasureddampingderivativesontheresponseofaLARwing,thenonlinearsolutioniscomputedusingtheexperimentallyobtaineddampingderivativesandiscomparedwiththenonlinearresponsewiththedampingtermssettozero.Eliminatingthedampingderivativeshaslittleeffectonthetheresponsewhentheangleofattackisheldconstant(Figs. 4-10 and 4-12 )orwhentheangleofattackchangesatahightrimangle(Fig. 4-13 ).Whenthevaryingangleofattackapproacheszerointhepresenceofhighlateralrates(,p),however,thedifferenceintherollresonancemodeisnoticeable(Fig. 4-11 ).Thisindicateshowtwoconditionsarenecessaryforthedampingderivativestoparticipatesignicantlyintheresponseofthewing.First,asthemagnitudeofthedampingderivativesissmallerthanthatofthestaticderivatives,theangularratesmustbehighfortheirproducttohaveanyimpact.Thisistrueforeitherconstantorvaryinganglesofattack.Thedifferencebetweenthesecasesariseswhentheangleofattackoscillationsapproach=0whentherolloryawratesarehigh.Aslateraldampingderivativesat=0wereuniversally 129

PAGE 130

foundtobenonexistent,therestoringmomentsduetopandraresignicantlyreduced.ThismuchisseeninFig. 4-11C attimet>1.5s,wheretherollmomentdoesnotreturntop=0deg/safterreachingalargeminimumofp)]TJ /F8 11.955 Tf 22.18 0 Td[(250deg/s.Atthistime,(t)isatitsminimumvalueof=2;assuch,therestoringrollmomentissignicantlyreducedandtherollrateremainsnegative. 4.10DiscussionofResultsThedominanceofthedivergentDutchrollandrollresonancemodeshavecriticalramicationsforthestabilityandcontrolpropertiesofLARwings.Firstandforemost,thepresenceofthesemodescanbeuniquelyattributedtothederivativescreatedbyrollstallandthelowmomentsofinertiawhichareinherenttothesewings.Thesubsequentcreationofasignicantrollstabilityderivative,l,duetothetipvortexasymmetryofaLARwinginsideslipisseenheretoinduceunstableoscillationsofalllateralvariables;thedependenceofthisresponseuponangleofattackperturbationsthroughrollresonancedemonstratesafundamentalcouplingofthelateralandlongitudinalstabilityaxeswhichisnotpresentforconventionalaircraftyingatequilibriumconditions.ThetendencyofLARwingstosubmittothismodeafteronlyminorperturbationsfromtrimconditionsindicatestheirvulnerabilitytoinstabilitieswhensmall,nonzerosideslipconditionsexist.Thisnewlyconsideredmodeisadditionallyinterestingasitcanbedescribedaspurelyaerodynamic;unlikeconventionalstabilitymodesforhighaspectratioaircraft,itisentirelyduetotheloadingasymmetriesofrollstallasopposedtothesizeandorientationofgeometricfeaturessuchastailsurfaces.ConsiderationofthismodehassignicantimpactonfuturevehicledesignofMAVs,ascontrolsurfacesmustbesizedtocompensateforwingswiththeirowninherentlyunstabledynamics.Previousworkbytheauthorshasindicatedthepotentialforwingletscenteredbelowthecenterofgravityofthewingtoreducethemagnitudeofrollstallandthustheassociatedstabilityderivativel[ 105 ].AsweightisalwaysaconcernforMAVdesigners,anotherpotentialsolution 130

PAGE 131

istheimplementationofayawdamperwhichwouldactivelyservetoreducetheNderivative.Themostcrucialimpactofthisinvestigation,however,istheunderstandingofthecreationandinuenceofthedivergentDutchrollandrollresonancemodesexperiencedbyLARwings,whichhavenotpreviouslybeenaccountedforinMAVdesign.Inessence,thisexplainstheineffectivenessofconventionaldesigntoolswhichdonotcompensatefortheunstabledynamicsofthewingitself. 131

PAGE 132

A B C D E F G H I J K LFigure4-6. Comparisonoflinearandnonlinearsolutionswith(t)=3sin(!t)at0=5and0=5anddivergentlateralmodeinitialconditions;solidline:nonlinearsolution,dashedline:linearmodel.A)(t)for!=!n,lat=0.5,B)(t)for!=!n,lat=1,C)(t)for!=!n,lat=2,D)(t)for!=!n,lat=0.5,E)(t)for!=!n,lat=1,F)(t)for!=!n,lat=2,G)p(t)for!=!n,lat=0.5,H)p(t)for!=!n,lat=1,I)p(t)for!=!n,lat=2,J)r(t)for!=!n,lat=0.5,K)r(t)for!=!n,lat=1,L)r(t)for!=!n,lat=2. 132

PAGE 133

A B C D E F G HFigure4-7. Diagramofrollresonancemotion;rstrow:topviewdepictingsideslipangleandtrailingtipvortices,secondrow:bodyaxisviewfromdownstreamdepictingrollangle,thirdview:sideviewdepictingangleofattack.Lateraldisplacementisnotshown.Anglesarealltoscalewiththeexceptionofangleofattack,whichisexaggeratedforclarity.A)t=0,B)t=T/4,C)t=T/2,D)t=3T/4,E)t=T,F)t=5T/4,G)t=3T/2,H)t=7T/4. 133

PAGE 134

A B C DFigure4-8. Variationsinnonlineartimehistoriesof~xlatforvaryingphasesoftheprescribedangleofattacktrajectories,(t)=3sin(!t+),anddivergentlateralmodeinitialconditions.;phaselagsof=0,90and180representpositiveandnegativeangleofattackperturbations.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 134

PAGE 135

A B C DFigure4-9. Approximationofdivergentresponseusingalineartimevariant(LTV)modeltrimmedat0=5and=5;blackline:nonlinearmodel,dashedline:LTVmodel.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 135

PAGE 136

A B C DFigure4-10. Effectsofdampingderivativesonthenonlinearlateralresponsesolutiontodivergentlateralmodeinitialconditionsatatrimangleof=5and=0;solidline:experimentaldampingderivatives,dashedline:nodampingderivatives.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 136

PAGE 137

A B C DFigure4-11. Effectsofdampingderivativesonthenonlinearrollresonancesolutiontodivergentlateralmodeinitialconditionsatatrimangleof=5and=0with(t)=3sin(!t);solidline:experimentaldampingderivatives,dashedline:nodampingderivatives.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 137

PAGE 138

A B C DFigure4-12. Effectsofdampingderivativesonthenonlinearlateralresponsesolutiontodivergentlateralmodeinitialconditionsatatrimangleof=15and=0;solidline:experimentaldampingderivatives,dashedline:nodampingderivatives.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 138

PAGE 139

A B C DFigure4-13. Effectsofdampingderivativesonthenonlinearrollresonancesolutiontodivergentlateralmodeinitialconditionsatatrimangleof=15and=0with(t)=3sin(!t);solidline:experimentaldampingderivatives,dashedline:nodampingderivatives.A)Sideslipangleresponse,B)Bankangleresponse,C)Rollrateresponse,D)Yawrateresponse. 139

PAGE 140

CHAPTER5ROLLSTALLMODELINGANDCOMPARISONWITHHIGHASPECTRATIOSTheprevioustwochaptershavepresentedextensiveexperimentaldatawhichcharacterizethenewphenomenonofrollstallforlowaspectratiowings,andthenusestheseresultstosimulatethezero-inputlateralresponseofawingtovariousperturbationsfromequilibrium.ThepresenceofrollstallinLARwingswasshowntoinciteaninherentlyunstable,purelyaerodynamicmode;furthermore,theclaimwasmadethatthismodeisuniquelycreatedbylowaspectratios.Inordertojustifythisassertion,amodelforrollstallisdevelopedwhichisusedtopredicttherollstabilityderivative,Cl,forbothhighandlowaspectratios.AsimpliedlateralsystemisconsideredinwhichtheonlyrelevantstabilityderivativesareCl,andCl,p;unlikethesysteminChapter 4 ,inwhichthetimehistoriesofonlythelowaspectratiowingswereofinterest,inthischapterthesystemwillbepresentedinnondimensionalformtofacilitatethecomparisonwithhighaspectratiowings.Byincreasingtheaspectratioofthewing,theinuenceoftherolldampingderivativegrowswhilethatoftherollstabilityderivativeisreduced.Atransitiontoastable,lightlydampedDutchrollmodeisobservedasthestable(real)eigenvaluebecomesdominant,asexpectedforconventionalwings. 5.1StabilityFormulationWhilea44linearsystemwasshowntosatisfactorilyrepresentthenonlinearresponseofLARwings,thisremainsahighlycoupledsystem;assuch,itisdifculttopreciselyquantifytheimpactofcriticalstabilityderivatives.AsimpliedapproximationmaybeusedtoidentifythemostsignicantstabilityderivativesforLARwingsandmaythenbeextendedtohigheraspectratios(withoutusingexperimentalresults)forcomparativepurposes. 140

PAGE 141

Alinear44systemforthelateraldynamicsofasymmetricwingmaybewritteninnondimensionalformas[ 13 ]: 8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:___p_r9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;=26666666666666641CY,gccos0 2U201CY,p+w0c U0b1CY,r)]TJ /F3 11.955 Tf 13.15 8.08 Td[(c b00c b02Cl,02Cl,p2Cl,r3Cn,03Cn,p3Cn,r37777777777777758>>>>>>>>>>>>>><>>>>>>>>>>>>>>:pr9>>>>>>>>>>>>>>=>>>>>>>>>>>>>>;(5)inwhich1=Sc=4m,2=c=bIx,3=c=bIz,misthemassofthewing,gisthegravitationalacceleration,U0isthefreestreamvelocity,w0istheverticalvelocityatagivenangleofattack,IxandIzaretherollandyawmomentsofinertia,0istheequilibriumpitchangle(equaltotheangleofattackforcruiseight),,,pandrarethestatevariablesforsideslip,bank,rollrateandyawrate,andCY,ClandCnarethesideforce,rollmomentandyawmomentcoefcients.AquantitysuchasCY,representsastabilityderivative.Nondimensionalizationswerecarriedoutwiththeusualscalingfactorst=t2U0=c,(porr)=(porr)b=2U0,I(xorz)=(8I(xorz)=Sb3,CY=Y=1 2U20S,andC(lorn)=(lorn)=1 2U20Sb.PreviouslypublishedexperimentalresultsbytheauthorshaveindicatedthatthederivativesCY,p,CY,r,Cl,p,Cl,r,Cn,pandCn,rarezeroornegligibleforLARwings[ 83 94 ].Furthermore,whiletheCY,derivativeisnotnegligible,itisclosetoanorderofmagnitudesmallerthantheothertermsinthe_equation;assuch,itmayalsoberemovedfromEq. 5 .Finally,byconsideringtheprevioussimulationresults,whilethesideslipandbankanglesareofsimilaramplitudes,itisclearthattherollrateisanorderofmagnitudelargerthantheyawrate(p>>r)[ 94 ].AsthecontributionoftherollresponseisthusmoresignicanttotheunstableDutchrollmode,theapproximationthat_r=0permitstheyawrateequationtoberemovedfromthesystem.Thisseriesofassumptions,basedonexperimentalobservationsandorderofmagnitudearguments, 141

PAGE 142

reducesthedynamicmodeltoEq. 5 8>>>>>>>>><>>>>>>>>>:___p9>>>>>>>>>=>>>>>>>>>;=266666666640gccos0 2U20w0c U0b00c b2Cl,02Cl,p377777777758>>>>>>>>><>>>>>>>>>:p9>>>>>>>>>=>>>>>>>>>;.(5)Notethat,althoughthemagnitudeofCl,pwasexperimentallyfoundtobezeroforLARwings,itwillberequiredinlatersectionsfortheanalysisofhighaspectratios;assuch,itisretainedinEq. 5 Table5-1. Comparisonofstabilityparametersoffull44systemwith33approximation. Systemjj!n[rad/s] 441.477.49i7.63-0.197.62)]TJ /F8 11.955 Tf 9.3 0 Td[(2.471.47i2.880.862.88331.816.85i7.09-.267.09-3.623.6213.62 Acomparisonofthestabilityparameters(ie,eigenvalues,dampingratio,naturalfrequency)ofthesystemsinEq. 5 and 5 isprovidedinTable 5-1 foranillustrativecaseofanAR=1wingtrimmedat=10;stabilityderivativesaretakenfrom[ 83 94 ].Ofmostinterestisthebehaviorofthedominantunstablemode,asthestablemode(complexforthe44systemandrealforthe33system)isheavilydampedandfeaturesasmallermagnitudeeigenvalue.Theslightchangestothestabilityparametersafterthereductiontotheapproximatesystemarenotsignicant,andtheresponseofthestatevariablesremainswellrepresented.Thenaturalfrequencyismatchedtowithin7%,andwhilethedampingratiovariesbyover30%duetotheremovaloftherestoringCN,termintheyawrateequation,thenatureoftheresponseisqualitativelymatchedandisplottedinFig. 5-1 .Itisclearfromthegurethatthesignicantlyreduced33 142

PAGE 143

systemcapturesthekeyfeaturesoftheresponseofthewing;byextension,asCl,istheonlyremainingstabilityderivativeinEq. 5 ,therollstabilityderivativecanbeconsideredtobethemostsignicantderivativeforLARwinglateraldynamics. A B C DFigure5-1. Comparisonofnonlinearandlinearmodelsatatrimangleof=10and=0;solidline:nonlinearsolution,dashedline:44linearmodel,x-symbols:33linearmodel.A)Sideslipangleresponse,B)Bankangleresponse,C)Nondimensionalrollrateresponse,D)Nondimensionalyawrateresponse. ThetwodistinctfeaturesofLARwingswhichdifferfromtheirhighaspectratiocounterpartsarethepresenceoftherollstabilityderivativeandtheabsenceofanyrolldampingatequilibriumightconditions[ 83 94 ].Theconverseofthisistrueforconventionalwings,inwhichthespanwiseloadingconditionsarenotdrasticallyaffected 143

PAGE 144

bysideslipperturbationsandrolldampingisaninherentproductoftheinducedvelocityalongthespanofarollingwing.AdirectcomparisonofthestabilitypropertiesofhighandlowaspectratiowingscanbeusedtoidentifyhowtheimpactoftherespectiveCl,andCl,pderivativesfundamentallychangesthedynamicresponse.Whilepublisheddatafromtheauthorsisavailableforthelowaspectratiocases,experimentalchallengesrestricttheabilitiestocollectreliableresults(particularlyforrolldamping)withthecurrentapparatus.Furthermore,amodelfortherollstabilityderivativeisdesirabletoidentifytheunderlyingphysicsbehindthedevelopmentofrollstall.Inthefollowingsection,basicmodelswillbepresentedforCl,andCl,pandwillbeusedtoevaluateEq. 5 forhighandlowaspectratios.Comparisonoftheresultingeigenvalueswilldemonstratehowdecreasingtheaspectratio,andchangingtheaerodynamicregime,resultsinatransitionfromastabletoanunstabledynamicresponse. 5.2AModelforRollStallAnumberoftheuniqueloadingcharacteristicsofLARwingsatlowReynoldsnumbers,suchasnonlinear/reducedliftcurveslopes,highstallangles,androllinglimitcycles,havebeenattributedtotheinuenceofthetipvorticesinrecentyears[ 68 69 76 81 ];thenoticeableasymmetryofthevorticesduringasideslipperturbationsuggeststhattheyarealsoresponsibleforrollstall[ 83 ].Asatruetrailingvortexisainherentlydifcultowstructuretomodelduetocorewandering,turbulence,andthree-dimensionaleffects[ 46 106 107 ],noattemptismadetocharacterizethelocalowdynamicsofLARtipvortices;instead,aglobalmodelwhichusesasimpliedrepresentationofthetrailingvorticestopredicttheimpactofsideslipperturbationsonthesurfacepressuredistribution(and,thus,therollmoment)ispresented.Thetrailingvorticesaredenedtobelinevorticescoincidentwiththeleadingedgewingtips,andarerepresentedbydiscreteirrotationalvorticesateachdownstreamlocation.Thispermitsthepressuredistributiononthesurfacetobecomputedrelativetothebaselineconditionat=0andthesubsequentrollmomenttobecomputed. 144

PAGE 145

5.2.1DenitionofTermsandCoordinateSystemThetwolinevorticesoriginateupstreamofthewingatx!1andextendtoalocationfardownstreamofthewinginlinewiththefreestreamow.Inordertousepotentialowtheorytoobtainanestimateofthecirculation,)]TJ /F1 11.955 Tf 6.78 0 Td[(,acomponentofthevortexmustbeorientedorthogonaltothevelocityvector;toaccomplishthis,thetrailingvorticesarejoinedtogetherfardownstreambythestartingvortexofthewing.Bydenition,thestartingvortexhasthesamecirculationmagnitudeastheboundvortexoftenusedtorepresenttheliftingsurfaceofthewinginaconventionalhorseshoevortexformulationforwinglift[ 98 ].Byusingittoevaluatethestrengthofthetrailingvortexpair,aphysicallyrealisticestimateofthetipvortexstrengthmaybeobtainedwithoutplacinganadditionalvortexlamentnearthesurfaceofthewing.Asaresult,itispossibletoisolatetheeffectsofonlythetrailingvorticesonthesurfacepressuredistribution.AschematicofthisinvertedhorseshoevortexsystemshowninFig. 5-2 aforawingatzerosideslip.Byassumingconstantvorticityalongthevortexsystem,thestrengthofthestartingvortexmaybecomputedfrompotentialowtheory: L=Zb=2)]TJ /F4 7.97 Tf 6.59 0 Td[(b=2U0)]TJ /F3 11.955 Tf 6.78 0 Td[(dy=U0)]TJ /F3 11.955 Tf 6.77 0 Td[(b,(5)whichcanberearrangedandexpressedintermsofthenondimensionalliftcoefcientCLtoyield: )-278(=CLU0S 2bcos,(5)inwhichSisthewingarea,bisthewingspan,andisthesideslipangle.Thebcostermisincludedbecause,asthesideslipangleofthewingincreases,thedistancebetweenthetrailingvorticesdecreasesastheyareconstrainedtoremainattachedtotheleadingedgewingtipswhilepropagatinginthedirectionofthefreestreamow;thisadjustedsystemisshowninFig. 5-2 b.Foragivenangleofattack,theliftcoefcientremainsconstantaspreviouslypublishedexperimentaldatahasdemonstrated[ 75 ]. 145

PAGE 146

Thus,Eq. 5 showsthatforhighersideslipangles,thecirculationofthestartingvortexmustincreasetomaintainaconstantvalueofCL.Thestrengthofthetrailingvorticesthusincreasesaccordinglytosatisfyvorticityconservation. A BFigure5-2. Denitionoftrailingvortexmodelcoordinatesystemandrelevantterms.TopviewsofaLARwingtrailingvortexsystem,A)=0,B)<0. 5.2.2EffectiveAngleofTrailingVorticesTheheightofthetrailingvorticesabovethewingisanimportantvariabletoconsiderasitsignicantlyaffectsthepressuredistribution,andbecauseoftheshortwingspanofLARwings,theleftandrightvorticesarecloseenoughtoinduceadownwardvelocityuponeachother.Thisresultsinaneffectiveincidenceangle,e,whichislessthanthegeometricangleofattack,g.TheinduceddownwardvelocitywicanbecomputedusingtheBiot-Savartlaw;therequiredvariablesaredenedinFig. 146

PAGE 147

5-2 b.Anelementdxoftherighttrailingvortexwillactuponapointxp,whichisaradialdistancea,verticaldistanced,andhorizontaldistancesaway.Theanglebetweenthetwolengthvectorsisdenedtobe,andisnegativebyconvention.UsingtheBiot-Savartlaw,thetotalinuenceoftherighttrailingvortexonthedownwashatxpcanbewrittenas wi(xp)=)]TJ ET q 0.478 w 213.42 -140.46 m 226.77 -140.46 l S Q BT /F8 11.955 Tf 213.42 -151.65 Td[(4Z+1cos a2dx.(5)Eq. 5 canbemoreeasilyintegratedalongtheangle@,whichrequiresthefollowingsubstitutionstobemade: s=bcos,d=stan,d=xp)]TJ /F3 11.955 Tf 11.96 0 Td[(x,a=bseccos.(5)ThesepermitEq. 5 tobeexpressedas wi(xp)=)]TJ ET q 0.478 w 164.49 -383.57 m 177.84 -383.57 l S Q BT /F8 11.955 Tf 164.49 -394.76 Td[(4Z)]TJ /F6 7.97 Tf 6.59 0 Td[(=2=2cos (bcossec)2(bsec)d,=)]TJ /F8 11.955 Tf 9.3 0 Td[()]TJ ET q 0.478 w 164.49 -417.07 m 217.16 -417.07 l S Q BT /F8 11.955 Tf 164.49 -428.26 Td[(4bcos2(sin)j)]TJ /F6 7.97 Tf 6.58 0 Td[(=2=2,=)]TJ ET q 0.478 w 164.49 -448.32 m 217.16 -448.32 l S Q BT /F8 11.955 Tf 164.49 -459.51 Td[(2bcos2,(5)whichistheinduced(downward)velocityontherighttrailingvortexbythelefttrailingvortex.AsEq. 5 isintegratedbetweenpositiveandnegativeinnity,thevalueofwiinducedontheleftvortexbyitscounterpartwillbeequivalenttotheresultinEq. 5 .Theeffectiveangleofattackcannowbecomputedas e=g)]TJ /F7 11.955 Tf 11.95 0 Td[(i=sin)]TJ /F5 7.97 Tf 6.59 0 Td[(1wg)]TJ /F3 11.955 Tf 11.95 0 Td[(wi U0.(5) 147

PAGE 148

Eq. 5 yieldstheheightofthetrailingvortexcoreabovethesurfaceofthewing.ThisisrequiredtocomputetheinducedpressureusingBernoulli'sequation. 5.2.3InducedPressureonWingSurfaceInordertoobtainasimpliedrepresentationofthepressureinducedonthesurfaceofthewing,thetrailingvorticesareeachrepresentedateveryxlocationonthewingbyatwodimensionalirrotationalvortex.Whilethisapproximationdoesnotfullycapturethephysicaluidstructure,itdoesprovideasimplemethodforestimatingtherelativeoweldpressurerelativetothevortexcore.Evaluatingthispressureeldonthesurfaceofthewingrequiresknowingtheheightofthetrailingvortexabovethewingandthelateraldisplacementcausedbythesideslipangle.TheformerquantityisgivenbyEq. 5 andthelatterisdenedbythesineofthesideslipangle,asdepictedinFig. 5-2 b;botharefunctionsofthechordwisedisplacementx.Acrosssectionofthewingatagivendownstreamstation(lookingupstream)whichshowsthepositioningoftheleftandrighttrailingvorticesabovethewingsurface,alongwiththerequiredquantitiestocomputethepressureeldonthewing,isshowninFig. 5-3 Figure5-3. Variablesforcomputingthepressuredistributionfromthetrailingvorticestothewingsurfaceatagivenchordwisestation;viewfromdownstreamwithfreestreamowcomingoutoftheplane. Intheabsenceofexternalforces,theradialpressuregradient@p=@rofanirrotationalvortexbalancesthecentrifugalaccelerationexperiencedbyacirculatinguidparticle.Knowingthatthetangentialvelocity,u,ataradialdistancerfromthe 148

PAGE 149

centerofanirrotationalvortexisdenedas u=)]TJ ET q 0.478 w 236.23 -44.83 m 255.08 -44.83 l S Q BT /F8 11.955 Tf 236.23 -56.02 Td[(2r,(5)themomentumequationincylindricalcoordinates,assumingnoaxialacceleration,maybewrittenas @p @r=u2 r=)]TJ /F5 7.97 Tf 6.77 4.34 Td[(2 42r3.(5)Withoutanabsolutepressurereference,Eq. 5 mayonlybeusedtocomputepressuresrelativetoanotherpointintheoweld.Anarbitrarypressure,p0,isdenedataradius,r0,anincrementallysmalldistancefromthevortexcore.IntegratingEq. 5 makesitpossibletocomputethepressurealongthewing,p(x,y),intermsofp0andr0;bysubtractingtheresultsfromthebaseline=0conditionfromtheresultswithnonzerosideslip,therelativepressuredistributioninducedbytheasymmetrictrailingvorticesmaybeestimated.Theintegralforthepressureinducedbythelefttrailingvortex,pL(x,y),isgivenby ZpL(x,y)p0@p=)]TJ /F5 7.97 Tf 6.77 4.34 Td[(2 42ZrLr0r)]TJ /F5 7.97 Tf 6.59 0 Td[(3@r.(5)Theexpressionfortheeffectsoftherighttrailingvortex,pR(x,y),isidenticalbutwillnotbelistedhereforbrevity.IntegratingEq. 5 yieldsBernoulli'sequation: pL(x,y)=p0+)]TJ /F5 7.97 Tf 6.78 4.33 Td[(2 821 r20)]TJ /F8 11.955 Tf 15.23 8.09 Td[(1 r2L,(5)inwhich)]TJ /F1 11.955 Tf 10.09 0 Td[(isthestrengthofthetrailingvortexandrListheradialdistancetothelocation(x,y)onthewing(depictedinFig. 5-3 ).Equation 5 maybeusedtodeterminehowthepressuredistributiononthetopofthewingisaffectedbythelateraldisplacementofthelefttipvortexatagivendownstreamlocationx,andthedisplacedpressuredistributionmaythenbeusedtocomputetherelativelocallift.Liftisgeneratedbythepressuredifferentialbetweenthe 149

PAGE 150

bottomandtopsurfacesofthewing, p(x,y)=pb)]TJ /F3 11.955 Tf 11.95 0 Td[(pt.(5)Assumingthatpbdoesnotchangewithsideslipangle,therelativeshiftinlocalliftasthesideslipangleincreasesis: p(x,y)6=0=(pb)]TJ /F3 11.955 Tf 11.95 0 Td[(pt)6=0)]TJ /F8 11.955 Tf 11.96 0 Td[((pb)]TJ /F3 11.955 Tf 11.96 0 Td[(pt)=0,p(x,y)6=0=pt,=0)]TJ /F3 11.955 Tf 11.95 0 Td[(pt,6=0.(5)Thisassumptionisbaseduponthephysicalconceptthattheincidentowuponthebottomsurfaceofthewingisnotdrasticallyalteredasthewingyaws,whichwillresultinsimilarpressuredistributions.AsseeninEq. 5 ,thiscausesthepbtermstocancelandthustherelativelocalliftinsidesliponlyrequiresknowledgeofthepressureonthetopsurface.SubstitutingtheresultsofEq. 5 intoEq. 5 andcancelingtermsprovidesthefollowingexpression: pL(x,y)6=0=p0+)]TJ /F5 7.97 Tf 6.78 4.34 Td[(2 821 r20)]TJ /F8 11.955 Tf 15.23 8.09 Td[(1 r2L=0)]TJ /F10 11.955 Tf 11.96 16.86 Td[(p0+)]TJ /F5 7.97 Tf 6.77 4.34 Td[(2 821 r20)]TJ /F8 11.955 Tf 15.23 8.09 Td[(1 r2L6=0,=)]TJ /F5 7.97 Tf 6.78 4.33 Td[(2 82"1 r2L=0)]TJ /F10 11.955 Tf 11.95 16.86 Td[(1 r2L6=0#.(5)ThesimplicationinEq. 5 requirestheassumptionthatthepressureatr0remainsconstantasthesideslipanglevaries;however,Eq. 5 showsthatthestrengthofthetrailingvortexchangeswithsideslipinordertomaintainconstantlift.Thiswillaffectthetranslationalvelocityatr0(givenbyEq. 5 )and,asaresult,p0isnottrulyconstant;instead,usingEq. 5 andintegratingfrominnitytor0,itmaybeshownthatp0variesinverselywithcos2.Assuch,theaccuracyofthisrepresentationwilldegradeathighersideslipanglesbutisreasonableforsmallerlateralperturbations;forexample,at=10,thepressurep0isaccuratetowithin3%ofthevalueat=0. 150

PAGE 151

Theradialdistancealongthetopsurfaceofthewingfromthecoreofthelefttrailingvortex,rL,iscomputedusingthegeometrictermsdenedinFig. 5-3 : rL=q h(x)2+y2L=q (xsin(e)2+(b=2+y)]TJ /F3 11.955 Tf 11.96 0 Td[(xtan)2.(5)SubstitutingthistermintoEq. 5 yieldstheexpressionfortherelativelocalpressureatagiven(x,y)locationonthesideslippingwing.Notethatifthesideslipangleissetto=0,thetermscancelandtheskewedpressurebecomesidenticallyzero. pL(x,y)6=0=)]TJ /F5 7.97 Tf 6.77 4.34 Td[(2 821 (xsin(e))2+(b=2+y)2)]TJ /F8 11.955 Tf 99.84 8.09 Td[(1 (xsin(e))2+(b=2+y)]TJ /F3 11.955 Tf 11.96 0 Td[(xtan)2.(5) 5.2.4ComputationofRollMomentEquation 5 maynowbeusedtocomputetherollmomentaboutthecenterspanofthewing.Multiplyingtheexpressionbyy,whichisthemomentarmtotherollaxis,integratingoverthesurfaceofthewing,andnondimensionalizingtheresultby1 2U20Sbprovidestherollmomentcoefcientcausedbytheskewedpressuredistribution: ClL=)]TJ /F5 7.97 Tf 6.78 4.34 Td[(2 42U20SbZ)]TJ /F4 7.97 Tf 6.59 0 Td[(c0Zb=2)]TJ /F4 7.97 Tf 6.59 0 Td[(b=2y (xsin(e))2+(b=2+y)2)]TJ /F8 11.955 Tf 11.96 0 Td[(......y (xsin(e))2+(b=2+y)]TJ /F3 11.955 Tf 11.95 0 Td[(xtan)2@y@x(5)WhileananalyticintegrationofEq. 5 ispossibleusingpolylogarithmexpansions,theensuingsolutionislengthyandcomplex;asaresult,itdoesnotprovidefurtherphysicalinsighttotheproblemandwillnotbediscussedhere.Inthissection,Eq. 5 issolvenumericallyforthediscussedcaseoftheleftvortex,andadditionallyfortheinducedpressurefromtherighttipvortexusingrRinplaceofrL.Combiningthetworesultsyieldsthetotalwingrollmomentcoefcientasafunctionofand. 5.2.5ValidationwithExperimentalDataThemodelforrollstallpresentedinEq. 5 canbenumericallyintegratedandplottedagainstwindtunnelresultspreviouslypublishedbytheauthorsforrectangular 151

PAGE 152

wingsofAR=0.75,1,1.5and3ataReynoldsnumberof7.5104[ 83 ];resultsareshowninFig. 5-4 a5-4 d.Inaddition,therollstabilityderivativeCl,computedfromthemodeledrollmomentiscomparedwithexperimentallyobtainedvaluesinTable 5-2 A B C DFigure5-4. Rollmomentmodel(numericallyintegratedEq. 5 )comparedwithexperimentaldata.A)AR=0.75,B)AR=1,C)AR=1.5,D)AR=3. Thebestagreementwiththeexperimentaldataisexhibitedbytheloweraspectratiosatlowtomoderatesideslipangles.Intheseregions,thepressuregradientofthesideslippingwingisbetterrepresentedastheassumptionthatthenear-centerpressurep0remainsconstantismoreaccurate.Animportantpointtonoteisthattheangleatwhichrollstalloccurs,typicallybetween20)]TJ /F8 11.955 Tf 10.89 0 Td[(30,isqualitativelypredictedbythetrailingvortexmodelforaspectratiosuptoAR=1.5;thisisthecaseeventhoughtheliftcurve 152

PAGE 153

usedtoestimatethevortexstrengthstallsat>40forAR=0.75and1[ 75 ].Thisindicatesthatalthoughcirculatoryowaroundthetipsispresentandhelpstopreventliftstall[ 76 ],theincreasedheightofthevortexsystemabovethewingreducestheimpactontherollmomentattheseanglesofattack.Thisrepresentationislesseffectiveforthehigheraspectratiowing,whichsuggeststhattherepresentationofthewingasasinglehorseshoevortexismoreapplicabletoLARwings;higherwingspansaremorelikelytoapproachtheclassicellipticvorticitydistribution[ 98 ].Itshouldbenotedthatthemodeldoestendtooverpredicttherollmomentforanglesofattackofupto=10forAR=3. Table5-2. ComparisonofCl,valuesobtainedfromexperimentaldataandmodel. Cl,AR=0.75AR=1AR=1.5AR=3AR=100 =5Experiment-.008-.048-.0556-.0403N/AModel-.0796-.073-.056-.046-.0016=10Experiment-.0704-.101-.1102-.0675N/AModel-.1197-.115-.0921-.0653-.003=15Experiment-.1423-.161-.146-.1399N/AModel-.1553-.154-.097-.0451-.0039=20Experiment-.197-.214-.196-.2797N/AModel-.1792-.201-.1083-.0294-.0047 TherollstabilityderivativespresentedinTable 5-2 arecomputedusingexperimentalandpredictedvaluesofClinEq. 5 ;astheassumptionsmadeinthederivationofthetrailingvortexmodelaremostapplicableatlowersideslipvalues,bothcasesusevaluesoftherollmomentat=5tocomputeCl,. Cl,=Cl =Clj=5)]TJ /F3 11.955 Tf 11.96 0 Td[(Clj=)]TJ /F5 7.97 Tf 6.59 0 Td[(5 5)]TJ /F8 11.955 Tf 11.96 0 Td[(()]TJ /F8 11.955 Tf 9.3 0 Td[(5).(5)ThevaluesofCl,computedfromtheexperimentaldataaretypicallywellrepresentedbythepredictedvalues.Whilecertainregionsarelesseffectivethanothers,suchastheAR=0.75caseat=5ortheAR=3caseathighanglesofattack,mostestimates 153

PAGE 154

nearthenominalangleofattackrangeforMAVs(10)providegoodresults.Atworst,thepredictedvaluescanbeconsideredorderofmagnitudeestimates;atbest,suchasfortheAR=1.5wingat=5,thevaluesarecorrecttowithin1%.Thevaluesoftherollstabilityderivativeforatheoreticalhighaspectratiowing(AR=100)arealsoincludedinTable 5-2 .Typically,therollmomentofconventionalaircraftisattributedtothetailsurfacesorwingdihedral,nottheinherentaerodynamicforcegeneratedbythetipvortices.TheresultsforAR=100illustratethat,whilealoadingasymmetryandassociatedrollmomentaregenerated,thevaluesaretwoordersofmagnitudesmallerthantheAR=1results;thiseffectivelyjustiestheinherentassumptioninconventionalaircraftdesignthatarectangularwingdoesnotcontributetotherollstabilityderivative.TheeffectsoftipvortexdisplacementonLARwingsareseentobefarmoresignicantasthesewingsareassociatedwithafundamentallydifferent,tipvortexdominatedowregime.Furthermore,despitetheapproximateandidealizednatureofthetrailingvortexmodel,reasonableestimatesoftherollstabilityderivativemaybeobtained.ThepredictivemodelisseentobecapableofreproducingexperimentalresultsforLARwings,andindicatesasignicantlyreducedmagnitudeforhighaspectratiowings. 5.3RollDampingatLowAspectRatios 5.3.1HighAspectRatioModelWhilethemodelforrollstallandtheassociatedCl,derivativeneededtobenewlydeveloped,atechniqueforobtaininganorderofmagnitudeestimateoftherolldampingforhighaspectratiowingsisreadilyavailable[ 14 ].Asawingrollsaboutitscenterofgravity,thedownwardsvelocityactingonthehalfofthewingrotatingupwardsreducestheeffectiveangleofattackonthathalfofthewing.Conversely,astheotherhalfofthewingrotatesdownwards,itexperiencesanincreaseinupwardsvelocityandacorrespondingincreaseineffectiveangleofattack.Theseresultscombinetoreducetheliftontheupwards-movingwinghalfandtoamplifytheliftonthedownward-movingwing 154

PAGE 155

half,dampingoutthemotion.ThisdevelopinginducedvelocityisdepictedinFig. 5-5 ,andcontributesstronglytotherollsubsidencemode. Figure5-5. Spanwise-asymmetricinducedlocalangleofattackponanairplanewinginrollingight;adaptedfrom[ 14 ]. GivenarollingmotionasshowninFig 5-5 ,anestimatefortherestoringrollmomentcreatedbytheeffectivesectionalangleofattackcanbeestimated: Cl,p=)]TJ /F3 11.955 Tf 14.42 8.09 Td[(b 2SZ10cl()c()d(5)whereisthenondimensionallaterallocationonthewing,2y=b;cl()isthewingsectionalliftcoefcientspanwisevariationduetothelocalangleofattack;pistheinducedantisymmetricangleofattackduetounitvalueofnondimensionalrollratepb=2U;c()isthelocalwingchordwhichmaybeafunctionof;andSisthewingarea.Thiscanbeexpressedintermsoftheliftcurveslopeofthewing(whichisassumedtobeconstantalongthespan): Cl,p=)]TJ /F3 11.955 Tf 14.42 8.08 Td[(b 2SZ10@CL @c()2d.(5) 155

PAGE 156

Foratheoretical(highaspectratio/liftingline)liftcurveslopeof2andawingofconstantchord,Eq. 5 reducesto Cl,p=)]TJ /F7 11.955 Tf 10.5 8.09 Td[( 3.(5)Whilethisexpressionprovidesreasonableapproximationsathigheraspectratios[ 14 ],experimentalresultsrecentlypublishedbytheauthorshaveshownthatatplatewingswithAR2donotexperienceanyrolldampingintheReynoldsnumberregimeofMAVs[ 94 ].Eq. 5 indicatesthat,foranynonzeroliftcurveslope,Cl,pwillalsobenonzero;assuch,theequationmustbemodiedtoaddresslowaspectratioconditions. 5.3.2EffectsofAspectRatioonLiftCurveSlopeInordertoinvestigatethedependenceoftherolldampingderivativeonaspectratio,arelationshipbetweenCL,andARmustbeobtained;thismaybeusedtoevaluateEq. 5 atanyaspectratioandwillalsodetermineatwhatvalueofARtheliftcurveslopeapproachesthetheoreticalvalueof2.Whilethenonlinearandthree-dimensionaleffectsofthetipvorticeshaveastrongimpactontheliftofLARwings[ 68 69 ],sufcientexperimentalevidenceexiststobuildasatisfactorymodel.Fig. 5-6 comparestheliftcurveslopeinthelinearregion(<10)withtheaspectratioofrectangularatplatewingsataReynoldsnumberofRe=8104;dataaretakenfrompreviouslypublishedresults[ 68 75 ].Anominallylinearrelationshipisnoticeable,whichmaybeextrapolatedtothetheoreticalliftcurveslopevalueof2forHARwings;Fig. 5-6 indicatesthatthiswilloccurnearanaspectratioofAR=10forthisReynoldsnumber.WhencomputingtherolldampinginEq. 5 ,thislinearrelationshipmaybeappliedtoobtainCL,;forAR10,thetheoreticalvalueof2willbeimplemented. 5.3.3TheNatureofRollDampingWhilevaluesofCL,predictedusingthelineartofFig 5-6 maybeusedtoestimatetherolldampingatanyaspectratio,thisstilldoesnotcorrelatewithexperimentalresultsforAR2asEq. 5 showsthatevenareducedCL,willcreatesomedamping.A 156

PAGE 157

Figure5-6. ComparisonoftheliftcurveslopeandaspectratioofatplatewingsatRe=8104.Therst5datapointsaretakenfrompublishedexperimentaldataandareusedtocomputethebesttline[ 68 75 ];thisisextrapolatedtothetheoreticalliftcurvevalueof2,whichoccursnearAR=10. newmetricwhichdeterminesatwhataspectratiorolldampingbecomessignicantisnecessarytoeffectivelycomparehighandlowaspectratiolateralstability.Fig. 5-5 demonstrateshowtherollrotationofawingwillinducesomerestoringmomentduetotheinducedvelocityalongthespan;whilethisloadwillalwaysexist,itisproposedherethatforlowaspectratiositislesssignicantthanthemomentumoftherotatingwing.Thesetwoquantitiesmaybecomparedusingthedenitionoftorque,whichstatesthattorqueisthetimederivativeoftheangularmomentumMofanobject: =dM dt.(5)Thevariationinsectionalangleofattackduetorollrate,p,maybecomputedusinggeometricargumentsasshowninFig. 5-5 .WiththemodelofCL,,itispossibletoestimatethesectionalvariationinliftL(y)foranyaspectratioand,integratingthisalongthewingspan,computetherestoringmoment(depictedinFig. 5-7 ).Theangular 157

PAGE 158

momentumis,bydenition,theproductoftherollmomentofinertiaandtherollrate,ie,M=Ixp.SolvingEq. 5 willyieldthetimetakenfortherestoringrollmomenttoovercometheinitialangularmomentum.ThefollowingderivationwillbecompletedinnondimensionalunitsandwillproduceaparameterTfwhichessentiallybalancestheinertialmomentumoftherollingwingwiththerestoringmomentscreatedbythesectionallift.Tfisthe(nondimensional)timewhichisrequiredfortherestoringmomenttostopthemotionofthewing,andwillthusbereferredtoasthedamptime. Figure5-7. SectionalliftforceL(y)generatedbyrotatingwingwith(positive)rollratep;integratingthisliftalongthespangivestherestoringmoment. 5.3.4DerivationoftheDampTimeParameterThefollowingnondimensionalizationswillberequiredtoformthesolutionofEq. 5 : t=tb 2U,p=p2U b,IxIxwingb 25,M=M1 16wingUb4,L=1 2airU2SCL. Notethattwodifferentdensitiesareused:theairdensity,air,andthedensityofthewing,wing.Thesectionalliftmaybedescribedasfollows,assumingthattheliftcurveslopeisconstantalongthespan: L(y)=@L @p=@L @py 2U.(5)IntegratingEq. 5 providesanexpressionforthetotalrestoringmomentexertedontherollingwingbythevariationinsectionallift.InEq. 5 ,isnegativeasit 158

PAGE 159

opposestherollratep. =)]TJ /F8 11.955 Tf 9.29 0 Td[(2Zb=20L(y)dy,=)]TJ /F8 11.955 Tf 9.29 0 Td[(2@L @p 2UZb=20ydy,=)]TJ /F7 11.955 Tf 10.86 8.09 Td[(@L @p U1 2y2b=20,=)]TJ /F7 11.955 Tf 10.86 8.09 Td[(@L @pb2 8U.(5)TheexpressionforinEq. 5 maynowbesubstitutedintoEq. 5 ,nondimensionalizedandsimplied.ThesubstitutionS=bcisusedtodenetheplanformareaofrectangularwings. )]TJ /F7 11.955 Tf 10.87 8.09 Td[(@L @pb2 8U=dM dt,)]TJ /F10 11.955 Tf 11.29 16.86 Td[(1 2airU2bc2U b@CL @pb2 8U=1 16wingUb42U bdM dt,)]TJ /F7 11.955 Tf 10.49 8.09 Td[(airU2cb2 8CL,p=wingU2b3 8dM dt,)]TJ /F3 11.955 Tf 9.3 0 Td[(CL,p=wing airb cdM dt,)]TJ /F3 11.955 Tf 9.3 0 Td[(CL,p=wing airARdM dt,(5)wherethenalsteprecognizesthattheaspectratioofrectangularwingsisAR=b=c.Eq. 5 isnowintegratedfrom0tTftodeterminethenondimensionaltimerequiredfortherestoringmomenttohaltthemotionofthewing;forsimplicity,therollrateisassumedtodecaylinearlywithtimefromaninitialrollratep0,ie,p(t)=)]TJ /F3 11.955 Tf 11.68 8.09 Td[(p0 Tft+p0.Furthermore,notethattheinitialangularmomentumisgivenbyM0= 159

PAGE 160

Ixp0=1 16wingUb4.SubstitutingtheseexpressionsintoEq. 5 andintegratingyields: )]TJ /F3 11.955 Tf 9.3 0 Td[(CL,ZTf0)]TJ /F3 11.955 Tf 11.68 8.09 Td[(p0 Tft+p0dt=wing airARZ0M0dM,)]TJ /F3 11.955 Tf 9.3 0 Td[(CL,)]TJ /F3 11.955 Tf 14.81 8.08 Td[(p0 2Tft2+p0tTf0=wing airAR(M)j0M0,)]TJ /F3 11.955 Tf 9.29 0 Td[(CL,)]TJ /F3 11.955 Tf 10.49 8.09 Td[(p0T2f 2Tf+p0Tf=)]TJ /F7 11.955 Tf 10.5 8.23 Td[(wing airAR0B@Ixp0 1 16wingUb41CA,)]TJ /F3 11.955 Tf 9.29 0 Td[(CL,)]TJ /F3 11.955 Tf 10.49 8.08 Td[(p0T2f 2Tf+p0Tf=)]TJ /F3 11.955 Tf 9.3 0 Td[(AR0BBB@Ixp0 airb 251CCCA,CL,Tf 2=IxAR,Tf=2IxAR CL,.(5)ThedamptimeTfshowninEq. 5 isessentiallyabalancebetweentheangularmomentumoftherotatingwing(theIxARterm)andthedegreetowhichtherestoringrollmomentprovidedbythesectionallift(representedbyCL,)canovercomethewinginertia.Assuch,itfollowsthatifTf>1,therestoringmomentissmallerthantheinertialcomponentandtherotationofthewingwillnotexperiencesignicantdamping.Conversely,ifTf<1thentherestoringmomentissignicantandtherolldampingwillhaveacontribution.Itisinterestingtonotethattheinitialrollratep0doesnotappearinthenalequationforTf,sotheseresultsareindependentoftheactualrollrate.Higherrollrateswillincreaseboththedampingandthemomentumbythesamefactor,sothephysicsoftheproblemdonotchange.ResultsforTfasafunctionofaspectratioareplottedinFig. 5-8 .TheresultsforTfindicatethatwhentheaspectratioislessthan3.4,therestoringmomentscreatedbytheinducedliftalongthespanarelesssignicantthantheinertiaoftherotatingwing;asaresult,Tf>1.Thiscorrespondswiththerecentexperimental 160

PAGE 161

Figure5-8. NondimensionalparameterTfforincreasingangleofattack.ForAR<2,experimentaldatashowednorolldamping;thiscorrespondstoanareainthegurewhereTf>1.AtAR=10,therestoringmomenttermisanorderofmagnitudesmallerthantheinertialterm(Tf0.1). data,takenatAR2,inwhichnorolldampingwasobserved[ 94 ].Whentheliftcurveslopeapproaches2nearAR=10,therestoringmomenttermisanorderofmagnitudegreaterthantheinertialterm(Tf0.1);thismaybeconsideredtorepresentthehighaspectratiocasewhererolldampingissignicant.ThisinsightintothedecreaseofrolldampingatlowaspectratiosprovidesacriterionwhichmaybeusedtodeterminethecasesinwhichCl,pmustbeaccountedfor;forAR<3.4,thedamptimeisgreaterthanoneandtherolldampingderivativeisconsideredtobenegligible.Forhigheraspectratios,Cl,pmaybeestimatedusingEq. 5 andtheliftcurveslopecomputedfromthelinearrelationshipofFig. 5-6 .ThismakesitpossibletoevaluateEq. 5 andtocomputetheeigenvaluesofrectangularwingsforanyaspectratio. 5.4StabilityAnalysisatVaryingAspectRatioTheeigenvaluesoftheapproximatelateralsystemdescribedbyEq. 5 canbeusedtoevaluatetheopen-loopstabilitycharacteristicsofhighandlowaspect 161

PAGE 162

ratiowings;aspreviouslydiscussed,thissimpliedrstordersystemispopulatedwithonlytwostabilityderivatives,Cl,andCl,p,whichwereshowntobetheprimarycontributorstothedynamicresponse.ThemodelspresentedinEqs. 5 and 5 providereasonableestimatesforthesederivativesatanyaspectratio,whichpermitstheeigenvalues()tobecomputedandcompared.The33systemwillproducethreeeigenvalues,oneofwhichisalwaysrealandtheothertwoofwhichformacomplexpair.Aconventional(44)lateralstabilitymatrixtypicallyproducesfoureigenvalueswhichdescribethespiral,rollsubsidence,andDutchrollmodes;asthespiralmoderequiresparticipationbytheyawrateandheadingangle,bothofwhichareneglectedinEq. 5 ,theapproximatedmodesarerollsubsidence(realeigenvalue)andDutchroll(complexpair).Thesemodesarewelldenedforconventionalaircraftandarediscussedinanytextbookonightdynamics[ 13 14 ];thedirectinuenceofhighandlowaspectratiowings(neglectinganyeffectsfromcontrolsurfacesorverticaltails)willbecharacterizedinthissection.Tocomputetheeigenvaluesoftheapproximatelateralsystem,Eq. 5 ispopulatedwithanalyticallydeterminedvaluesofCl,andCl,patatrimangleofattackof=10;theReynoldsnumberisheldconstantforallaspectratiosandthemasspropertiesareobtainedassumingthatthewingretainsaconstantdensity.Fortheeigenvaluesatagivenaspectratio,theratioZisdenedas Z=krealk kcomplexk.(5)ThequantityZcomparesthemagnitudesoftherealandcomplexeigenvaluestodeterminewhichismoredominant;physically,thisdetermineswhethertherollsubsidenceortheDutchrollmodecontributesmoresignicantlytotheinitialresponseofthewingtoanarbitraryperturbation.IfZ>1,rollsubsidenceismoredominantwhereasifZ<1,theoscillatoryresponseoftheDutchrollwillbemoreprevalent.TheeigenvaluesandtheZ-ratioarecalculatedforarangeofaspectratios(0.75
PAGE 163

100),permittingacontinuousdistributionofthestabilityparameterstobecomputedandplottedinFig. 5-9 and 5-10 ;theformerplotdisplaystherealandimaginarycomponentsofthecomplexpair,andthelattershowstheZ-ratio,realandthemagnitudeofcomplexasafunctionofaspectratio. Figure5-9. ComplexeigenvaluesforhighandlowaspectratiowingscomputedusingEq. 5 .Aspectratioincreasescontinuouslyfromrighttoleft. ThebehavioroftheeigenvaluesrevealsasubstantialdegreeofinformationaboutthelateraldynamicsofwingsatthisReynoldsnumber,whichareseentoinhabitcontrastingstabilityregimesathighandlowaspectratios.Experimentalresultspertainingtothelattercasewerediscussedinarecentpublicationbytheauthors;anunstableDutchrollresponsewasshowntodominatetheresponseofthewingtoperturbationsfromequilibriumandwasprimarilyattributedtothemagnitudeofCl,.Furthermore,thecharacteristicsoftheresponseweredemonstratedtobecoupledtothelongitudinalstabilityofthewingduetothedependenceofrollstallonangleofattack[ 94 ].ThemagnitudeofZ,whichisshowninFig. 5-10 tobelessthanunityforthelowaspectratiocases,indicatesthatreal(whichrepresentsthedampedrollsubsidence 163

PAGE 164

Figure5-10. Realandcomplexeigenvaluemagnitudes(leftaxis)andZ-ratio(rightaxis)asafunctionofaspectratio. mode)isovercomebythemagnitudeoftheDutchrolleigenvalue.Therefore,therangeofaspectratiosrelevanttoMAVs(AR1)isdominatedbytheunstablecomplexshowninFig. 5-9 ,whichisgeneratedbythepresenceoftherollstabilityderivativeandthenegligibleamountofrolldamping.ThisistrueforaspectratioslessthanAR4.5,althoughastheaspectratioincreasesthemagnitudeofthereal(stable)termbecomeslarger,indicatingthataLARwingsubjecttoaperturbationwillexperienceaninitiallydampedmotionpriortothedivergentDutchroll.Foraspectratiosgreaterthan4.6,therealcomponentofthecomplexeigenvalueisnegativewhichmeansthattheaerodynamicloadingpropertiesofthewingnowcontributetothestableDutchrollmodecommonlyassociatedwithhighaspectratioaircraft.ThedecreasingmagnitudeoftheeigenvaluesuggeststhattheactualimpactofthewingontheDutchrollmodewillbelesssignicantthanthatofthetailsurfaces,wingdihedral,andothermoreconventionalfactors[ 104 ].Furthermore,theZ-ratiocontinuestoincrease,indicatingthetendencyofthewingtoexhibitrollsubsidencebehaviorrather 164

PAGE 165

thanDutchroll.Forveryhighaspectratios(anumericallysolventcaseofAR=100isplottedinFig. 5-9 torepresenttheinnitewing),thecomplexeigenvalueapproacheszeroandshowsthattheDutchrollmodeisnolongerpresent.Thisisbecausean`innitely'longwinghasnotrailingvorticesand,thus,norollstallexiststoexcitetheaerodynamicDutchrollresponse.Itisinterestingtonotethat,inthissense,theaspectratioofthewingessentiallyactsasastabilizinggainfactor(acompensator)whichdrivesthesystemtowardsstabilityasthe`gain'isincreased. 5.5FutureResearchDirectionsThisstabilityanalysisprovidesanewperspectiveonMAVdesign.Whilethesevehicleshavebeenanecdotallyreferredtoas`gustsensitive'or`jittery',theirexistenceinafundamentallydifferentstabilityregimethanconventionalaircraftwithhighaspectratiowingshasnotbeenpreviouslyconsidered.ThisimprovedunderstandingoftheinherentstabilitypropertiesoftheLARwingsusedinMAVdesignmaybeappliedforanumberofpotentialresearchapplications(owcontrol,maximizingendurance,etc),althoughtwoparalleldirectionsareofmostinterest:improvedmaneuverabilityorimprovedstability.Bothwillbediscussedinthefollowingsections,undertheassumptionthataMAVwingwillincorporateawingwithanear-unityaspectratio. 5.5.1ImplementationofRollStallasaLateralControlMechanismAcommonafictionofmanyMAVdesignsistheineffectivenessofconventionalcontrolsurfaces.Thehighlycambered/reexedairfoilsoftenincorporatedintoMAVwings,coupledwiththeinherentpropertiesoflowReynoldsnumberaerodynamics,resultinlargeregionsofseparatedowtowardsthetrailingedgeofthewing.Consequently,controlsurfaces(ailerons,elevators,rudder,elevons,etc)whichareplacedinthisregioncanonlyimpartalimiteddegreeofauthority.MAVighttestingexperiencehasindicatedthatthisimpedestheabilityofapilottocontrolaMAVduringaggressivemaneuvers(orevenasimpleclimbingight),orwhenrespondingtoagustperturbation.Conventionaltechniques,suchasincreasingthesizeofthecontrolsurfaces,areoftennotfeasibleas 165

PAGE 166

MAVwingshaveonlyalimitedamountofavailablespace;addingnewsurfacesisnotidealasavailablemassmarginsareslimornonexistentforMAVs.ThepresenceofrollstallprovidesaninherentmechanismbywhichthelateraldynamicsofaLARwingmaybecontrolled;ifasideslipangleisintentionallyprovidedtothewing,thegeneratedrollmomentmaybeusedtocontrolthevehicle.(ItshouldalsobenotedherethattheresultsofChapter 3.7 indicatethatMAVwings,inadditiontothecanonicaltestcases,alsoexperiencerollstall).UsingtheexperimentaldataandtrailingvortexmodelofthisthesisforabaselineunderstandingofhowrollstallaffectsLARstabilityproperties,acontrolschemecanbedesignedwhichinitiatesaspecicsideslipangletocreateanassociatedrollmoment,andusesthismomenttomaneuver.AninherentchallengeherewouldbeovercomingthedivergentDutchrollmodesseeninthisresearch,whichwoulddevelopduringsideslip;however,theprovideddatashowsthattheyarenominallylinearoverareasonablecontroltime(t<3s),whichwouldgreatlysimplifythedemandsonanautopilot.Aconceptforutilizingrollstallasacontrolmechanismwouldinvolvedevelopinganautopilotspecicallyforayingwingplanform;thedynamicmodelusedintheautopilotwouldexactlydeterminetherollmomentforagivenincidenceangleofattackandsideslip.TheautopilotcouldbeplacedonayingwingMAVwithnocontrolsurfaces,notailsurfaces,andnoservos-simplyawingandagimbaledpropeller.Byslightlyadjustingthethrustvector,arollmomentcouldbegeneratedthroughrollstalltomaneuverthevehicle.Theadvantagestothisuniquedesignaresignicant.AcommonissuewithMAVdevelopmentisthatslightvariationsintail-surfacemountingcanhavedrasticeffectsonvehiclestability.Withthisdesign,notailsurfacesarerequired;thus,alargenumberofrepeatableMAVScouldberapidlyproducedandusedtodevelopaswarmwheretheightdynamicsofeachmemberareidentical.Furthermore,eachmembercouldpotentiallybemoremaneuverablethanaMAVwhichusesconventionalcontrolsurfaces 166

PAGE 167

thatarelimitedbytheseparatedow.Finally,thelackofsuperuoustailsurfacesandservoswouldallowsignicantweightsavingsandthusincreasepayloadcapacity.Whilethedevelopmentofsuchavehicleandautopilotsystemwouldrequireasubstantialdegreeofwork,thenumerousbenetswouldgreatlyincreasethecapabilitiesofsuchaMAV. 5.5.2PassiveandActiveStrategiesforIncreasedStabilityWhiletheprevioussectiondemonstratedanoveltechniquebywhilerollstallcouldbeimplementedforaggressivemaneuvers,anumberofMAVapplicationsrequireamorestablevehicle.Attenuatingthemagnitudeofrollstallprovidesameanstoimprovethelateralstabilityofthewing,andcanbeaccomplishedusingeitheractiveorpassivestrategies.Thepremisebehindthisconceptisbaseduponthetrailingvortexmodel,whichshowsthatthecloseproximityofthetipvorticestothewinginducesrollstall,whichinturncreatesunstablemodes.Bydisplacingordisruptingthesetipvortices,themagnitudeofrollstallmaybedecreasedandthestabilitypropertiesmodied;somepreliminarydatashowingtheimpactofwingletsonreducingrollstallwerediscussedinChapter 3.6 .Theseresultsprovidethebasisforfutureresearchdirections.PassivetechniquesforreducingtheimpactofrollstallonLARwingswouldincorporatewingplanformdesign,verticalgeometriessuchaswinglets,andnextlevelstudiesintothematerialsusedtofabricatethewings.First,bydesigningawinginwhichthedownstreamareafromtheleadingedgewingtipisminimizedwillreducetheareaaffectedbythetrailingvortex.Thiswouldinvolvesignicantwindtunneltestingtoensurethatthenewdesignsdonotsignicantlyaltertheliftcharacteristics.Additionalstudiesintowingletsmayprovideanoptimaldesignwhichdisruptstheformationofthetrailingvortexwithoutadverselyaffectingthedragorweightdistributioncharacteristics.Finally,activewingmorphingcouldbeusedtodeecttheupstreamwingtipduringasideslipperturbationtorelocatethetrailingvortexawayfromthemainliftingsurface. 167

PAGE 168

Activeowcontroltechniquesprovideaninterestingavenueofexploration;anumberofcurrentstudieshaveinvestigatedtheuseofsyntheticjetactuatorsforenforcinganattachedboundarylayerforlowReynoldsnumberwings.Theymayalsobeutilizedtodisruptordeectthetrailingvorticesand,thus,reducethemagnitudeofrollstall.Twoseparatedirectionsmaybepursued;rst,thedisruptionofthetipvorticesmaybeachievedbyalteringthecirculation(ortangentialvelocity)ofthevortices.AsseeninEq. 5 ,thiswillultimatelyreducethepressuregradientemanatingfromthecoreofthevortexandwillreducethepressuredistributiononthewingsurface.ByarranginganarrayofsyntheticjetsnearthewingtipsofaLARwing,andcommandingthemtoreintothetrailingvortexduringsideslipperturbations,thecirculationofthevortexwillbereducedandwillresultinareducedrollmoment.Asecondapproacheffectofthistechniqueisthat,byinjectingupwardsvelocityfromthewingtowardsthecoreofthetrailingvortex,theeffectiveangleofattackbetweenthewingandthevortexwillbeincreasedasseeninEq. 5 .Thisincreasestheradialdistancefromthecoretothewingandwillalsoreducethepressureonthewingsurface.BothofthesemethodswouldrequireextensivePIVtestingtobetterdescribethetipvorticesinsideslipandtodeterminehowtheinjectionofmomentumfromthesyntheticjetscouldpotentiallyaffecttheirformation.TheseexamplesprovideanexplanationofthecriticalnatureofrollstallforLARwingsandhowanimprovedunderstandingmaybeusedtoprovidetwoparallelavenuesofinvestigationwhichmayeitherresultinmorestableormoremaneuverablefutureMAVs. 168

PAGE 169

CHAPTER6SUMMARYANDCONCLUSIONSThecapabilitiesofMicroAerialVehicles(MAVs)haveadvancedconsiderablyinrecentyearsandhaveledtotheirwidespreaduseinanumberofapplications.Whilethedesignersoftheseaircrafthavegrownincreasinglyadeptatbuildingyingvehicles,thenatureoftheaerodynamicloadinganditsassociatedimpactonthestabilitycharacteristicsremainpoorlyunderstood.Withoutaviabledynamicmodelfortheforceandmomentdependenciesonthestaticanddynamicpositioningofthevehicle,MAVsremainsusceptibletoinstabilitiesduetoperturbationsfromequilibriumasitisdifculttopredicttheirresponsetogusts.Theauthor'sexperienceofighttestingMAVsdemonstratedthepotentialforunrecoverablerollsafterexperiencingsmallsideslipperturbations;thechallengesofalleviatingtheseinstabilitieswiththeuseofconventionaltailsurfacedesignssuggestedthattheproblemwasnotentirelyduetogeometriceffects,butthattheaerodynamicsoflowaspectratio(LAR)wingcreatefundamentallydifferentresponsecharacteristicsthantraditionalaircraft.Abetterunderstandingofhowtheshortwingspaninuencestheloadingandstabilitypropertiesofthewingitselfisnecessarytodevelopareasonabledynamicmodelwhichcanbeusedtoimprovevehicledesignanddevelopefcientcontrollawsfortheseuniqueiers.Theapproachofthisthesisisanexperimentallybasedwindtunnelinvestigationwhichpermits(indeed,requires)ahighdegreeofversatilityinthetypesoftestingbeingperformed.Aspreviouslymentioned,itisnecessarytocharacterizetheaerodynamicloadingforLARwingsforvariousstaticanddynamicpositions.Toachievethis,theModelPositioningSystem(MPS)wasdesignedandfabricatedtoprovideuniquemaneuveringcapabilitiesinthegroup'slowReynoldsnumberwindtunnel(thePrototunnel).TheMPSallowsactuationinfourdegreesoffreedom(roll,pitch,yaw,plunge)andthuspermitsstatictestingatincidenceanglesofattack,sideslipandbank;furthermore,itcanbeusedtoconductdynamictestingcorrespondingtoaerodynamic 169

PAGE 170

ratesofroll,pitch,yaw,andtranslationalaccelerations.Themicroloadingtechnology(MLT)forcebalancewascalibratedandimplementedintotheMPSsetup,permittingthemeasurementsofallsixaerodynamiccomponents(normal/axial/sideforceandroll/pitch/yawmoments).TheowenvironmentofthePrototunnelwascharacterizedusinghotwireanemometryandwasfoundtohavesatisfactoryowhomogeneity.Experimentalproceduresforstaticanddynamictestingweredeveloped,validatedandimplementedalongwithpostprocessingtechniquesrequiredtoextractthedata.Finally,owvisualizationwasconductedtodescribethenatureoftheaerodynamicloadingwithaparticularemphasisonthebehaviorofthetipvorticesduringsideslipperturbations,whichwereobservedtoskewasymmetricallywiththe`upstream'vortexbeingcenteredoveraninboardsectionofthewingandthe`downstream'vortexbeingconvectedawayfromthewingbythefreestreamow.Withtheexperimentalfacilitiesdeveloped,thelongitudinalloading(lift,drag,pitchingmoment)ofaseriesofatplate(0%camber)wingsofvaryingaspectratiosandtaperratioswasinvestigatedatReynoldsnumbersbetween5104and1105.InitialtestswerefavorablycomparedwithclassicalresultsandconrmedanumberofpreviouslypublishedcharacteristicsofLARaerodynamics,suchasthehighstallangleofattackandthedeparturefromthetheoretical2liftcurveslope.Furthermore,testsatincreasingsideslipanglesindicatedthatthelongitudinalloadsdidnotvarysignicantlyatanglesofattackbelow20.ThemostsignicantchangewasinthepitchstabilityderivativeCM,whichwasnonzeroatlowanglesofattackandincreasedfor>10atthehighestsideslipangletested(=35).Thesendingsindicatethat,fromacontrolsperspective,moderatelateraldisturbancesfromequilibriumarenotexpectedtosignicantlyinuencetheliftordragcharacteristicsofthewing.Anentirelydifferentdevelopmentwasuncoveredbymeasuringthelateralloads(sideforce,rollmoment,yawmoment)ofthesamewingsinsideslipataReynoldsnumberof7.5104.Therollmomentofthewingswasfoundtoincreaselinearlyupto 170

PAGE 171

amaximumatanglesofattacknear20beforedecreasinginmagnitudeinasimilarmannertothewellknownliftstall.Thegradientoftherollmomentresponseincreasedatgreatersideslipangles,resultinginhigherrollmoments.Thisbehavior,referredtoas`rollstall',hasnotpreviouslybeendescribedforLARwingsatlowReynoldsnumbers;itwasattributedtotheaforementionedtipvortexasymmetryinsideslip,whichinducesanasymmetricspanwiseloadingandanincreasedlocalangleofattackattheupstreamwingtip.TheafliatedrollstabilityderivativeCl,wastypicallyontheorderof-0.15fortrimanglesofattackbetween5<<10,whichisasignicantmagnitude;conventionalaircrafttailsurfacesaredesignedtoproducearollstabilityderivativeoflessthan-0.10toproduce`goodhandlingcharacteristics'.ThefactthatthislargevalueofCl,existsforatplateLARwingswithnoneofthegeometricfeaturestowhichtherollstabilityderivativeisattributed(verticaltailsurfaces,wingsweep/taper,fuselage,etc)isremarkable;thisindicatesthatwingsofanaspectrationearunityoperatingatlowReynoldsnumbersinherentlygeneratetheirownlateralloadspurelyduetothedominanteffectsofthetipvortices.Rollstallwasalsofoundtoaffecttheyawmomentandsideforcetoalesserextentduetotheinduceddragoftheattachedtipvortex,whichcreatedasmallbutrestoringmagnitudeofCn,.AdditionaltestsconductedwithwingletsmountedtothewingtipsofthemodelsindicatedthepotentialtoreduceCl,bydisruptingtheformationofthetipvortex;furthermore,thePIPERMAVwastestedwithandwithouttailsurfacestoconrmthattheeffectsofrollstallarestillprevalentforacompletevehicleaswellasthecanonicalatplatewings.ThenatureofthelateralstabilityderivativesCl,andCn,(negativeandpositive,respectively)correspondstothesignsofthesevaluesforconventionalaircraftandsuggestthepossiblepresenceofperiodicstabilitymodes.Acompletestabilityanalysisorformulationofadynamicmodelrequiresestimatesoftheloadingdependenciesonaerodynamicrates,commonlyreferredtoasthedampingderivatives.Asystematicexperimentalinvestigationintothesevaluesforatplatewingshasnotbeenconducted; 171

PAGE 172

hence,aforcedoscillationtechniquewasimplementedtomeasuretheeffectsofrotational(p,q,r)andtranslational(_and_)velocitiesontheaerodynamicloads.Testswereconductedatawiderangeofincidenceangles(050and035)andfrequencies(0Hzf3Hz)tocharacterizethedependenciesofthedampingderivatives.Derivativesdueto_and_werefoundtobenegligible;however,derivativesduetotherotationalratescouldbeconsistentlyestimated.Aclearfrequencydependencewasnoticeablewiththelowestfrequenciesdemonstratingthelargestmagnitudeswhichconvergedtowardssmallervaluesatf=3Hz.Anumberofcross-coupledderivatives,suchasCZ,pandCl,qatincreasedsideslipanglesastheinteractionsbetweenboundcirculationandleadingedgeeffectswithtipvortexpropagationcouplethelateralandlongitudinalloading.Still,themagnitudesofthedampingderivativesweretypicallysmallrelativetothestaticderivativesandwillhavelesseffectonthestabilityresponseofthewings.Inmanycasestheabsenceofaderivativeismoresignicantthanthosewhichweremeasured;forexample,thelackofarolldampingderivativeatequilibrium=0ightconditionsisindicativeofthe`jittery'natureofMAVight.ThesystematicinvestigationintothedampingderivativespermitstheformulationofadynamicmodelforLARwingsand,forthersttime,allowsaninvestigationintotheinherentstabilitycharacteristicsofthesewings.TheequationsofmotionforarotatingrigidbodywerederivedforLARaspectratiowings;thebasicgeometry(ie,thecrossproductofinertiaIxziszero)permitsasimpliedversionoftheequationstobeusedwhichdirectlyreectstheimpactoftheloadingduetorollstall.TheresponseofLARwingstoperturbationsfromvariousequilibriumconditionswasdeterminedbynumericallyintegratingtheequationsofmotiontoobtainthetimehistoryofthelateralstatevariables~xlat=fprgT.Theinstantaneousvaluesoftheaerodynamicforcesandmomentsweredeterminedfromtheexperimentaldatausingatwo-dimensionalinterpolationschemebasedonthecurrentvaluesoftheanglesofattackandsideslip;initially,theangleofattackwasheldconstantatvaluesbetween 172

PAGE 173

5<<15.Thelateralloadingcreatedbyrollstallwasfoundtoinduceadivergentoscillatoryresponsetoasideslipperturbationof=1.All~xlatvariableswerefoundtorapidlygrowwithoutboundandreachlargeamplitudes(forexample,max>20within3seconds).Alineartimeinvariant(LTI)modelwasdevelopedfromthecoupledequationsofmotionandwasfoundtoaccuratelyrepresentthenonlinearsolutionfortherst3secondsofmotion.Thispermittedalinearanalysisofthestabilitycharacteristicsofthesystem;complexeigenvaluescorrespondingtothedivergentmodeandaheavilydampedstablemodewerepresentatallanglesofattackconsidered;thenatureoftheformermodewasnominallysimilartoadivergentDutchrollmodebasedontherelativephasingbetweenthestatevariables.Simulationsoftheresponseofthewingtoinitialconditionsspecictothemode,asopposedtotheoriginal,arbitrarysideslipperturbation,yieldedasimilarresponse;thisindicateshowthepurelyaerodynamic,divergentmodeinducedbyrollstalldominatesthelateralbehaviorofLARwings.Thestaticrollstallresultsindicatedacleardependenceofthelateralloadingontheangleofattack;thissuggeststhatthedivergentDutchrollmodewouldbeaffectediftheangleofattackisallowedtovaryalongwiththelateralvariables.Assymmetricwingswithoutthebenetofahorizontaltailareinherentlyunstable,insteadofassigninganinitialangleofattackperturbationandsolvingtheequationsofmotion,asinusoidalangleofattacktrajectorywithasmallamplitudeof3wasprescribedwhichrepresentsashortperiodtyperesponse.Whenthefrequencyoftheangleofattackoscillationswasclosetothenaturalfrequencyofthelateralmode,thenatureofthebankangleresponsewasfoundtofundamentallychange;insteadofoscillatingaboutitsequilibriumvalueof0=0,itwasobservedtodriftawayfrom0whilestilloscillating.Furthermore,thedirectionofthedivergencewasfoundtodependupontheinitialphaseoftheangleofattackperturbation(ie,ifinitiallyincreasedordecreased).Asthenatureofthismodeexhibitsadependenceupontherelativefrequenciesoftheandoscillations,itisreferredtoasthe`rollresonance'mode.Rollresonancehasnotpreviouslybeen 173

PAGE 174

consideredforLARwingsasitrequiresknowledgeoftheeffectsofrollstall;however,itillustrateshowthelateralandlongitudinalaxesarecoupledinthepresenceofminorsideslipperturbationsandhowthisresultsinafundamentalchangeintheresponseofthelateralvariables.ThistypeofresponseisnotwellmodeledbytheLTImodel,whichinherentlyassumesthattheharmonicbehavioriscenteredabouttheequilibriumvalue;however,alineartimevariant(LTV)modelwasdevelopedwhichignores@=@dependenciesbutallowstherollstabilityderivativeCl,tovarywithtime.TheLTVmodelwasfoundtocapturethekeyaspectsofrollresonance,particularlythebankangledivergence;thisindicatesthesignicanceofCl,tothestabilitypropertiesofLARwings,andtheimpactthatrollstallhasoncreatingpreviouslyunconsideredstabilitymodeswhichmayeasilydevelopfromequilibriumightconditions.Todevelopamorequantitativedescriptionoftherollstabilityderivativeatarangeofaspectratios,amodelwasdevelopedforrollstallwhichrepresentsthetrailingvorticesastwosemi-innitelinevorticesandcomputestheirinuenceonthesurfacepressuredistributionofasideslippingwing.Theanalyticalresultswerefoundtomatchwellwiththeaforementionedexperimentaldata,validatingtheassumptionthatrollstallisgeneratedbythetipvorticesofLARwings.Additionally,apre-existingmodelfortherolldampingderivative,Cl,pwasmodiedtocorrelatewithlowaspectratioexperimentaldata(wheredampingwasfoundtobenegligible).Usingonlythesetwoderivatives,asimplieddynamicmodelforthelateralresponseofLARwingswasdevelopedandwasshowntoaccuratelyreectthefull44system.Usingthetheoreticalvaluesoftherollstabilityandrolldampingderivatives,theeigenvaluesofthe33systemcouldbeobtainedforarangeofeigenvaluesencompassingtheshortwingspansofMAVsandhypothetical,nominallyinnitewings(0.75
PAGE 175

realcomponent;thiscorrespondstothedivergentDutchrollmodeseenfromtheexperimentaldata,andismostrelevanttotheightregimesMAVsandtheirnear-unityaspectratios.IftheaspectratioisincreaseduptoavalueofAR4.5,themotionofthewingwillinitiallyexperienceahighlydamped,realresponseandwillprogressintothedivergentDutchroll.Iftheaspectratioincreaseabove4.6,botheigenvalues(realandcomplex)arestable;furthermore,astheaspectratiocontinuestoincrease,thecomplexeigenvaluetendstowardzero.Thisindicateshowhighaspectratiowingscontributemostsignicantlytotherollsubsidencemotionandhaveonlyasmall,typicallystable,contributiontotheDutchrollmode.Ofmostcriticalimportanceisthecleardemarcationbetweenthestable,highaspectratioandtheunstable,lowaspectratioregimes.TheexistenceofthesewingsinafundamentallyunstableregimemaypotentiallyexplainsomelateralinstabilitiesexperiencedbyMAVs.Thendingsofthisresearchsuggestanumberofinterestingfuturedirections: DesignofMAVwhichusesrollstallasacontrolmechanisminsteadofconventionalcontrolsurfaces. Designofanactiveowcontrolstrategywhichdisruptsordisplacesthetrailingvorticestoattenuaterollstallandincreasestability. DesignofaMAVwingplanformwhichmitigatesrollstallbydisruptingtheformationofthetipvortices. UsingPIVtestingtoimprovethetrailingvortexmodelforrollstall. DevelopmentofaMAVdesigntoolwhichpredictsthecontributionofthewingtotherollstabilityderivativeandsizestailsurfacesappropriatelytocompensateforrollstall. ImplementationofrobustMAVautopilotswhichincludedynamicmodelsofrollstall. DesignofagustinsensitiveMAVwhichbalancesgeometricandaerodynamicsymmetrytoobtainazeroCl, In-depthstudyofhowbiologicalyerswithlowaspectratiowings(ie,bumblebees)areaffectedbytipvorticesandwhatnaturalmitigationstrategiestheyuse. 175

PAGE 176

Comparisonofighttestdatawiththedynamicmodels(includingdampingderivatives)obtainedfromwindtunnelexperimentation;identicationofdivergentmodesfromightdata. 176

PAGE 177

APPENDIXADERIVATIONOFTHEAIRCRAFTEQUATIONSOFMOTIONThederivationoftheaircraftequationsofmotioncanbefoundinseveralstandardtextbooks;thisderivationfollowsthatofPhillips[ 13 ],towhichthereaderisreferredforgreaterdetail.WebeginwithNewton'ssecondlawforarigidbody: FS+W=d dt(mV)+!(mV)MS=d dt([I]V)+!([I]V)(A)InEq. A ,FSisthenetsurfaceforce,MSistheresultingmomentvectoraboutthecg,Wistheweightvectorand[I]istheinertiatensor: [I]=26666666664Ixxb)]TJ /F3 11.955 Tf 9.29 0 Td[(Ixyb)]TJ /F3 11.955 Tf 9.3 0 Td[(Ixzb)]TJ /F3 11.955 Tf 9.3 0 Td[(IyxbIyyb)]TJ /F3 11.955 Tf 9.3 0 Td[(Iyzb)]TJ /F3 11.955 Tf 9.3 0 Td[(IzxbIzybIzzb37777777775(A)wherethemomentsofinertiainEq. A aredenedintheusualway[ 13 ].Theseequationscanbesolvedforthetranslationalandrotationalaccelerationstoprovideasystemofsixnonlinearrstorderdifferentialequations.Inaircraftnotation(thatis,with 177

PAGE 178

translationalvelocitiesu,v,wandrotationalvelocitiesp,q,r):26666666664W=g000W=g000W=g377777777758>>>>>>>>><>>>>>>>>>:_u_v_w9>>>>>>>>>=>>>>>>>>>;=8>>>>>>>>><>>>>>>>>>:Fxb+Wxb+(rv)]TJ /F3 11.955 Tf 11.95 0 Td[(qw)W=gFyb+Wyb+(pw)]TJ /F3 11.955 Tf 11.96 0 Td[(ru)W=gFzb+Wzb+(qu)]TJ /F3 11.955 Tf 11.96 0 Td[(pv)W=g9>>>>>>>>>=>>>>>>>>>; (A)26666666664Ixxb0)]TJ /F3 11.955 Tf 9.3 0 Td[(Ixzb0Iyyb0)]TJ /F3 11.955 Tf 9.3 0 Td[(Izxb0Izzb377777777758>>>>>>>>><>>>>>>>>>:_p_q_r9>>>>>>>>>=>>>>>>>>>;=8>>>>>>>>><>>>>>>>>>:Mxb+(Iyyb)]TJ /F3 11.955 Tf 11.95 0 Td[(Izzb)qr+IxzbpqMyb+(Izzb)]TJ /F3 11.955 Tf 11.96 0 Td[(Ixxb)pr+Ixzb(r2)]TJ /F3 11.955 Tf 11.96 0 Td[(p2)Mzb+(Ixxb)]TJ /F3 11.955 Tf 11.96 0 Td[(Iyyb)pq+Ixzbqr9>>>>>>>>>=>>>>>>>>>; (A)WhiletheformulationgiveninEq. A modelsthedynamicsoftheaircraftinacoordinatesystemattachedtotheairframe,itismoreconvenienttoexpressthepositionandorientationofthevehicleusingEuleranglesinanEarth-xedframe.TheEulerbank(),elevation()andazimuth( )anglesdescribetheaircraft'sorientationrelativetoEarth-xedaxeswiththex-axispointingnorth,they-axispointingeastandthez-axispointingdown.ToexpressthepositionandorientationofanaircraftintheEarth-xed 178

PAGE 179

frame,thefollowingkinematictransformationisused: 8>>>>>>>>><>>>>>>>>>:_xf_yf_zf9>>>>>>>>>=>>>>>>>>>;=26666666664coscos sinsincos )]TJ /F8 11.955 Tf 11.96 0 Td[(cossin cossincos +sinsin cossin sinsinsin +coscos cossinsin )]TJ /F8 11.955 Tf 11.96 0 Td[(sincos )]TJ /F8 11.955 Tf 11.29 0 Td[(sinsincoscoscos377777777758>>>>>>>>><>>>>>>>>>:uvw9>>>>>>>>>=>>>>>>>>>;...+8>>>>>>>>><>>>>>>>>>:VwxfVwyfVwzf9>>>>>>>>>=>>>>>>>>>;(A) 8>>>>>>>>><>>>>>>>>>:___ 9>>>>>>>>>=>>>>>>>>>;=266666666641sinsin=coscossin=cos0cos)]TJ /F8 11.955 Tf 11.29 0 Td[(sin0sin=coscos=cos377777777758>>>>>>>>><>>>>>>>>>: 9>>>>>>>>>=>>>>>>>>>;(A)InEq. A theVtermsontherighthandsiderepresentthecomponentsofthewindvectorintheEarth-xedframe.Itshouldbenotedthatasingularityoccurswhencos()=0onthelefthandsideofthesecondequation;whiletherearetechniquesfordealingwiththissituation,theyarenotrelevantforMAVsandwillnotbeaddressedhere.Equations A A areknownastheEulerangleformulationoftheequationsofmotionforarigidbody,spanwisesymmetricaircraft.Theyrepresentasystemof12coupled,nonlinear,rstorderdifferentialequationswhichcanbeintegratedusinganumericalscheme(suchasafourthorderRunge-Kutta)toyieldthepositionvector,translational/rotationalvelocities,andEuleranglesasafunctionoftime.Itisstill 179

PAGE 180

desirabletolinearizetheequationstoobtainaclosedformsolutionwhichcapturesthedynamicsoftheaircraftwithasimplersolvingtechnique.Thisisaccomplishedusingsmalldisturbancetheory.Thestatevariables,theEuleranglesandtheforcesandmomentsarewrittenintheformofaequilibriumvalueplusasmalldisturbance;ie,u=u0+u.Theequilibrium(or`cruise')ightconditionisassumedtohavezerosideslipandbankanglesinadditiontozeroangularvelocity.Consideringthatthesumofforcesandmomentsonabodyinequilibriummustbezero(andthustheforceandweightcomponentscancel),theequilibriumvaluesofthetermsintheEulerangleformulationbecome: Fxb0=)]TJ /F3 11.955 Tf 9.3 0 Td[(Wxb0,Fzb0=)]TJ /F3 11.955 Tf 9.3 0 Td[(Wzb0,v0=w0=p0=q0=r0=0= 0=Fyb0=Wyb0=M=0u0=U0,x0=(Vwxf+U0cos0)t,y0=Vwyft,z0=(Vwzf+U0sin0)t(A)EachforceandmomentcanberepresentedasarstorderTaylorseriesexpansionabouttheequilibriumcondition.Traditionally,itisassumedthattheforcesandmomentsarefunctionsofthetranslationandrotationalvelocities,thetranslationalacceleration, 180

PAGE 181

andthedeectionofthecontrolsurfaces(a,e,r)asshowninEq. A 8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:FxbFybFzbMxbMybMzb9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;=26666666666666666666666664Fxb,uFxb,vFxb,wFyb,uFyb,vFyb,wFzb,uFzb,vFzb,wMxb,uMxb,vMxb,wMyb,uMyb,vMyb,wMzb,uMzb,vMzb,w377777777777777777777777758>>>>>>>>><>>>>>>>>>:uvw9>>>>>>>>>=>>>>>>>>>;+26666666666666666666666664Fxb,pFxb,qFxb,rFyb,pFyb,qFyb,rFzb,pFzb,qFzb,rMxb,pMxb,qMxb,rMyb,pMyb,qMyb,rMzb,pMzb,qMzb,r377777777777777777777777758>>>>>>>>><>>>>>>>>>:pqr9>>>>>>>>>=>>>>>>>>>;+26666666666666666666666664Fxb,_uFxb,_vFxb,_wFyb,_uFyb,_vFyb,_wFzb,_uFzb,_vFzb,_wMxb,_uMxb,_vMxb,_wMyb,_uMyb,_vMyb,_wMzb,_uMzb,_vMzb,_w377777777777777777777777758>>>>>>>>><>>>>>>>>>:_u_v_w9>>>>>>>>>=>>>>>>>>>;+26666666666666666666666664Fxb,aFxb,eFxb,rFyb,aFyb,eFyb,rFzb,aFzb,eFzb,rMxb,aMxb,eMxb,rMyb,aMyb,eMyb,rMzb,aMzb,eMzb,r377777777777777777777777758>>>>>>>>><>>>>>>>>>:aer9>>>>>>>>>=>>>>>>>>>;(A)Atthispointaseriesofassumptionsaremadeaboutthedependenciesoftheforcesandmoments;manyofthesearenotvalidforMAVs,butwillbepresentedhereasareference.Itisassumedthattheonlyaffectoftranslationalacceleration(specicallytherateofchangeofangleofattack)istoalterthetimetakefordownwashfromthemainwingtoreachthehorizontaltail;therefore,theliftandpitchingmomentaredependentupon_uand_w;allothertranslationalaccelerationdependenciesareassumedtobezero.Inaddition,thesymmetryoftheaircraftisassumedtonegateseveralother 181

PAGE 182

derivatives.Theseassumptionssetthefollowingpartialderivativesequaltozero: Fyb,_u=Fxb,_v=Fyb,_v=Fzb,_v=Fyb,_w=0Mxb,_u=Mzb,_u=Mxb,_v=Myb,_v=Mzb,_v=Mxb,_w0=Mzb,_w=0Fxb,v=Fyb,u=Fyb,w=Fzb,v=0Mxb,u=Mxb,w=Myb,v=Mzb,u=Mzb,w=0Fxb,p=Fxb,r=Fyb,q=Fzb,p=Fzb,r=0Mxb,q=Myb,p=Myb,r=Mzb,q=0Fxb,a=Fxb,r=Fyb,e=Fzb,a=Fzb,r=0Mxb,e=Myb,a=Myb,r=Mzb,e=0(A)BymakingthesesimplicationsinEq. A and A ,makingthesmallangleapproximations, sin',sin'sin(0)+cos(0),sin cos'1,cos'cos(0))]TJ /F8 11.955 Tf 11.96 0 Td[(sin(0),cos '1(A)andneglectingsecondordertermsinEq. A (whichisvalidduetothesmalldisturbanceapproximation),itispossibletodecoupletheEulerformulationintolateraland 182

PAGE 183

longitudinalequations,bothcomprisedof6coupledrstorderdifferentialequations: 26666666666666666666666664W g)]TJ /F3 11.955 Tf 11.96 0 Td[(Fxb,_u)]TJ /F3 11.955 Tf 9.3 0 Td[(Fxb,_w0000)]TJ /F3 11.955 Tf 9.3 0 Td[(Fzb,_uW g)]TJ /F3 11.955 Tf 11.96 0 Td[(Fzb,_w0000)]TJ /F3 11.955 Tf 9.29 0 Td[(Myb,_u)]TJ /F3 11.955 Tf 9.3 0 Td[(Myb,_wIyyb000000100000010000001377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:_u_w_q_xf_zf_9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;=8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:Fxb,eFzb,eMyb,e0009>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;e+26666666666666666666666664Fxb,uFxb,wFxb,q00)]TJ /F3 11.955 Tf 9.29 0 Td[(Wcos0Fzb,uFzb,wFzb,q+U0W g00)]TJ /F3 11.955 Tf 9.3 0 Td[(Wsin0Myb,uMyb,wMyb,q000cos0sin0000)]TJ /F3 11.955 Tf 9.3 0 Td[(U0sin0)]TJ /F8 11.955 Tf 11.29 0 Td[(sin0cos0000)]TJ /F3 11.955 Tf 9.3 0 Td[(U0cos0001000377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:uwqxfzf9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;(A) 183

PAGE 184

26666666666666666666666664W g000000Ixxb)]TJ /F3 11.955 Tf 9.3 0 Td[(Ixzb0000)]TJ /F3 11.955 Tf 9.3 0 Td[(Ixzb)]TJ /F3 11.955 Tf 9.3 0 Td[(Izzb000000100000010000001377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:_v_p_r_yf__ 9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;=8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:Fyb,aFyb,rMxb,aMxb,rMzb,aMzb,r0000009>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;8>>>><>>>>:ar9>>>>=>>>>;+26666666666666666666666664Fyb,vFyb,pFyb,r)]TJ /F3 11.955 Tf 11.96 0 Td[(U0W g0Wcos00Mxb,vMxb,pMxb,r000Mzb,vMzb,pMzb,r00010000U0cos001tan000000sec0000377777777777777777777777758>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:vpryf 9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;(A)ThelongitudinalandlateralequationsinEq. A andEq. A ,respectively,canbefurthernondimensionalizedandexpressedinstabilityaxesinsteadofbodyaxes(ie,anglesofattackandsideslipinsteadofu,v,wvelocitycomponents);thisadditionalmanipulationcanbefoundinPhillips'text[ 13 ]. 184

PAGE 185

APPENDIXBMLTBALANCEOPERATIONTheoperationofthetheMLTbalancereliesontheinteractionbetweenstrainmeasurementson6Wheatstonebridges(oneperchannel)comprising24totalstraingauges.Thestraingaugesforeachbridgeareplacedatknownlocationsalongthebodyofthebalancetoallowcomputationoftheappliedloads.Fig. B-1A showsasimpliedmodelofthebalanceonlyincorporatingthenormalforce(NF)sensorsforillustrativepurposes;thewiringschematicforaWheatstonebridgeisshowninFig. B-1B .Inthisappendix,therelationshipbetweenthephysicalstrainsonthebalanceandthevoltageoutputoftheNFWheatstonebridgewillbederived. A BFigureB-1. SystemrepresentationoftheMLTbalance.A)PlacementofNFstraingaugesontheMLTbalance,B)WiringdiagramforWheatstonebridgeconguration. ForadiscreteforceFappliedtothebalanceatsomelocationalongthexaxis,thestructurewillactlikeabeamandwilldeectaccordingly;thus,thebottomsideofthebeamwillbeincompressionandthetopsidewillbeintension.Thebendingmomentcreatedbytheforceatthelocationsofthestraingauges(NF1,NF2,NF3,NF4)isdenedasM=F(Dx).Deningtensilestressaspositive,thestressesateachofthe 185

PAGE 186

fourgaugelocationsare: NF1=My I=kM1=)]TJ /F3 11.955 Tf 9.3 0 Td[(kF(D+x)=1 CR1 R1NF2=My I=kM2=+kF(D+x)=1 CR2 R2NF3=My I=kM3=)]TJ /F3 11.955 Tf 9.3 0 Td[(kF(D)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=1 CR3 R3NF4=My I=kM4=+kF(D)]TJ /F3 11.955 Tf 11.96 0 Td[(x)=1 CR4 R4,(B)wherek=y=IandCisanarbitraryconstant.FromFig. B-1B ,itispossibletodenethecurrentsI1andI2throughthedifferentlegsofthecircuitas: I1=Vext R1+R2I2=Vext R3+R4,(B)UsingKirchoff'slaws,theoutputvoltagecanthenbeexpressedintermsofthecurrentandthestraingaugeresistances. Vout=I1R2)]TJ /F3 11.955 Tf 11.95 0 Td[(I2R3.(B)SubstitutingEq. B intoEq. B allowstheoutputvoltagetobeexpressedentirelyintermsofthestraingaugeresistances: Vout Vext=R2 R1+R2)]TJ /F3 11.955 Tf 26.94 8.09 Td[(R3 R3+R4=R2(R3+R4))]TJ /F3 11.955 Tf 11.96 0 Td[(R3(R1+R2) (R1+R2)(R3+R4)=R2R4)]TJ /F3 11.955 Tf 11.96 0 Td[(R1R3 (R1+R2)(R3+R4).(B)ItcanbeseenfromEq. B thatiftheresistanceofthefourstraingaugesareequal,theoutputvoltagewillbezero.WhentheforceFisappliedtothebalancethedifferentdeectionsofthegaugeswillresultintheirresistanceschangingindependently; 186

PAGE 187

thus,Eq. B willnolongerbezeroandwillequal: Vout Vext=(R2+R2)(R4+R4))]TJ /F8 11.955 Tf 11.96 0 Td[((R1+R1)(R3+R3) (R1+R1+R2+R2)(R3+R3+R4+R4).(B)Now,forrelativelysmalldeectionsofthebalance(ie,forloadslessthantheelasticyieldstrengthofthebalance),thechangesinresistanceofthestraingaugeswillbecorrespondinglysmall.Thus,wecanneglecttheRtermsinthedenominatorandtheproductofRtermsinthenumeratorofEq. B ;theequationbecomes: Vout Vext=R2R4+R2R4+R4R2)]TJ /F3 11.955 Tf 11.96 0 Td[(R1R3)]TJ /F3 11.955 Tf 11.95 0 Td[(R3R1)]TJ /F3 11.955 Tf 11.95 0 Td[(R1R3 (R1+R2)(R3+R4).(B)Subtractingtheinitial,balancedvoltageoutput(Eq. B )fromEq. B givesthedifferentialoutputfromthestraingauge.Aftersomealgebra,thisisseentobe: Vout Vext=R2R4R4 R4+R2 R2)]TJ /F3 11.955 Tf 11.95 0 Td[(R1R3R1 R1+R3 R3 (R1+R2)(R3+R4).(B)TherelationsbetweenthebendingstressandthechangeinresistancegivenbyEq. B cannowbeincorporated: Vout Vext=R2R4[CkF(D)]TJ /F3 11.955 Tf 11.96 0 Td[(x)+CkF(D+x)])]TJ /F3 11.955 Tf 11.95 0 Td[(R1R3[)]TJ /F3 11.955 Tf 9.3 0 Td[(CkF(D+x))]TJ /F3 11.955 Tf 11.95 0 Td[(CkF(D)]TJ /F3 11.955 Tf 11.96 0 Td[(x)] (R1+R2)(R3+R4)=CkFR2R4(D)]TJ /F3 11.955 Tf 11.96 0 Td[(x+D+x)+R1R3(D+x+D)]TJ /F3 11.955 Tf 11.96 0 Td[(x) (R1+R2)=2DCkFR2R4+R1R3 (R1+R2).(B)Thus,theoutputvoltageislinearlyproportionaltotheappliedforceandthexdropsoutoftheequation.ThevaluesofCandkforeachbridgeareobtainedbythecompanyduringanextensivecalibrationprocedure.ItshouldbenotedthatthepitchingmomentcanalsobeobtainedusingthesameprocedurewiththesignarrangementswappedforNF1andNF2;thextermthendoesnotdropoutoftheequationandtheoutputvoltageisseentobeproportionaltoFx,orthepitchingmoment. 187

PAGE 188

Thederivationshownhereisforonlyoneoftheaxes(normalforce).TheactualMLTbalanceisasixcomponentbalance,anddeectionsofoneaxiswillinteractwiththeother5tovaryingdegrees.Asaresult,itisnotenoughtosimplyconvertthevoltagesdirectlytophysicalloads.TheinteractionsbetweenthechannelsmustbeaccountedforusingtheiterativeproceduredescribedinChapter 2 188

PAGE 189

APPENDIXCALTERNATEREPRESENTATIONOFDAMPINGDERIVATIVESTheseguresrecreatethedataplottedinthethree-dimensionalguresofFigs. 3-15 3-17 foreaseofinterpretation(attheexpenseoftherequisitenumberofgures).Toprovidebetterresolutionoftheindividualvaluesatdifferentincidenceangles,thedataareplottedinFigs. C-1 C-7 foreachrelevantdegreeoffreedomtested.Theindependentvariablesandareplottedonthexandyaxesandthecontourmagnitudescorrespondtothenondimensionaldampingderivatives.Thetrendsofeachplotareseentobetypicallysimilarforanyfrequencyofmotiontestedalthoughthemagnitudesdecreaseathighervaluesoff,asseeninFigs. 3-15 3-17 .Thelargestmagnitudestypicallyoccuratnonzerobutmoderatevaluesofandwheretheasymmetrictipvorticeshaveastrongerimpactbutbeforerollstallhasoccurred.Aninterestingexampleofthisisthecrosscoupledderivative@CZ=@p(normalforcedampingduetorollrate)seeninFig. C-6 ;nodampingexistsatzerosideslipandamaximumvaluesisseennear=20and=35wherethetipvortexasymmetryismostpronounced.Thisbehaviorissomewhatsimilartotherollmomentdampinginpitch,@Cl=@q,anothercrosscoupledderivative.Similartosomestaticloads,thisindicatesthatathighersideslipanglestheimpactofrollmomentandnormalforcebecomesmoresimilarastheinteractionsbetweentheleadingedgeseparationandthetipvortexbecomestronger.Finally,aspreviouslymentioned,therolldampingderivative@Cl=@piszerowhennosideslipexists,andonlyexperiencessmallmagnitudesatthehighestsideslipangletested.TheresultinglackofrolldampingsuggestsanexplanationforthejitterynatureofMAVight,asnorestoringmomentiscreatedbytherollrateduetothesmallinducedvelocitiesalongtheshortwingspan.AstheaspectratioisincreasedtoAR=2,therolldampingresults(evenatincreasedsideslipangles)werefoundtobenegligible,suggestingthattheskewedtipvorticesatloweraspectratiosmaycontributeaminordegreeofdampingatincreasedsideslipangles.Asthemagnitudeof 189

PAGE 190

rollstalldecreasesforhigherwingspans,thisdampingiseliminatedandthespanisnotyetlongenoughtointroducedampingthroughspanwiseinducedvelocity. A B C D E FFigureC-1. Normalforcedampinginpitch@CZ @qforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 190

PAGE 191

A B C D E FFigureC-2. Pitchmomentdampinginpitch@Cm @qforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 191

PAGE 192

A B C D E FFigureC-3. Rollmomentdampinginpitch@Cl @qforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 192

PAGE 193

A B C D E FFigureC-4. Rollmomentdampinginyaw)]TJ /F6 7.97 Tf 6.68 -4.57 Td[(@Cl @rforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 193

PAGE 194

A B C D E FFigureC-5. Rollmomentdampinginroll@Cl @pforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 194

PAGE 195

A B C D E FFigureC-6. Normalforcedampinginroll@CZ @pforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 195

PAGE 196

A B C D E FFigureC-7. Normalforcedampinginroll@Cm @pforanAR=1wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 196

PAGE 197

A B C D E FFigureC-8. Normalforcedampinginpitch@CZ @qforanAR=2wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 197

PAGE 198

A B C D E FFigureC-9. Pitchmomentdampinginpitch@Cm @qforanAR=2wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 198

PAGE 199

A B C D E FFigureC-10. Rollmomentdampinginpitch@Cl @qforanAR=2wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 199

PAGE 200

A B C D E FFigureC-11. Rollmomentdampinginyaw)]TJ /F6 7.97 Tf 6.67 -4.58 Td[(@Cl @rforanAR=2wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 200

PAGE 201

A B C D E FFigureC-12. Normalforcedampinginroll@CZ @pforanAR=2wingatRe=7.5104.A)f=0.5Hz,B)f=1Hz,C)f=1.5Hz,D)f=2Hz,E)f=2Hz,F)f=3Hz. 201

PAGE 202

A B C D EFigureC-13. Normalforcedampinginroll@Cm @pforanAR=2wingatRe=7.5104.Resultsforf=0.5Hzwerenegligibleatallincidenceanglestested.A)f=1Hz,B)f=1.5Hz,C)f=2Hz,D)f=2Hz,E)f=3Hz. 202

PAGE 203

REFERENCES [1] Hundley,R.andGritton,E.,FutureTechnology-DrivenRevolutionsinMilitaryOperations,RANDCorporationDocumentNo.DB-110-ARPA,1994. [2] Grasmeyer,J.andKeennon,M.,DevelopmentoftheBlackWidowmicroairvehicle,AIAApaper2001-0127,Reno,NV,2001,39thAIAAAerospaceSciencesMeeting&Exhibit. [3] Wasak,R.,Jenkins,D.,andIfju,P.,Stabilityandcontrolpropertiesofanaeroelasticxedwingmicroaerialvehicle,AIAApaper2002-4005,2001. [4] Ifju,P.,Jenkins,D.,Ettinger,S.,Lian,Y.,Shyy,W.,andWaszak,M.,Flexible-wing-basedmicroairvehicles,AIAApaper2002-0705,January2002,40thAerospaceSciencesMeeting&Exhibit,Reno,Nevada. [5] Kellog,J.,Boavis,C.,andet.al.,D.C.,NonconventionalaerodynamicsforMAVs,No.26.1-12,Bristol,UK,2001,Proc.Int.Conf.UnmannevairVeh.Syst.16th. [6] Kellog,J.,Bovais,C.,Dahlburg,J.,andet.al.,R.F.,TheNRLMITEAirVehicle,Tech.rep.,NavalResearchLaborotory,2001. [7] Krashanitsa,R.,Silin,D.,Shkarayev,S.,andAbate,G.,FlightDynamicsofaFlapping-WingAirVehicle,InternationalJournalofMicroAirVehicles,Vol.1,No.1,2009,pp.35. [8] Thipyopas,C.andMoschetta,J.,AFixed-WingBiplaneMAVforLowSpeedMissions,InternationalJournalofMicroAirVehicles,Vol.1,No.1,2009,pp.13. [9] deCroon,G.,Groen,M.,Wagter,C.D.,Remes,B.,Ruijsink,R.,andvanOudheusden,B.,Design,aerodynamicsandautonomyoftheDelFly,Bioin-spiration&Biomimetics,Vol.7,May2012,doi:10.1088/1748-3182/7/2/025003. [10] Shields,M.andMohseni,K.,StaticAerodynamicLoadingandStabilityConsiderationsforaMicroAerialVehicle,AIAApaper2010-4389,28thAIAAAppliedAerodynamicsConference,Chicago,IL,June28July12010. [11] Lanchester,F.,Aerodonetics,Vol.1,ArchibaldConstable&Co,Ltd,London,1908. [12] Bryan,G.,StabilityinAviation,London,MacMillanandCo.,Ltd,StMartinsSt,London,1911. [13] Phillips,W.,MechanicsofFlight,JohnWiley&Sons,2nded.,2010. [14] Schmidt,L.,IntroductiontoAircraftFlightDynamics,AIAA,1998. 203

PAGE 204

[15] Phillips,W.,EffectofSteadyRollingonLongitudinalandDirectionalStability,Tech.rep.,June1948,NACATN627. [16] Tobak,M.,Reese,D.,andBeam,B.,ExperimentalDampinginPitchof45TriangularWings,Tech.rep.,Decemberl1950,NACARMA50J26. [17] Tobak,M.,DampinginPitchofLow-Aspect-RatioWingsatSubsonicandSupersonicSpeeds,Tech.rep.,April1953,NACARMA52L04a. [18] Fromme,J.andGoldberg,M.,UnsteadyTwoDimensionalAirloadsActingonOscillatingThinAirfoilsinSubsonicVentilatedWindTunnels,Tech.rep.,May1978,NASAContractorReport1978. [19] Campbell,J.andPaulson,J.,TheEffectsofStaticMarginandRotationalDampinginPitchontheLongitudinalStabilityCharacteristicsofanAirplaneasDeterminedbyTestsofaModelintheNACAFree-FlightTunnel,Tech.rep.,June1944,NACAAdvanceRestrictedReportL4F02. [20] Fisher,L.andWolhard,W.,SomeEffectsofAmplitudeandFrequencyontheAerodynamicDampingofaModelOscillatingContinuouslyinYaw,Tech.rep.,September1952,NACATN2766. [21] Bird,J.,Fisher,L.,andHubbard,S.,SomeEffectsofFrequencyontheContributionofaVerticalTailtotheFreeAerodynamicDampingofaModelOscillatinginYaw,Tech.rep.,April1953,NACAReport1130. [22] Fisher,L.andFletcher,H.,Effectoflagofsidewashonthevertical-tailcontributiontooscillatorydampinginyawofairplanemodels,Tech.rep.,October1954,NACATN3356. [23] Averett,B.,DynamicStabilityCharacteristicsinPitchofModelsofProposedApolloCongurationsatMachNumbersfrom0.30to4.63,Tech.rep.,August1965,NASATMX-1127. [24] Boyden,R.andFreeman,D.,Subsonicandtransonicdynamicstabilitycharacter-isticsofaspaceshuttleorbiter,NASAtechnicalnote,NationalAeronauticsandSpaceAdministration,1975. [25] Tomek,D.andBoyden,R.,SubsonicandTransonicDynamicStabilityCharacteristicsoftheX-33,AIAApaper2000-0266,38thAIAAAerospaceSciencesMeeting,Reno,NV,January10-132000. [26] Day,R.,CouplingDynamicsinAircraft:AHistoricalPerspective,Tech.rep.,1997,NASASpecialPublication532. [27] Albertani,R.,Stanford,B.,DeLoach,R.,Hubner,J.,andIfju,P.,Wind-TunnelTestingandModelingofaMicroAirVehiclewithFlexibleWings,JournalofAircraft,Vol.45,No.3,2008,pp.1025. 204

PAGE 205

[28] Babcock,J.,Albertani,R.,andAbate,G.,ExperimentalEstimationoftheRotaryDampingCoefcientsofaPliantWing,JournalofAircraft,Vol.49,No.2,2012,pp.390. [29] Garcia,H.,Abdulrahim,M.,andLind,R.,RollcontrolforaMicroAirVehicleusingactivewingmorphing,AIAAPaper2003-5347,AIAAGuidance,Navigation,andControlConference,11-14August2003. [30] Morelli,E.,GlobalNonlinearAerodynamicModelingUsingMultivariateOrthogonalFunctions,JournalofAircraft,Vol.32,No.2,1995,pp.270. [31] Schmitz,F.,AerodynamikdesFlugmodells,C.J.E.VolckmannNachf.E.Wette,Berlin-Charlottenburg,1942. [32] Jones,B.,Stalling,JournaloftheRoyalAeronauticSociety,Vol.38,pp.747. [33] Maekawa,T.andAtsumi,S.,TransitionCausedbyLaminarFlowSeparation,NACAtechnicalreportNACA-TM-1352,September1952. [34] Gault,D.,AnExperimentalInvestigationofRegionsofSeparatedLaminarFlow,NACAtechnicalreportNACA-TM-3505,September1955. [35] Tani,I.,LowSpeedFlowsInvolvingBubbleSeparations,Prog.inAeronautical.Sci.,Vol.5,1964,pp.70. [36] Arena,A.andMueller,T.,LaminarSeparation,Transition,andTurbulentReattachmentneartheLeadingEdgeofAirfoils,AIAAJ.,Vol.18,No.5,1980,pp.747. [37] Brendel,M.andMueller,T.,Boundary-LayerMeasurementsonanAirfoilatLowReynoldsNumbers,JournalofAircraft,Vol.25,No.7,1988,pp.612. [38] Cole,G.andMueller,T.,ExperimentalmeasurementsofthelaminarseparationbubbleonanEppler387airfoilatlowReynoldsnumber,FinalReportUNDAS-1419-FR,1990. [39] O'Meara,M.andMueller,T.,LaminarSeparationBubbleCharacteristicsofanAirfoilatLowReynoldsNumbers,AIAAJ.,Vol.25,No.8,1987,pp.1033. [40] Lian,Y.andShyy,W.,Laminar-TurbulentTransitionofaLowReynoldsNumberRigidorFlexibleAirfoil,AIAAJ.,Vol.45,July2007,pp.1501. [41] Ol,M.,McAuliffe,B.,Hanff,E.,Scholz,U.,andKaehler,C.,ComparisonofLaminarSeparationBubbleMeasurementsonaLowReynoldsNumberAirfoilinThreeFacilities,AIAApaper2005-5149,35thAIAAFluidDynamicsConferenceandExhibit,Toronto,Ontario,Canada,June6-92005. 205

PAGE 206

[42] Hu,H.andYang,Z.,AnExperimentalStudyoftheLaminarFlowSeparationonaLow-Reynolds-NumberAirfoil,J.ofFluidsEngineering,Vol.130,May2008,pp.051101051101. [43] Burgmann,S.,Brucker,C.,andSchroder,W.,ScanningPIVmeasurementsofalaminarseparationbubble,ExperimentsinFluids,Vol.41,2006,pp.319. [44] Lambourne,N.andBryer,D.,Somemeasurementsinthevortexowgeneratedbyasharpleading-edgehaving65sweep,Tech.Rep.477,London,1959,AeronauticalResearchCouncil. [45] Earnshaw,P.,Anexperimentalinvestigationofthestructureofaleadingedgevortex,Tech.Rep.3281,London,1961,AeronauticalResearchCouncil. [46] Hall,M.,Atheoryforthecoreofaleading-edgevortex,J.FluidMech.,Vol.11,1961,pp.209. [47] Payne,F.,Ng,T.,Nelson,R.,andSchiff,L.,VisualizationandWakeSurveysofVorticalFlowsOveraDeltaWing,AIAAJ.,Vol.26,No.2,1988,pp.137. [48] Miau,J.,Kuo,K.,Liu,W.,Hsieh,S.,Chou,J.,andLin,C.,FlowDevelopmentsAbove50-DegreeSweepDeltaWingswithDifferentLeadingEdgeProles,JournalofAircraft,Vol.32,No.4,1995,pp.787. [49] Ol,M.andGharib,M.,LeadingEdgeVortexStructureofNonslenderDeltaWingsatLowReynoldsNumbers,AIAAJ.,Vol.41,No.1,2003,pp.16. [50] Hall,M.,VortexBreakdown,Ann.Rev.FluidMech.,Vol.4,1972,pp.195. [51] Ellington,C.,TheAerodynamicsofHoveringInsectFlight.IV.AerodynamicMechanisms,Phil.Trans.R.Soc.Lond.B,Vol.305,1984,pp.79. [52] Ellington,C.,vanderBerg,C.,,Willmott,A.,andThomas,A.,Leading-edgevorticesininsectight,Nature,Vol.384,1996,pp.626. [53] Saffman,P.andShefeld,J.,Flowoverawingwithanattachedfreevortex,StudiesinAppliedMathematics,Vol.57,1977,pp.107. [54] Lentink,D.andDickinson,M.,Rotationalaccelerationsstabilizeleadingedgevorticesonrevolvingywings,JournalofExperimentalBiology,Vol.212,August2009,pp.2705. [55] Dickinson,M.andGotz,K.,UnsteadyAerodynamicPerformanceofModelWingsatLowReynoldsNumbers,J.Exp.Biol.,Vol.174,1993,pp.45. [56] Spedding,G.andMaxworthy,T.,Thegenerationofcirculationandliftinarigidtwo-dimensionaling,J.FluidMech.,Vol.165,1986,pp.247. 206

PAGE 207

[57] Wang,Z.,Birch,J.,andDickinson,M.,UnsteadyforcesandowsinalowReynoldsnumberhoveringight:Two-dimensionalcomputationsvs.roboticwingexperiments,J.Exp.Biol.,Vol.207,2004,pp.449. [58] Ol,M.,Eldredge,J.,andWang,C.,High-AmplitudePitchofaFlatPlate:anAbstractionofPerchingandFlapping,Int.J.ofMicroAirVehicles,Vol.1,No.3,September2009,pp.203. [59] Sun,M.andTang,J.,Unsteadyaerodynamicforcegenerationbyamodelfruitywinginappingmotion,J.Exp.Biol.,Vol.205,August2002,pp.55. [60] Sun,M.andTang,J.,LiftandpowerrequirementsofhoveringightinDrosophilavirilis,J.Exp.Biol.,Vol.205,August2002,pp.2413. [61] Ansari,S.,Zbikowski,R.,andKnowles,K.,Non-linearunsteadyaerodynamicmodelforinsect-likeappingwingsinthehover.Part2:methodologyandanalysis,ProceedingsIMechEPartG:JournalofAerospaceEngineering,Vol.220,2006,pp.61,DOI:10.1243/09544100JAERO50. [62] Ansari,S.,Zbikowski,R.,andKnowles,K.,Non-linearunsteadyaerodynamicmodelforinsect-likeappingwingsinthehover.Part2:implementationandvalidation,ProceedingsIMechEPartG:JournalofAerospaceEngineering,Vol.220,2006,pp.169,DOI:10.1243/09544100JAERO50. [63] Ramesh,K.,Gopalarathnam,A.,Edwards,J.,Ol,M.,andGranlund,K.,Theoretical,ComputationalandExperimentalStudiesofaFlatPlateUndergoingHigh-AmplitudePitchingMotion,AIAApaper2011-217,49thAIAAAerospaceSciencesMeeting,Orlando,FL,January4-72011. [64] Wang,C.andEldredge,J.,Low-orderphenomenologicalmodelingofleading-edgevortexformation,Theor.Comput.FluidDyn.,August2012,pp.1,DOI:10.1007/s00162-012-0279-5. [65] Lam,K.andLeung,M.,AsymmetricVortexSheddingFlowPastanInclinedFlatPlateatHighIncidence,EuropeanJournalofMechanicsB/Fluids,Vol.24,2005,pp.33. [66] Laitone,E.,WindtunneltestsofwingsatReynoldsnumberbelow70000,ExperimentsinFluids,Vol.23,1997,pp.405. [67] Selig,M.,Guglielmo,J.,Broeren,A.,andGiguere,P.,SummaryofLowSpeedAirfoilData,Volume1,SoarTech,VirginiaBeach,VA,1996. [68] Pelletier,A.andMueller,T.,LowReynoldsNumberAerodynamicsofLow-Aspect-Ratio,Thin/Flat/Cambered-PlateWings,JournalofAircraft,Vol.37,No.5,2000,pp.825. [69] Torres,G.andMueller,T.,Low-Aspect-RatioWingAerodynamicsatLowReynoldsNumbers,AIAAJournal,Vol.42,2004,pp.865. 207

PAGE 208

[70] Roadman,J.andMohseni,K.,LargeScaleGustGenerationforSmallScaleWindTunnelTestingofAtmosphericTurbulence,AIAApaper2009-4166,39thAIAAFluidDynamicsConference,sanAntonio,TX,June22-252009. [71] Zimmerman,C.,CharacteristicsofClarkYAirfoilsofSmallAspectRatios,Tech.rep.,May1932,NACATM431. [72] Winter,H.,FlowPhenomenaonPlatesandAirfoilsofShortSpan,Tech.rep.,July1936,NACATM798. [73] McCormick,B.,Tangler,J.,andSherrieb,H.,StructureofTrailingVortices,JournalofAircraft,Vol.5,No.3,1968,pp.260. [74] Viieru,D.,Albertani,R.,Shyy,W.,andIfju,P.,EffectofTipVortexonWingAerodynamicsofMicroAirVehicles,JournalofAircraft,Vol.42,No.6,2005,pp.1530. [75] Shields,M.andMohseni,K.,EffectsofsideslipontheaerodynamicsoflowaspectratiowingsatlowReynoldsnumbers,AIAAJ.,Vol.50,No.1,2012,pp.85. [76] Taira,K.andColonius,T.,Three-DimensionalFlowsAroundLow-Aspect-RatioFlat-PlateWingsatLowReynoldsNumbers,JournalofFluidMechanics,Vol.623,2009,pp.187. [77] Freymuth,P.,Finaish,P.,andBank,W.,Furthervisualizationofcombinedwingtipandstartingvortexsystems,AIAAJ.,Vol.25,No.9,1987,pp.1153. [78] Hart,A.andUkeiley,L.,TipVortexDevelopmentonaPitching-PlungingLowAspectRatioFlatPlate,AIAApaper2011-3580,41stAIAAFluidDynamicsConferenceandExhibit,Honolulu,HI,June27-302011. [79] Williams,D.,Quach,V.,Kerstens,W.,Buntain,S.,Tadmor,G.,Rowley,C.,andColonius,T.,Low-ReynoldsNumberWingResponsetoanOscillatingFreestreamwithandwithoutFeedForwardControl,AIAApaperAIAA2009-0143,Orlando,FL,5-8January2009,47thAIAAAerospaceSciencesMeetingincludingtheNewHorizonsForumandAerospaceExposition. [80] Tang,J.andZhu,K.,NumericalandExperimentalStudyofFlowStructureofLowAspectRatioWing,JournalofAircraft,Vol.41,2004,pp.1196. [81] Gresham,N.,Wang,Z.,andGursul,I.,LowReynoldsnumberaerodynamicsoffree-to-rolllowaspectratiowings,ExperimentsinFluids,Vol.49,2010,pp.11. [82] Tregidgo,L.,Wang,Z.,andGursul,I.,Frequencylock-inphenomenonforself-sustainedrolloscillationsofrectangularwingsundergoingaforcedperiodicpitchingmotion,Phys.Fluids,Vol.24,2012,pp.117101117101. 208

PAGE 209

[83] Shields,M.andMohseni,K.,RollStallforLow-Aspect-RatioWings,J.Aircraft,Vol.50,No.4,July2013,pp.1060,doi:10.2514/1.C031933. [84] Lawrence,D.,Frew,E.,andPisano,W.,LyapunovVectorFieldsforAutonomousUnmannedAircraftFlightControl,JournalofGuidance,Control,andDynamics,Vol.31,No.5,2008,pp.1220. [85] Rae,W.andPope,A.,Low-SpeedWindTunnelTesting(Secondedition),JohnWiley&Sons,1984. [86] Williams,D.,Quach,V.,Kerstens,W.,Buntain,S.,Tadmor,G.,Rowley,C.,andColonius,T.,Low-ReynoldsNumberWingResponsetoanOscillatingFreestreamwithandwithoutFeedForwardControl,Proceedingsofthe47thAIAAAerospaceSciencesMeetingincludingtheNewHorizonsForumandAerospaceExposition,AIAA,Orlando,FL,2009,AIAApaper2009-143. [87] Maskell,E.,ATheoryoftheBlockageEffectsonBluffBodiesandStalledWingsinaClosedWindTunnel,Tech.rep.,November1963,R.A.EReportNo.3400. [88] Jones,W.,Wind-TunnelInterferenceEffectsonMeasurementsofAerodynamicCoefcientsforOscillatingAerofoils,Tech.rep.,September1950,A.R.CTechnicalReportR&MNo.2786. [89] Runyan,H.andWatkins,C.,ConsiderationsontheEffectsofWind-TunnelWAllsonOscillatingAirForcesforTwo-DimensionalSubsonicCompressibleFlow,Tech.rep.,December1951,NACATechnicalNote2552. [90] Duraisamy,K.,McCroskey,W.,andBaeder,J.,AnalysisofWindTunnelWallInterferenceEffectsonSubsonicUnsteadyAirfoilFlows,J.Aircraft,Vol.44,No.5,2007. [91] CalibrationandUseofInternalStrain-GageBalanceswithApplicationtoWindTunnelTesting,AIAArecommendedpracticeR-091-2003,AIAA,1801AlexanderBellDrive,Suite500,Reston,VA,20191,2003. [92] Bendat,J.andPiersol,A.,RandomData:AnalysisandMeasurementProcedures,JohnWileyandSons,Hoboken,NewJersey,USA,4thed.,2011. [93] Owens,D.,Brandon,J.,Croom,M.,Fremaux,C.,Heim,E.,andVicroy,D.,OverviewofDynamicTestTechniquesforFlightDynamicsResearchatNASALaRC(Invited),AIAApaper2006-3146,25thAIAAAerodynamicMeasurementTechnologyandGroundTestingConference,SanFrancisco,CA,June5-82006. [94] Shields,M.andMohseni,K.,InherentStabilityModesofLow-Aspect-RatioWings,J.Aircraft,acceptedforpublication. [95] Schueler,C.,Ward,L.,andA.E.Hodapp,J.,TechniquesforMeasurementofDynamicStabilityDerivativesinGroundTestFacilities,AGARDograph121,Paris,France,October1967. 209

PAGE 210

[96] Colonius,T.andWilliams,D.,ControlofVortexSheddingonTwo-andThree-DimensionalAirfoils,PhilosophicalTransactionsoftheRoyalSocietyA,Vol.369,2011,pp.1525. [97] Lamar,J.,PredictionofVortexFlowCharacteristicsofWingsatSubsonicandSupersonicSpeeds,JournalofAircraft,Vol.13,No.7,1975,pp.490. [98] Prandtl,L.,Applicationsofmodernhydrodynamicstoaeronautics,NACAtechnicalreportNACA-TM-116,September1923. [99] Bastedo,W.andMueller,T.,SpanwiseVariationofLaminarSeparationBubblesonWingsatLowReynoldsNumbers,JournalofAircraft,Vol.23,No.9,1986,pp.687. [100] Hall,J.,Lawrence,D.,andMohseni,K.,Lateralcontrolofataillessmicroaerialvehicle,AIAApaper2006-6689,AIAAGuidance,Navigation,andControlConferenceandExhibit,Keystone,Colorado,August21-242006. [101] Morris,S.andM.Holden,DesignofMicroAirVehiclesandFlightTestValidation,ProceedingoftheConferenceonFixed,FlappingandRotaryWingVehiclesatVeryLowReynoldsNumbers,NotreDame,Indiana,June2000. [102] Viieru,D.,Lian,Y.,Shyy,W.,andIfju,P.,Investigationoftipvortexonaerodynamicperformanceofamicroairvehicle,AIAApaper2003-3597,2003. [103] Torres,G.andMueller,T.,AerodynamicscharacteristicsoflowaspectratiowingsatlowReynoldsnumbers,ProceedingsoftheConferenceontheFixed,FlappingandRotaryWingVehiclesatveryLowReynoldsNumbers,Univ.ofNotreDame,NotreDame,IN,5-7June2002,pp.228. [104] Phillips,W.,ImprovedClosed-FormApproximationforDutchRoll,JournalofAircraft,Vol.37,No.3,2000,pp.484. [105] Shields,M.andMohseni,K.,Passivemitigationofrollstallforlowaspectratiowings,AdvancedRobotics:SpecialissueonAerialRobots,Vol.27,April2013,doi:10.1080/01691864.2013.778941. [106] Devenport,W.,Rife,M.,Liapis,S.,andFollin,G.,TheStructureandDevelopmentofaWing-TipVortex,J.FluidMech.,Vol.312,1996,pp.67. [107] Hall,M.,Thestructureofconcentratedvortexcores,Prog.Aerospace.Sci.,Vol.7,1966,pp.53. 210

PAGE 211

BIOGRAPHICALSKETCH MattShieldsearnedhisconcurrentBS/MSdegreefromtheUniversityofColoradointheDepartmentofAerospaceEngineeringSciencesin2010.HebeganworkinginDr.KamranMohseni'sgroupinthesummerof2007andhelpedbuildtherstyingMicroAerialVehicleatCU.ThechallengesofthedesignprocessinspiredMatttopursuegraduatestudiesdeterminingthecauseoftheinstabilitiesseensoofteninighttesting.Hisdoctoralstudies,mostlyattheUniversityofFlorida,involveddevelopingtheuniquelyversatilePrototunnel,measuringafullcontingentofstabilityderivativesoflowaspectratiowings,thendescribingandmodelingthestabilitymodesinherenttoMAVs.Matt'sfutureinterestsincluderenewableenergyresearchandengineeringeducationatbothcollegiateandsecondarylevels.MattcurrentlyabidesinSeattle,andwhennotpreoccupiedwithanengineeringproject,spendshistimehiking,camping,homebrewing,traveling,seeinglivemusic,throwingfrisbees,crashingsnowboards,watchingbaseball,reading,eatinghotwings,andbrazenlyinsertingwittypunsintounsuspectingconversations.HisbestadvicetofuturePh.D.studentsistolistentotheGratefulDeadearlyandoften. 211