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Aeroservoelastic Design for Closed-Loop Flight Dynamics of a MAV

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Title:
Aeroservoelastic Design for Closed-Loop Flight Dynamics of a MAV
Physical Description:
1 online resource (318 p.)
Language:
english
Creator:
Babcock, Judson T
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:
Lind Jr, Richard C
Committee Members:
Ifju, Peter G
Ukeiley, Lawrence S
Bloomquist, David G

Subjects

Subjects / Keywords:
aeroelastic -- aeroelasticity -- aeroservoelasticity -- aswing -- closed-loop -- elastic-axis -- flight-control -- flight-dynamics -- gust -- lqr -- micro-air-vehicle -- stiffness
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:
Some fixed-wing micro air vehicles (MAVs) have high levels of structural flexibility, a property which can change the flight dynamics and control characteristics of the vehicle. However, the exact level of flexibility is typically the result of a trial-and-error approach instead of being part of a rigorous design framework and may result in unknown aeroelastic effects on the flight dynamics. The current research investigates the nature of these aeroservoelastic effects by using a generic MAV configuration. The main parameter of interest is the stiffness of the wing. Bending and torsional stiffness of the wing are independently varied from 1.0 to 0.07 Newton meters squared while the trim conditions, flight dynamics, and structural dynamics are analyzed. Large changes in both the frequencies and damping ratios of the oscillatory flight modes are seen. The bending stiffness mainly affects the lateral-directional flight modes through an increase in the effective dihedral angle due to increased wing tip deflection. The direction and magnitude of the effect varies greatly between modes. Non-traditional mode shapes resulting from decreased bending stiffness are observed in the dutch roll mode and phugoid mode. The effects of torsional stiffness depend on the relative positioning of the elastic axis and center of pressure. When the elastic axis is near the center of pressure, changing torsional stiffness has only minor effects on the flight dynamics. Elastic axis locations which are further away from the center of pressure result in stronger effects from changes in torsional stiffness. In general, the torsional stiffness affects the longitudinal modes more than the lateral directional modes because of the changing angle of attack and pitching moment. Aeroservoelastic effects of wing stiffness on the tracking performance of the aircraft are investigated. For an LQR controller with fixed weightings, the tracking performance decreases as stiffness decreases. Changes in the phugoid mode damping and shape at low bending stiffness are found to have a very strong effect on the longitudinal tracking performance. The possibility of virtually changing the stiffness of the wing by using a model-following control scheme is investigated. It is observed that the stiff aircraft can approximate the response of the flexible but the flexible aircraft is unable to adequately approximate the performance of the stiff aircraft. An important consideration for micro air vehicles is their response to a wind gust. A frequency-domain approach is used to evaluate the aircraft's longitudinal gust response in the presence of aeroservoelastic effects. The level of wing bending stiffness is found to have an important effect on the gust sensitivity and gust rejection properties of the aircraft. The direction and frequency of the gust can drastically change the gust sensitivity of the aircraft. Lowering wing stiffness can reduce the gust sensitivity at low gust frequencies but can increase it at high frequencies. Changes in modal damping and shape due to decreasing wing stiffness have a strong influence on the gust sensitivity. For a basic LQR controller with fixed weighting matrices, the gust rejection properties are very good across the range of stiffness values.
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 Judson T Babcock.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Lind Jr, Richard C.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045699:00001

MISSING IMAGE

Material Information

Title:
Aeroservoelastic Design for Closed-Loop Flight Dynamics of a MAV
Physical Description:
1 online resource (318 p.)
Language:
english
Creator:
Babcock, Judson T
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:
Lind Jr, Richard C
Committee Members:
Ifju, Peter G
Ukeiley, Lawrence S
Bloomquist, David G

Subjects

Subjects / Keywords:
aeroelastic -- aeroelasticity -- aeroservoelasticity -- aswing -- closed-loop -- elastic-axis -- flight-control -- flight-dynamics -- gust -- lqr -- micro-air-vehicle -- stiffness
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:
Some fixed-wing micro air vehicles (MAVs) have high levels of structural flexibility, a property which can change the flight dynamics and control characteristics of the vehicle. However, the exact level of flexibility is typically the result of a trial-and-error approach instead of being part of a rigorous design framework and may result in unknown aeroelastic effects on the flight dynamics. The current research investigates the nature of these aeroservoelastic effects by using a generic MAV configuration. The main parameter of interest is the stiffness of the wing. Bending and torsional stiffness of the wing are independently varied from 1.0 to 0.07 Newton meters squared while the trim conditions, flight dynamics, and structural dynamics are analyzed. Large changes in both the frequencies and damping ratios of the oscillatory flight modes are seen. The bending stiffness mainly affects the lateral-directional flight modes through an increase in the effective dihedral angle due to increased wing tip deflection. The direction and magnitude of the effect varies greatly between modes. Non-traditional mode shapes resulting from decreased bending stiffness are observed in the dutch roll mode and phugoid mode. The effects of torsional stiffness depend on the relative positioning of the elastic axis and center of pressure. When the elastic axis is near the center of pressure, changing torsional stiffness has only minor effects on the flight dynamics. Elastic axis locations which are further away from the center of pressure result in stronger effects from changes in torsional stiffness. In general, the torsional stiffness affects the longitudinal modes more than the lateral directional modes because of the changing angle of attack and pitching moment. Aeroservoelastic effects of wing stiffness on the tracking performance of the aircraft are investigated. For an LQR controller with fixed weightings, the tracking performance decreases as stiffness decreases. Changes in the phugoid mode damping and shape at low bending stiffness are found to have a very strong effect on the longitudinal tracking performance. The possibility of virtually changing the stiffness of the wing by using a model-following control scheme is investigated. It is observed that the stiff aircraft can approximate the response of the flexible but the flexible aircraft is unable to adequately approximate the performance of the stiff aircraft. An important consideration for micro air vehicles is their response to a wind gust. A frequency-domain approach is used to evaluate the aircraft's longitudinal gust response in the presence of aeroservoelastic effects. The level of wing bending stiffness is found to have an important effect on the gust sensitivity and gust rejection properties of the aircraft. The direction and frequency of the gust can drastically change the gust sensitivity of the aircraft. Lowering wing stiffness can reduce the gust sensitivity at low gust frequencies but can increase it at high frequencies. Changes in modal damping and shape due to decreasing wing stiffness have a strong influence on the gust sensitivity. For a basic LQR controller with fixed weighting matrices, the gust rejection properties are very good across the range of stiffness values.
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 Judson T Babcock.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Lind Jr, Richard C.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045699:00001


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AEROSERVOELASTICDESIGNFORCLOSED-LOOPFLIGHTDYNAMICSO FAMAV By JUDSONT.BABCOCK ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2013

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c r 2013JudsonT.Babcock 2

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Tomybeautifulwifeandtheloveofmylife,Elisha 3

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ACKNOWLEDGMENTS Firstandforemost,IwouldliketothankmywifeElishaforhe rprayersandsupport throughoutmydoctoralprogramandourmarriage.Duringthe seyearsshehasexcelled inparentingourchildrenonadailybasis,ataskwhichisoft enmorestrenuousand formidablethanaeroservoelasticity. Secondly,Iwouldliketosincerelythankmyadvisor,Dr.Ric kLind,forsupporting myeducationwithhisguidanceandmentorship.Iamalsograt efultomycommittee membersDr.LarryUkeiley,Dr.PeterIfju,andDr.DaveBloom quistfortheirtime,insight, andadvice. I'mverythankfultotheDepartmentofAeronauticsattheUni tedStatesAirForce Academyfortheirsponsorshipinthisprogram.Iwouldespec iallyliketoextendmy thankstoDr.TomYechoutforhisadviceandencouragement,n otonlyduringmytime asoneofhisundergraduatestudentsbutalsoduringmydocto ralprogram. I'mindebtedtoDr.GreggAbateforhismentorshipduringmyy earsattheAirForce ResearchLaboratory.ItwasunderhisguidancethatIbecome involvedinexperimental aerodynamicsanddecidedtopursuemydoctorate.Duringtha ttimeinthewindtunnel, IgainedtremendousinsightfromDr.RobertoAlbertaniandD r.LarryUkeileyandthe resultingresearchwasasignicantcontributiontothisdi ssertation. IwouldliketothankmyfellowstudentsAbePachikara,Ahmed Jorge,andAdam Hartfortheiradviceandstimulatingdiscussionsduringou rconcurrentdoctoral programs.IamalsoverygratefultoProf.MarkDrelaforhisa dviceandtheuseof hissoftware,withoutwhichthisworkwouldnothavebeenpos sibleinitscurrentform. Finallyandmostimportantly,Iwouldliketothankmyperson alSavior,JesusChrist, notonlyforHissacriceformebutalsoforHisguidanceandb lessingsinmylife. NowuntotheKingeternal,immortal,invisible,theonly wiseGod,behonorandgloryforeverandever.Amen. 1Timothy1:17 4

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Theviewsexpressedinthisdocumentarethoseoftheauthora nddonotreectthe ofcialpolicyorpositionoftheUnitedStatesAirForce,De partmentofDefense,orthe U.S.Government. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................11 LISTOFFIGURES .....................................13 ABSTRACT .........................................20 CHAPTER 1INTRODUCTION ...................................22 1.1PriorResearch .................................26 1.1.1AeroelasticModelDevelopment ....................26 1.1.2ModelingAeroelasticFlightDynamicsandControl .........30 1.1.3ModelingandSimulationofAeroelasticMAVs ............32 1.2ProblemStatement ...............................33 1.3DocumentOrganization ............................37 2THEORY .......................................39 2.1FlightEquationsofMotion ...........................39 2.1.1ReferenceFrames ...........................39 2.1.1.1Earthreferenceframe ....................39 2.1.1.2Bodyreferenceframe ....................39 2.1.2CoordinateTransformations ......................41 2.1.3NonlinearEquationsofMotion .....................43 2.1.3.1Translationaldynamics ...................44 2.1.3.2Rotationaldynamics .....................47 2.1.3.3Kinematics ..........................52 2.1.3.4Therigidbodyequationsofmotion .............55 2.1.4LinearizedEquationsofMotion ....................55 2.1.5SolutionApproach ...........................59 2.2StructuralDynamics ..............................60 2.2.1SingleDOFSystem:TheMass-Spring-Damper ...........60 2.2.3Multi-DOFMass-Spring-DamperSystem ...............64 2.2.5StructuralDamping ...........................67 2.2.6StructuralStiffness ...........................68 2.3Aeroelasticity ..................................69 2.3.1StaticAeroelasticity ...........................70 2.3.3DynamicAeroelasticity .........................72 2.3.3.1Unsteadyaerodynamics ...................73 2.3.3.2Aeroelasticequationsofmotion ..............76 2.3.3.3Elasticstabilityderivatives .................78 6

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2.4OptimalControlTheory ............................80 3METHODOLOGY ..................................83 3.1ASWING ....................................83 3.1.1ASWINGCoordinateSystems .....................84 3.1.2StructuralModeling ...........................85 3.1.3Aerodynamics ..............................88 3.1.4StallModeling ..............................91 3.1.5SolutionMethod ............................91 3.1.6ValidationandApplication .......................93 3.2TheGenMAVAircraft .............................94 3.2.1AerodynamicProperties ........................94 3.2.3StructuralProperties ..........................98 3.2.3.1Wingstiffness ........................99 3.2.3.2Wingelasticaxis .......................101 3.2.3.3Wingtensionandmasscentroidaxes ...........102 3.2.4ASWINGModel .............................103 3.2.4.1Structural ...........................104 3.2.4.2Aerodynamic .........................104 3.2.4.3Massproperties .......................105 3.2.4.4Controlsurfaces .......................105 3.2.5Validation ................................106 3.2.6StructuralModeling ...........................111 4EXPERIMENTALAEROELASTICITYOFAMEMBRANEWING ........115 4.1Background ...................................116 4.2ExperimentalSet-UpandProcedure .....................117 4.2.1Low-speedWindTunnel ........................118 4.2.22-DOFMotionRig ...........................118 4.2.3DataAcquisition ............................119 4.2.4StrainMeasurement ..........................120 4.3Methodology ..................................120 4.3.1DesignofExperiments .........................121 4.3.2MotionDevelopmentandControl ...................122 4.3.3MAVWingTension ...........................123 4.4ResultsandDiscussion ............................125 4.4.1DynamicBehavior ...........................127 4.4.2PitchDampingDerivatives .......................130 4.5SummaryandConcludingRemarks .....................131 5AEROELASTICITYANDFLIGHTDYNAMICS:UNIFORMWINGSTIFFNE SS .133 5.1Methodology ..................................135 5.1.1DesignSpaceOverview ........................135 5.1.2ModelingProcedure ..........................135 7

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5.2Case1:MediumtoHighStiffness ......................136 5.2.1TrimConditions .............................137 5.2.2FlightDynamics .............................139 5.2.2.1Lateral-directionaldynamics ................139 5.2.2.2Longitudinaldynamics ....................141 5.2.3StructuralDynamics ..........................141 5.2.4Summary ................................143 5.3Case2:LowtoMediumStiffness .......................143 5.3.1TrimConditions .............................145 5.3.2FlightDynamics .............................147 5.3.2.1Lateral-directionaldynamics ................147 5.3.2.2Longitudinaldynamics ....................152 5.3.3StructuralDynamics ..........................157 5.3.4Summary ................................161 5.4Case3:AirspeedEffectsontheRigidAircraft ................162 5.4.1TrimConditions .............................162 5.4.2FlightDynamics .............................162 5.4.3Summary ................................166 5.5Case4:AirspeedEffectsonFlexibleCongurations ............166 5.5.1TrimConditions .............................166 5.5.2FlightDynamics .............................167 5.5.3StructuralDynamics ..........................170 5.5.4Summary ................................170 5.6Case5:EffectsoftheElasticAxis ......................171 5.6.1TrimConditions .............................171 5.6.2FlightDynamics .............................172 5.6.3StructuralDynamics ..........................174 5.6.4Flutter ..................................183 5.6.5Summary ................................183 5.7ConcludingRemarks ..............................185 6AEROELASTICITYANDFLIGHTDYNAMICS:NON-UNIFORMWINGSTI FFNESS 188 6.1LinearlyVaryingDistribution ..........................189 6.1.1Results .................................189 6.1.2Summary ................................191 6.2RootvsTipStiffness ..............................194 6.2.1Results .................................195 6.3AeroelasticSpan ................................197 6.3.1StepwiseDistributionsintheAeroelasticSpan ............198 6.3.2Non-StepwiseDistributionsintheAeroelasticSpan .........202 6.4AeroelasticRootStiffness ...........................205 6.4.1AeroelasticRootStiffnessforDecreasingStiffness .........206 6.4.2AeroelasticRootStiffnessforIncreasingStiffness ..........209 6.5ConcludingRemarks ..............................214 8

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7AEROELASTICEFFECTSOFWINGBATTENS .................216 7.1Methodology ..................................217 7.2ResultsandDiscussion ............................219 7.2.1TrimCharacteristics ..........................219 7.2.2SensitivityofLiftandDrag .......................221 7.2.3StaticStability ..............................222 7.2.4ControlEffectiveness ..........................227 7.2.5LongitudinalFlightDynamics .....................228 7.2.6Lateral-DirectionalFlightDynamics ..................230 7.2.7StructuralDynamics ..........................233 7.2.8Interpretation ..............................235 7.3Summary ....................................236 7.4ConcludingRemarks ..............................237 8AEROSERVOELASTICDESIGNUSINGWINGSTIFFNESS ..........238 8.1Methodology ..................................240 8.1.1ModelReduction ............................240 8.1.2ROMValidation .............................243 8.1.3DesignSpace ..............................245 8.2OpenLoop ...................................245 8.2.1FrequencyDomainResults ......................246 8.2.1.1Longitudinal .........................246 8.2.1.2Lateral-directional ......................247 8.2.2TimeDomainResults .........................250 8.2.2.1Longitudinal .........................250 8.2.2.2Lateral-directional ......................251 8.2.3Summary ................................252 8.3ClosedLoop ..................................253 8.3.1ControlDesign .............................254 8.3.2Longitudinal ...............................256 8.3.3Lateral-Directional ...........................258 8.3.4Summary ................................259 8.4ModelFollowing ................................260 8.4.1ControlDesign .............................262 8.4.2Results .................................262 8.4.3Summary ................................264 8.5ConcludingRemarks ..............................266 9AEROSERVOELASTICGUSTALLEVIATION ...................267 9.1Methodology ..................................270 9.1.1GustModeling .............................270 9.1.2GenMAVModel .............................272 9.1.3ControlDesign .............................274 9

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9.1.4DesignSpace ..............................275 9.2GustSensitivity .................................276 9.2.1FrequencyResponse ..........................277 9.2.2TimeResponse .............................284 9.2.3AerodynamicDerivatives ........................285 9.3GustRejection .................................287 9.3.1FrequencyResponse ..........................287 9.3.2TimeResponse .............................290 9.4Summary ....................................291 9.5ConcludingRemarks ..............................291 10CONCLUSIONS ...................................293 10.1ResearchSummary ..............................293 10.2FutureResearch ................................296 APPENDIX:EXTENDEDTHEORY ............................299 A.1DyadicProduct .................................299 A.2RateofChangeTransportTheorem .....................299 A.3Lagrange'sEnergyEquations .........................300 REFERENCES .......................................302 BIOGRAPHICALSKETCH ................................318 10

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LISTOFTABLES Table page 3-1AerodynamicparametersspeciedalongabeaminASWING ..........95 3-2Experimentaldatafromthetwo-pointbendingstiffness test ...........101 3-3Experimentaldatafromthetorsionalstiffnesstest ................101 3-4User-speciedstructuralparametersinASWING .................104 3-5AerodynamicparametersspeciedalongabeaminASWING ..........105 3-6MasspropertiesofASWINGmodelcomparedtoAVLmodel ..........106 3-7ControlderivativesfortheGenMAVASWINGmodel ...............106 3-8ComparisonofstabilityderivativesbetweenASWINGand AVL .........108 3-9Flighttestmaneuvers ................................108 4-1Experimentalfactorsandtheirrespectiveranges .................122 4-2Characterizationofmembranepre-tensionforthethree levelstested ......125 4-3Valueofconstantmodelparameters(codedunits) ................127 5-1Overviewoftheruncasesandtheindependentvariablesi neachcase .....136 5-2Case2:Normalizedeigenvectorcomponentsofthespiral mode ........148 5-3Case2:Normalizedeigenvectorcomponentsoftherollco nvergence ......150 5-4Case2:Naturalfrequenciesoftheoscillatoryightmod es ............152 5-5Case2:Dampingratiosoftheoscillatoryightmodes ..............152 5-6Case2:Normalizedeigenvectorcomponentsofthedutchr ollmode ......153 5-7Case2:Normalizedeigenvectorcomponentsofthephugoi dmode .......155 5-8Case2:Normalizedeigenvectorcomponentsoftheshortp eriodmode ....156 5-9Case2:Naturalfrequenciesofthestructuralmodes ...............160 5-10Case2:Dampingratiosofthestructuralmodes ..................161 5-11Case4:Rangeoffactors ..............................166 6-1Stepwisechangesin EI oninner/outer25%span:naturalfrequency ......195 6-2Stepwisechangesin EI oninner/outer25%span:dampingratio ........196 11

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6-3Resultsfromstepwisechangesin EI oninner/outer25%span .........196 6-4Comparisonofchangesinrootstiffnesstooverallstiff ness:naturalfrequencies 197 6-5Comparisonofchangesinrootstiffnesstooverallstiff ness:dampingratios ..197 6-6Comparisonofchangesinrootstiffnesstooverallstiff ness:timeconstants ..197 7-1Normalizedeigenvectorcomponentsofthephugoidmode ............229 7-2Normalizedeigenvectorcomponentsoftheshortperiodm ode .........229 7-3Normalizedeigenvectorcomponentsofthedutchrollmod e ...........231 7-4Normalizedeigenvectorcomponentsofthespiralconver gence .........232 7-5Normalizedeigenvectorcomponentsoftherollconverge nce ..........234 9-1Percentchangesfromhightolow EI inresponsetoa w gust ..........282 9-2Percentchangesfromhightolow EI inresponsetoa u gust ..........283 12

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LISTOFFIGURES Figure page 1-1Overviewofaeroelasticinteractions ........................24 2-1Earthcoordinatesystem ...............................40 2-2Bodycoordinatesystem ...............................40 2-3Stabilitycoordinatesystem .............................41 2-4Overviewofcoordinatetransformations ......................41 2-5AngularmomentumofaparticlePrelativetopointO ...............48 2-6Angularmomentumofadifferentialelementonarigidbod y ...........49 2-7SingleDOFmass-spring-dampersystem .....................60 2-8Freevibrationresponseofanunderdampedmass-springdampersystem ...63 2-9TwoDOFmass-spring-dampersystem .......................64 2-10Aeroelasticpitchingairfoil ..............................71 2-11Harmonicallyoscillatingairfoil ............................76 3-1AircraftbodyandlocalbeamcoordinatesystemsinASWIN G ..........85 3-2Awingrepresentedbyaliftinglinecomposedofthreehor seshoevortices ...89 3-3TheGenMAVaircraft .................................94 3-4TheGenMAVairfoilatthreelocationsalongthespan ...............95 3-5Centerofpressurelocationonanairfoil ......................96 3-6CenterofpressurealongtheGenMAVwingfor =0 ..............98 3-7TheGenMAVwing ..................................98 3-8Two-pointbendingstiffnesstest ...........................99 3-9Experimentaltwo-pointbendingstiffnesstestsetup ................100 3-10Wingdeectioninresponsetoanappliedload ..................100 3-11Wingtwistvs.excitationlocation ..........................102 3-12ElasticaxisoftheGenMAVwing ..........................103 3-13TensionaxisoftheGenMAVwing .........................103 13

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3-14GeometryoftheGenMAVinASWING .......................104 3-15ComparisonofightmodesbetweenASWINGandAVL .............107 3-16ComparisonofASWINGmodelwithighttestmaneuverA ...........109 3-17ComparisonofASWINGmodelwithighttestmaneuverB ...........110 3-18ComparisonofASWINGmodelwithighttestmaneuverC ...........110 3-19ComparisonofASWINGmodelwithighttestmaneuverD ...........111 3-20Modalfrequenciesofthestiff-chordandexible-chor dwingcongurations ...112 3-21FirstbendingmodeshapesfromGVTtest .....................113 3-22FirsttorsionmodeshapesfromGVTtest .....................113 3-23Modeshapeofchordat y =0 : 65 b ..........................114 4-1Windtunnelsetupandmodel ............................118 4-2Twodegrees-of-freedomtestrig ..........................119 4-3Kinematicplotsofan and motion ........................121 4-4Contourplotsofthewind-offmembranestrainstate ...............124 4-5Plotof 2 inthetwodirectionsasafunctionofvelocity ..............125 4-6Liftcoefcientofthelow-tensionexiblewing ...................126 4-7Modelcomparisontostaticwindtunneldata ....................128 4-8Themedium-tensionexiblewing:liftanddrag ..................128 4-9Therigidwing:liftanddrag .............................129 4-10Lift-to-dragratiosoftherigidwingandexiblewing ................130 4-11Pitchingmomentcoefcientoftherigidwingandexibl ewing ..........131 5-1Case1:Designspaceofbendingandtorsionalstiffness .............137 5-2Case1:Aircraft and e attrim ...........................138 5-3Case1:Wingdeformationsattrim .........................138 5-4Case1:Allpolescorrespondingtotheightmodesforthe designspace ....139 5-5Case1:Rollconvergencetimeconstant ......................140 5-6Case1:Dutchrollmodalcharacteristics ......................140 14

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5-7Case1:Phugoidmodalcharacteristics .......................141 5-8Case1:Shortperiodmodalcharacteristics ....................142 5-9Case1:Firstsymmetricbendingmodalcharacteristics ..............142 5-10Case2:Designspaceofbendingandtorsionalstiffness .............144 5-11Case2:Aircraft and e attrim ...........................144 5-12Case2:Wingdeformationsattrim .........................145 5-13Elasticaxisandcenterofpressurealongthewing ................146 5-14Locationofthespan-averaged x cp andelasticaxisversusangleofattack ...147 5-15Case2:Allpolescorrespondingtotheightmodesforth edesignspace ....148 5-16Case2:Polesoftheightmodes ..........................149 5-17Case2:Rollconvergencetimeconstant ......................150 5-18Case2:Dutchrollmodalcharacteristics ......................151 5-19Animationofthedutchrollmodewithvarying EI and GJ =1.0 ..........153 5-20Case2:Phugoidmodalcharacteristics .......................155 5-21Case2:Shortperiodmodalcharacteristics ....................156 5-22Animationoftheshortperiodmodewithvarying EI and GJ =1.0 ........157 5-23Case2:Firstsymmetricbendingmodalcharacteristics ..............158 5-24Animationoftherstbendingmodewithvarying EI and GJ =1.0 ........159 5-25Case2:Firstsymmetrictorsionmodalcharacteristics ..............159 5-26Effectofairspeedontherstbendingnaturalfrequenc yatlow EI .......160 5-27Case3:Trimresults .................................163 5-28Case3:Allpolescorrespondingtotheightmodes ...............163 5-29Case3:Polesoftheightmodes ..........................164 5-30Case3:Naturalfrequencies .............................165 5-31Case3:Dampingratios ...............................165 5-32Case4:Trimresults .................................167 5-33Case4:Wingdeformationsattrim .........................167 15

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5-34Case4:Dutchrollnaturalfrequencyanddampingratio .............168 5-35Case4:Phugoidnaturalfrequencyanddampingratio ..............169 5-36Case4:Shortperiodnaturalfrequencyanddampingrati o ............169 5-37Case4:Firstbendingmode .............................170 5-38Wingtipdeection ..................................176 5-39Wingtwistat y =0 : 75 b ................................176 5-40Aircrafttrim .....................................177 5-41Aircrafttrim e .....................................177 5-42Dutchrollnaturalfrequency .............................178 5-43Dutchrolldampingratio ...............................178 5-44Phugoidnaturalfrequency ..............................179 5-45Phugoiddampingratio ................................179 5-46Shortperiodnaturalfrequency ...........................180 5-47Shortperioddampingratio .............................180 5-48Firstsymmetricbendingnaturalfrequency .....................181 5-49Firstsymmetricbendingdampingratio .......................181 5-50Firstsymmetrictorsionnaturalfrequency .....................182 5-51Firstsymmetrictorsiondampingratio .......................182 5-52Bending-torsionutter ................................184 6-1Samplelinearstiffnessdistributionwith EI 0 =1 ; EI =0 : 5 ...........189 6-2Trimresultswithlinearlyvaryingstiffness .....................190 6-3Wingtipdeectionwithlinearlyvaryingstiffness .................190 6-4Polesoftheightmodeswithlinearlyvaryingstiffness ..............192 6-5Naturalfrequencieswithlinearlyvaryingstiffness .................193 6-6Dampingratioswithlinearlyvaryingstiffness ...................193 6-7Firstbendingmodewithlinearlyvaryingstiffness .................194 6-8Stepwisechangescreatedintherootandtip EI .................195 16

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6-9Aeroelasticspananalysisoftrim. ..........................200 6-10Aeroelasticspananalysisofthedutchrollmode .................200 6-11Aeroelasticspananalysisofthephugoidmode ..................201 6-12Aeroelasticspananalysisoftheshortperiodmode ................201 6-13Aeroelasticspananalysisoftherstbendingmode ................202 6-14Stepwise,linear,andexponentialslopesoveraportio noftheinnerspan ....203 6-15Effectsofslopeintheaeroelasticspan:phugoidanddu tchroll .........204 6-16Effectsofslopeintheaeroelasticspan:shortperiod ...............204 6-17Effectsofslopeintheaeroelasticspan:rstbendingm ode ...........205 6-18Aeroelasticrootstiffnessfordecreasing EI :trimconditions ...........207 6-19Aeroelasticrootstiffnessfordecreasing EI :dutchrollmode ..........207 6-20Aeroelasticrootstiffnessfordecreasing EI :phugoidmode ...........208 6-21Aeroelasticrootstiffnessfordecreasing EI :shortperiodmode .........208 6-22Aeroelasticrootstiffnessfordecreasing EI :rstbendingmode .........209 6-23Aeroelasticrootstiffnessforincreasing EI :trimconditions ...........211 6-24Aeroelasticrootstiffnessforincreasing EI :dutchrollmode ...........212 6-25Aeroelasticrootstiffnessforincreasing EI :phugoidmode ............212 6-26Aeroelasticrootstiffnessforincreasing EI :shortperiodmode .........213 6-27Aeroelasticrootstiffnessforincreasing EI :rstbendingmode .........213 7-1GenMAVwinggeometrywiththreebattens ....................218 7-2Proleoftorsionalstiffnessacrossthehalf-spanfore achbattenconguration .219 7-3Trim and e .....................................220 7-4Span-averagedwingtwist ..............................220 7-5Effectiveangleofattackattrim ...........................221 7-6Aircraftliftanddrag .................................222 7-7Longitudinalstaticstability ..............................223 7-8Directionalstaticstability ..............................224 17

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7-9Lateralstaticstability .................................226 7-10Primarycontrolderivatives .............................227 7-11Naturalfrequencyanddampingofthelongitudinalmode s ............230 7-12Naturalfrequencyanddampingofthedutchrollmode ..............230 7-13Timeconstantofthespiralconvergence ......................232 7-14Timeconstantoftherollconvergence .......................233 7-15Naturalfrequencyanddampingofthestructuralmodes .............234 7-16Twistofbatten-reinforcedwingscomparedtoawingwit huniform GJ ......235 8-1ComparisonoftheROMtothefullmodel( EI =0 : 1 GJ =1 : 0 ) .........244 8-2Designspaceofbendingandtorsionalstiffness ..................245 8-3Elevatortoangleofattack ..............................246 8-4Elevatortopitchrate .................................247 8-5Ailerontobankangle ................................248 8-6Ailerontorollrate ..................................249 8-7Ruddertoyawrate ..................................249 8-8Open-loopangleofattackresponsetoastepinputinelev ator ..........250 8-9Open-loopbankangleresponsetoastepinputinaileron ............251 8-10Open-looprollrateresponsetoastepinputinaileron ..............252 8-11Open-loopyawrateresponsetoastepinputinrudder ..............253 8-12BlockdiagramofaLQRtrackingcontroller ....................254 8-13Outputofastiffaircrafttrackinga ref =5 : 4 command .............257 8-14Longitudinaltrackingperformanceusingxedweighti ngmatrices ........258 8-15Longitudinaltrackingperformancewithxedmodes ...............259 8-16Outputofastiffaircrafttrackinga =5 : 4 command ...............260 8-17Lateral-directionaltrackingperformanceusingxed weightingmatrices ....261 8-18Blockdiagramofamodel-followingcontrollerusingLQ R .............262 8-19Astiffaircraftfollowingtheresponseofaexibleair craft .............263 18

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8-20Aexibleaircraftfollowingtheresponseofastiffair craft .............265 9-1Exampledisturbancesfora1Hzgustwith g = 3 ...............271 9-2Blockdiagramofagustrejectioncontroldesign ..................275 9-3LQRgainsfortheGenMAVaircraftwithvaryinglevelsofs tiffness ........276 9-4Gustsensitivityinresponsetoa u -gust ......................277 9-5Gustsensitivityinresponsetoa w -gust ......................279 9-6Responseofwingtipvelocitytothegustdisturbance ...............279 9-7Gustsensitivityinresponsetoa e -gust ......................280 9-8Gustsensitivitytoa w gustwithconstantphugoidandshortperioddamping ..282 9-9Pitchrateresponsetoa u gustatthephugoidmode'snaturalfrequency ....285 9-10GustsensitivityRMSvaluesforthepitchrateresponse .............286 9-11Aerodynamicderivativesrelevanttogustsensitivity ................286 9-12Gustrejectioninresponsetoa u -gust .......................288 9-13Gustrejectioninresponsetoa w -gust .......................289 9-14Responseofwingtipvelocitytothegustdisturbance ...............289 9-15GustrejectionRMSvaluesforthepitchrateresponse ..............290 19

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy AEROSERVOELASTICDESIGNFORCLOSED-LOOPFLIGHTDYNAMICSO FAMAV By JudsonT.Babcock August2013 Chair:RickLindMajor:AerospaceEngineering Somexed-wingmicroairvehicles(MAVs)havehighlevelsof structuralexibility, apropertywhichcanchangetheightdynamicsandcontrolch aracteristicsofthe vehicle.However,theexactlevelofexibilityistypicall ytheresultofatrial-and-error approachinsteadofbeingpartofarigorousdesignframewor kandmayresultin unknownaeroelasticeffectsontheightdynamics.Thecurr entresearchinvestigates thenatureoftheseaeroservoelasticeffectsbyusingagene ricMAVconguration.The mainparameterofinterestisthestiffnessofthewing. Bendingandtorsionalstiffnessofthewingareindependent lyvariedfrom1.0Nm 2 to0.07Nm 2 whilethetrimconditions,ightdynamics,andstructurald ynamicsare analyzed.Largechangesinboththefrequenciesanddamping ratiosoftheoscillatory ightmodesareseen.Thebendingstiffnessmainlyaffectst helateral-directionalight modesthroughanincreaseintheeffectivedihedralangledu etoincreasedwingtip deection.Thedirectionandmagnitudeoftheeffectvaries greatlybetweenmodes. Non-traditionalmodeshapesresultingfromdecreasedbend ingstiffnessareobservedin thedutchrollmodeandphugoidmode. Theeffectsoftorsionalstiffnessdependontherelativepo sitioningoftheelasticaxis andcenterofpressure.Whentheelasticaxisisnearthecent erofpressure,changing torsionalstiffnesshasonlyminoreffectsontheightdyna mics.Elasticaxislocations whicharefurtherawayfromthecenterofpressureresultins trongereffectsfrom 20

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changesintorsionalstiffness.Ingeneral,thetorsionals tiffnessaffectsthelongitudinal modesmorethanthelateraldirectionalmodesbecauseofthe changingangleofattack andpitchingmoment. Aeroservoelasticeffectsofwingstiffnessonthetracking performanceoftheaircraft areinvestigated.ForanLQRcontrollerwithxedweighting s,thetrackingperformance decreasesasstiffnessdecreases.Changesinthephugoidmo dedampingandshapeat lowbendingstiffnessarefoundtohaveaverystrongeffecto nthelongitudinaltracking performance. Thepossibilityofvirtuallychangingthestiffnessofthew ingbyusingamodel-following controlschemeisinvestigated.Itisobservedthatthestif faircraftcanapproximatethe responseoftheexiblebuttheexibleaircraftisunableto adequatelyapproximatethe performanceofthestiffaircraft. Animportantconsiderationformicroairvehiclesistheirr esponsetoawindgust.A frequency-domainapproachisusedtoevaluatetheaircraft 'slongitudinalgustresponse inthepresenceofaeroservoelasticeffects.Thelevelofwi ngbendingstiffnessis foundtohaveanimportanteffectonthegustsensitivityand gustrejectionproperties oftheaircraft.Thedirectionandfrequencyofthegustcand rasticallychangethe gustsensitivityoftheaircraft.Loweringwingstiffnessc anreducethegustsensitivity atlowgustfrequenciesbutcanincreaseitathighfrequenci es.Changesinmodal dampingandshapeduetodecreasingwingstiffnesshaveastr onginuenceonthegust sensitivity.ForabasicLQRcontrollerwithxedweighting matrices,thegustrejection propertiesareverygoodacrosstherangeofstiffnessvalue s. 21

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CHAPTER1 INTRODUCTION Aeroelasticitystudiestheinteractionofaerodynamic,in ertial,andelasticforces actingonaexiblestructureexposedtoairow.Aeroelasti cityisrelevantforavarietyof elds,includingcivilengineering,automotiveengineeri ng,andmechanicalengineering. However,thescienceofaeroelasticityismostcommonlyapp liedasadisciplineof aeronauticalengineering.Aircraftstructuresareneverp erfectlyrigid;consequently,their behaviorinightwillalwaysbeaeroelasticinnature.When notproperlyconsideredin vehicleorcontroldesign,theaeroelasticphenomenathata riseareusuallyundesirable andcanrangeinseverityfrombenigntocatastrophic. Aeroelasticityhasplayedamajorroleinaeronauticssince thebeginningofpowered ightin1903.Earlierthatyear,beforetheWrightbrothers madetheirhistoricight, SamuelLangleymadetwounsuccessfulattemptsthatresulte dincatastrophicfailure ofhisaircraft'swings.Thisstructuralfailurewasduetoi nsufcienttorsionalstiffness, whichresultedintheaerodynamicforcesovercomingthestr ucturalforces[ 64 ].This aeroelasticphenomenonisknownasdivergenceandwasamajo rconcerninearly aircraftdesignuntilthe1930swhenaircraftstructureswe redesignedwithmetallicskins capableofprovidingadditionaltorsionalstiffness. Aeroelasticitywasamajorconcernthroughouttheearlyhis toryofaviation[ 36 ]. TherstdocumentedcaseofutterinvolvedtheHandleyPage O/400bomberin1916. FluttercontinuedtobeamajorconcernduringandaftertheF irstWorldWar.After theSecondWorldWar,theprevalenceofaeroelasticphenome nonincreasedfurther duetoincreasedightspeeds,thinnershapes,morecomplex designs,andmore demandingaircraftmissions[ 15 63 ].Thesetrendscontinuedintothemodernera wherenewaeroelasticapplicationsbegantosurface,sucha sexibleairships,missiles, windturbines,androtorcraft[ 29 56 86 96 ].Despiteoveracenturyofresearchand development,aeroelasticitycontinuestobeadynamicandc hallengingeld[ 57 89 ]. 22

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Twofrontiersofighttodayarehypersonics[ 62 ]andmicroornano-sizedaircraft[ 135 ]. Bothofthesefrontierswillchallengeandexpandthescienc eofaeroelasticity. Inaircraft,aeroelasticphenomenaarisewhenaerodynamic forcescausestructural deformations,whichcauseadditionalchangesintheaerody namicforces.This cycleisrepeated.Incertaincases,asteady-stateequilib riumisreachedwherethe elasticforcesbalanceouttheaerodynamicforces.Theseph enomenaaregrouped asstaticaeroelasticityandcanhaveimportantconsequenc esforthesteady-state ightofanaircraft.Theaircraftloads,controleffective ness,trimbehavior,andstatic stabilityalldependonthestaticaeroelasticbehavior.Ne gativeconsequencesofstatic aeroelasticitycanresultindivergence,reducedcontrole ffectiveness,orcontrolreversal. Insomesituations,oscillationsbetweentheaerodynamicf orcesandstructural forcescontinueovertime.Thisclassofphenomenaiscalled dynamicaeroelasticity. Dynamicaeroelasticityisconcernedwiththeoscillatoryn atureoftheinteraction betweenthestructureanduidow,namelytheextractionof energyfromtheow eldbythestructure[ 186 ].Themaininterestisthephenomenonofutter,although theeffectsofdynamicaeroelasticityontheightdynamics oftheaircraftarealsoof importance. OnepioneerintheeldofaeroelasticitywasArthurR.Colla r(1908-1986)[ 18 ].He iscreditedwithformingtheaeroelastictriangle[ 34 ]whichisreproducedinmodiedform inFig. 1-1 .Thegureillustratesthethree-wayinteractionbetweent heaerodynamic, structural,andinertialforceswhichresultsindynamicae roelasticity.Itfurtherdepicts staticaeroelasticityarisingfromtheinteractionofaero dynamicandelasticforces. Stabilityandcontrolphenomenaresultfromtheinteractio nofinertialandaerodynamic forcesandvibrationresultsfromtheinteractionofinerti alandelasticforces. Thedisciplineofaeroservoelasticityextendstheaeroela sticinteractionsinFig. 1-1 toincludetheinteractionofacontrolsystem.Sinceaight controlsystemisnow acommonfeatureamongaircraft,aeroservoelasticityhasb ecomeveryrelevantin 23

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Aerodynamic Forces Elastic Forces Inertial Forces Dynamic Aeroelasticity StaticAeroelasticityStability&Control StructuralVibration Figure1-1.Overviewofaeroelasticinteractionsaircraftdesign.Mostaeroservoelasticproblemsoccurwhe ntheaircraft'ssensors detectnotonlytherigid-bodymotionoftheaircraftbutals othemotionfromtheexible structure[ 186 ].Thesensormeasurementsfromthesestructuralvibration sarefedback intothecontrolsystem,whichmightreactinawaythatfurth erincreasesthevibrations. Theresultcanbereducedcontrolsystemperformance,reduc edhandlingqualities, increasedstructuralfatigue,orevencatastrophicfailur e. Flightdynamicsisabranchofappliedmechanicswhichdeals withthemotion ofvehiclesyingintheatmosphere[ 51 ].Flightdynamicsisabroadeldwhich incorporatesappliedmathematics,aerodynamics,rigidbo dymechanics,aeroelasticity, andthedynamicsofahumanpilot.Flightdynamicsisuniqueb ecauseitdealswith aerodynamicforcesintheabsenceofkinematicconstraints .Flightdynamicsproblems canincludeaircraftperformance,motiontrajectories,st ability,vehicleresponseto controlinputs,responsetoturbulence,handlingqualitie s,andaeroelasticconcerns. Theprimarygoalofanyightdynamicsanalysisistodetermi nethetrajectoryand orientationoftheaircraftbodyovertime. Microairvehicles(MAVs)haveemergedoverthepasttwodeca desassmall, unmannedaircraftwithcertaincharacteristicsthatenabl ethemtoaccomplishaunique 24

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setofmissions.ThedenitionofaMAVcanvarybutinthecurr entstudyisdened asbird-sizedorsmaller.Twoprominentcharacteristicsof MAVsareoperationin lowReynoldsnumberightregimesandsmallphysicaldimens ions.MAVsarealso characterizedbytheiragilityandlowcost.Theseproperti esmakethemidealtooperate inurbanareas,tunnels,caves,orotherconstrainedenviro nments.Militarymissions canincludesurveillance,reconnaissance,communication ,detection,tracking,chemical orbiologicalmonitoring,orprecisionstrike[ 95 ].Civilianmissionscanincludedisaster relief,agriculture,mapping,communications,andsurvei llanceactivitiessuchasforest remonitoring,scienticobservation,ormonitoringofel ectricpowerlines[ 118 ]. MAVsoperateinaverysensitiveReynoldsnumberregime,typ icallyontheorder of 10 3 10 5 ,whichresultsinafundamentalshiftintheaerodynamicbeh avior.Inthis regime,theowischaracterizedbycomplexfeaturesandint eractionssuchasunsteady three-dimensionalseparation,transitioninboundarylay ersandshearlayers,vortical ows,bluffbodyows,andunsteadyightenvironments[ 152 ]. Becauseofthesechallenges,MAVdesignershavetakenmuchi nspirationfrom biologicalightvehicleswhichsuccessfullyoperateinth esameReynoldsnumber regime[ 150 ].Smallbirds,bats,andyinginsectsalluseexiblewings toachieveight. Toobtainsuccessfulightwithsimilarexibledesigns,re searchersandengineers willhavetogainanewunderstandingoftheaeroelasticinte ractionsthatoccuratthis scale[ 89 120 ]. ThesinglemostimportantfactorthathasallowedtheUnited StatesAirForce tobecomethemosteffectiveandpowerfulairforceinthewor ldisitsunmatched technologicaladvantage[ 146 ].Maintainingthisadvantagewhileadvancedtechnologies proliferateisaprimefocusoftheAirForce[ 175 ].Inparticular,theAirForcehasa vestedinterestinsmall,micro,andnano-sizedairvehicle sbecauseofthelife-saving situationalawarenesscapabilitiesthattheycanprovidet othecommanderandthe individualsoldier.Thesebenetshavebeenampliedasmil itaryoperationsnow 25

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frequentlyconsistofsmallteamsoperatinginnon-traditi onalenvironmentslikeurban centers[ 95 ].Asaresult,thesesystemsaresomeofthemostin-demandca pabilities thattheAirForceprovides[ 106 ]. MicroairvehiclesareanintegralpartoftheAirForcestrat egicvisionforunmanned aircraftsystems[ 145 ].Theseaircraftareenvisionedtoconductawidevarietyof challengingindoorandoutdoormissions.TheAirForceRese archLaboratoryhasa goalofdemonstratinganano-sizedUAVplatformperforming missionsinanurban environmentbytheyear2030[ 106 ].Theirresearchsuggeststhatusingairframeswith inherentexibilitymayprovidesomeadvantagesinaccompl ishingthesemissions[ 1 ]. However,thisexibilityposesseveraltechnicalchalleng esintheareasofaeroelasticity, ightdynamics,andightcontrol.Thisresearchisaimedat addressingsomeofthese challengesandtheresultswillaidtheAirForceinthedesig noffuturemicroairvehicles. 1.1PriorResearch 1.1.1AeroelasticModelDevelopment Muchefforthasbeeninvestedindevelopingmodelsforinves tigatingtheaeroelastic behaviorofightvehicles.Earlyeffortsderiveequations ofmotionforanunrestrained exiblevehicle[ 19 ].Solutionstotheequationsofmotionuselinearizationab outan equilibriumbyassumingsmallperturbationsintheelastic andrigidbodydegreesof freedom[ 107 ]. Furtherdevelopmenteffortsfortheaeroelasticequations ofmotionaretheWaszak study[ 176 ]andtheButtrillstudy[ 27 ].Bothofthesedevelopmentsuseameanaxis bodyreferenceframeandutilizeLagrange'smethodwhereth eelasticstrainenergy ofthevehicleisincludedinthepotentialenergyterms.Bot hexpresstheaerodynamic forcesforanelasticaircraftinastabilityderivativefor m.Suchareferenceframe removesanyinertialcouplingbetweentherigid-bodyandel asticdegreesoffreedom. Bothmakeuseofseveralimportantassumptionssuchassmall structuraldeformations 26

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(whichallowsthebody'sinertiatobetreatedasconstant), synchronouselasticmotion, andassumingthatthestructurecanbetreatedasacollectio nofpointmasses. Aframeworkiscreatedfortheintegrationofanalyticaldyn amics,structural dynamics,aerodynamics,andcontrolforthesimulationofd ynamicaircraftresponse[ 97 – 104 ].Thetheoryusesareferenceframeattachedtoanundeforme daircraft.Sucha referenceframeavoidsthecomplicationsassociatedwitha meanaxisreference frame,suchasexpressingtheaerodynamicforcesandenforc ingtheconstraintsin themeanaxes.TheequationsofmotionarederivedusingLagr ange'sequations withquasi-coordinates.Aerodynamicsareestimatedusing striptheory,butitisnoted thatanewaerodynamicmethodforcomputingthewholeaircra fttimeresponseina rapidmannerisneeded.Aperturbationapproachisused,sep aratingtheproblem intoazero-orderproblemfortherigidbodymotionandarst -orderproblemforthe elasticdisplacements.Anemphasisisplacedonlowcomputa tionalcostforon-board computing. Theaeroelasticstabilityandresponseofanonlinearaeroe lasticwingisinvestigated usingageometricallyexactstructuralmodelcoupledwitha nonlinearaerodynamic modelincludingstalleffects[ 127 ].Finitestateaerodynamictheory[ 133 134 ]isusedto obtainastate-spacerepresentationoftheaerodynamicswi thalownumberofstates. Themethodaccountsforlargescaleairfoilmotionaswellas smalldeformationsofthe airfoilsuchastrailing-edgeapdeection.Themethodisv alidatedagainsttheGoland wing[ 58 ]viaautteranalysis. ThepreviousworkledtothecreationoftheNonlinearAeroel asticTrimandStability ofHALEAircraft(NATASHA)program[ 30 65 125 126 189 ],whichwasdeveloped toanalyzetheaeroelasticcharacteristicsofhighlyexib leyingwings,specicallythe high-altitudelong-endurance(HALE)classofaircraft.Ex amplesofHALEaircraftinclude NASA'sPathnder,PathnderPlus,Helios,andtheEuropean HeliPlatUAV[ 50 142 ]. HALEaircraftarechallengingtomodelbecauseoftime-vary inginertiaproperties, 27

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coupledinertialforcesduetoarotatingcoordinatesystem andrelativevelocityofexible members,andexternalforcesandmomentswhicharenolonger basedonarigidbody geometry.ResultsfromNATASHAcomparedfavorablywithext ernaldata[ 153 ]. AntherapproachtomodelHALEaircraftisundertakenbysepa ratingthedynamics intonominalandperturbationdynamics[ 174 ].Thenominaldynamicsareusedto simulatethelarge-scalemotionofthemaneuverandthepert urbationdynamicsare usedtoaddressthestabilityoftheaircraftalongtheight path. Acomparisonbetweenmodelsforaveryexible,highaspectratiowingis conductedusingacommonframework[ 124 ].Themodelsincludeanintrinsicmodel, strain-basedmodel,andniteelementmodel.Thevelocitie sandwingtipdisplacements ofanaircraftarecompared.Resultsshowgoodagreementbet weenthemodelsandthe intrinsicandstrain-basedformulationsarefoundtohavea lowcomputationalcostas comparedtotheniteelementmethod. FLEXSTABwasacomputerprogramdevelopedfortheanalysiso felasticaircraft congurationsatsubsonicandsupersonicspeeds[ 49 171 ].Theprogramuseslinear methodstoevaluatestaticanddynamicstability,thetrims tate,aerodynamics,and elasticdeformationswithanemphasisonthestabilityandc ontrolcharacteristics. AnAutomatedSTRucturalOptimizationSystem(ASTROS)wasd evelopedto performautomatedpreliminarystructuraldesignforanaer oelasticvehicleusinga nite-elementapproachcoupledwithsteadyandunsteadyae rodynamics.Thecode iscapableofstaticaeroelasticanalysis,utteranalysis ,andlimitedcontrolresponse analysis[ 114 ]. TheAeroelasticDesignOptimizationProgram(ADOP)wasasi milartooldeveloped incompetitionwithASTROS[ 42 ].AlthoughtheADOPbidwasunsuccessful,theireffort focusedonminimizingstructuralweightwithoutviolating threestaticaeroelasticdesign constraints:lifteffectiveness,rolleffectiveness,and divergence. 28

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ASWINGisanintegratedtoolforaerodynamic,structural,a ndcontrollawanalysis anddesigninafullyandnonlinearlycoupledmanner[ 44 45 ].Themethodallowsrapid conceptualanalysisandisespeciallysuitedtotheearlyph asesofaircraftdesign.The formulationisbasedonanonlinearbeamapproachwithlifti ng-lineaerodynamicswhich includescorrectionsforaerodynamiclag. TheUniversityofMichigan'sNonlinearAeroelasticSimula tionToolbox(UM/NAST) isacomputationalframeworkfortheaeroelasticanalyseso fveryexibleaircraft[ 24 148 165 166 ].Itusesareducedorder,nonlinear,strain-basedbeamfor mulationto modelthestructureandincorporatesunsteadyaerodynamic sandnonlinearcontrol dynamics.Itisalsoabletomodelcompositebeamstructures withembeddedactive piezoelectricmaterials. ResearchersatTexasA&Mdevelopedandtestedanexperiment alaeroelastictest apparatuswhichallowedinvestigationintononlinearaero elasticresponses[ 121 – 123 ]. Theirapparatushasnonlinearspringsandinterchangeable camswhichcanalter thelinearityoftheresponse.Numericaluttersimulation scomparewellwiththe experiments.Applicationsarealsomadefornonlinearcont rollawdevelopment[ 77 78 ]. TheBoeingX-53ActiveAeroelasticWing(AAW)developmentp rogram[ 21 33 39 73 ]wasundertakenjointlybytheAirForceResearchLaborator y,BoeingPhantom Works,andNASADrydenFlightResearchCentertoactivelyco ntrolaeroelasticwing twistforthepurposeofaircraftcontrol.Theprogrammodi edaF/A-18Aghteraircraft tocontroltheaeroelasticwingtwistthroughmultiplelead ingandtrailingedgeaps,thus obtainingthedesiredamountofwingcontrolpower.TheF/A18Awaschosenbecause ofitsrelativelyhighaspect-ratio,thin,exiblewing.In fact,thepreproductionaircraft, basedontheoriginalNorthropYF-17design,showeddegrade drollperformance becauseoflowtorsionalstiffness.Althoughthisdegradat ionwasxedbystiffeningthe wingfortheproductionaircraft,theX-53aircraftwasmodi edtoreturntotheoriginal, preproductiontorsionalstiffnesslevels.TheX-53succes sfullyprovedtheAAWconcept 29

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duringrollmaneuversintestights.TheAAWtechnologywas alsoappliedtoanF-16 aeroelasticmodelwithfavorableresultsforwingcontrolp ower[ 131 ]. 1.1.2ModelingAeroelasticFlightDynamicsandControl Thesubjectofaircraftdynamicshasoftenbeendividedbetw eenightdynamics andaeroelasticity[ 101 ].Ingeneral,ightdynamicshastodowitharigid-bodyairc raft undergoingmaneuvers.Aeroelasticityisusuallyconcerne dwiththeinteractions betweentheaerodynamicsandstructureofanon-maneuverin gexibleaircraft.In thisregard,ightdynamicsandaeroelasticityhavedevelo pedseparately.However,the importanceofconsideringtheirinteractionhasoftenbeen noted[ 34 35 94 115 139 ]. Investigationsintotheshort-periodmodeofanaircraftwi thanelasticwingand varyinglevelsofsweepareconducted[ 94 ].Themethodisrestrictedtoalongitudinal investigationtoreducethemodelsize.Theauthorsreporta lossofstaticstabilitydueto wingexibilityforallwingcongurationsandadecreasein dynamicstabilityforthe0 sweepconguration. Aforward-sweptcongurationisanalyzedtoshowtheimport anceofincluding aircraftrigid-bodymodesintheaeroelasticanalysis[ 181 ].Couplingoftheelastic andrigidbodymodesisfoundtodependontheinertial,aerod ynamic,andstiffness characteristicsoftheaircraft.Possibleconsequencesof thiscouplingarebody-freedom utterordivergence. AnaeroelasticmodeldevelopedfromLagrange'sequationsi sappliedtoahigh speedtransportwithamoderatelevelofexibility[ 176 ].Resultsshowanunstable phugoidmodefortheexibleaircraft.Theshortperiodfreq uencyanddampingofthe aeroelasticmodelare55%and14%differentfromtherigidmo del,respectively. AdynamicaeroelasticmodelisusedtoevaluateCooper-Harp erpilotratingsto demonstratetheimportantrelationshipbetweenightdyna micsandaeroelasticity[ 143 ]. Bothlongitudinalandlateralrigid-bodydynamicsareincl udedinthemodel.Asevere 30

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degradationofthehandlingqualitiesresultsasthelowest structuralfrequency decreased. Considerationoftheinteractionsbetweenaeroelasticity andightdynamicsis veryimportantforHALEaircraftbecauseoftheiruniquecha racteristics[ 188 ].These aircraftoperatewithunusuallylargewingdeections;con sequently,traditionallinear theorywillnotprovideaccurateestimationsoftheightdy namics.Furthermore,the lowstructuralfrequenciesoftheseaircraftarewithinthe rangeoftherigidbodymodes. TheseuniquecharacteristicscausedaprototypeHELIOSair crafttoexperiencean in-ightmishaponJune26,2003[ 117 ].Afterencounteringturbulence,aeroelastic effectscausedaveryhighdihedralangletodevelopwhichle dtoadivergentpitching mode.Oscillationsofincreasingamplituderesultedinhig hairspeedandhighdynamic pressures,whichcausedthewingstructuretofail.Themish apinvestigationshowed thattheliftdistributionofsuchanaircraftcanbeverysen sitivetosmallamplitudegusts, especiallywhenundergoinglargedeformationsthatinvolv eahighdihedralangle.The mishapinvestigationattributedthecrashtoalackofadequ ateanalysismethodswhich ledtoaninaccurateriskassessmentandaninappropriatede cisiontoytheaircraft. StudiesusingtheNATASHAprogramfoundasignicantchange intheight dynamicscharacteristicsofHALEaircraftduetowingexib ility,specicallythe phugoidandshortperiodmodes[ 30 125 127 128 ].Inonestudy,thepairofcomplex short-periodrootsmergestobecometworealrootsandtheph ugoidmodegoes unstablewhentheaircraftisunderloading[ 126 ].Nonlinearightsimulationofthe aircraftindicatesthatthephugoidinstabilityledtocata strophicconsequences.Thetrim shapeisfoundtobehighlydependentontheightmissionand ightconditions.This resultiscriticalbecausetheightdynamicresponseforea chtrimshapecanbequite different. Theightdynamicsofaveryexiblevehiclearesuccessfull ycharacterizedusing UM/NAST[ 147 ].Itisfoundthattherigid-bodymodeldidnotadequatelyca pturethe 31

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dynamicsoftheexiblevehicle.Forsymmetricmaneuvers,r esultsshowedthata linearizedaeroelasticmodelisadequate.However,forasy mmetricmaneuvering,a nonlinearapproachisnecessarytocapturethevehicleresp onse.Thisworkledto thedevelopmentofaspecializedUAVforighttestingofver yexible,aspect-ratio wings[ 28 ]. 1.1.3ModelingandSimulationofAeroelasticMAVs Increasingeffortisbeingplacedonaeroelasticanalyseso fmicroairvehiclesin theiruniqueightregime.AlthoughhighlyexibleMAVwing shavebeenusedinpractice withnotableimprovementsintheyingqualities[ 71 ],littlehasbeendonetocharacterize themanalytically. Aexiblemembraneairfoilinteractingwiththeuidowism odeledbycouplinga two-dimensionalelasticmembranemodelbasedonnormaland shearstresseswitha two-dimensionalcomputationaluiddynamics(CFD)code[ 149 ].Themodelisapplied toarigidwing,exiblewing,andhybridwing.Theresultssh owanincreaseinpeak aerodynamicperformanceoftheexiblewing. Thepreviousworkisextendedtonitemembranewingsofvary ingconguration[ 159 ]. Athinmembranemodelcombinedwithacompositestructuralm odelisusedinorderto modelthemixedmembrane/carbonberstructure.Fourdiffe rentnumericalapproaches tomodelingthemembraneareimplementedandreviewed,incl udinghighdelity nonlinearandhyperelasticmembranemodels.TheCFDcodeis expandedtoathree dimensionalincompressibleNavier-StokesCFDmodelinclu dinga k viscous turbulencemodel.Theexperimentalandnumericalworkshow theadvantagesand disadvantagesofamembranewing.Someoftheadvantagesare increasedcamber anddelayedstall.Additionally,improvedlift,drag,pitc hingmomentsleadtoimproved staticstabilityandgustrejection.Onedisadvantageisap ossiblerollinstabilityfromthe exibledesign.Thestudyconcludeswitharecommendationf orahigherdelitymodel inordertocaptureunsteadyphenomenalikevortexshedding ,vibration. 32

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Anexperimentalefforttocharacterizethestaticstabilit yofanelasticMAVwith variousstructuralcongurationsisconducted[ 72 ].Thestructuralcongurationsconsist ofvaryingthenumberofbattensusedtoconstrainthemembra newing.Adelayin stallisseendueadecreaseintheeffectiveangleofattackc ausedbythepassive deformationofthewing.Theexiblewingshadweakerwingti pvortices,lowerliftto dragratios,andextensivemembranevibrationsduringtest ing.Overall,thevehicleis staticallystable.Astatespacesimulationmodelisdevelo pedtoassesstheightcontrol characteristics[ 178 ].Thesimulationmodelsarebasedonaeroelasticexperimen taldata butarenotfullycoupledstructural/aerodynamicmodels. TheightdynamicsofagenericMAVwithavariouslevelsofwi ngexibilityis analyzed[ 162 – 164 ].Flighttestmaneuversshowalargedifferenceinthedynam ic responseofveryexiblecongurationscomparedtoveryrig idwingcongurations.A reductioninthetransientresponseoftheexiblewingcon gurationascomparedtothe rigidwingcongurationisnoted.Flighttestresultsareco mparedtorigidbodyanalytical predictionsandshowsomedifferences. 1.2ProblemStatement Typicalmethodsofanalyzingtheightdynamicsofanaircra fttreattheaircraftas arigidbody.Foraircraftwhichhavealargeamountoffreque ncyseparationbetween theightandstructuralmodes,thisassumptionhasbeenval id[ 180 ].However,as aircraftstructuresbecomelighterandmoreexible,thefr equencyseparationreduces, especiallybetweentheshort-periodandrstbendingmodes [ 45 ].Insuchsituations, theightdynamicscanbesignicantlyinuencedbytheelas ticstructure[ 167 177 ], possiblyleadingtodynamicinstability[ 181 ]. Aircraftdesigntypicallyonlyevaluateschangesinconven tionaldesignparameters withoutconsideringchangestoparameterswhichwouldhave adirecteffectonthe aeroelasticresponseoftheaircraft,suchasstructuralst iffness.Asaresult,thenature oftheaeroelasticinuenceontheightdynamicsisnotcapt ured. 33

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Manyengineersandresearchersdesiretoexploitexibilit yinmicroairvehicles, butalackofunderstandingoftheinteractionbetweenaeroe lasticityandightdynamics canleadtosub-optimaldesigns.Theuniquestructuralchar acteristicsandightregimes ofMAVsimpliesthattheinteractioneffectsatthesesmalls calesmaynotfollowthe interactioneffectsatlargerscales.Therelativelysmall amountofresearchdoneon aeroelasticMAVshasfocusedonstaticaeroelasticityandd oesnotadequatelyaddress thedynamicinteractionsbetweenaeroelasticityandight dynamicsfromadesign perspective[ 71 135 159 ]. Considerationoftheseeffectsmayprovidetheaircraftdes ignerwithanewdesign parametertoaltertheopen-loopbehavioroftheaircraftin thepreliminarydesign process.Thepurposemaybetoavoidundesirableeffectsont heightdynamicsorto takeadvantageofeffectsthatarebenecialtothemission. Itmayalsobedesirableto rapidlytailorormorphthestructureofanexistingMAVtoal tertheightdynamicsina benecialway.Anunderstandingofthestructure'sinuenc eontheightdynamicsis criticaltoachievethesegoals,andathoroughstudyofthee ffectsofaeroelasticityon theightdynamicsofanMAVisneeded. AuniqueclassofMAVstructureisthemembranewing.Theinte ractionofthe membranewiththeaerodynamicsisdifculttocharacterize .Someworkhasbeen donetonumericallyandexperimentallycharacterizethest aticstabilityandcontrol characteristicsofmembranewings.Nopriorresearchhasex perimentallyinvestigated thedynamicstabilityandcontrolcharacteristics.Suchan investigationcouldleadto improvedMAVvehicledesigns. Battensaresometimesusedtostiffenandconstrainahighly exiblewing.Common designparametersarethenumber,size,andorientationoft hewingbattens.Some limitedinvestigationshavebeensuccessfullyconductedi nthisarea[ 87 88 157 159 ],however,noresearchhasinvestigatedtheaeroelasticim pactofthesedesign parametersonthevehicle'sightdynamics. 34

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Aeroservoelasticdesignsynthesisinvolvesexaminingthe interactionofthe structuralandcontroldesignintheaircraftdesignproces s.Thisapproachisuncommon inaircraftdesignbuthasthepotentialtorevealimportant trade-offsbetweenthe controlsystem,structure,andaerodynamicstothedesigne r.ForaexibleMAV, theseinteractionsarenotwellunderstoodbutmayhaveacri ticalimpactonaircraft performanceandmissionsuccess. TheconstrainedandclutteredenvironmentsinwhichMAVsof tenoperatepresent uniqueenvironmentalchallengesintheformofwindgusts.A sanMAViesdowna streetinanurbancanyon,forexample,itcouldbesubjectto gustswhosevelocitiesand spatialdimensionsmaybeontheorderoftheaircraft'svelo cityanddimensions.When combinedwiththelowinertiaofMAVs,thesegust-inducedfo rcesandmomentscould easilyupsettheaircraft.Thesegustsbecomeevenmoreimpo rtantwhenconsidering theaeroelasticeffectsthatarepossibleinexibleMAVs.A sthewingbecomesmore exible,itcouldabsorbsomeenergyofthegust,possiblyde creasingthesensitivityof theoverallvehicletothegust.Atthesametime,highlevels ofexibilitycoulddelaya controller-inducedreactiontothegust,whichmightreduc ethegustrejectionability. However,theightdynamicsofanaeroelasticaircraftcanc hangedrasticallyanditisnot clearhowtheyinteractwithaMAV'sgustsensitivityandgus trejectionproperties. Thecurrentresearchproposestocontributetothebodyofkn owledgebysystematically investigatingtheeffectsofaeroelasticityontheightdy namicsofaxed-wing,exible MAV.Inparticular,thebendingandtorsionalstiffnessoft hewingarechosenasthetwo mainparametersofinterest.Theeffectsonthetrimconditi ons,rigidbodymodes,and structuralmodesarecharacterized.Directcorrelationsb etweenthestiffnessandthe ightdynamicsareobtained,increasingtheabilitytoutil izeormitigatetheseeffects. Researchisconductedtoinvestigatethepossibilityofin uencingtheight dynamicsthroughsmall,specializedchangesinthestructu ralcongurationofthe aircraft.Knowingtheeffectsofsuchchangesontheightdy namicscouldenable 35

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designerstoeasilytailortheMAVstructureforaparticula rgoalwithoutalteringthe entirestructure. Auniqueexperimentalfacility,theUniversityofFlorida' slowspeedwindtunnel[ 9 ], isusedtoexperimentallyinvestigateanddeterminethesta ticanddynamicstabilityand controlcharacteristicsofamembrane-wingMAV.Thisdynam iccharacterizationofa membranewingisarstofitskind. Theeffectofwingbattensontheightdynamicsisstudied.B attensareapproximated usingstepwisechangesintorsionalstiffnessacrossthewi ng.Characterizingthe directeffectofbattensontheightdynamicsisauniquecon tributiontothebodyof knowledge. Afterunderstandingtheeffectsofwingstiffnessontheig htdynamics,optimal controlisappliedtotheexibleaircraft.Theimpactofcha ngingthewingstiffnesson theclosed-loopaircraftperformanceandcontrolactuatio nisstudied.Amodel-following controlapproachisusedinanovelattempttovirtuallychan gethestiffnessofthewing. Differenttypesofgustsatvaryingfrequenciesareapplied totheexibleaircraft models.Theaeroelasticeffectofthewingstiffnessontheg ustsensitivityoftheaircraft isanalyzed.Aregulationcontrollerisusedtoanalyzethee ffectsofwingstiffnesson thegustrejectionproperties.Theresultsarecomparedtot raditionalmetricsforgust sensitivitytodeterminetheirusefulnessforaexibleair craft. TheprimarynumericaltoolusedisASWING[ 45 ].Itsfocusonrapidconceptual analysisofaeroelasticvehiclesiswell-suitedtothisres earch.Theprimaryvehicleused forthisstudyisthegenericMAV(GenMAV),developedattheM unitionsDirectorate oftheAirForceResearchLaboratoryatEglinAirForceBase, Florida[ 163 164 ].The GenMAVwasspecicallydesignedtoprovideaversatileplat formforMAVresearchand development. 36

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1.3DocumentOrganization Chapter 1 introducesthedocumentbyaddressingtherelevantscienti cmethods andsurveyingthecurrentbodyofknowledge.Theresearchgo alsareintroducedand discussedinlightofthecurrentstateoftheart. Chapter 2 recallsthetheorynecessarytoproceedwiththeresearch.F irst,the classiclinearizedrigidbodyequationsofmotionaredevel oped.Second,fundamental structuraldynamicstheoryisintroducedbywayofamass-sp ring-dampersystem.The topicofaeroelasticityisintroduced,includingstatican ddynamicaeroelasticitywithan overviewofunsteadyaerodynamics.Finally,thenecessary controltheoryisreviewed. Chapter 3 introducesASWING,theprimarymodelingandsimulationtoo lusedfor thenecessaryportionsoftheresearch.Areviewofitstheor eticalbasisisundertaken withanemphasisonapplicationtothecurrentresearch.Ina ddition,theprimaryvehicle ofinterest,theGenMAV,isintroduced. Chapter 4 reportsonthemethodologyandresultsofanexperimentalin vestigation todeterminethestaticanddynamicstabilityandcontrolch aracteristicsofamembrane-wing MAV. Chapter 5 givesadetailedaccountoftheinvestigationintotherelat ionship betweenwingstiffnessandtheightdynamicsofthevehicle .Theemphasisisona uniformstiffnessdistributionacrossthewing.Chapter 6 relatestheresultsofasimilar investigationwithselectednon-uniformdistributionsof wingstiffness. Chapter 7 introducestheconceptofwingbattensandreportsonthemet hodology andresultsforaninvestigationintotheeffectofwingbatt ensontheightdynamicsof anMAV. Chapter 8 appliestheresultsofChapter 5 andexaminestheirinteractionwitha typicaloptimalightcontrolscheme.Amodel-followingco ntrolschemeisalsoanalyzed, whichattemptstovirtuallychangethestiffnessofthewing 37

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Chapter 9 takestheaircraftdevelopedinChapter 5 andanalyzestheirgust sensitivityandgustrejectionpropertiesinlightoftheae roelasticeffects. Chapter 10 concludesthedissertationandpresentsrecommendationsf orfuture researchopportunities. 38

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CHAPTER2 THEORY 2.1FlightEquationsofMotion 2.1.1ReferenceFrames Therststepinanalyzingthekinematicsorkineticsofasys temistochoosea referenceframe.Areferenceframeistheperspectivefromw hichmotionisobserved andconsistsofatleastthreenoncolinearpointsthatmovei nthree-dimensional Euclideanspace( R 3 ).Thedistancebetweenpointsinareferenceframedoesnot changeastheframemoves.2.1.1.1Earthreferenceframe TheEarthisassumedtobeatandstationaryforthepurposes oflocalatmospheric ight.Areferenceframe F E isattachedtotheearthandisconsideredaninertialframe inwhichNewton'slawsofmotionarevalid. Acoordinatesystem,calledtheEarthcoordinatesystem,is createdintheEarth frameusingaright-handedsetofbasisvectorswiththeiror iginarbitrarilylocatedon thesurfaceoftheEarth.Theverticalunitvectorin F E (denoted ^z E )pointstowardthe centeroftheearth.Theunitvectors ^x E and ^y E arechosentopointNorthandEast, respectively.Thecoordinatesystemisdenotedby F E ( O E ; ^x E ; ^y E ; ^z E ) andisillustrated inFig. 2-1 ,wheretheoriginofthecoordinatesystemin F E isdenoted O E 2.1.1.2Bodyreferenceframe Thebodyreferenceframe,denoted F B ,isxedtotheaircraft.Acoordinatesystem, calledthebodycoordinatesystem,isdenedinthebodyrefe renceframewithitsorigin atthecenterofgravity.Notethatgravityisassumedtobeun iformandthustheaircraft centerofgravity(CG)iscoincidentwiththeaircraftcente rofmass(CM).The ^x B axisis denedtorunfromtheCGoutthenoseandthe ^y B axisrunsparalleltotherightwing. The ^z B axisresultsfromthecrossproduct ^x B ^y B .Thecoordinatesystemisdenoted by F B ( O B ; ^x B ; ^y B ; ^z B ) andisillustratedinFig. 2-2 39

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O E ^x E ^y E ^z E Figure2-1.Earthcoordinatesystem Thebodycoordinatesystemisofprimaryconcernsinceitist hesysteminwhich thebody'sinertiaismosteasilydened.Theequationsofmo tionwillbedevelopedin thiscoordinatesystem. ^x B ^y B ^z B O B Figure2-2.Bodycoordinatesystem Asecondcoordinatesystem,calledthestabilitycoordinat esystem,isdenedin thebodyframe.Anaxis ^x S isalignedwiththeprojectionofrelativewindonthe ^x B ^z B planeandisfoundbyrotatingthebodycoordinatesystemaro und ^y B throughanangle 40

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,calledtheangleofattack.Thiscoordinatesystemisshown inFig. 2-3 andisdenoted by F B ( O S ; ^x S ; ^y S ; ^z S ) .Thissystemisusefulfordeterminingtheaerodynamicforc esand momentssincetheyaredependentontheorientationoftheai rcraftwithrespecttothe oncomingow. ^x B ^x S ^y B ; ^y S ^z S ^z B O B ; O S Figure2-3.Stabilitycoordinatesystem2.1.2CoordinateTransformations TheangularrelationsoftheEarth,body,andstabilitycoor dinatesystemsareshown inFig. 2-4 Earth Body Stability ; ; Figure2-4.Overviewofcoordinatetransformations AvectormaybetransformedfromtheEarthframeintothebody framewitha3-2-1 Eulerrotationsequencethroughtheheadingangle ,pitchangle ,androllangle TheseanglesarecommonlyknownastheEulerangles.Avector expressedintheEarth frame(denoted f a g E )canthusbetransformedtothebodyframethroughtheseries of rotationsshowninEq.( 2–1 ),wherearotationaboutthe x -axisthroughtheangle is 41

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denoted R 1 ( ) f a g B = R 1 ( ) R 2 ( ) R 3 ( ) f a g E (2–1) Arotationaboutthex-axisthroughtheangle isaccomplishedby: R 1 ( )= 266664 1000cos sin 0 sin cos 377775 (2–2) Arotationaboutthey-axisthroughtheangle isaccomplishedby: R 2 ( )= 266664 cos 0 sin 010 sin 0cos 377775 (2–3) Arotationaboutthez-axisthroughtheangle isaccomplishedby: R 3 ( )= 266664 cos sin 0 sin cos 0 001 377775 (2–4) TheresultingrotationmatrixfromtheEarthcoordinatesys temtothebodycoordinate system( R BE )isshowninEq.( 2–6 ). R BE = R 1 ( ) R 2 ( ) R 3 ( ) (2–5) = 266664 cos cos cos sin sin sin sin cos cos sin cos cos +sin sin sin sin cos sin sin +cos sin cos cos sin sin sin cos cos cos 377775 (2–6) Avectorinthestabilitycoordinatesystemmaybetransform edintothebody coordinatesystemthroughthetransformationshowninEq.( 2–7 ),where R BS isdened 42

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accordingtoEq.( 2–9 ). f a g B = R BS f a g S (2–7) R BS = R 2 ( ) (2–8) = 266664 cos 0 sin 010 sin 0cos 377775 (2–9) Forexample,theaerodynamicforcesofdragandliftarecomm onlyexpressedinthe stabilitycoordinatesystembutmustbetransformedtotheb odycoordinatesystem.This transformationcanbeaccomplishedbyapplyingEq.( 2–7 )asshowninEq.( 2–10 ). 8>>>><>>>>: 266664 F x F y F z 377775 9>>>>=>>>>; B = 266664 cos 0 sin 010 sin 0cos 377775 8>>>><>>>>: 266664 D F y L 377775 9>>>>=>>>>; S (2–10) Becausearotationmatrixisanorthogonalmatrix,ithasthe specialpropertythat R 1 = R T .Thus,thereversetransformationscanbeobtainedwitheit hertheinverseor transposedrotationmatrices.Forexample,avectorexpres sedinthebodyframecanbe transformedintotheearthframeviaEq.( 2–11 )orEq.( 2–12 ). f a g E = R TBE f a g B (2–11) = R 1 BE f a g B (2–12) 2.1.3NonlinearEquationsofMotion Twelvequantitiescompletelydescribetheaircraftmotion overtime:position( x;y;z ), translationalvelocity( u;v;w ),orientation( ;; ),andangularvelocity( p;q;r ).Unless otherwisestated,allquantitiesareexpressedinthebodyc oordinatesystem.Twelve equationsareneededtoobtainasolutionforthesequantiti es. 43

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2.1.3.1Translationaldynamics Equationsforthetranslationalmotionofthecenterofmass arederivedfrom Newton'ssecondlawwhichstatesthatthetimerateofchange ofmomentumisequalto theforcesactingonthebody. E d dt ( m V )= F (2–13) Assumingmassisconstantandknowingthat E d dt V = E a ,Eq.( 2–13 )canbewrittenas Eq.( 2–14 ).Thenotation E f a g B denotestheaccelerationvectorexpressedinthebody coordinatesystemasviewedbyanobserverintheinertialre ferenceframeand,sinceall quantitiesareexpressedinthebodycoordinatesystemunle ssotherwisenoted,canbe writtenas E a m E a = F (2–14) Letthepositionofthebody'scenterofmassasmeasuredwith respecttothebody framebedenoted r = B r r = 266664 x y z 377775 (2–15) Thevelocityasobservedinthebodyframe,giveninEq.( 2–16 ),issimplythetime derivative. B V = B d dt r = 266664 x y z 377775 (2–16) ThevelocityasobservedintheEarthframeisdenoted B V andshowninEq.( 2–17 ). E V = E d dt B V = B V + E B r (2–17) 44

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Notethatthetransporttheoremmustbeemployedsincetheti merateofchangeof thevectorisbeingobservedinadifferentreferenceframe( thetransporttheoremis introducedinAppendix A.2 ).Theangularvelocityvector E B istheangularvelocityof referenceframe F B asviewedbyanobserverinreferenceframe F E andconsistsofthe individualrates p;q;r .TheresultisdenedinEq.( 2–18 ). E V = 266664 u v w 377775 (2–18) NowtheaccelerationofthebodyintheEarthframemaybeform ulated.Notethatthe transporttheoremmustbeemployedagain.Theresult,shown inEq.( 2–21 ),represents theaccelerationoftheaircraftasviewedbyanobserverint heEarthframe,expressed inthebodycoordinatesystem. E a = E d dt E V = B d dt E V + E B E V (2–19) = 266664 u v w 377775 + 266664 0 rq r 0 p qp 0 377775 266664 u v w 377775 (2–20) E a = 266664 u + qw rv v + ru pw w + pv qu 377775 (2–21) Nowtheright-handsideofEq.( 2–14 )isdeveloped.Theforcesthatwillbeincluded aregravitational( F g ),aerodynamic( F a ),andpropulsive( F T ),asshowninEq.( 2–22 ). Eachoftheseforcesmustbeexpressedinthebodycoordinate systemtomaintain 45

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consistencywithEq.( 2–21 ). f F g B = 266664 F g x + F A x + F T x F g y + F A y + F T y F g z + F A z + F T z 377775 (2–22) First,thegravitationforceiseasilyexpressedintheEart hcoordinatesystemasshown inEq.( 2–23 ). f F g g E = 266664 00 mg 377775 (2–23) Thegravitationalforcemaythenbeexpressedinthebodycoo rdinatesystemby applyingtherotationdescribedinEq.( 2–6 ),resultinginEq.( 2–24 ). f F g g B = 266664 mg sin mg sin cos mg cos cos 377775 (2–24) Theaerodynamicforcesoflift,drag,andsideforcearenati velyexpressedinthestability coordinatesystemasshowninEq.( 2–25 ). f F a g S = 266664 D F a y L 377775 (2–25) Theseforcescanbeexpressedinthebodycoordinatesystemb yapplyingEq.( 2–9 ), resultinginEq.( 2–26 ). f F a g B = 266664 D cos + L sin F a y D sin L cos 377775 (2–26) 46

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Finally,thethrustforcesactinthebodyreferenceframewi th T denedasthe anglebetweenthethrustvectorand x B .Thethrustforcescanberotatedintothe bodycoordinatesystemthroughthestandardrotation R T2 ( T ) .Theresultisgivenin Eq.( 2–27 ). f F T g B = 266664 T cos T 0 T sin T 377775 (2–27) ThenalsetofbodyforcesisgroupedasshowninEq.( 2–28 ). 266664 F x F y F z 377775 = 266664 mg sin D cos + L sin + T cos T mg sin cos + F a y mg cos cos D sin L cos T sin T 377775 (2–28) TheresultsofEqs.( 2–21 ),( 2–24 ),( 2–26 )and( 2–27 )canbeassembledintothe formofEq.( 2–14 ),asshowninEq.( 2–29 ). m 266664 u + qw rv v + ru pw w + pv qu 377775 = 266664 mg sin D cos + L sin + T cos T mg sin cos + F a y mg cos cos D sin L cos T sin T 377775 (2–29) Equation( 2–29 )representsthetranslationalmotionoftheaircraft'scen terofmassin responsetotheforcesactingontheaircraft.2.1.3.2Rotationaldynamics Expressionsfortherotationalequationsofmotionoftheai rcraftwillnowbe obtained.TheseexpressionswillbeformulatedusingEuler 'ssecondlawofmotion, showninEq.( 2–30 ),whichstatesthatthetimerateofchangeoftheangularmom entum ofarigidbodyrelativetopointOintheinertialreferencef rame F E isequaltothe momentofthebodyaboutthesamepointinthesameinertialre ferenceframe[ 137 ]. Theangularmomentumrelativetopoint O asviewedbyanobserverin F E isdenoted 47

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E H O E d dt ( E H O )= M O (2–30) First,theleft-handsideofEq.( 2–30 )willbedeveloped.Recallthatthelinear momentumofaparticle,denoted p ,isdenedasshowninEq.( 2–31 ). p = m V (2–31) Theangularmomentumoftheparticlerelativetopoint O isdepictedinFig. 2-5 and giveninEq.( 2–32 ). h O = r p (2–32) r p O P Figure2-5.AngularmomentumofaparticlePrelativetopoin tO Asimilarprocessisusedtondtheangularmomentumoftheai rcraftusing adifferentialelementofmassandintegratingoverthebody .Tobegin,refertothe illustrationinFig. 2-6 First,thepositionofanelement dm isdenedinEq.( 2–33 ). R dm = R CM + r (2–33) Thevelocityofelement dm asviewedbyanobserverintheinertialreferenceframeis developedinEqs.( 2–34 )and( 2–35 ). E V dm = E d dt R dm (2–34) = E V CM + r + E B r (2–35) 48

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dm R CM R dm r F E Figure2-6.Angularmomentumofadifferentialelementonar igidbody Nowtheangularmomentumforthisdifferentialelementrela tivetothecenterofmass maybedevelopedbeginningwithEqs.( 2–36 )and( 2–37 ). E d h CM =( r E V dm ) dm (2–36) =( r E V CM ) dm +( r r ) dm +( r ( E B r )) dm (2–37) Integrationoverthebodyyieldsthetotalangularmomentum ofthebodyrelativeto thecenterofmassasviewedbyanobserverintheinertialref erenceframe,givenin Eq.( 2–39 ). E H CM = Z E d h CM (2–38) = Z ( r E V CM ) dm + Z ( r r ) dm + Z ( r ( E B r )) dm (2–39) ThethreetermsinEq.( 2–39 )deserveindividualanalysis.Becausethevelocityofthe centerofmassisconstantthroughoutthebody,therstterm mayberewrittenasshown inEq.( 2–40 ). Z ( r E V CM ) dm = Z r dm E V CM (2–40) Becausethereferencepointisthecenterofmass,theintegr alisequaltozeroandthis termfallsout.Thesecondtermisalsozeroundertherigidbo dyassumptionbecause 49

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_ r =0 .Thethirdtermmayberewrittenbyapplyingthevectortripl eproductasseenin Eq.( 2–41 ),resultinginEq.( 2–42 ). a ( b c )= b ( a c ) c ( a b ) (2–41) Z ( r ( E B r )) dm = Z ( E B ( r r ) r ( r E B )) dm (2–42) Theangularvelocityvectorisdenedintermsoftheindivid ualbodyratesasshownin Eq.( 2–43 ). E B = 266664 p q r 377775 (2–43) Theright-handsideofEq.( 2–42 )mayberewrittenusingthedyadicproductasdened inAppendix A.1 .TheresultisshowninEq.( 2–44 ). Z ( E B ( r r ) r ( r E B )) dm = Z [( r r ) 1 rr ] E B dm (2–44) Uponexpansionoftheright-handside,Eq.( 2–45 )isobtained. Z [( r r ) 1 rr ] E B dm = Z 266664 ( y 2 + z 2 ) xy xz xy ( x 2 + z 2 ) yz xz yz ( x 2 + y 2 ) 377775 E B dm (2–45) Theinertiamatrixisdenoted I anddenedasshowninEq.( 2–46 ). I = 266664 R ( y 2 + z 2 ) dm R xydm R xzdm R xydm R ( x 2 + z 2 ) dm R yzdm R xzdm R yzdm R ( x 2 + y 2 ) dm 377775 (2–46) 50

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ThediagonaltermsarethemomentsofinertiaasshowninEq.( 2–47 )andthe off-diagonaltermsarecalledtheproductsofinertiaassho wninEq.( 2–48 ). I xx = Z ( y 2 + z 2 ) dmI yy = Z ( x 2 + z 2 ) dmI zz = Z ( x 2 + y 2 ) dm (2–47) I xy = Z xydmI xz = Z xzdmI yz = Z yzdm (2–48) Formostaircraft,itisacceptabletoassumesymmetryabout the ^x B ^z B plane[ 187 ], causing I xy = I yz =0 .Also,itisimportanttonotethattheinertiamatrixmust beexpressedinacoordinatesystemandtheeasiestchoiceis thebodysystem. Coordinate-freeequationsofmotioncannotbeobtainedunl essatensorapproachis employed[ 190 ]. NowEqs.( 2–39 )and( 2–45 )maybeassembledtoobtaintheexpressionforthe angularmomentumofthebodyshowninEq.( 2–49 ). E H CM = I E B (2–49) ToapplyEq.( 2–49 )toEq.( 2–30 ),thetimerateofchangeoftheangularmomentum asviewedbyanobserverintheinertialreferenceframemust beobtained,shownin Eq.( 2–50 ). E d dt E H CM = B d dt ( I E B )+ E I E B (2–50) Theinertiamatrixisassumedtobeconstant.Thisassumptio nisnotalwaystrue,since anaircraft'smasscanchangeduringight.Forexample,mas sislostthroughfuel consumptionorcanbegainedthroughmid-airrefueling.How ever,anychangesare consideredtobesmallcomparedtotheoverallmass.Further more,forelectric-motor drivenmicroairvehiclesthisassumptionisespeciallyval id.Applyingthisassumptionto Eq.( 2–50 )resultinginEq.( 2–51 ). E d dt E H CM = I E B + E B I E B (2–51) 51

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Themomentsactingontheaircraftrelativetothecenterofm assinthebodycoordinate systemareaerodynamicandpropulsiveinnatureandareden edinEq.( 2–52 ). M CM = 266664 M A x + M T x M A y + M T y M A z + M T z 377775 = 266664 M x M y M z 377775 (2–52) Finally,thesetofequationsdescribingtherotationaldyn amicsusingEuler'ssecond lawcanbeobtainedbycombiningEqs.( 2–51 )and( 2–52 )intoEq.( 2–30 ),resultingin Eq.( 2–53 ).AmorecompactformisgiveninEq.( 2–54 ). 266664 I xx 0 I xz 0 I yy 0 I xz 0 I zz 377775 266664 p q r 377775 + 266664 0 rq r 0 p qp 0 377775 266664 I xx 0 I xz 0 I yy 0 I xz 0 I zz 377775 266664 p qr 377775 = 266664 M x M y M z 377775 (2–53) 266664 pI xx + qr ( I zz I yy ) ( pq +_ r ) I xz qI yy + pr ( I xx I zz ) ( r 2 + p 2 ) I xz rI zz + pq ( I yy I xx )+( qr p ) I xz 377775 = 266664 M x M y M z 377775 (2–54) 2.1.3.3Kinematics ThesixequationsobtainedthusfarareEqs.( 2–29 )and( 2–54 ).Sixmoreequations areneeded.Threemorecanbeobtainedbyrelatingtheangula rratesinthebody coordinatesystem( p;q;r )totheEulerrates( ; ; )throughtheappropriatecoordinate transformation. Whenrotatingfromtheearthcoordinatesystemtothebodyco ordinatesystem throughthethreeEulerangles,therearetwointermediatec oordinatesystems.The rstrotation isperformedintheEarthsystem,thesecondrotation isperformed intherstintermediatesystem,andthethirdrotation isperformedinthesecond intermediatesystem.Eachofthesesystemsmustbetransfor medinthebody coordinatesystemwherethebodyratesarecoordinatized.T histransformationis 52

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accomplishedbyrotatingeachsystemasshowninEq.( 2–55 ). E B B = R 1 ( ) R 2 ( ) R 3 ( ) + R 1 ( ) R 2 ( ) + R 1 ( ) (2–55) Notethatbecauseneighboringcoordinatesystemsshareaco mmonaxisofrotation, Eq.( 2–55 )canbesimpliedtoEq.( 2–56 ). E B B = R 1 ( ) R 2 ( ) + R 1 ( ) + (2–56) EvaluatingEq.( 2–56 )usingEqs.( 2–2 )and( 2–3 )alongwithEq.( 2–43 )yieldsthe additionalthreeequationsshowninEq.( 2–57 ). 266664 p qr 377775 = 266664 _ sin cos + sin cos sin + cos cos 377775 (2–57) Equation( 2–57 )mayberewrittenasEq.( 2–58 ). 266664 p qr 377775 = 266664 10 sin 0cos sin cos 0 sin cos cos 377775 266664 _ 377775 (2–58) ThematrixinEq.( 2–58 )maybeinvertedtoobtaintheEulerratesasafunctionofthe bodyrates,showninEq.( 2–59 ).Notethatasingularitymayoccurinthisinversionif = = 2 ,aphenomenoncommonlycalledgimblelock. 266664 _ 377775 = 266664 1sin tan cos tan 0cos sin 0sin sec cos sec 377775 266664 p qr 377775 (2–59) Thenalsetofthreeequationsisfoundbyrelatingtheveloc ityinthebodyframe tothevelocityintheEarthframe.First,thevelocityinthe Earthframeisdenedin Eq.( 2–60 )andthevelocityinthebodyframeisdenedinEq.( 2–18 ).Thetwoare 53

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relatedthrough R EB = R TBE ,asshowninEq.( 2–61 ). E V E = 266664 x y z 377775 (2–60) 266664 x y z 377775 = 266664 cos cos sin sin cos cos sin sin sin +cos sin cos cos sin cos cos +sin sin sin cos sin sin sin cos sin sin cos cos cos 377775 266664 u v w 377775 (2–61) Equation( 2–61 )isthenalsetofequationsneededtofullydeterminetheai rcraft motion. 54

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2.1.3.4Therigidbodyequationsofmotion Thenonlinear,rigidbodyequationsofmotionfromEqs.( 2–29 ),( 2–54 ),( 2–59 ) and( 2–61 )arecollectedintothesetshowninEqs.( 2–62 )to( 2–65 ). m (_ u + qw rv )= mg sin D cos + L sin + T cos T m (_ v + ru pw )= mg sin cos + F a y m (_ w + pv qu )= mg cos cos D sin L cos T sin T (2–62) M x =_ pI xx + qr ( I zz I yy ) ( pq +_ r ) I xz M y =_ qI yy + pr ( I xx I zz ) ( r 2 + p 2 ) I xz M z =_ rI zz + pq ( I yy I xx )+( qr p ) I xz (2–63) = p + q sin tan + r cos tan = q cos r sin = q sin sec + r cos sec (2–64) x = u cos cos + v (sin sin cos cos sin )+ w (sin sin +cos sin cos ) y = u cos sin + v (cos cos +sin sin sin )+ w (cos sin sin sin cos ) z = u sin + v sin cos + w cos cos (2–65) Undersomeightconditionstheseequationsmaybedecouple dintolongitudinaland lateral-directionalequationsofmotion. Onahistoricalnote,thegoverningequationsofmotionfora rigidaircraftwithsix degreesoffreedomwererstdevelopedbyG.H.Bryanin1911a ndareessentiallythe sameasthesetofequationspresentedhere[ 25 ]. 2.1.4LinearizedEquationsofMotion Thenonlinearequationsgoverningtheaircraftmotion,sho wninEqs.( 2–62 ) to( 2–65 ),arecomplexandanalyticalsolutionsareverydifcult[ 51 ].Although someattemptsatfullnonlinearanalyseshavebeenmade[ 92 ],theseequationsare 55

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commonlylinearizedaboutareferenceconditionknownastr im.Thislinearization isonlyconsideredtobevalidwhenthedeviationsfromthetr imconditionaresmall. Theresultingsetofhomogeneouslineardifferentialequat ionsismoreeasilysolved. Valuableinsightintotheaircraftdynamicscanbeobtained fromtheresultingclosed solutiontotheseequations. Thelinearizationapproachfollowsthesebasicsteps: 1.Denethereferenceightcondition2.Deneareferenceandperturbationvalueforeachvariabl es 3.Linearizeandsimplifytheequations4.Obtainrst-orderapproximationsfortherelevantforce sandmoments Atypicalreferenceightconditionissteadyightatsomea irspeed u 0 withno sideslip( =0 ),nobank( =0 ),andnoangularvelocity( p = q = r =0 ).Additionally, usingthestabilitycoordinatesystemxedinthebodyrefer enceframeresultsin v = w = 0 Typically,thereferenceandperturbationvaluesaredene dinthebodycoordinate systemaccordingtoEq.( 2–66 ).Eachreferenceconditionisdenotedwithasubscript 0 andtheperturbationsaredenotedwith u = u o + uv = v 0 + vw = w 0 + w p = p 0 + pq = q 0 + qr = r 0 + r = 0 + = 0 + = 0 + (2–66) Thelinearequationsareobtainedbyrecastingthenonlinea requationsintermsof thereferenceandperturbationvaluesatthedenedreferen cecondition.Theresulting equationsarethensimpliedaccordingtoperturbationthe ory,whichpostulatesthatthe effectofhigher-ordertermsismuchlessthantheeffectoft herst-ordertermsandthat anyproductsofperturbationsareconsideredtobenegligib lecomparedtotheindividual 56

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perturbations.Forangularvariables,thetrigonometrici dentitiesinEq.( 2–67 )apply. sin ( 0 + )=sin 0 cos | {z } 1 +cos 0 sin | {z } =sin 0 + cos 0 cos ( 0 + )=cos 0 cos | {z } 1 sin 0 sin | {z } =cos 0 sin 0 (2–67) Thenonlinearlongitudinalequationsofmotion,showninEq s.( 2–68 )to( 2–70 ),will belinearizedasanexample. m (_ u + qw rv )= mg sin D cos + L sin + T cos T (2–68) qI yy + pr ( I xx I zz ) ( r 2 + p 2 ) I xz = M y (2–69) m (_ w + pv qu )= mg cos cos D sin L cos T sin T (2–70) First,thenewlydenedvariablesareinsertedintoEqs.( 2–68 )to( 2–70 ),resultingin Eqs.( 2–71 )to( 2–73 ). m (_ u +( q 0 + q )( w 0 + w ) ( r 0 + r )( v 0 + v ))= mg sin D cos + L sin + T cos T (2–71) qI yy +( p 0 + p )( r 0 + r )( I xx I zz ) (( r 0 + r ) 2 +( p 0 + p ) 2 ) I xz = M y (2–72) m (_ w +( p 0 + p )( v 0 + v ) ( q 0 + q )( u o + u ))= mg cos cos D sin L cos T sin T (2–73) AfterexpandingEqs.( 2–71 )to( 2–73 ),removinganyproductsoftheperturbations,and simplifyingaccordingtothereferenceightcondition( p 0 = q 0 = r 0 = v 0 = w 0 =0 ),the linearizedlongitudinalequationsofmotioninEqs.( 2–74 )to( 2–76 )areobtained. m u = mg sin D cos + L sin + T cos T (2–74) qI yy = M y (2–75) m (_ w u 0 q )= mg cos cos D sin L cos T sin T (2–76) Theaerodynamicforceandmomenttermsmustalsobelineariz edabouta referencecondition.Strictlyspeaking,aerodynamicforc esandmomentsarefunctionals oftheindependentvariables.Thesefunctionalsdependont hevariables'completepast historyinadditiontotheircurrentvalues.Foraerodynami cfunctionals,thisrelationship 57

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isduetohysteresisintheowseparationprocess.Thisfunc tionalrelationshipisgiven inEq.( 2–77 ). L ( t )= L [ ( )] 1 t (2–77) ATaylorseriesexpansioncanbeusedtoexpress ( ) ,asshowninEq.( 2–78 ). ( )= ( )+( t ) @ @ t 0 + ( t ) 2 2! @ 2 @ 2 t 0 + ::: + ( t ) N N @ N @ N t 0 (2–78) Inthisexample,theliftattime t 0 isdeterminedby andallitsderivativesevaluated attime t 0 .Theclassicassumptionoflinearaerodynamicsisthatthe rst-ordertermin Eq.( 2–78 )issufcienttorepresenttheaerodynamicforce[ 25 ].Equation( 2–77 )can thenbewrittenasEq.( 2–79 ),where L 0 = L ( ( t 0 )) andthestabilityderivative L is denedinEq.( 2–80 ). L ( t )= L 0 + L (2–79) L = @L @ ( t 0 ) (2–80) Therst-orderapproximationinEq.( 2–79 )representsthechangeintheliftforcedue toaperturbationinangleofattack.Continuingtheexample ,thelongitudinalforcesand momentsareprimarilyafunctionoftheveparametersshown inEq.( 2–81 ). L;D;M y = f ( u;; ;q; e ) (2–81) Areferenceconditionandperturbationvalueisdenedfore achofthesevariablesin Eq.( 2–82 ). u = u 0 + u = 0 + =_ 0 +_ q = q 0 + q e = e 0 + e (2–82) 58

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Afterapplyingtherst-orderTaylorseriesexpansion,lin earexpressionsfor L;D;M y areobtainedinEq.( 2–81 ). L = L 0 + L u u + L + L _ + L q q + L e e (2–83) M y = M y 0 + M y u u + M y + M y _ + M y q q + M y e e (2–84) D = D 0 + D u u + D + D _ + D q q + D e e (2–85) Anal,importantstepforaircraftightdynamicsistoputt hestabilityderivativesin non-dimensionalform.Non-dimensionalquantitiesallowf ordirectcomparisonsbetween aircraftbyautomaticallyaccountingforthemajoreffects ofairspeed,size,anddensity. Forexample,toobtainthenon-dimensionalcoefcientofli ft( C L ),thelift L isdivided by 1 2 V 2 S ,where denotestheatmosphericdensity.Toobtainthenon-dimensi onal coefcientofmomentaboutthey-axis( C m ),themoment M y isdividedby 1 2 V 2 Sc Applyingthisnon-dimensionalizationtoEqs.( 2–83 )to( 2–85 )yieldsEqs.( 2–86 ) to( 2–88 ),where C D isthenon-dimensionalaircraftcoefcientofdrag. C L = C L 0 + C L u u + C L + C L _ + C L q q + C L e e (2–86) C m = C m 0 + C m u u + C m + C m _ + C m q q + C m e e (2–87) C D = C D 0 + C D u u + C D + C D _ + C D q q + C D e e (2–88) 2.1.5SolutionApproach Thelinearizedequationsofmotioncanbewritteninstatesp aceform,shownin Eq.( 2–89 ). _x = Ax + Bu (2–89) Thestatevectorisdenoted x ,thecontrolvectoris u ,andthematrices A and B arethe statespacematrices.ThemodelinEq.( 2–89 )representsthelinearizedightdynamics oftheaircraftaboutthetrimcondition. 59

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2.2StructuralDynamics Anunderstandingofstructuraldynamicsisessentialtopre dictinganaeroelastic response.Ingeneral,astructure'smotioncanbecharacter izedbythemechanical oscillations(vibrations)aboutanequilibriumpoint.Ama ss-spring-dampersystemis usedtodeveloptheessentialtheoryofstructuraldynamics 2.2.1SingleDOFSystem:TheMass-Spring-Damper Thefundamentalsofvibrationanalysiscanbeunderstoodby studyingasimple onedegree-of-freedom(DOF)mass-spring-dampersystem.I nfact,manymodelswith higherdegreesoffreedomcanbeconvertedtosingleDOFprob lems. Amass-spring-dampersystemisillustratedinFig. 2-7 andconsistsofamass m aspringofstiffness k ,andaviscousdamper.Theviscousdamperactsasanenergy dissipationdeviceandoutputsaforcelinearlyproportion altothevelocityofthemass accordingtothedampingcoefcient c ,whichhasunitsof N= m s .Gravityisignored.A force f ( t ) isappliedtothemass;theresultingmotionofthemassisafu nctionofthe time t andisdenedbythedisplacement x ( t ) m x ( t ) f ( t ) k c Figure2-7.SingleDOFmass-spring-dampersystem Theequationsofmotionforthemass-spring-dampersystema reeasilyderived usingNewton's2 nd LaworLagrange'senergyequations.Lagrange'sapproachwi llbe usedherebecauseofitsusefulnessinmoredifcultproblem ssuchasmultipledegree offreedomstructures.Lagrange'sapproachforanon-conse rvativesystemcontaining 60

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dissipativeforcesisdevelopedinAppendix A.3 andisgiveninEq.( 2–90 ). d dt @ L @ q i @ L @q i + @F d @ q i = Q i (2–90) First,theLagrangian( L )ofthesystemisformedasthedifferenceofthekinetic energyandthepotentialenergyofthesystem: L = T V .Thekineticenergyofthe systemiswritteninEq.( 2–91 )andthepotentialenergyisshowninEq.( 2–92 ). T = 1 2 m x 2 (2–91) V = 1 2 kx 2 (2–92) Thecontributionofthedamperisincludedbythedissipativ eforce F d ,denedin Eq.( 2–93 ). F d = 1 2 c x 2 (2–93) Setting Q i = f ( t ) andtreating x ( t ) asthegeneralizedcoordinate q i ,Eq.( 2–90 )yields theordinary,second-orderdifferentialequationgoverni ngthesystem'smotion,shownin Eq.( 2–94 ). m x + c x + kx = f ( t ) (2–94) ManysingleDOFsystemshaveagoverningequationofsimilar formtoEq.( 2–94 ). 2.2.2Freevibrationofa1-DOFsystem Vibrationsmaybedividedintofreevibrationsorforcedvib rations.Freevibrations actinresponsetoaninitialconditionwithnoexternalforc es.Forcedvibrationsactin responsetoaforcingfunctionactingonthesystemovertime Ingeneral,aresponsetoafreevibrationcouldbeeitherosc illatoryornon-oscillatory decay.Inanaircraftstructurewithlowdamping,oscillato ryismorecommonandwillbe 61

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addressedhere.Thesolutionisassumedtotaketheformgive ninEq.( 2–95 ). x ( t )= X e t (2–95) Theinitialamplitudeisgivenby X andtheexponent denesthedecayrate.Equation( 2–95 ) isinsertedintoEq.( 2–94 )with f ( t )=0 toobtainEq.( 2–96 ),whichissimpliedto Eq.( 2–97 ). m 2 X e t + cX e t + kX e t =0 (2–96) m 2 + c + k =0 (2–97) ThesolutiontoEq.( 2–97 )producestwocomplexrootswiththeformofEq.( 2–98 ). i = c 2 m i r k m c 2 m 2 (2–98) Theundampednaturalfrequency n andthedampednaturalfrequency d aredened inEqs.( 2–99 )and( 2–100 ).Thedampingratio expressesthedampingasaratioofthe criticaldampingrequiredforthesystemtobecomenon-osci llatory. n = p k=m (2–99) d = n p 1 2 (2–100) Usingthesedenitions,Eq.( 2–98 )mayberewrittenasEqs.( 2–101 )and( 2–102 ). i = n n p 1 2 (2–101) = n d (2–102) Thesolutiontothemass-spring-dampermotionforafreevib rationisgivenbythe sumofthecontributionfromthetworoots,asshowninEq.( 2–103 ). x ( t )= X 1 e 1 t + X 2 e 2 t (2–103) 62

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TherootsinEq.( 2–101 )aresubstitutedintoEq.( 2–103 )toobtainEq.( 2–104 ) x ( t )= X 1 e ( n + d ) t + X 2 e ( n d ) t (2–104) AfterusingEuler'sformula,Eq.( 2–104 )reducestoEq.( 2–105 ). x ( t )= e n t [( X 1 + X 2 )cos d t + i ( X 1 X 2 )sin d t ] (2–105) Since X 1 and X 2 arecomplex-conjugatepairs,Eq.( 2–105 )furthersimpliestothe classicalforminEq.( 2–106 ). x ( t )= A e n t sin( d t + ) (2–106) where A istheamplitudeand isthephaseoftheresponse.Thesevaluesare determinedfromtheinitialconditions x (0) ; x (0) .Anexampleofanoscillatoryresponse ofanunderdampedsystem( 0 << 1 )isshowninFig. 2-8 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 x(t)t [s] Figure2-8.Freevibrationresponseofanunderdampedmassspring-damperwith =0 : 5 ;! n =10 ;! d =20 ; =0 : 5 63

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m 1 x 1 ( t ) f 1 ( t ) k 1 c 1 m 2 x 2 ( t ) f 2 ( t ) k 2 c 2 Figure2-9.TwoDOFmass-spring-dampersystem2.2.3Multi-DOFMass-Spring-DamperSystem ExpandingthesingleDOFsystemtoamulti-DOFsystemcanrev ealmore principlesofstructuraldynamics.Aclassictwo-DOFmassspring-damperinachain conguration,illustratedinFig. 2-9 ,willbeanalyzedandisapplicabletohigherDOF systems.Thesystemconsistsoftwomasses,twosprings,two dampers,andtwoforcing functions.Themotionofthesystemconsistsofthetwodispl acements x 1 ( t ) and x 2 ( t ) Theequationsofmotionsaredevelopedinthesamemannerast hesingleDOF system.First,thekineticenergy( T )andpotentialenergy( V )termsaregivenin Eqs.( 2–107 )and( 2–108 ).TheLagrangianisgiveninEq.( 2–109 ). T = 1 2 m 1 x 21 + 1 2 m 2 x 22 (2–107) V = 1 2 k 1 x 21 + 1 2 k 2 ( x 2 x 1 ) 2 (2–108) L = 1 2 m 1 x 21 + 1 2 m 2 x 22 1 2 k 1 x 21 1 2 k 2 ( x 2 x 1 ) 2 (2–109) Thedissipativecontributionofthetwodampersdependsont herelativevelocityandis showninEq.( 2–110 ). F d = 1 2 c 1 x 21 + 1 2 c 2 (_ x 2 x 1 ) 2 (2–110) 64

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ThegeneralformofLagrange'sequationforasystemwithdis sipativeforcesand N degreesoffreedomisshowninEq.( 2–111 ). d dt @ L @ q i @ L @q i + @F d @ q i = Q i i =1 ; 2 ;:::;N (2–111) EvaluatingthepartialderivativesinEq.( 2–111 )yieldsEqs.( 2–112 )to( 2–117 ). @ L @x 1 = k 1 x 1 k 2 ( x 2 x 1 ) (2–112) @ L @ x 1 = m 1 x 1 (2–113) @F d @ x 1 = c 1 x 1 c 2 (_ x 2 x 1 ) (2–114) @ L @x 2 = k 2 x 2 k 2 x 1 (2–115) @ L @ x 2 = m 2 x 2 (2–116) @F d @ x 2 = c 2 x 2 c 2 x 1 (2–117) InsertingEqs.( 2–112 )to( 2–117 )inEq.( 2–111 )yieldsthetwogoverningequationsin Eqs.( 2–118 )and( 2–119 ). m 1 x 1 +( c 1 + c 2 )_ x 1 c 2 x 2 +( k 1 + k 2 ) x 1 k 2 x 2 = f 1 ( t ) (2–118) m 2 x 2 c 2 x 1 + c 2 x 2 k 2 x 1 + k 2 x 2 = f 2 ( t ) (2–119) ThegoverningequationsinEqs.( 2–118 )and( 2–119 )maybeplacedintomatrixform asshowninEqs.( 2–120 )and( 2–121 ),where M isthemassmatrix, C isthedamping matrix,and K isthestiffnessmatrix.Notethatallthreearesymmetricma trices. 264 m 1 0 0 m 2 375 264 x 1 x 2 375 + 264 c 1 + c 2 c 2 c 2 c 2 375 264 x 1 x 2 375 + 264 k 1 + k 2 k 2 k 2 k 2 375 264 x 1 x 2 375 = 264 f 1 f 2 375 (2–120) Mx + C_x + Kx = f ( t ) (2–121) 65

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ThematrixformofthegoverningequationsinEq.( 2–121 )allowstheequationsto easilyexpandtoaccommodatesystemsofhigherdegreesoffr eedom.Mostmulti-DOF systemshaveasimilarformofthegoverningequationswitha similarsolution. 2.2.4Freevibrationsofamulti-DOFsystem Similartotheonedegree-of-freedomsystem,asolutionisa ssumedhavingtheform inEq.( 2–122 ). x ( t )= X sin !t (2–122) Consideredtheundampedcaserst( C = 0 ),insertingEq.( 2–122 )intoEq.( 2–121 ) yieldsthesolutioninEq.( 2–123 ). K 2 M X =0 (2–123) Since X isnonzero,thedeterminant j K 2 M j mustequalzerotosatisfytheequation. Thedeterminantyieldsan N th orderpolynomialin ,therootsofwhicharethe (undamped)naturalfrequenciesofthesystem.Amorefamili arformofthesolution isshowninEq.( 2–124 ),whichisequivalenttotheclassiceigenvalueproblemsho wnin Eq.( 2–125 ). KX = 2 MX (2–124) AX = BX (2–125) Forthesolutiontotheundampedmulti-DOFsysteminEq.( 2–124 ),theeigenvalues i arethenaturalfrequencies 2 i .Thecorrespondingvector X i istheeigenvector whichdescribesthemodeshapecorrespondingtoeacheigenv alue.Eachmodeshape describesthe relative displacementsofeachcoordinateinthesystemwhenthesyst em isvibratingatthatnaturalfrequency.Variousmethodsare usedtonormalizethemode shape,suchasnormalizingtoamaximumvalueofunityofavec tornormofunity.The 66

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totalmotionisthesumofallthemodeshapes.Theamplitudea ndphaseofthemotion resultsfromtheinitialconditions. Inthedampedcase,thesolutionisassumedtohavetheformin Eq.( 2–126 )which resultsinEq.( 2–127 ). x ( t )= X e t (2–126) 2 M + C + K X =0 (2–127) Equation( 2–127 )resultsina 2 N th ordercharacteristicpolynomialwith N complex roots,assumingthereisoscillatorymotionineachmode.Ea cheigenvaluehasthe form j = a j + i b j for j =1 ; 2 ;:::;N ,whichcanbeusedtoobtainvaluesforthe naturalfrequencyanddampingofthecorrespondingmodeacc ordingtoEqs.( 2–128 ) and( 2–129 ). j = q a 2j + b 2j (2–128) j = a j j (2–129) 2.2.5StructuralDamping TheviscousdampingtermintroducedinEq.( 2–94 )assumesthatthedampingforce islinearlyproportionaltovelocity.However,energyisal soabsorbedbyinternalfriction generatedbythestructureasitdeforms[ 136 ].Thisadditionaldampingisknownas hystereticorstructuraldamping. Experiencehasshownthatthestructuraldampingisindepen dentoffrequency(and thereforevelocity)butactsinquadrature(ataphaseof90 )tothedisplacementofthe system[ 186 ].Inthiscase,acomplexstiffnesscanbeusedtodescribeth ecombined dampingasshowninEq.( 2–130 ). k = k (1+ i g ) (2–130) 67

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Usingthiscomplexstiffnessresultsinaslightlydifferen tformofthegoverningequation forthesystem,giveninEq.( 2–131 ). m x + k (1+ i g ) x = f ( t ) (2–131) Equation( 2–131 )canberewrittenwithanequivalentviscousdamping c eq ,resultingin Eq.( 2–132 ). m x + c eq x + kx = f ( t ) (2–132) Theequivalentviscousdampingisdenedas c eq = gk=! andresultsinanequivalent dampingratioof eq = g 2 n (2–133) Valuesofstructuraldampingareoftenfoundthroughexperi mentaltesting. 2.2.6StructuralStiffness Thestiffnessofastructureisacriticalpropertytoconsid erwhenevaluatingthe structuraldynamics.Stiffnessisameasureoftheresistan ceofanelasticbodyto deformationwhileunderloading.Itrelatestheappliedfor cesandmomentstothe deformationsofthestructure.Consideringthesingle-DOF mass-spring-dampershown inFig. 2-7 ,thestiffnessofthespringwilldirectlyaffectthedispla cementofthemass inresponsetoasteadyforce.Inadynamicresponse,thestif fnessdirectlyaffectsthe naturalfrequencyasshowninEq.( 2–99 ). Thesameprincipleholdsinmorecomplicated,higherDOFstr uctures.Thereis astiffnessparameterforeachdegreeoffreedominthesyste maswellascoupling stiffnessparameters.Inanaeroelasticapplication,theb endingandtorsionalstiffnessof thewingareofprimaryconcern. Thebendingstiffness,sometimescalledexuralrigidity, ofabeamisdenoted EI andiscalculatedinonedimensionaccordingtoEq.( 2–134 ),where M istheapplied 68

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momentand istheresultingcurvatureofthebeam. EI = M (2–134) Thetorsionalstiffnessofabeam,sometimescalledtorsion alrigidity,isdenoted GJ andiscalculatedaccordingtoEq.( 2–135 ),where T istheappliedtorqueand isthe resultingtwistofthebeam. GJ = T (2–135) Inthepresentresearcheffort,thebendingstiffness EI referstothestiffness associatedwithabendingmomentaroundthechordaxis,resu ltinginadeectionofthe wingtipinthe ^z B direction.Torsionalstiffnessreferstothestiffnessass ociatedwithan appliedmomentaroundthespanwiseaxiswhichresultsinwin gtwist. Notethatbendingstiffness EI isequaltotheproductoftheelasticmodulus E and thearea(orsecond)momentofinertia I .Likewise,thetorsionalstiffness GJ isthe productoftheshearmodulus G andthepolarmomentofinertia J .Consequently,the stiffnessofastructureisdirectlyproportionaltotheela sticmodulusofthematerial chosenforthestructuralcomponents.Ahighelasticmodulu swillresultinastiff structurewhereasalowelasticmoduluswillresultinamore exiblestructure.Because theelasticmodulusandshearmodulushaveunitsofN/m 2 andthemomentofinertia hasunitsofm 4 ,stiffnesshasunitsofNm 2 2.3Aeroelasticity Aeroelasticitydescribestheinteractionbetweentheiner tial,structural,and aerodynamicforcesactinguponanaircraft.Theseinteract ionsarenon-trivialand canstronglyinuencethestabilityandcontrolcharacteri sticsandotheraspectsofthe aircraft.Acompleteaeroelasticmodelisobtainedbycoupl ingastructuralmodelwithan aerodynamicsmodel. 69

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2.3.1StaticAeroelasticity Staticaeroelasticitystudiesthesteady-statedeection ofaexiblestructureunder anaerodynamicload.Theforcesandmomentsactingonthestr uctureareconsidered tobeindependentoftime.Asanaerodynamicloadactsonaex iblestructuresuchas awing,thewingwilldeform,whichwillaltertheaerodynami cforces,whichwillalterthe deformationofthewing.Thiscyclecontinuesuntilasteady -stateequilibriumisreached betweentheaerodynamicforcesandthedeformationofthewi ng. Inparticular,staticaeroelasticityencompassesthetwoi mportantphenomena ofdivergenceandcontrolreversal.Divergenceresultsins tructuralfailureduetothe aerodynamicmomentsovercomingtherestoringmomentsofth estructure,whichmost commonlyhappensinwingtorsion.Controlreversalhappens whentheforcefromthe controlsurfacetwiststhewingenoughtoresultinanetforc eoppositeofwhatwas desired.Forexample,anaileronmaybedeecteddownwardin ordertoincreaseliftand movethewingup.Butifthewingisofverylowtorsionalstiff ness,themomentfromthe aileronmayresultinnegativetwist,generatingnegativel iftandmovingthewingdown. Thestiffnessofthestructureisofcriticalimportancefor aeroelasticconsiderations becausethestiffnessdirectlyrelatestheforcesactingon thebodytothedeformation ofthebody.Stiffnessismuchmoreimportanttoaeroelastic itythanotherstructural parameterssuchasyieldstrengthortensilestrength.2.3.2Aeroelasticpitchingairfoil Apedagogicaldeviceforintroducingaeroelasticproblems isasimpletwo-dimensional pitchingairfoil[ 43 186 ].Thesystem,illustratedinFig. 2-10 ,consistsofarigid, symmetricairfoilofchordlength c mountedonatorsionalspringwhichhasastiffness denoted k .Thedistancefromtheaerodynamiccenter(atthequarterch ord)tothe elasticaxisisexpressedas ec ,where e representsafractionofthechordlength c .The liftcurveslopeis C l .Theangleoftheairfoilwithrespecttotheoncomingowis ,with 0 representingtheinitialangle. 70

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V 1 L 0 ec k Figure2-10.Aeroelasticpitchingairfoil Thepitchingmoment M aboutthespringattachmentpointisduetotheliftatthe aerodynamiccenter,asexpressedinEq.( 2–136 ). M = L ( e )( c ) (2–136) Thepuremomentabouttheaerodynamiccenterisassumedtobe zeroforsimplicity. Fromaerodynamictheory,theliftattheaerodynamiccenter isgiveninEq.( 2–137 ), where C l = C l 0 + C l 0 .Thenon-dimensionalairfoilcoefcientofliftisdenoted C l ,its valueatzeroangleofattackisdenoted C l 0 ,and C l denotestheairfoillift-curveslope. L = qSC l (2–137) InEq.( 2–137 ), q referstothedynamicpressureandisfoundthrough q = 1 2 V 2 and S referstotheplanformareaandissimply S = c .Assuming C l 0 =0 ,thetotalmoment abouttheelasticaxiscanbeexpressedasEq.( 2–138 ). M = qec 2 C l 0 (2–138) Lagrange'sequationcanbeappliedtoobtainthegoverninge quationforthesystem. Sincethisanalysisisonlystaticwithnodissipativeforce s,thekineticenergyand dissipativetermsareignoredandLagrange'sequationredu cesfromEq.( 2–90 )to Eq.( 2–139 ). @V @ = Q (2–139) 71

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where V isthepotentialenergy 1 2 k 2 and Q isthemomentactingonthesystem. UsingEq.( 2–138 )as Q ,thegoverningequationisEq.( 2–140 ),whichcanalsobe writtenasEq.( 2–141 ). k = qec 2 C l 0 (2–140) = qec 2 C l k 0 (2–141) Equation( 2–141 )canbesolvedusinganiterativeapproachoradirectapproa ch. Thedirectapproachwillbeusedhereandentailssettingthe initialangleto 0 = 0 + Usingthisapproach,Eq.( 2–141 )mayberewrittenasEq.( 2–142 ). k = qec 2 C l ( 0 + ) (2–142) SolvingEq.( 2–142 )for yieldsEq.( 2–143 ). = qec 2 C l k qec 2 C l 0 (2–143) OnecanimmediatelyseefromEq.( 2–143 )thatthereisacertaindynamicpressure q wherethetwist becomesinnitelylarge.Thevalueofthisdynamicpressure forthis systemisshowninEq.( 2–144 ). q = k ec 2 C l (2–144) Thisdynamicpressuredenesthedivergenceconditionfort hissystem.Atthis condition,theaerodynamicpitchingmomentovercomesthes tructuralrestoringmoment andresultsinstructuralfailure.2.3.3DynamicAeroelasticity Dynamicaeroelasticityencompassesanyaeroelasticeffec tthatenduresfora periodoftime.Flutteristhemostcommondynamicaeroelast iceffectandisanunstable vibrationofthestructureinwhichtheaerodynamicforcesi mpartenergytothestructural vibrations,potentiallycausingcatastrophicfailure.In modernaircraft,theonsetofutter 72

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usuallyhappensbeforetheonsetofdivergenceorcontrolre versalandthusisofprimary concern.Belowsomecriticalspeed,calledtheutterspeed ,structuraloscillations followinganinitialdisturbancearedamped.Abovetheutt erspeed,oneormoreof thestructuralorinertialmodesbecomenegativelydampeda ndresultintheunstable oscillations.Flutterhasbeencalledthemostimportantae roelasticphenomenonandis alsothemostdifculttopredict[ 186 ]. Althoughutterisusuallyofprimaryconcern,aeroelastic ityalsohasavery importanteffectontheightdynamicsofanelasticaircraf t.Theexiblemodesof anelasticaircraftcaninuencetherigidbodyightmodes, resultinginchangestothe performanceorhandlingqualitiesoftheaircraft.Inaddit ion,thestructuralexibilitycan affectthedynamicstabilityderivatives.2.3.3.1Unsteadyaerodynamics Animportantconsiderationfordynamicaeroelasticity,es peciallyutterandgust response,istheeffectofunsteadyaerodynamics.Unsteady aerodynamicsdealswith thefactthattheforcesandmomentsactingonamovingaerody namicsurfacevarywith time. Consideranairfoilatrestatsomeangle infreestreamowat V 1 withforce F 1 andmoment M 1 actingonit.Attime t = t 1 theairfoilisrotatedto + .Thenew forcesandmomentsresultingfromtheincreaseinangleofat tack( F 2 and M 2 )donot reachtheirsteady-statevalueuntilsomeadditionaltime t haspassed.Thisdelayis termedaerodynamiclagandisduetothetimeittakesforthec irculationaroundthe airfoilandthewakeairowtoreachtheirnewsteadystateco nditions. Theclassicquasi-steadyassumptionisthattheresultingf orcesandmoments F 2 and M 2 occurattime t = t 1 .Inotherwords,atanyinstantintimetheforcesand momentsactingontheairfoilarethesameastheforcesandmo mentsactingona stationaryairfoilatthesameorientationandowspeed.Th isassumptionimpliesthat therearenofrequency-dependentaerodynamiceffects. 73

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Thequasi-steadyassumptionisattractiveinitssimplicit ybutishighlyinaccurate forutterandgust-responsecalculationsbecauseofthesi gnicantinuencethat unsteadyeffectscanhaveontheoverallforcesandmoments. Inreality,thechangesin circulationandwakeassociatedwithairfoilmotiontaketi metodevelop.Therefore,for accuratedynamicaeroelasticmodeling,considerationoft heunsteadyaerodynamicsis imperative. Fortheairfoilundergoinganinstantaneouschangeinangle ofattack,Wagner's functionisusedtomodelhowtheliftoftheairfoilchangeso vertimeafterachange inangleofattack.Itisformulatedintermsofthenon-dimen sionaltime ,denedin Eq.( 2–145 ). = 2 Vt c = Vt b (2–145) Thisnon-dimensionaltime isconvenientlydenedintermsofthesemi-chord b andairspeedandthusisindependentofboth.Theincreasein liftperunitspanis expressedasafunctionofthenon-dimensionaltime usingWagner'sfunction,givenin Eq.( 2–146 ). L = 1 2 V 2 cC l ( ) (2–146) Equation( 2–146 )canberewrittenintermsofthedownwashontheairfoilassh ownin Eq.( 2–147 ),wherethedownwash w isdescribedas w = V sin L = 1 2 VcC l w ( ) (2–147) InEq.( 2–147 ), ( ) representsWagner'sfunction,whichisdenedinEq.( 2–148 )for theincompressiblecasewhere > 0 ( )= +2 +4 (2–148) 74

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Wagner'sfunctionrelatestheadditionaltimerequiredfor themagnitudeoflifttoreachits steady-statevalueafteranairfoilundergoesastepchange inangleofattack. Foranairfoilundergoingharmonicoscillations,thereisb othareductioninthe magnitudeofliftandaphaselagbetweentheairfoilmotiona ndtheaerodynamic forces.Asthefrequencyoftheoscillationsincrease,thea mplitudeoftheunsteady aerodynamicforcedecreasesfurtherandthephaselagchang es.Theamplitude attenuationandphaselagaregivenasafunctionofthereduc edfrequency k intermsof thechordorsemi-chordinEq.( 2–149 ). k = !c 2 V = !b V (2–149) Theodorsen'sfunction C ( k ) ,giveninEq.( 2–150 ),modelsthechangesinthe amplitudeandphaseoftheunsteadyaerodynamicforcesrela tivetothequasi-steady forcesasafunctionofthereducedfrequency[ 168 ]wherethe K j ( i k ) aremodied Besselfunctionsofthesecondkind. C ( k )= F ( k )+ i G ( k )= K 1 ( i k ) K 0 ( i k )+ K 1 ( i k ) (2–150) ApproximationsofTheodorsen'sfunctionhavebeenfoundan dareshowninEq.( 2–151 )[ 19 ]. C ( k )=1 0 : 165 1 0 : 045 k i 0 : 335 1 0 : 30 k i ;k 0 : 5 =1 0 : 165 1 0 : 041 k i 0 : 335 1 0 : 32 k i ;k 0 : 5 (2–151) Theodorsen'sfunctioniseffectivelytheFouriertransfor mofWagner'sfunction andismathematicallyequivalenttoWagner'sfunctionwhen usedtosatisfythesame boundaryconditions[ 84 ].However,itispreferabletothetimedomainsolutionfor utter computations.Forgustresponse,theunsteadyaerodynamic behaviorinthetime domainisstillofinterest. 75

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z elasticaxis L V 1 c 4 c 4 ab ec M Figure2-11.Harmonicallyoscillatingairfoil2.3.3.2Aeroelasticequationsofmotion ConsiderthesystemdepictedinFig. 2-11 .Asymmetricairfoilofchord c undergoes harmonicoscillationsinheaveandpitch,wheretheheavemo tionisdenedby z = z 0 e i !t andthepitchmotionissimilarlydenedas = 0 e i !t .Theelasticaxisispositioned somedistance ab aftofthemidchord(where b = c= 2 )whichcorrespondstoaposition ec behindtheaerodynamiccenter.Theliftandpitchingmoment abouttheelasticaxisare expressedasshowninEqs.( 2–152 )and( 2–153 ). L = b 2 h z + V ab i +2 VbC ( k ) z + V + b ( 1 2 a ) (2–152) M = b 2 ab z Vb ( 1 2 a ) b 2 ( 1 8 + a 2 ) +2 Vb 2 ( a + 1 2 ) C ( k ) z + V + b ( 1 2 a ) (2–153) ThersttermsinEqs.( 2–152 )and( 2–153 )arethenoncirculatorytermsresultingfrom themovementoftheairfoilacceleratingthemassofair.The secondtermsarethe circulatorytermsandexpresstheliftandpitchingmomentd uetothevorticityoftheow. ThesecirculatorytermsaredependentonTheodorsen'sfunc tion C ( k ) Equations( 2–152 )and( 2–153 )canbewrittenincomplexalgebraduetothe harmonicnatureofthemotion.ThisapproachisshowninEqs. ( 2–154 )and( 2–155 )and 76

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isappropriatesinceTheodorsen'sfunctionisalsowritten incomplexform. L = V 2 b h ( L z + i kL z ) z 0 b +( L + i kL ) 0 i e i !t (2–154) M = V 2 b 2 h ( M z + i kM z ) z 0 b +( M + i kM ) 0 i e i !t (2–155) RecallingtheexpressionsshowninEq.( 2–156 ),Eqs.( 2–154 )and( 2–155 )canbe rewrittenasshowninEqs.( 2–157 )and( 2–158 ). k = !b V z = z 0 e i !t z = i !z 0 e i !t = 0 e i !t = i !z 0 e i !t (2–156) L = V 2 L z z + L z b z V + L b + L b 2 V (2–157) M = V 2 M z bz + M z b 2 z V + M b 2 + M b 3 V (2–158) Equations( 2–157 )and( 2–158 )maybeexpressedinmatrixformasEq.( 2–159 ),which reducestoEq.( 2–160 ),where B and C aretheaerodynamicdampingandstiffness matrices,respectively. 264 L M 375 = V 264 bL z b 2 L b 2 M z b 3 M 375 264 z 375 + V 2 264 L z bL bM z b 2 M 375 264 z 375 (2–159) 264 L M 375 = V B 264 z 375 + V 2 C 264 z 375 (2–160) Thesematricesarenon-symmetric,whichcontributestothe utterinstability. Equation( 2–160 )maybecombinedwithexpressionsforthestructuraldynami cs, showninEq.( 2–121 ),toobtainaeroelasticequationsofmotionasshowninEq.( 2–161 ). A q +( V B + D ) q +( V 2 C + E ) q =0 (2–161) wherethestructuralinertia,damping,andstiffnessmatri ceshavebeenwritten A ; D ; E insteadof M ; C ; K asinEq.( 2–121 ).Itisimportanttoknowthattheaerodynamic 77

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dampingandstiffnessmatrices( B ; C )arevalidonlyatthereducedfrequencyforwhich theyweredened. TheresultinEq.( 2–161 )isimportantbecauseitdescribesthefundamental interactionbetweentheexiblestructureandtheaerodyna micforces.Solutionsofthe systemareobtainedusinganeigenvalueapproach.Anaircra ftinightisconsidered afree-freesystem;thatis,asystemfreelyoatinginspace andnotconnectedtothe ground.Assuch,theaeroelasticsolutionwillbecomprised ofrigidbodymodesand exiblemodes.Stabilitycanbeanalyzedfromtheeigenvalu esandisthusalinear concept-validforsmalldisturbancesaboutsteadystate. Thestructuraldamping D isoftenignored,butingeneralanystructuraldamping isbenecialinthatitwilldelaytheonsetofutterbyincre asingtheoveralldamping. Onecommonapproachtomodelingthestructuraldampingisto transformthesystem intomodalformandincludedampingratioswhicharedenedb asedonexperienceor groundvibrationtesting. Ingeneral,theequationsofmotionofelasticaircraftareq uitecomplicated[ 177 ]. Theresultingmodelsareofhighdynamicorder.Furthermore ,therelationshipsbetween variousmodelparametersforanelasticaircraftarenotasw ellunderstandasforrigid aircraftandaccuratenumericalvaluesaredifculttoobta in.Therearealsosignicant uncertaintiesassociatedwiththeelasticparameters. Theclassicalstabilityderivativeshavelimitedutilityf oraveryexibleaircraft.For suchanaircraft,linearstabilityisbestexaminedthrough aneigenmodeanalysisatthe desiredightcondition.Suchananalysiswillincludethee lasticeffectsofdivergenceor utter.Thenonlinearstabilityofaexibleaircraftinig htcanonlybefullyascertainedby atime-domainsolutionofthatightscenario.2.3.3.3Elasticstabilityderivatives Animportantconsequenceofaeroelasticityistheabilityo ftheexiblebodyto affectthedynamicandstaticstabilityderivativesofthea ircraft,thusaffectingthe 78

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ightdynamicsandhandlingqualities.Thiseffectofthee xiblemodesonthestability derivativescanbeseenwithasimpleexampleusingawingand aerodynamicstrip theory[ 176 ].Itisassumedthatthefreevibrationmodesofthewingarea vailableand thatthedisplacements d ofthewingcanbeexpressedasafunctionofthemodeshapes i ( x;y;z ) andthegeneralizeddisplacementcoordinate i ( t ) aswritteninEq.( 2–162 ). d = 1 X i =1 i ( x;y;z ) i ( t ) (2–162) Striptheorytreatstheliftofthewingasmadeupfromthelif tofmanytwo-dimensional airfoil“strips”.Theliftofeachstripisisfoundusingthe relationshipshowninEq.( 2–163 ), where s istheangleofattackofthesection. l = 1 2 V 2 cC l s (2–163) ThesectionangleofattackisshowninEq.( 2–164 )asthesumofthecontributionfrom thevehicleangleofattack v ,thestructuralangleofincidence i s ,contributionsfromthe pitchrate q androllrate p ,contributionsfromwingbendinginthe ^z B directiondueto thei th mode(expressedas bi ),andcontributionsfromwingtorsionduetothei th mode (expressedas d bi =dx ).Thedistancefromtheelasticaxistothewingaerodynamic centerisdenoted e s = v + i s q x + e U + p y U + 1 X i =1 d bi dx i + 1 U bi i (2–164) Theliftonthewing L w isassumedtobeexpressedbyEq.( 2–165 ). L w = Z b= 2 b= 2 ldy (2–165) CombiningEqs.( 2–163 )and( 2–164 )intoEq.( 2–165 )yieldsanexpressionfortheliftof thewing,showninEq.( 2–166 ). L w = 1 2 V 2 S C L 0 + C L v v + C L p p + C L q q + 1 X i =1 C L i i + C L i i (2–166) 79

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Theinuenceoftheelasticmodeshapesonthestabilityderi vativecomesthroughthe C L i ;C L i derivatives,whicharedenedinEqs.( 2–166 )and( 2–168 ). C L i = 1 S Z b= 2 b= 2 C l d bi dx cdy (2–167) C L i = 1 S Z b= 2 b= 2 C l 1 U bi dy (2–168) TheeffectofEqs.( 2–167 )and( 2–168 )istoaccountforadditionalliftduetothebending andtorsionofthewingusingthebasiclift-curveslopeofth eairfoil.Similaranalysescan beperformedontheotherforcesandmomentstoobtainacompl etedescriptionofthe elasticstabilityderivatives.Itiscustomarytoincludeo nlyafewofthemostsignicant modesandassumethattheeffectofothermodesisnegligible (amethodknownas modetruncation). 2.4OptimalControlTheory Thepurposeofoptimalcontrolistoprovidethedesiredoper atingperformance ofthesystemasrepresentedbyintegralperformancemetric s.Thedesignofthe systemisbasedonminimizingaperformanceindexaccording toasetofuser-specied weightings.2.4.1LQR Alinear-quadraticregulator(LQR)controllerreferstoth eoptimallinearregulator withrespecttoaquadraticperformanceindex[ 16 ].Thisperformanceindexisexpressed asacostfunctionalinEq.( 2–169 ).Theweightingmatrices Q and R aresymmetric matrices,where Q mustbepositivesemidenite( Q 0 )and R mustbepositivedenite ( R > 0 ). J = Z 1 0 x T ( t ) Qx ( t )+ u T ( t ) Ru ( t ) dt (2–169) Thecontrolproblemisassumedtobearegulationproblem,wh erethegoalisto designacontrolsignal u ( t ) tominimizetheperformanceindex J whiledrivingthestates tozero.ThersttermofEq.( 2–169 )penalizesdeparturesin x fromzeroaccordingto 80

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theweightingspeciedbytheengineerin Q .Thesecondtermpenalizesthemagnitude ofthecontrolactuationin u accordingtotheweightingsin R ,alsospeciedbythe engineer. Usingthestatefeedbackcontrollaw u = Kx ,thestatespacesystemin Eq.( 2–89 )canbedescribedinclosed-loopformasshowninEq.( 2–170 ). x =( A BK ) x (2–170) SubstitutingEq.( 2–170 )intoEq.( 2–169 )resultsinEq.( 2–171 ). J = Z 1 0 x T ( t ) Qx ( t )+ x T ( t ) K T RKx ( t ) dt = Z 1 0 x T ( t ) Q + K T RK x ( t ) dt (2–171) AsolutiontoEqs.( 2–170 )and( 2–171 )requiresthatthesystembestabilizable, whichmeansthatanyunstablemodesinthesystemarecontrol lable.Afurther requirementisthatthesystemisdetectable,whichmeansth atanystatewhichis notobservableisstable.Stabilizabilityanddetectabili tyareimpliedwhenthesystemis controllableandobservable,respectively. ToobtaintheminimumvalueoftheperformanceindexinEq.( 2–171 ),theexistence ofanexactdifferentialasshowninEq.( 2–172 )ispostulated.Thematrix P isassumed tobesymmetricforsimplicity. d dt x T Px = x T Q + K T RK x (2–172) CompletingthedifferentiationinEq.( 2–172 )leadstoEq.( 2–173 ). x T Px + x T P x = x T Q + K T RK x (2–173) KnowingthatthesolutionisdescribedbyEq.( 2–170 )resultsinEq.( 2–174 ). x T ( A BK ) T Px + x T P ( A BK ) x = x T Q + K T RK x (2–174) 81

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FurthersimplicationofEq.( 2–174 )yieldsEq.( 2–175 ). x T ( A BK ) T P + P ( A BK ) x = x T Q + K T RK x (2–175) EquatingtheinteriorofeachsideofEq.( 2–175 )yieldsEq.( 2–176 ). ( A BK ) T P + P ( A BK )+ Q + K T RK =0 A T P K T B T P + PA PBK + Q + K T RK =0 (2–176) Foragain K tominimize J forallinitialconditions,itmustsatisfythenecessary conditioninEq.( 2–177 )[ 16 ]. K = R 1 B T P (2–177) SubstitutingEq.( 2–177 )intoEq.( 2–176 )andrecallingthat P = P T yieldsthe continuous-timealgebraicRiccatiequationshowninEq.( 2–178 ). A T P PBR 1 B T P + PA PBR 1 B T P + Q + PBR 1 RR 1 B T P =0 A T P ( ( ( ( ( ( ( PBR 1 B T P + PA PBR 1 B T P + Q + ( ( ( ( ( ( ( PBR 1 B T P =0 A T P + PA PBR 1 B T P + Q =0 (2–178) Theoptimalcontrolgainisobtainedfromtheunique,symmet ric,positive-denite solution P ofthealgebraicRicattiequation,whichcanbeobtainedvia numerical methods. 82

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CHAPTER3 METHODOLOGY 3.1ASWING TheprimarynumericaltoolusedinthecurrentresearchisAS WING.ASWINGis anintegratedtoolforthemodelingandsimulationofaerody namics,structuraldynamics, andcontrolofexibleaircraft. ASWINGbeganinthelate1980sbyProfMarkDrelawiththedeve lopmentofa methodforthesimultaneoussolutionofaerodynamicandstr ucturalloadsonahigh aspect-ratiowingwithexternalstrutsorwires[ 44 ].Themethodwasexpandedin 1997toincludemultiplebeams,whichallowedacompleteair craftcongurationtobe modeled[ 48 ].In1999,ASWINGversion5.40waspresentedasanintegrate dsimulation toolforpreliminaryaerodynamic,structural,andcontrol -lawdesignofafullaircraft conguration,includingunsteadyeffects[ 45 ].ASWINGhassinceexpandedtoversion 5.95,theversionpresentlyinuse. Acommonapproachtoaeroelasticanalysisistouseinuence matricestocouple aerodynamicandstructuralanalyses.Thismethodoftenuse samodalapproachthat reducesthesizeoftheproblembyonlyconsideringthelowes tfrequencystructural modes;however,thismethodcanbetime-consumingtosetupa ndintroduces uncertaintybecauseofmodetruncationandmodecoupling.F urthermore,itmakes theanalysisoflarge-deection,nonlinearstructuresdif cultbecausetheinuence coefcientsonlyaccountforlinearizedstructuraldeect ions.Someaircraft,including microairvehicles,mayhavelargestructuraldeectionsth ataltertheaerodynamicloads inanonlinearmanner.Othersolutionmethodscanachievesu chnonlinearanalyses byusinganiterativeapproachtocoupletheaeroelasticint eractions,buttheiterative approachcanbeexpensiveandisnotguaranteedtoconverge. TheapproachdevelopedinASWINGfullycouplestheaerodyna mics,structural dynamics,ightdynamics,andcontroldynamicsbycombinin gtheminasingle, 83

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nonlinearsystemofequations.Allstructuralcomponentsa remodeledinASWING asone-dimensionalEuler-Bernoullibeamswithliftinglin es.Usingaone-dimensional beamrepresentationresultsinasmallstatevectorandfast solutionconvergence.Lifting linetheoryprovidesefcientestimatesoftheaerodynamic sbutlimitsthemethodto moderateorhighaspect-ratioaircraft.State-feedbackco ntrollawsdriveactuatorinputs, whichconsistofcontrolsurfacesandenginethrust. Thefullycoupledsystemofnonlinearequationsaresolvedw ithafullNewton method.Frequency-domaincomputationsareaccomplishedw ithaneigenmode analysisandallowforinvestigationsoftheightdynamics ,controlresponse,andutter. Theresultingimplementationisusefulfordeterminingdiv ergence,controlreversal, utter,elasticeffectsonstabilityandcontrolderivativ es,beamstressdistributions, elasticeffectsoninduceddrag,andloadestimatesonthede formedgeometry.Solutions fromthismethodarerapidlyobtained,makingtheapproachw ell-suitedforconceptual aircraftdesignanddetailed,rapidexplorationoftheV-n ightenvelope. 3.1.1ASWINGCoordinateSystems ASWINGusesabodycoordinatesystemwhichhasitsorigin O B attachedtothe aircraftatsomelocationalongthefuselagecenterlinewit h ^x runningalongthefuselage inthedownstreamdirection, ^y runningparalleltotheundeformedrightwing,and ^z pointingupwards.Thiscoordinatesystemisillustratedin Fig. 3-1 .Thegeometry, velocities,accelerations,andforcesareallexpressedin thebodycoordinatesystem. Alocalbeamcoordinatesystem,denoted O b ,isdenedateachstructuralnode alongthebeamforthepurposeofdeninglocalstress/strai nandaerodynamicforces. Thiscoordinatesystemconsistsofthe ^c axiswhichrunsparalleltothechord,the ^s axis whichrunsparalleltothespan,and ^n whichisnormaltothesurfaceofthebeam.These axesareillustratedinFig. 3-1 84

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^x ^y ^z ^c ^s ^n O B O b Figure3-1.Aircraftbodyandlocalbeamcoordinatesystems inASWING 3.1.2StructuralModeling Aircraftstructuralcomponentssuchasthewing,fuselage, andempennageare modeledasnonlinearbeamsusingEuler-Bernoullibeamtheo rymodiedforlarge deections.Thebasicmoment/curvaturerelationshipfrom Euler-Bernoullibeamtheory isshowninEq.( 3–1 )where w isthebeamdeection, x isthespatialcoordinate, isthe curvature, EI isthebendingstiffness,and M isthemoment. M = EI d 2 w dx 2 = EI (3–1) Theshearforces F c and F n ,tensileforce F s ,bendingmoments M c and M n ,and torsionalmoment M s aredenedaccordingtotheirrespectivestress-strainrel ationships inthelocalcoordinatesystemasshowninEq.( 3–2 ),where istheshearstress. F c = ZZ cs d ^c d ^n M c = ZZ ss ^n d ^c d ^n F s = ZZ ss d ^c d ^n M s = ZZ ( cs ^n ns ^c ) d ^c d ^n F n = ZZ ns d ^c d ^n M n = ZZ ss ^c d ^c d ^n (3–2) Theforcesandmomentsarerelatedtothebeamstrainsandcur vaturesviathe symmetricstiffnessmatrixinEq.( 3–3 ),where r istheshearstrain, s isthetensilestrain 85

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atthe ^c ; ^n origin,and 0 isthecurvatureoftheunloadedbeam. 2666666666666664 F c F s F n M c M s M n 3777777777777775 = 2666666666666664 11 12 13 14 15 16 22 23 24 25 26 33 34 35 36 44 45 46 SYM 55 56 66 3777777777777775 2666666666666664 r c s r n c c 0 s s 0 n n 0 3777777777777775 (3–3) Fortypicalbeambendingapplications,thestiffnessmatri xisassumedtotakethe simplerformshowninEq.( 3–4 ),where GK istheshearstiffnessand EA isthe extensionalstiffness.Thissimplicationassumesthatce rtaincomponentsofbending aresmallrelativetoothercomponentsofbending. 2666666666666664 11 12 13 14 15 16 22 23 24 25 26 33 34 35 36 44 45 46 55 56 66 3777777777777775 = 2666666666666664 GK c 000 GK c n ea 0 EA 0 EAn ta 0 EAc ta GK n 0 GK n c ea 0 44 45 46 55 56 66 3777777777777775 (3–4) Thelocationsoftheelasticandtensionaxesaredescribedi nthelocalbeamcoordinate systembytheoffsets c ea ;n ea and c ta ;n ta ,respectively. Anewmomentvectorabouttheelasticandtensionaxes M 0 isdenedbyEq.( 3–5 ). 266664 M 0 c M 0 s M 0 n 377775 = 266664 M c M s M n 377775 + 266664 0 n ea 0 n ta 0 c ta 0 c ea 0 377775 266664 F c F s F n 377775 (3–5) 86

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TherelationinEq.( 3–3 )canthenbeinvertedanddecoupledintoEqs.( 3–6 ) and( 3–7 ). 266664 r c s r n 377775 = 266664 F c =GK c F s =EA F n =GK n 377775 + 266664 0 n ea 0 n ta 0 c ta 0 c ea 0 377775 266664 EI cc EI cs EI cn GJEI sn EI nn 377775 266664 M 0 c M 0 s M 0 n 377775 (3–6) 266664 c c 0 s s 0 n n 0 377775 = 266664 EI cc EI cs EI cn GJEI sn EI nn 377775 266664 M 0 c M 0 s M 0 n 377775 (3–7) ThebendingstiffnessesaredenedinEqs.( 3–8 )to( 3–10 ). EI cc = ZZ E ( n n ta ) 2 dcdn (3–8) EI nn = ZZ E ( c c ta ) 2 dcdn (3–9) EI cn = ZZ E ( n n ta )( c c ta ) dcdn (3–10) Thestructuralrepresentationisdiscretizedalongthebea mbyspecifyingthe positionvector r i ,orientationvector i ,momentvector M i ,andforcevector F i at eachnode i alongthebeam.Fortheunsteadycase,thelocalvelocity u i androtation rate w i vectorsarealsoincluded.Thegoverningstructuralequati onsasafunctionof thesevariablesaregiveninEqs.( 3–11 )to( 3–14 )where r =[ r c 1+ s r n ] T T is thetransformationmatrixfromthe c;s;n coordinatesystemtothe x;y;z coordinate system, K isthecurvaturedenitionmatrix, K 0 isthecurvaturedenitionmatrixofthe unloadedbeam, E isthemoment/curvaturestiffnessmatrix,and s 0 istheunloaded beamarc-length.Thedistributedforcesandmomentsduetol ift,drag,acceleration,and apparentmassarerepresentedby f ; m andtheconcentratedforcesandmomentssuch aspointmasses,struts,orjointsarerepresentedby F ; M .Thesubscript () a represents aquantitythatisaveragedbetweennodes i and i +1 and representsthedifference betweennodes i and i +1 .Aperfectlyrigidbeamcanberepresentedbysettingthe 87

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diagonalelementsof E toinnity,making E 1 =0 inEq.( 3–12 )[ 46 47 ]. r = T Ta r s 0 (3–11) K a K a 0 0 = E 1 a M 0 s (3–12) M i +1 M i + m a s + M + r F a =0 (3–13) F i +1 F i + f a s + F =0 (3–14) Thisdiscretizationapproachallowsanonlinearityinthef ormofastepwisechange alongthebeam.Thestepwisechangeisaccomplishedbyplaci ngazero-length structuralintervalwhere r =0 and s =0 acrosstheinterval.Anyloadsor discontinuitiesinthesolutionarecapturedperfectly. StructuraldampinginASWINGismodeledbyassumingthatthe stressesandloads aredependentonthestrainrateinadditiontothestrain.As impleone-dimensional exampleisshowninEq.( 3–15 ),where t d isaspeciedcharacteristicdampingtime. xx = E ( xx + t d xx ) (3–15) Dampingtimesforeachbeamcomponentareincorporatedinto thebeamrelationships inEqs.( 3–6 )and( 3–7 ).Thesixdampingtimesareconsolidatedtoonlytwo( t and t r ) byassumingthatthetwomajorcomponentsofstructuraldamp ingareinspanwisestrain andshearing.3.1.3Aerodynamics Aerodynamicmodelingisbasedonacompressiblevortex/sou rcelatticemethod enhancedwithwing-alignedtrailingedgevortices,Prandt l-Glauertcompressibility corrections,andlimitedstallmodeling.Yawandrollratee ffectsareincludedinthe lifting-lineformulation.Unsteadyaerodynamiceffectsa reincludedbyrepresentingthe circulationinatime-laggedFourierseries. Liftinglinetheoryuseshorseshoevorticestomodelthecir culationofawing. Recallthatasinglehorseshoevortexconsistsofaboundvor texlamentalignedwith 88

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thewingspan b withtwofreevorticesextendingdownstreamtoinnity[ 10 ].Awing canberepresentedbysuperimposinganumberofhorseshoevo rtices,eachwitha differentlengthoftheboundvortexandtwotrailingvortic esextendingtoinnity.A wingrepresentedbythreehorseshoevorticesisillustrate dinFig. 3-2 ,where isthe circulationofavortex.Themultipletrailingvorticeseff ectivelybecomeacontinuous vortexsheet. d 1 d 2 d 3 d 3 d 2 d 1 d 1 d 1 + d 2 d 1 + d 2 + d 3 d 1 + d 2 d 1 Figure3-2.Awingrepresentedbyaliftinglinecomposedoft hreehorseshoevortices Considerthespanwiselocationofthewingasspeciedinter msoftheGlauert coordinate asshowninEq.( 3–16 ),where 0 2 y = b 2 cos (3–16) Theellipticalcirculationdistributionalongthewingcan thenbeexpressedasshownin Eq.( 3–17 ),where 0 isthetotalcirculationatthemid-span. ( )= 0 sin (3–17) Equation( 3–17 )canbewrittenasaFourierseriesasshowninEq.( 3–18 ),wherethe numberofterms K intheseriesischosendependingontheaccuracydesired. ( y )= K X k =1 A k sin( k ) (3–18) 89

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The K resultingequationscanbesolvedforthe A k circulationcoefcients. Unsteadyeffectsaremodeledbyincludingatime-dependent parameter .The unsteadycirculationdistributionbecomestheformshowni nEq.( 3–19 ),where = t l=V 1 and l isthestreamwisecoordinateonthefreevortexsheet. ( ; )= K X k =1 A k ( )sin( k ) (3–19) Inatime-domainsolution,the A k ( ) areaccumulatedbeginningwiththeinitialstate.Ina frequency-domainsolution,thesolutionisassumedtobeof theform A k ( )= A k e Thesteadyorunsteadycontributionsofthecirculationtot heliftforcecannowbe expressedusingthesteadyorunsteadyversionoftheKuttaJoukowskytheoremas showninEqs.( 3–20 )and( 3–21 ),respectively. F L s = V 1 (3–20) F L u = @ @t c j V j V ^s (3–21) Thetotalliftisthesumofthesteadyandunsteadycontribut ionsasshowninEq.( 3–22 ). F L = F L s + F L u (3–22) Themomentisapproximatedbyassumingthatonlythevelocit ycomponent perpendiculartothewingcontributestothemoment,asshow ninEq.( 3–23 )where r c= 4 isthelocationofthequarter-chordpointinthebeamcoordi natesystemand V ? is thevelocitynormaltothebeamasdenedinEq.( 3–24 ). M = r c= 4 F L s + r c= 4 F L u + 1 2 c 2 j V ? j 2 C m (3–23) V ? = V ( V ^s ) ^s (3–24) Dragisexpressedasthesumoffrictionandproledragassho wninthersttwo termsofEq.( 3–25 ).ThethirdterminEq.( 3–25 )representsasimplestallmodeland contributestodragonlywhentheowtangentvelocity V ^n isnon-zero,whichhappens 90

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whenthelocal C l exceedsthespeciedstalllimit. F D = 1 2 c j V j V C d f + 1 2 c j V ? j V ? C d p +2 c V ? j V ? j ( V ^n ) 2 (3–25) 3.1.4StallModeling Constraintsarerequiredonthecirculationcoefcientsto enforceowtangency alongeachsurface.ASWINGmodelsstallbehaviorbymodifyi ngtheowtangency requirementwitha“leakage”velocitywhichmodelsthesuct ion-sidedisplacementeffect oftheseparatingboundarylayer.Thismodicationonlyocc ursoutsidethe C l min or C l max limitswhicharespeciedbytheuser.Theresultingeffecti storeducethelift-curve slopebyapproximately98%outsidethestalllimits. Thisapproachisdifferentthanthephysicalnatureofwings tall,wherethevalue of C l usuallydecreasesrapidlyafterreachingthestallangleof attack.Thisapproach maynotbephysicallyrealistic,butithasthebenecialpro pertyofmaking C l ( ) unique, whichcandecreasethesolutionconvergencetimeandmaketh esolutionmethodmore robust.3.1.5SolutionMethod Thestatevectorfortheoverallunsteadyproblemisshownin Eq.( 3–26 ),where thesubscript i referstothebeamnodenumber, n s referstothenumberofstructural nodes,thesubscript J referstothejointnumber,thesubscript K referstothenumberof circulationnodes, n F referstothenumberofaps, n e referstothenumberofengines, and n g referstothenumberofgustvariables. R ; describethepositionandorientation oftheaircraft, U ; n describetheaircraftvelocityandrotationrate, a o describesthe aircraftacceleration, o describestheaircraftangularacceleration,and e represents 91

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errorintegrals.Thetotalstatevectorsizeisdenoted n tot x = h r i i M i F i u i w i ::: r n s n s M n s F n s u n s w n s r J J M J F J A 1 A 2 ::: A K RUna o o F 1 F 2 ::: n F e 1 e 2 ::: e n e g 1 g 2 ::: g n g e i T (3–26) TheresidualformoftheproblemisshowninEq.( 3–27 )andissolvedusingNewton iterations. r ( x ; x ; u )= 0 (3–27) Aftertheiterationshaveconverged,producing x and u ,thesmallperturbations x ; x ; u abouttheconvergedsolutioncanbeconsidered.Thesepertu rbationsresult inaformcontainingJacobianmatricesevaluatedattheconv ergedstate,shownin Eq.( 3–28 ). @ r @ x x + @ r @ x x + @ r @ u u =0 (3–28) Equation( 3–28 )canberewrittenasEq.( 3–29 ),wheretheJacobianmatricesare denedinEq.( 3–30 ). M x = A x + B u (3–29) M = @ r @ x A = @ r @ x B = @ r @ u (3–30) Equation( 3–29 )isthegeneralizedstate-spaceform,alsocalledthedescr iptor state-spaceform,where M isthedescriptormatrixandisnotinvertibledueto x beingof differentdimensionthanthestatevector x Intheunforcedcase,theperturbationsolutionisassumedt obeoftheformshown inEq.( 3–31 ). x ( t )= ^ x e t (3–31) 92

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ThesystemofequationsinEq.( 3–29 )thenreducestotheformshowninEq.( 3–32 ), whosesolutionresultsin n tot eigenmodes. A ^ x = M ^ x (3–32) 3.1.6ValidationandApplication CompletevalidationofASWINGagainstothernumericalsour cesisdifcultsince itcontainsanumberofuniquecapabilitiesnotcommonlyfou ndinasinglesimulation package,suchaslargedeformationswithunsteadyowandac tivecontrollaws. However,componentsofASWINGhavebeenvalidatedagainsta nalyticalsolutionsin thepast[ 44 45 ].Thesevalidationcasesincludedellipticallyloadedwin gs,frequency analysesoffreebeams,andclassicaltwo-dimensionalutt er[ 168 ].Aileronreversal anddivergencehavebeenvalidatedagainstNASTRANsolutio ns.ASWINGwasalso qualitativelyvalidatedagainstightdatafortheDaedalu shuman-poweredaircraft[ 44 ]. ASWINGwasusedtomodelthecouplingofrigidbodyandelasti cmodesinacase ofbodyfreedomutterofasmallUAV[ 90 ].Structuralpropertieswerederivedfrom NASTRANandinputintoASWINGforautteranalysis.Variati onsinstiffness,altitude, andcenterofmasslocationwereexploredasameansofpassiv elyincreasingtheutter speed. ASWINGwasusedintheNASAmishapinvestigationofthe2003H elioscrash toanalyzeandpredictightdata[ 117 ].Aftertheaircraftencounteredturbulence, aeroelasticeffectscausedaveryhighdihedralangletodev elopwhichledtoadivergent pitchingmode.Oscillationsofincreasingamplituderesul tedinhighairspeedand highdynamicpressures,whichcausedthewingstructuretof ail.ASWINGwasused toestimatethespanwiseliftdistributioninthepresenceo fverticalgustsandto estimatethebehaviorofthephugoidmode.ASWINGcorrectly indicatednegative phugoiddampingwithmorethan30feetofwingdihedral.Alth oughASWINGslightly under-predictedtheamountofdihedralneededbeforethene gativedampingwould 93

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occur,themishapreportdidnotconsiderthedifferencesto besignicantbecauseofthe uncertaintyintheighttestdata.Below30feetofwingtipd eection,ASWINGpredicted astablephugoidmode,whichagreedwiththeighttestdata. 3.2TheGenMAVAircraft TheaircraftusedinthisstudyisthegenericMAV(GenMAV),d evelopedatthe MunitionsDirectorateoftheAirForceResearchLaboratory atEglinAirForceBase, Florida[ 162 – 164 ].TheGenMAVwasspecicallydesignedtoprovideaversatil e baselineplatformforMAVresearchanddevelopmentwhichca nbeopenlysharedina collaborativeenvironment.Thisdesignphilosophyhasled tomanyresearchendeavors basedontheGenMAV[ 66 85 91 129 140 141 170 ]. TheGenMAV,showninFig. 3-3 ,isaconventionalaircraftwithaxed,high-wing designandbothhorizontalandverticalstabilizers.Ithas anall-carbonberconstruction withawingspanof61cmandafuselagelengthof42cm.Theairc raftisabank-to-turn vehicleusingelevonstocontrolpitchandroll.Itusesanel ectricmotorwithanose-mounted propellerandwasdesignedforightspeedsaround15m/s. Figure3-3.TheGenMAVaircraft3.2.1AerodynamicProperties ThewingoftheGenMAVisdesignedwitha7 dihedralanda5 incidenceangle withseveraldifferentairfoilsalongthespan,threeofwhi chareshowninFig. 3-4 94

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c =12 : 7 cm y =0 b c =11 : 59 cm y =0 : 80 b c =7 : 52 cm y =0 : 95 b Figure3-4.TheGenMAVairfoilatthreelocationsalongthes pan Theinviscidaerodynamicsofeachairfoilwasestimatedusi ngXFOILandusedto buildtheaerodynamicproleoftheentirewing.Therelevan tparametersarelistedin Table 3-1 alongwiththerangeofvaluesforeachparameteralongthesp an. Table3-1.AerodynamicparametersspeciedalongabeaminA SWING ParameterDenitionWing 0 zero-liftangleofattack-5.0to-1.6 C m c= 4 pitchingmomentcoefcientabout c= 4 -0.029to-0.065 C l lift-curveslope6.1to7.29rad 1 C l max liftcoefcientatpositivestalllimit1.25to1.65 C l min liftcoefcientatnegativestalllimit-1.10to-0.60 3.2.2Centerofpressure Ofparticularimportancetothecurrentresearchistheloca tionofthecenterof pressureforeachairfoil,denoted x cp .Recallthedenitionofthecenterofpressure isthelocationatwhichreplacingthedistributedpressure loadswitharesultantforce producesthesameoverallforceonthebody.Thisdenitioni sexpressedinEq.( 3–33 ) anddepictedinFig. 3-5 ,where N isthenormalforce, A istheaxialforce,and M LE is theaerodynamicmomentabouttheleadingedgeoftheairfoil x cp = M LE N (3–33) 95

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Forsmallangles, x cp canbewrittenasshowninEq.( 3–34 ). x cp M LE L (3–34) M LE N A x cp Figure3-5.Centerofpressurelocationonanairfoil Thecenterofpressurelocationisestimatedusingthinairf oiltheory[ 10 ].The fundamentalequationofthinairfoiltheory,giveninEq.( 3–35 ),statesthatthecamber lineoftheairfoilisalsoastreamlineoftheow. 1 2 Z c 0 r ( ) d x = V 1 ( dz dx ) (3–35) Equation( 3–35 )iswrittenatagivenpoint x alongthechordwhere dz=dx isevaluated atthesamepoint.Thevariable isadummyvariableofintegrationwhichvariesfrom 0to c .Thevortexstrength r isawrittenasafunctionof andistheonlyunknownin Eq.( 3–35 ).Solvingfor r ( ) isoneoftheprimaryproblemsinthinairfoiltheory. Thedummyvariable canbetransformedintopolarcoordinatesviaEq.( 3–36 ). TheCartesiancoordinate x canalsobetransformedintopolarcoordinatesaccordingto Eq.( 3–37 ). = c 2 (1 cos( )) (3–36) x = c 2 (1 cos 0 ) (3–37) 96

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CombiningthesetransformationswithEq.( 3–35 )resultsinthefundamentalthinairfoil theoryequationintermsof ,asshowninEq.( 3–38 ). 1 2 Z 0 r ( )sin d cos cos 0 = V 1 ( dz dx ) (3–38) Asolutionfor r ( ) subjecttotheKuttacondition r ( )=0 isgiveninEq.( 3–39 ). r ( )=2 V 1 A 0 1+cos sin + 1 X n =1 A n sin n (3–39) Thecoefcients A 0 and A n aregiveninEqs.( 3–40 )and( 3–41 ). A 0 = 1 Z 0 dz dx d 0 (3–40) A n = 2 Z 0 dz dx cos n 0 d 0 (3–41) Usingtheresultthat C l =2 ,thecoefcientofliftisfoundthroughEq.( 3–42 ),where L =0 istheangleofattacknecessaryforzeroliftandisfoundthr oughEq.( 3–43 ). c l =2 ( L =0 ) (3–42) L =0 = 1 Z 0 dz dx (cos 0 1) d 0 (3–43) Themomentcoefcientabouttheleadingedgecanbeestimate dthroughEq.( 3–44 ), wherethecoefcients A 1 and A 2 aredevelopedaccordingtoEq.( 3–41 ). c m LE = c l 4 + 4 ( A 1 A 2 ) (3–44) Thecenterofpressurelocationcanthenbeestimatedusingt heresultsfromthin airfoiltheoryinEqs.( 3–42 )and( 3–44 )asshowninEq.( 3–45 ). x cp = c m LE c c l (3–45) ThisprocessisappliedtotheGenMAVairfoilsusingthemean camberline z=c .The resultinglocusoftheairfoilcentersofpressureisdepict edinFig. 3-6 for =0 97

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-14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 x [cm]y [cm] c/4 x cp at a =0 Figure3-6.CenterofpressurealongtheGenMAVwingfor =0 3.2.3StructuralProperties ThestructuralpropertiesoftheGenMAVwingareofprimaryc oncern.Thecarbon berlayuptapersfroma [ 45] 3 carbonberlayupinthefrontquarter-chordtoasingle [ 45] plyatthetrailingedge,exceptintheinnerspanwhereittap erstoa [ 45] 2 layupat thetrailingedge.Thewingalsohasthreebattensofuni-dir ectionalcarbonberaligned inthechorddirectionatspanlocationsof0.45 b ,0.66 b ,and0.85 b .Theactualwingis picturedinFig. 3-7 Figure3-7.TheGenMAVwing,bottomview 98

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3.2.3.1Wingstiffness Thebendingandtorsionalstiffnessofthewingaredetermin edbymeasuringthe deformationofthewinginresponsetoanappliedloadusinga dynamicstereoscopic visualimagecorrelation(VIC)system[ 38 ].Thesystemconsistsoftwincalibrated, synchronizedcamerasacquiringimageswitharesolutionof 2448x2048pixels.Itthen usesstereotriangulationtorecover3Ddatafromtheimages .Theoverallsystemis capableofmeasuringfrom0.05%-500%strainonaspecimensi zeassmallas1mm. Atwo-pointbendingstiffnesstestisused.Thetwo-pointte stcalculatesthebending stiffness EI bymeasuringthetipdeectioninresponsetoanappliedforc e.Themethod isillustratedinFig. 3-8 andthebendingstiffnessiscalculatedusingEq.( 3–46 ). EI = Fl 3 3 (3–46) l F Figure3-8.Two-pointbendingstiffnesstest Aspecklepatternwasappliedtotheuppersurfaceofthewing andthewingwas clampedattherootinviewofthetwocameras.Thistestsetup isshowninFig. 3-9 Referenceimagesweretakenoftheundeformedwing,followe dbyimagestakenafter applyinganupwardloadataparticularpointonthebottomof thewing.Thesequential imagesarecomparedusingcross-correlationtechniquesto obtainthedisplacementat allpointsontheinteriorofthewingasshowninFig. 3-10 Usingthisapproach,thewingdeectionwasmeasuredatthec orresponding spanlocationtowhichtheforcewasapplied.Forceswereapp liedatsevenspanwise locationsaswellasmultiplechordwiselocationswithinea chspanwiselocation.The 99

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twincameras clampedwing Figure3-9.Experimentaltwo-pointbendingstiffnesstest setupwiththeVICsystem z [mm] 0 1.2 2.4 3.6 Figure3-10.Wingdeectioninresponsetoanappliedloadof 0.98N resultsforbendingstiffnessalongthespanareshowninTab le 3-2 .Theoverallmean bendingstiffnessofthewingis1.77Nm 2 Thesamedatawasusedtoestimatethetorsionalstiffnessof thewing.Theforce, whenappliedatsomechordwisedistancefromtheelasticaxi s,causesanapplied torque.TheresultingtwistofthewingwasdeducedfromtheV ICdataandusedto 100

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Table3-2.Experimentaldatafromthetwo-pointbendingsti ffnesstest b [cm] EI [Nm 2 ]std.dev.[Nm 2 ] 5 : 10 : 970 : 51 7 : 11 : 230 : 47 10 : 91 : 750 : 33 15 : 52 : 080 : 29 20 : 42 : 220 : 19 26 : 82 : 010 : 09 28 : 61 : 900 : 14 estimatethetorsionalstiffnessaccordingtoEq.( 3–47 ). GJ = Tl (3–47) ThespanwisevaluesoftorsionalstiffnessareshowninTabl e 3-3 .Theouterspanof thewinghasalowertorsionalstiffnesssincethewingtaper sfromthreepliesofcarbon bertooneplyatthetrailingedge.Atthewingroot,thewing tapersfromthreepliesto twopliesatthetrailingedgeandthushasahighertorsional stiffness.Theoverallmean valueoftorsionalstiffnessis0.32Nm 2 Table3-3.Experimentaldatafromthetorsionalstiffnesst est b [cm] GJ [Nm 2 ]std.dev.[Nm 2 ] 5 : 10 : 830 : 16 7 : 10 : 470 : 11 10 : 90 : 350 : 00 15 : 50 : 280 : 03 20 : 40 : 180 : 01 26 : 80 : 130 : 01 28 : 60 : 090 : 02 3.2.3.2Wingelasticaxis Thevisualmeasurementsofthewingdeformationswereusedt oestimatethe positionoftheshearcenterateachofthesevenspanwiseloc ations.Ateachspanwise location,aforcewasappliedatvariouslocationsalongthe chordtoproducevarying 101

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amountsofwingtwist.Thewingtwistwasplottedagainstthe chordwiselocationof theappliedforce;arepresentativeplotforaspanlocation of y =15 : 5 cmisshownin Fig. 3-11 .Thepointwherethewingtwistcrosseszeroisthechordwise locationofthe shearcenteratthatspanwiselocation. -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 q [deg]% chord Figure3-11.Wingtwistvs.excitationlocationfor F =0 : 98 Nat y =15 : 5 cm Theelasticaxisisalineconnectingtheshearcentersofeac hspanwiselocation. TheelasticaxisoftheGenMAVwingisshowninFig. 3-12 andliesslightlyaftofthe quarter-chordlineatapproximately28%chord.Intheregio nwherenodatawas available(atthewingrootorwingtip),themeasureddatawa sextrapolatedtoparallel thequarter-chordline.Thisextrapolationisconsidereda validestimateoftheelastic axislocationsincethewingstructureisuniforminthesere gions. 3.2.3.3Wingtensionandmasscentroidaxes Thetensionaxisisthelocusofthecross-sectionalareacen troidsalongthespan whichareeffectiveincarryingtension[ 55 ].Ateachspanwiselocation,thecentroidof thecross-sectionalareawascalculatedbymeasuringtheta peringofeachcarbonber ply.TheresultingtensionaxisisshowninFig. 3-13 102

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-14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 x [cm]y [cm] c/4 Elastic Axis measured Elastic Axis extrapolated Figure3-12.ElasticaxisoftheGenMAVwing SincetheGenMAVwingissolidcarbonber,thelocationofte nsionaxiscorresponds tothelocationofthemasscentroidaxisbydenition. -14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 x [cm]y [cm] c/4 Tension Axis Figure3-13.TensionaxisoftheGenMAVwing3.2.4ASWINGModel TheASWINGmodeloftheGenMAVwasderivedfromapreviouslye xisting rigid-bodyAVLmodelandthenumericalresultsforthewings tiffness.TheGenMAV isrepresentedinASWINGbyvariousparametersspeciedalo ngfourbeamsjoinedwith kinematicconstraints.AnillustrationoftheGenMAVgeome tryasmodeledinASWING isshowninFig. 3-14 .NotethattheengineandpropellerarenotmodeledinASWING 103

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Figure3-14.GeometryoftheGenMAVinASWING3.2.4.1Structural Therelevantparametersforspecifyingthestiffnessofthe structurearelistedin Table 3-4 .Thetailcomponentsandfuselageareheldrigidwhileonlyt hebendingand torsionalstiffnessofthewingaremadeelastic.Theaverag evaluesofthewingbending andtorsionalstiffnessaretakenfromtheexperimentaltes ts. Table3-4.User-speciedstructuralparametersinASWING ParameterDenitionWing HorizontalVertical Fuselage TailTail EI cc bendingstiffnessabout c axis1.77Nm 2 111 EI nn bendingstiffnessabout n axis 1111 EI cn c n couplingstiffness 1111 EI cs bending/torsioncouplingstiffness 1111 EI sn bending/torsioncouplingstiffness 1111 GJ torsionalstiffness0.32Nm 2 111 EA extensionalstiffness 1111 GK c c shearstiffness 1111 GK n n shearstiffess 1111 3.2.4.2Aerodynamic TheGenMAVwingisdenedbyseveraldifferentairfoilsalon gthespan.The inviscidaerodynamicsofeachairfoilwasestimatedusingX FOILandusedtobuildthe 104

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aerodynamicproleoftheentirewing.Therelevantparamet ersarelistedinTable 3-5 alongwiththerangeofvaluesforeachparameteralongthesp an.Thefrictiondrag coefcientaccountsforsomeviscouseffectsandwasestima tedusingXFOIL.The pressuredragcoefcientcomesdirectlyfromtheAVLmodel. Theprolepressureandskinfrictiondragcoefcientsarea lsoappliedtothe horizontalandverticalstabilizersandfuselage.Thehori zontalandverticalstabilizers arebothatplatesandresultsfromthinairfoiltheoryareu sedtoestimatetheir aerodynamics.Table3-5.AerodynamicparametersspeciedalongabeaminA SWING ParameterDenitionWing c chordlength12.7cm(root) 0 zero-liftangleofattack-5.0to-1.6 C m c= 4 pitchingmomentcoefcientabout c= 4 -0.029to-0.065 C l lift-curveslope6.1to7.29rad 1 C l max liftcoefcientatpositivestalllimit1.25to1.65 C l min liftcoefcientatnegativestalllimit-1.10to-0.60 C d f frictiondragcoefcient0.007 C d p pressuredragcoefcient0.0101 3.2.4.3Massproperties Themasspropertieswereinputthroughthedensityspecied alongthebeams anddumbbellsmadeoftwopointsmasseswithsomedistancebe tweenthem.The dumbbellsweremanipulatedsothattheoverallaircraftmas spropertiesmatchedthe previouslyvalidatedAVLmodel.Acomparisonbetweenthema sspropertiesofthe resultingASWINGmodelandtheAVLmodelisgiveninTable 3-6 3.2.4.4Controlsurfaces TheelevonsoftheGenMAVaremodeledinASWINGbycreatingel evators andaileronsonthehorizontalstabilizerofthesamesizean dshape.Themapping betweenelevondeectionandelevatorplusailerondeecti onisshowninEqs.( 3–48 ) and( 3–49 ),where a istheailerondeection, e istheelevatordeection, le istheleft 105

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ParameterASWINGAVL mass[kg]1.0181.018 x cg [m]0.16310.1631 y cg [m]-0.683 10 10 0.104 10 10 z cg [m]0.903 10 2 0.903 10 2 I xx [kgm 2 ]0.182 10 2 0.182 10 2 I yy [kgm 2 ]0.753 10 2 0.753 10 2 I zz [kgm 2 ]0.850 10 2 0.852 10 2 I xy [kgm 2 ]0.526 10 10 0.249 10 10 I xz [kgm 2 ]0.134 10 3 0.135 10 3 I yz [kgm 2 ]0.116 10 10 0.269 10 11 Table3-6.MasspropertiesoftheASWINGmodelcomparedtoth eAVLmodel elevondeection,and re istherightelevondeection. 2 a = le re (3–48) 2 e = le + re (3–49) ArudderwasaddedtotheASWINGmodeltomaketheconguratio nmore conventional.Thenalcontrolderivativesusedareshowni nTable 3-7 Table3-7.ControlderivativesfortheGenMAVASWINGmodel CoefcientValue C L a 0 : 04 C m a 0 : 02 C L e 0 : 1 C m e 0 : 05 C L r 0 : 05 C m r 0 : 03 3.2.5Validation TheASWINGmodeloftheGenMAVwasvalidatedusingthreemeth ods.Therst wasacomparisonofthetrimconditionandightmodestothee xistingAVLmodel. TheAVLmodelwasheavilyusedandvalidatedinpriorresearc h[ 163 164 ].Forthis comparison,theGenMAVASWINGmodelwasmadetobeperfectly rigid.Atan 106

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airspeedof15m/s,theAVLmodeltrimmedat6.25 angleofattackand-4.4 elevator deectionwhiletheASWINGmodeltrimmedatanangleofattac kof7.47 andan elevatordeectionof-4.05 .Sincethegeometryisidentical,thestructureisrigid,an d themasspropertiesmatchveryclosely,thedifferencesint rimangleofattackmust comefromaerodynamicdifferencesbetweenthemodels. StabilityderivativesarecomparedinTable 3-8 andtheeigenvaluesoftheight modesarecomparedinFig. 3-15 .Theresultsareverysimilar;however,theASWING modelshowsaslightincreaseindampingandthenaturalfreq uencyofthedutchroll andshortperiodmodes.Mostofthestabilityderivativesar everyclosetoAVL,except for C l whichdiffersby29%. -35 -30 -25 -20 -15 -10 -5 0 5 0 5 10 15 20 ImagRe AVL ASWING Figure3-15.ComparisonofightmodesbetweenASWINGandAV L Thesecondmethodofvalidationuseddataavailablefromig httestmaneuvers[ 164 ]. ThecontroldeectionsfromtheighttestwereinputtotheA SWINGmodelwhilethe solutionwasexecutedinatime-marchingfashion.Thestiff nessvaluesweresetas showninTables 3-2 and 3-3 FlightequipmentontheGenMAVincludedaGPSandKestrel2.2 autopilotwith telemetrycapability.Theautopilotprovidedsensordataa tanominalrateof2Hzand 107

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Table3-8.ComparisonofstabilityderivativesbetweenASW INGandAVL ASWINGAVL C L 5 : 056 : 55 C m 2 : 16 1 : 8 C l 0 : 0822 0 : 102 C n 0 : 08810 : 0916 C l r 0 : 2790 : 306 C n r 0 : 212 0 : 216 C l e 0 : 02050 : 0233 C m e 0 : 0471 0 : 0469 anincreasedrateof5Hzduringighttestmaneuvers.Allig htmaneuverswereown openloop.Table3-9.Flighttestmaneuvers PointManeuverDescription APitchDoublet-Full Positivefulldeectionfor1sec,negativefulldeectionfor1sec,releasecontrols BRollDoublet-Full Left1/2deectionfor1sec,right1/2deectionfor1sec,releasecontrols CPitchPulseFwd-1/2Positive1/2deectionfor1sec,relea secontrols DPitchPulseFwd-FullPositivefulldeectionfor1sec,rel easecontrols Thecomparisonoftherstmaneuver,afullpitchdoublet,is showninFig. 3-16 .The trimconditionvariesslightlybetweenASWINGandtheight databutthemagnitudeand trendsofthecontrolinputaretakendirectlyfromtheight data.Thepitchrateresponse showsgoodagreement.TheresponseofASWINGleadstheight testresponseslightly butthemagnitudeoftheresponseissimilar,exceptat3.75s econdswhenreverse controlinputisapplied,ASWINGover-predictstherespons eofthevehicle. Thesecondmaneuver,arolldoublet,isshowninFig. 3-17 .Theresponsesvary slightly,whichmaybeduetonoiseintheighttestdata,but ingeneraltheASWING responsefollowstheighttestresponseverywell. 108

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Thethirdmaneuver,ahalfpitchpulse,isshowninFig. 3-18 .Theagreementin phaseandmagnitudeoftheresponseisverygood,althoughth eASWINGresponse doesover-predictthemagnitudeofthepitchrateduringrel easeofthecontrolinput. Thefourthmaneuver,afullpitchpulse,isshowninFig. 3-19 .Theresponseof ASWINGslightlyleadstheighttestsdataandalargeover-p redictionofthepitchrateis seenduringthereleaseofthecontrols. Overall,thesimulationsconductedinASWINGshowreasonab leagreementwith theighttestdata.TheresponseofASWINGisfasteringener althantheighttestdata andthemagnitudeoftheresponsetendstobegreaterinASWIN G.Therearemany sourcesoferrorthatcouldaccountforthesedifferences,s uchasinviscidaerodynamics, differencesinthemasspropertiesandinertiaoftheightv ehicleandtheASWING model,ornoiseintheighttestdata. -20 -10 0 10 20 2 2.5 3 3.5 4 4.5 5 d e [deg]Time [s] Flight ASWING A -400 -300 -200 -100 0 100 200 300 400 2 2.5 3 3.5 4 4.5 5 Pitch Rate [deg/s]Time [s] Flight ASWING B Figure3-16.ComparisonofASWINGmodelwithighttestmane uverA.A)Control input.B)Aircraftresponse. 109

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-20 -10 0 10 20 0 0.5 1 1.5 2 2.5 3 d a [deg]Time [s] Flight ASWING A -150 -100 -50 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 Roll Rate [deg/s]Time [s] Flight ASWING B Figure3-17.ComparisonofASWINGmodelwithighttestmane uverB.A)Control input.B)Aircraftresponse. -20 -10 0 10 20 2 2.5 3 3.5 4 4.5 5 d e [deg]Time [s] Flight ASWING A -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 2 2.5 3 3.5 4 4.5 5 Pitch Rate [deg/s]Time [s] Flight ASWING B Figure3-18.ComparisonofASWINGmodelwithighttestmane uverC.A)Control input.B)Aircraftresponse. 110

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-20 -10 0 10 20 0 0.5 1 1.5 2 2.5 3 d e [deg]Time [s] Flight ASWING A -300 -250 -200 -150 -100 -50 0 50 100 150 200 0 0.5 1 1.5 2 2.5 3 Pitch Rate [deg/s]Time [s] Flight ASWING B Figure3-19.ComparisonofASWINGmodelwithighttestmane uverD.A)Control input.B)Aircraftresponse. 3.2.6StructuralModeling ThestructuralcharacteristicsoftheGenMAVaredemonstra tedtobewellsuited formodelingbyASWING.Theone-dimensionalcomputational approachassumes chord-wiserigiditysuchthatthecamberisunchanged.Theb endingstiffnessand torsionalstiffnessareallowedtovaryalongthespanbutth echordremainsrigid.Such anapproachlimitsthetypesofplatformsthatmaybeaccurat elymodeled;however,itis asuitableapproachfortheGenMAV. Amodalanalysisindicatestheassumptionofchord-wiserig iditydoesnotcause signicantinaccuraciesfortherstbendingmodeandrstt orsionmodesofinterest. Theanalysisutilizesdatafromgroundvibrationtestingof apairofGenMAVwings. Onewinghasastiffnesswhichmatchestheightvehiclewhil etheotherwinghas signicantlyincreasedstiffnessinthechorddirection.I neachcase,theweightofeach wingisthesame.Thiswasachievedbyusingsmallunconnecte dweightsalongthe battensintherstwingcongurationandusingsmallmetalr odsadheredalongthe 111

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chordinthehigh-stiffnessconguration.TheoriginalGen MAVwingweighed23.8gbut thesetestcongurationsweighedanextra4.7gduetotheadd edmass. Thevibrationtestusesawingmountedhorizontallywiththe entirerootxedas acantilever.Ashakerisattachedtotheroottoprovideforc eexcitation.Theresulting motioniscapturedusingascanninglaserDopplervibromete r[ 154 155 ]at207points acrosstheouterspanofthewing.Asthelaserbeamshinesont hevibratingstructure, thesystemobservesthereectedlaserlightandmeasuresth ephaseshiftcausedby theDopplereffect.Thephaseshiftisdirectlyproportiona ltotheobject'svelocityand thebeamwavelengthandistherebyusedtodeterminethefreq uencyresponseofthe structure.Asinesweepfrom2Hzto200Hzover8secondsprovi dedtheinputwhilethe responsewasmeasuredat1600Hz. ThespectrumisshowninFig. 3-20 forthelow-frequencyrange.Inthiscase,atrio ofpeaksareevidentcorrespondingtothespan-wiserstben dingmodenear17.75Hz, arsttorsionmodenear33.1Hz,andaspan-wisesecond-bend ingmodenear75Hz. Thefrequenciesanddampingarenearlyidenticalbetweenth etwowingcongurations fortherstbendingandrsttorsionmodes. -110 -100 -90 -80 -70 -60 -50 -40 0.1 1 10 100 Magnitude [dB]w [Hz] Stiff chord Flexible chord Figure3-20.Modalfrequenciesofthestiff-chordandexib le-chordwingcongurations 112

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0 5 10 15 20 25 30 -10 -5 0 0 1 x [cm] y [cm] x10 -4Inst. z-velocity [m/s] Figure3-21.Firstbendingmodeshapesfromthestiffchord( red)andexiblechord (green)congurations Themodeshapesfortherstbendingandrsttorsionmodes,s howninFigs. 3-21 and 3-22 ,areunchangedbytheincreaseinchord-wisestiffness.The modeshapes areindistinguishablefortherstbendingmodewhilethemo desshapesforthe second-bendingmodeshowonlyslightdifferencesneartheo utboardtrailing-edge. Atwo-dimensionalslicealongalineofconstantspanisextr actedfromthemode shapesFigs. 3-21 and 3-22 togivetheairfoilshapesinFig. 3-23 .Inthiscase,the 0 5 10 15 20 25 30 -10 -5 0 -1 0 1 x [cm] y [cm] x10 -3Inst. z-velocity [m/s] Figure3-22.Firsttorsionmodeshapesfromthestiffchord( red)andexiblechord (green)congurations 113

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-0.003 -0.002 -0.001 0 0.001 0.002 0.003 0.004 0 0.2 0.4 0.6 0.8 1 Inst. z-velocity [m/s]x/c [-] 1 st Torsion stiff chord 1 st Torsion flex chord 1 st Bending stiff chord 1 st Bending flex chord Figure3-23.Modeshapeofchordat y =0 : 65 b sliceistakenat y =0 : 65 b tohighlighttheregionthatshowedthelargestdifferences inFigs. 3-21 and 3-22 .Thesimilarityoftherstbendingmodeshapeisclearlysho wn bythissliceasisarethedifferencesinthersttorsionmod eshape.Forthersttorsion mode,thetrue-stiffnesswinghasaworst-casedeectionab out28%higherthanthe high-stiffnesswingwhencomparingmotionrelativetothen odeline.Thisworst-case deectionisactuallylocalizedtotheregionsof : 065 c : 075 c and : 085 c : 095 c nearthe trailing-edgewhileovertwo-thirdsoftheairfoildiffers bylessthan5%duetothechange instiffness. Theapproximationofchord-wiserigidityisthusconsidere dacceptableformodeling therstbendingandrsttorsionmodesoftheGenMAV.Thevar iationsfromchord-wise rigiditytotruechord-wisestiffnessresultinonlyslight changestoasmallportionofthe rsttorsionmodeshape.Theeffectsofthesemodeshapesont heaeroelasticdynamics aresimilarlyaccurate. 114

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CHAPTER4 EXPERIMENTALAEROELASTICITYOFAMEMBRANEWING TwodeningcharacteristicsofMAVsistheirsmallsizeandc hallengingregimeof ight.Thisresultsindesignsinspiredbyhighlyexiblebi ologicalightvehicleswhich successfullyoperateinthesameregime.Basedontheoretic al[ 150 ]andexperimental investigations[ 69 112 130 ],apreferredMAVcongurationwasidentiedasalow aspect-ratioying-wingwithamicropropellerandperimet er-reinforcedelastomeric membranewing.Thewingcanhelptheaerodynamicperformanc ebydelayingstalland enhancingthestaticstabilitycharacteristics[ 7 71 ]throughtheintroductionofnonlinear uid-structureinteractions[ 88 149 ]. Windtunnelexperimentsareperformedtoinvestigatetheae rodynamiccharacteristics ofmicroairvehicleswithexiblewingsinunsteadyconditi ons,alargelyunknown phenomenon.Themaintargetofthisresearchistheexperime ntalcharacterizationof thedynamicderivatives,especiallythepitchdampingderi vatives,inthepresenceof signicantuid-structureinteractionatlowReynoldsnum bers.Variablewingstiffness, obtainedbytuningthestatictensionoftheelasticwingmem brane,wasalsoconsidered. Unsteadyightconditionssuchaswindgustswouldmanifest asarapidchangeinthe wingshapebutwerenotconsideredinthepresentstudy. Theexperimentalconditionsincludethethreefundamental motionsnecessary tostudythelongitudinaldynamics:apurerotationalmotio n,apureplungingmotion, andapurepitchingmotion.Theformerisamotioninwhichthe angleofattackequals therotationalangle.Thepureplungingmotionisamotionin whichtheangleofattack variescontinuouslyandthechangeinlift,drag,andmoment areonlyduetotherateof changeofangleofattack.Thepurepitchingmotionisamotio ninwhichrotationaland translationalcomponentsarecombinedsuchthattheangleo fattackremainsconstant Babcock,JudsonandAlbertani,Roberto.“ExperimentalEst imationoftheRotaryDampingCoefcients ofaPliantWing.” JournalofAircraft 49(2012).2:390–397. 115

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andaerodynamicchangesonlyarisefromtherotational(orp itch)rate.Auniquetwo degree-of-freedompitch-plungerigisusedtoaccomplisht hesemotions. Moderndesignofexperiment(MDOE)techniquesareusedtoef cientlyformulate reliablebutnotoverly-complicatednumericalmodelswhic hcapturerelevantinteractions amongthefactors.Realisticexperimentalvaluesofthedyn amicpitchdamping derivativesareobtained,alongwithdynamicliftanddragc oefcients.Similarconsiderations couldbemadewithdirectional-lateraldynamicsifthemode lisrotatedby90 aroundthe balance'slongitudinalaxis. 4.1Background Interestingcanonicalformulationsfortheestimationofs tructuralexibilityeffectson localdynamicderivativesareproposedforlargeconventio nalaircraft[ 52 172 ]. Overallnumericalvaluesforthedynamicderivativescoef cientsandslopescanbe valuatedusingtheoreticalmethodssuchasTornado[ 105 ]orPMARC[ 13 ].An exampleofadynamicsimulationmodelofamicroairvehicle[ 178 ]isformulatedusing estimatesfromPMARCandtypicalpublishedvalues[ 20 ].Thestabilityandcontrol linearpropertiesforanaeroelasticfamilyofMAVsisexper imentallycharacterized insteadyconditions[ 72 ].ExperimentalcharacterizationofaMAV'saerodynamican d structuralpropertiesisdoneinstaticconditionswithvar yinglevelsofwingtension, horizontalcontrolsurfacedeections,andpropellerspee dswhileformulatingthe aerodynamiccoefcientsusingmultiplelinearregression [ 7 8 ]. Thetensionofaperimeter-reinforcedmembranewinghasasi gnicanteffecton thewing'saerodynamics[ 8 88 149 159 ].Inthepresenceofdynamicpressure,the membraneskininateswhileconstrainedattheperimeterby thestiffcarbonber.Lift, drag,andpitchingmomentarefoundtobeconsiderablystron gerthanthosegenerated byanequivalentrigidwing.Inparticular,theslopeofthep itchingmomentcurveismuch steeper.Aperimeter-reinforcedsuffersfromincreaseddr agascomparedtoarigidwing duetothegreateramountofseparatedow. 116

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presenceofbattens.Itisshown,bothnumericallyandexper imentally,that unconventionalaeroelastictailoringcanbeusedtoimprov eMAVwingperformance. Thechordwiseandspanwisemembranepre-tension,numberof plainweavecarbon berlayers,laminate TheapplicationofMDOEtechniquestowindtunneltestscane ffectivelyincrease theexperiment'sprecisionbyremovingblockeffects(unwa ntedvariation)fromthe unexplainedvarianceinthetestresultswhilerevealingcr iticalinteractions[ 40 108 ]. Thetechniquesareappliedtothedevelopmentofamathemati caldescriptionof theaerodynamiccoefcientsoftheFree-yingAirplanefor Sub-scaleExperimental Research(FASER)[ 109 ].Theresultingmodelscomprisedasetofsmooth,different iable functionsforthenon-dimensionalaerodynamiccoefcient sintermsofordinary polynomialsintheindependentvariables,suitablefornon linearaircraftsimulation. ModelsforthebodyaxisaerodynamiccoefcientsoftheF-18 HighAngleofAttack ResearchVehiclearealsoformulatedwithMDOEtechniques[ 110 ]. Anewsystemfordynamicwindtunneltestingwaspresentedwh ichallowsthe measurementoftheseparatecomponentsoftherotarydampin gcoefcientsby decouplingthemomentduetorateofchangeofangleofattack andpitchangle[ 160 ].A threedegree-of-freedomgimbaledmechanismisalsousedto determineaircraftstatic anddynamicstabilityderivatives.Methodologiesusingmu ltipledegrees-of-freedom planarkinematicsarerelativelysimpleinoperationandco stwithrespecttoother methodsascablemountedmodels[ 5 17 ]androtaryrigs[ 75 ]. 4.2ExperimentalSet-UpandProcedure ThecurrentMAVwing,illustratedinFig. 4-1A ,isatypicalMAVwingwitha Zimmerman-typeplanformconsistingoftwoellipsesmeetin gatthequarterchord. Therigidmodelismadeentirelyofcarbonberwhiletheexi blewingisconstructed ofacarbonberperimeterwithalatexmembranestretchedov ertheinterior.Therigid andexiblewingssharethesameshape,size,andcurvature, withachordlengthof 117

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A B Figure4-1.ApliantMAVwingattachedtothemotionrigviath estingbalance.A)Pliant MAVwingB)Pitch-plungerig 125mm,wingspanof150mm,relativecamberattherootof2.8% ,andanareaof 0.0178m 2 .Thedening3 rd orderpolynomialforthethinairfoilshapeatthehalf-span is y =0 : 5621 x 3 1 : 0125 x 2 +0 : 4504 x .Allpitchingmomentdataisreferencedtothe quarter-chordpoint.4.2.1Low-speedWindTunnel Thelow-speedwindtunnelattheUniversityofFlorida'sRes earch,Engineering, andEducationFacility(REEF)wasusedtoconducttheseexpe riments.Theopen-loop, open-jetwindtunneliscapableofspeedsrangingfrom0-22m /swithturbulencelevels below0.16%.Thetestsectionhasanaxiallengthof3.05mwit h1.07m 2 opening surroundedbyastructuralenclosure.Thespecicsofthewi ndtunnelcapabilities,ow uniformity,andturbulencehavebeenextensivelydocument ed[ 9 ].Atypicalinstallationof theMAVwinginthewindtunnelisillustratedinFig. 4-1B 4.2.22-DOFMotionRig Atwo-degrees-offreedommotionrigwasdesignedforthereq uirementsofwind tunneltestingwiththemodelangleofattacknotequaltothe pitchangle(simulating dynamicmotionssuchasrotation,pitching,andplunging). Themaincomponentsare 118

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twoiron-lessmagneticlinearmotorsconnectedtotwodrive rsandaGalil R r motion controlleroperatingasaclosedloopsystem.Thelinearmot orshaveacarriagetravel distanceof0.76m.Themaximumtravelingspeedis5m/switha positioningresolution of5m,amaximumaccelerationof6g,withacontinuousforce of663.7Nandapeak forceof2967.2N.Therig,asillustratedinFig. 4-2 ,hastwo1.5maluminumvertical armsholdinga0.38mhigh-modulussteelrodwithasleevecon nectiontoallowforrod rotation.Thestingbalanceismountedonthesteelrod.Them utualpositionofthetwo linearmotorsdeterminestherodanglethusthemodelattitu de.Thenominalangletest rangeof 30degreescanbeextendedbymechanicaladd-ondevices. Figure4-2.Twodegrees-of-freedomtestrig4.2.3DataAcquisition Forceandmomentdataaremeasuredthroughasix-components train-gaugesting balance.Twodifferentsizestingbalancesareavailablein thefacility.Thelargerbalance iscapableofmeasuringamaximumbalanceloadof44.5Nnorma lforceand22.25N axialforcewitharesolutionof1.112 10 1 Nand8.900 10 3 Nrespectivelyandisused inthecurrentexperiment.Thesmallerbalancecanmeasureu pto13.35Nnormalforce and8.90Naxialforcewitharesolutionof3.560 10 2 Nand4.450 10 3 Nrespectively. Bothbalanceswerecalibratedtoobtain6x39calibrationma tricesthatresolvesecond 119

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andthirdordercomponentinteractions.Thewindtunnelow velocityismonitoredbya pitotprobeinstalledintheinletofthetestsection,andth eairtemperatureismonitored byaresistancetemperaturedetector(RTD)sensormountedo ntheinsideofthetest section.Readingsfromthosesensorsarestoredintheresul tsle. 4.2.4StrainMeasurement Adynamicstereoscopicvisualimagecorrelation(VIC)syst em[ 38 ]usingsynchronized twincamerasisusedtomeasurethetensionintheexiblewin gs.Thesystemuses stereotriangulationtorecover3Ddatafromthecalibrated two-camerasystemandis capableofmeasuringfrom0.05%-500%strainonaspecimensi zeassmallas1mm. Referenceimagesweretakenoftheunstretchedlatexmembra ne,followedbyimages takenafterthestretchedlatexmembranewasappliedtothew ing.Thesequential imagesarecomparedusingcross-correlationtechniquesto obtainthedisplacement eldandstrainofthelatexmembraneinboththexandydirect ionatallpointsonthe interiorofthewing.Thesedatawereaveragedtoobtainthem eanstraininthexandy directionsforvariouslevelsofwingtension. 4.3Methodology Thetwodegrees-of-freedomdynamicrigallowsforvariousd ynamiccasesof pitchingandplungingwithdistinctvaluesof and .Inthe case,illustratedin Fig. 4-3A ,thepitchangleofthemodelisheldconstantandthemodelis subjected toaplungingmotionwithalinearacceleration,thusproduc ingalinearvariationinthe angleofattack.Differentslopesareachievedbysweepingc ombinationsofthevertical velocityprole(linearacceleration)andthewindtunnelf reestreamvelocity.Inthe case,illustratedinFig. 4-3B ,thepitchangleisvariedduringtheplungeinrelationship totheplungeacceleration,resultinginaconstantangleof attackduringtheplunge. Aproofofthisexperimentaltechniquewasthefocusofprevi ousresearch[ 6 ]and thecurrentworkwillextendtheexperimentaltechniquetod eterminetherelationship 120

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1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 Angle [deg] Position [m]Time [s] a a Vinf q Y Motor 1 Y Motor 2 A -6 -4 -2 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 Angle [deg] Position [m]Time [s] a a Vinf q Y Motor 1 Y Motor 2 B Figure4-3.Kinematicplotsofan and motion.A) =10 =s B) = 10 =s betweentherelevantaerodynamicparametersandtherateof changeofangleofattack andpitchangle.4.3.1DesignofExperiments Amoderndesignofexperimentsapproachwasusedindetermin ingtheeffectof thedesignvariables,including and ,ontheresponsevariables( C L C D ,and C m ). Additionally,theinteractioneffectsbetweenfactorswas discoveredandanalyzed.A face-centeredcentralcompositedesignwasdevisedwithei ghtcenterpointstotestfor nonlinearity[ 108 ].TherstexperimentwasdesignedtoanalyzetherigidMAVw ing andincludedfourfactors( _ ,and q )withtherangeofvalueslistedinTable 4-1 Sinceitwasnottheobjectiveofthisresearchtodeterminea ndcomparestallangles andcorresponding C Lmax values,theexperimentremainedinthelinearportionofthe lift curve.Therangeofthedynamicpressure q correspondstoafreestreamvelocityfrom8 to13m/sandReynoldsnumbersfrom68,000to110,000.Theses electionsresultedin anorthogonaldesignconsistingof30runswithnoaliasingb etweenfactors. Withtheexiblewing,thepre-tension ofthemembranewasintroducedasafth factor.Theaveragestrainacrossthelatexmembranewasuse dasthewingtension 121

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Table4-1.Experimentalfactorsandtheirrespectiverange s FactorLowValueHighValue [deg] 015 [deg/s] 020 [deg/s] 200 q [Pa] 39 : 2103 : 5 [m/m] 1300060000 factor,atlevelsof13,000,33,000,and60,000microstrain (m/m).Thisexperiment designconsistedof50runs. Thegeneralequationforthemodelresponseasafunctionoft hedesignvariablesis showninEq.( 4–1 ).Effectshigherthansecondorderwerenotobservedormode led. f ; ; ; q; = a 0 + a 1 + a 2 + a 3 + a 4 q + a 5 + a 6 + a 7 + a 8 q + a 9 + a 10 _ + a 11 q + a 12 + a 13 q + a 14 + a 15 q + a 16 2 + a 17 2 + a 18 2 + a 19 q 2 + a 20 2 + E (4–1) 4.3.2MotionDevelopmentandControl Areversekinematicmodelwasusedtoobtainthedesired or foreachrun. Sincetheresultisdependentuponfreestreamvelocity,ani terativeapproachwasused. Forthe motion,themodelwasplungedwithalinearaccelerationtop roducealinear changeinangleofattack.Forthe motion,theangleofthemodelwasvariedduring theplungetocounterthecomponentofvelocityinducedbyth eplungeacceleration. Thisprocesswasusedtonumericallydeterminethetimehist oryofeachmotorposition duringthemotion.Thetimehistorywasthenconvertedintoa motioncontrollerprogram thatcanbeexecutedbytheuser. ALabview R r programwasdevelopedtocontroltheexecutionoftheindivi dualruns. Actingasaninterfacebetweenthedynamicmotionrig,thest ing-balance,andthewind tunnel,Labview R r wasusedtoautomaticallycontroltheexecutionofthemotio nwhile ahardwaretriggerwasusedtostartthebalancedataacquisi tionatthesamepointin eachmotion.Inthismanner,multiplecyclesofeachtestwer econductedwithminimal 122

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operatorinterventionwhilemaintainingtheabilitytorel iablysynchronizedatafrom multiplecycles.Datawasacquiredfromthesting-balancea tasamplingrateof1kHz. Forthewind-offtare,eachmotionwasrunusinganequivalen tmassatthesamecenter ofgravitypositionastherespectiveMAVwing.Inboththewi nd-onandwind-offcases, eachmotionisrepeatednumeroustimestoenableensembleav eraging.Specically,the motioncycleisrepeatedvetimesinthewind-oncaseandthr eetimesinthewind-off tare.Thenumberofcyclerepetitionswastheresultofanana lysisoftheconvergenceof themeanvaluewiththenumberoftestcyclesandanefforttom inimizethetotalnumber ofrequiredcyclesfortheexperiment.Includingtherepeti tions,thecombinedtestmatrix forallwingsconsistedof80distinctmotionsand640runs.A fterensembleaveragingis applied,thedataarelteredusinga4thorderButterworth lterwithanormalizedcutoff frequencyof10Hz.4.3.3MAVWingTension Theelasticdeformationofthewings,thustheirshapeandae rodynamics,are affectedbythemechanicalcharacteristicsandpre-strain stateofthemembrane. Whenaloadisappliedtothemembrane,causedbythepre-tens ionappliedduring assembly,itintroducesa de-facto newvariablethatneedstobeaccountedby measuringtheplane-strainconditionsofthepliantmembra ne.Thequanticationof theelasticpre-tensioninthelatexmembrane,acriticalfa ctorinthepresentwork, wasobtainedthroughvisualimagecorrelation.Thestrain eldisdenedbythethree strains xx yy ,and xy actingparalleltothewing'ssurface.Thewing'sthinmembr ane ischaracterizedbyaplane-stressstateinwhichthetwo-di mensionalstresstensor generatesathree-dimensionalstraintensor.Effortwasma detoquantifythestrainstate inwind-offconditionswiththreelevelsofstrain.Theresp ectivestraindistributionsare illustratedinFig. 4-4 showingthestraindistributioninthexdirectionfortheav erage strainvaluesof xx ontheorderof60,219and12,985m/minthecaseofhighandlo w pre-tensionmembranewings,respectively.Thewind-offst raindistribution,considered 123

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A B Figure4-4.Contourplotsofthewind-offmembranestrainst ateinthe x direction.A) LowtensioncaseB)Hightensioncase asastructuralboundarycondition,isusedtocharacterize thewingstiffnessandfor correlationswithitsaerodynamiccharacteristics. Forreferencepurposethepre-strainwascharacterizedbyu singadimensionless numberindicatedcalled 2 [ 149 ].The 2 factorisexpressedbythefollowingequation: 2 = h qc (4–2) where istheaveragestress, h isthemembranethicknessand c isthemean aerodynamicchord.Thenumberexpressestherelevanceofth epre-stressstate inthemembraneinrelationtothefreeowdynamicpressurea nditisde-coupled inthetwodirectionsx(chordwise)andy(spanwise).Thelev elsofpre-strainare reportedinTable 4-2 andaplotshowingthevalueof 2 asafunctionofthedynamic pressureforthethreelevelsofpre-strainusedintheexper imentsisillustratedin Fig. 4-5 .ThemembranestresswasevaluatedusingHooke'slawwithas implelinear stress-strainlinearrelationwith2.0MPaasthevalueofth eYoungmodulusofthelatex membrane[ 159 ].Highervaluesof 2 correspondtoahigherlevelsofpre-strainatthe samedynamicpressureor,forthesamelevelofpre-strain,r epresentalowervalueof 124

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Table4-2.Characterizationofmembranepre-tensionforth ethreelevelstested,along withassociated 2 valuesat q =39 : 5 Pa Level xx [%] xx std x [Pa] yy [%] yy std y [Pa] 2 x 2 y Low 1 : 461 : 11151925 : 471 : 130 : 9688952 : 974 : 622 : 70 Medium 3 : 881 : 6977574 : 302 : 671 : 4653463 : 262 : 361 : 62 High 7 : 605 : 6829270 : 954 : 451 : 6622670 : 770 : 890 : 68 0 5 10 15 20 4 6 8 10 12 14 P 2V [m/s] Low E y Medium E y High E y Low E x Medium E x High E x Figure4-5.Plotof 2 inthetwodirectionsasafunctionofvelocity dynamicpressuredenotinganeffectivepredominanceofthe membranestiffnessover thedynamicpressureeffects. 4.4ResultsandDiscussion Afterfollowingtheproceduresoutlinedabove,thetrendso fthedataobtainedfor theindividualrunsarecharacterizedbylownoiselevelsan dsmoothtimehistories.A sampleoftheresultsfortheliftcoefcientoftheexiblew ingisshowninFig. 4-6 .From theleft,thetimehistoryoftheliftcoefcientforthelowtensionexiblewingduringan caseisshown,withtheangleofattackvaryinglinearlyfrom -4 to5 atafreestream velocityof13m/s(Re=110,000).Theelasticpre-straincon ditionshaveavalueofthe 2 x factorof0.34and 2 y of0.26.Thecaserepresentsan of20 /sovertheduration oftheplunge(0.45s).Thelinearresponseoftheliftcoefc ientisclearlyseen,reaching amaximumvalueof0.77around5 angleofattack.Fig. 4-6B showsthetimehistory oftheliftcoefcientduringthe case,wheretheangleofattackisheldapproximately 125

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-6 -4 -2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 Angle [deg] Coefficient [-]Time [s] a q C L A 4 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Angle [deg] Coefficient [-]Time [s] a q C L B Figure4-6.Liftcoefcienttimehistoriesofthelow-tensi onexiblewingforthetwo cases.A) =20 =s B) = 20 =s constantat15 duringaplungewithalinearacceleration.Thismotionrepr esentsthe =-20 /scaseandresultsinarelativelysteadyliftcoefcientof 1.2duringtherelevant portionofthemotion. Afterthedatawerecollectedfromalltheexperimentalruns ,alinearmodelwas usedtoanalyzetheliftandpitchingmomentcoefcientsand aquadraticmodel withasquareroottransformationwasusedtoanalyzethedra gcoefcient.Forboth wings,theregressionmodelsforlift,drag,andpitchingmo menttthedataverywell. Residualswererandomlydistributedandstayedwithinacce ptablelimits.Valuesforthe mathematicalcoefcientsaregiveninTable 4-3 forboththerigidandexiblewings.The inuenceof and areseeninthe a 2 and a 3 coefcients,respectively.Additionally,the mostprominentinteractioneffectisbetween and ,representedbythea10coefcient. Afterobtainingthecoefcientsfortherigidandexiblewi ngs,themodelwas evaluatedatstaticconditions( ; =0 )forreferencepurposesandcomparedtothe resultsfromstaticwindtunneltesting.AsshowninFig. 4-7 ,themodel(createdfrom staticanddynamictests)comparescloselytothestaticdat aforbothrigidandexible wings.Especiallynotableistheagreementbetweenmodelan dstaticpitchingmoment 126

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Table4-3.Valueofconstantmodelparameters(codedunits) RigidWingFlexWing C L C D C m C L C D C m a 0 6 : 94 10 1 7 : 68 10 1 4 : 17 10 2 9 : 86 10 1 7 : 77 10 1 1 : 84 10 1 a 1 3 : 72 10 1 6 : 48 10 2 1 : 44 10 2 3 : 69 10 1 7 : 46 10 2 4 : 90 10 2 a 2 4 : 76 10 2 3 : 42 10 3 2 : 58 10 3 5 : 38 10 2 2 : 05 10 3 1 : 40 10 2 a 3 7 : 41 10 2 5 : 35 10 3 1 : 58 10 2 9 : 26 10 2 2 : 36 10 3 3 : 00 10 2 a 4 7 : 64 10 3 7 : 60 10 3 1 : 32 10 2 1 : 16 10 3 7 : 31 10 3 3 : 11 10 3 a 5 5 : 20 10 2 1 : 11 10 04 1 : 48 10 3 a 6 3 : 57 10 3 2 : 15 10 3 a 7 3 : 92 10 3 5 : 67 10 3 a 8 7 : 14 10 3 4 : 81 10 3 a 9 3 : 89 10 3 a 10 3 : 06 10 2 1 : 17 10 2 2 : 16 10 2 a 11 1 : 90 10 3 a 12 2 : 35 10 3 a 13 a 14 a 15 3 : 15 10 3 a 16 2 : 12 10 2 3 : 06 10 2 a 17 5 : 95 10 3 a 18 a 19 a 20 2 : 03 10 1 3 : 91 10 2 R 2 0 : 980 : 990 : 630 : 900 : 990 : 72 coefcients.Thebehaviorofbothwingswasfoundtobestron glydependentuponthe dynamicconditions.4.4.1DynamicBehavior Fortheexiblewing,showninFig. 4-8 ,astherateofchangeofangleofattack increases,themaximumliftcoefcientalsoincreases.For pure =20 /sthereisa8% increaseinliftoverthestaticcase.Forpure at-20 /sthereisa15%increaseinlift overthestaticcase.Withan =20 /sand =-20 /sthereisa24%increaseinlift overthestaticcase.Inthedragcoefcient,amildinteract ionwasobservedsuchthat 127

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 C LC D Flexible (Static) Flexible Rigid (Static) Rigid A -0.4 -0.3 -0.2 -0.1 0 0.1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 C mC L Flexible (Static) Flexible Rigid (Static) Rigid B Figure4-7.Coefcientsresultingfromthemodelcompareda gainststaticwindtunnel dataforthehigh-tensionexiblewingat q =103 : 5 Pa .A)LiftanddragB) Pitchingmomentcoefcient 0 5 10 15 0 5 10 15 20 0 0.5 1 1.5 2 C L 0 -10 -20 q [ /s] a [ ] a [ /s] C L A 0 5 10 15 0 5 10 15 20 0 0.1 0.2 0.3 0.4 C D 0 -10 -20 q [ /s] a [ ] a [ /s] C D B Figure4-8.Themedium-tensionexiblewingmodelevaluate dwith q =39 : 5 Pa .A) CoefcientofliftB)Coefcientofdrag thelowestdragatsmallanglesofattackoccursathigh rates,butthelowestdragat highanglesofattackoccursatlow rates. Therigidwingalsoexperienceshigherliftindynamiccondi tions(Fig. 4-9 ).With apure of20 /sthereisa17%increaseinliftoverthestaticcase,andfor apure 128

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0 5 10 15 0 5 10 15 20 0 0.5 1 1.5 2 C L 0 -10 -20 q [ /s] a [ ] a [ /s] C L A 0 5 10 15 0 5 10 15 20 0 0.1 0.2 0.3 0.4 C D 0 -10 -20 q [ /s] a [ ] a [ /s] C D B Figure4-9.Therigidwingmodelevaluatedwith q =39 : 5 Pa .A)CoefcientofliftB) Coefcientofdrag at-20 /sthereisa23%increase.Withan =20 /sand =-20 /sthereisa26% increaseinliftoverthestaticcase.Althoughthesepercen tagesaregreaterthanthe exiblewing,themagnitudeoftheliftcoefcientsofthee xiblewingishigher.There isamodestinteractioneffectbetween and ontheliftcoefcientoftherigidwing thatcanbeobservedviathepresenceofthea10coefcientin Table 4-3 .Considering thecoefcientofdrag,thereisnoexperimentalevidenceof aninteractionbetweenthe samefactors,implyingthattherateofchangeindragcoefc ientdueto isthesameat all rates,andvice-versa. Theincreasedperformanceinthepresenceofdynamicmotion isbestillustrated intheresponsesurfacesofthelift-to-dragratio(Fig. 4-10 ).Theresponseoftherigid wingexhibitssignicantinteractionsbetweenthedynamic parametersandproduces thehighestL/Dratioof16at =0 /sand =-20 /s.Themediumtensionexiblewing producesapeakL/Dofalmost40at =20 /sand =-20 /s.Apossibleexplanationis theoppositetrendsof C D and C L (Fig. 4-8 )with and thatincreasethe C L =C D ratio. 129

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0 5 10 15 0 5 10 15 20 0 10 20 30 40 L/D 0 -10 -20 q [ /s] a [ ] a [ /s] L/D A 0 5 10 15 0 5 10 15 20 0 10 20 30 40 L/D 0 -10 -20 q [ /s] a [ ] a [ /s] L/D B Figure4-10.Lift-to-dragratiosoftherigidwingandexib lewingat q =39 : 5 Pa .A)Rigid wingB)Medium-tensionexiblewing 4.4.2PitchDampingDerivatives Thedynamicpitchdampingderivativesmaynowbeobtainedby takingthepartial derivativeofEq.( 4–1 )andapplyingthemodelcoefcientsfromTable 4-3 .Forboth therigidandexiblewings,theresultingpitchdampingder ivativesarerepresentedin Eqs.( 4–3 )and( 4–4 ). C m = a 2 + a 10 (4–3) C m = a 3 + a 10 (4–4) Fromthisanalysis,itcanbeseenthatthepitchdampingderi vativesarenot constantbutincludeadependenceontheoppositedynamicpa rameter.Thisinteraction canbeobservedinFig. 4-11 .Fortherigidwingatlowvaluesof ,thehigh rates producethelargestrestoringmoment;athighvaluesof theoppositeistrue:the lowest producethelargestmoment.Alevelofinteractionbetween and inthe pitchingmomentoftheexiblewingisalsopresent.Thisint eractionhastheeffectof makingtheinuenceof onthepitchingmomentverysmallathigh ,whereasaslow 130

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0 5 10 15 0 5 10 15 20 -0.4 -0.3 -0.2 -0.1 0 C m 0 -10 -20 q [ /s] a [ ] a [ /s] C m A 0 5 10 15 0 5 10 15 20 -0.4 -0.3 -0.2 -0.1 0 C m 0 -10 -20 q [ /s] a [ ] a [ /s] C m B Figure4-11.Pitchingmomentcoefcientoftherigidwingan dexiblewingat q =39 : 5 Pa .A)RigidwingB)Medium-tensionexiblewing ,high hasalargeeffect(60%increase).Overall,theexiblewing producesalarger restoringmomentthantherigidwing. 4.5SummaryandConcludingRemarks Anexperimentalmethodhasbeendevelopedandtestedtodete rminethe dependenceoftheaerodynamicresponseofaexibleorrigid MAVwingtothedynamic parametersof and atlowReynoldsnumbers.Thedynamicderivatives,specica lly thepitch,lift,anddragdampingderivativesaretheorigin alresultspresentedinthis research.Furthermore,theexiblewingincludesapliantm embraneskinsetatthree levelsofpre-tension.Thepre-tensionstrainwasconsider edasafactorinthedesignof theexperimentsanditspotentialeffectsontheaerodynami ccoefcientsanddynamic derivativeswereinvestigated. Overall,thepresenceof and wereobservedtohaveasignicantinuence intheaerodynamicresponseoftheMAVwingsutilizedinther esearch.Anoticeable differenceofliftanddragbetweentherigidandexiblewin gsinstaticanddynamic conditionswasobserved.Themediumtensionexiblewingpr oducedthehighest lift-to-dragratioinbothstaticanddynamicconditions.T helift-to-dragratiowasalso 131

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inuencedbythedynamicparameters.Substantialchangesi nthepitchingmoment coefcientswereobservedinthepresenceofdynamicchange sinangleofattackor pitchangle.Thedynamicderivativeofthepitchingmomentw ithrespectto and containcross-correlationtermswitheachother.Theexib lewingproducedastronger restoringmomentwhencomparedtotherigidwing.Thus,thes eexperimentally-derived expressionsofpitchdampingderivativesmayhavearelevan tinuenceonthevehicle dynamiccharacteristicsandcontrol. Thepre-strainlevelsinvestigatedcoverarangeofthedime nsionlessfactor 2 between0.26and4.65,whencomparedwiththerespectivewin dtunneldynamic pressure.Thepre-strainlevelofthewingmembranewasfoun dnottobeafactorinthe pitchingmomentandliftdynamicderivativeswhereasitisp resentinthedragcoefcient dynamicderivativebutonlyrespectto .Additionalresearchisplannedtoinvestigate theinuenceof _ andwingmembranepre-strainontheaerodynamicresponseof a completeMAVincludingamicro-propeller. 132

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CHAPTER5 AEROELASTICITYANDFLIGHTDYNAMICS:UNIFORMWINGSTIFFNES S Adegreeofseparationhastraditionallyexistedbetweeni ghtdynamicsand aeroelasticity,twoofthedisciplineswhichcomprisemode rnightmechanics[ 101 144 ]. Traditionalightdynamicanalysesuseasimpliedapproac hbyjustconsideringa rigid-bodyaircraftundergoingsmallmaneuversaroundtri m,wherethestatesdescribe thesixdegree-of-freedommotionabouttheaircraftcenter ofgravity.Aeroelastic analysesareusuallyconcernedwiththeinteractionsbetwe entheunsteadyaerodynamics andstructureofanon-maneuveringexibleaircraft,where thestatesarenotaboutthe centerofgravity.Inthisregard,ightdynamicsandaeroel asticityhavedeveloped separately. Inearlyaircraftanalyses,notconsideringtheelasticeff ectsontheightdynamics wasanaturaloutcomefromthelackofavailabledataregardi ngtheelasticproperties ofthestructureandthetypicallylargefrequencyseparati onbetweenthestructural andightdynamics[ 27 34 ].However,increasedexibilityoftheaircraftstructure can resultinalowerfrequencyseparationandastrongaeroelas ticinuenceontheight dynamics,possiblyleadingtodynamicinstability[ 167 177 180 ]. Therehasbeenmuchresearchdirectedatcreatingnumerical modelswhichinclude aeroelasticeffectsontheightdynamics,asreviewedinSe ction 1.1.2 .Thesestudies arefocusedonthedevelopmentofmodelswhichcananalyzeae roelasticityandight dynamicsinacommonframework.Modeldevelopmentisavital stepinunderstanding theeffectsofaeroelasticityonightdynamics;however,i tdoesnotaddressthese effectsintheaircraftdesignprocess. Aircraftdesignisfocusedonnegotiatingthetradespacebe tweenthemultiple disciplinesofightmechanics.Aeroelasticeffectsarera relyconsideredinthe conceptualdesignphase,andiftheyare,itisusuallyforth epurposeofmitigating 133

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undesirablephenomenasuchasdivergence,controlreversa l,orutter[ 22 80 116 138 ]. Thelittleexistingresearchthataddressestheinuenceof aeroelasticityonthe ightdynamicsasitpertainstothedesignprocessisalsore viewedinChapter 1 Thosedesignstudiesonlyevaluatechangesinconventional designparameterswithout consideringchangestoparameterswhichwouldhaveadirect effectontheaeroelastic responseoftheaircraft,suchasstructuralstiffness.Asa result,thenatureofthe aeroelasticinuenceontheightdynamicsisnotcaptured. Manyengineersandresearchersdesiretoexploitexibilit yinmicroairvehicles, butalackofunderstandingoftheinteractionbetweenaeroe lasticityandightdynamics canleadtosub-optimaldesigns.Theuniquestructuralchar acteristicsandightregimes ofMAVsimpliesthattheinteractioneffectsatthesesmalls calesmaynotfollowthe interactioneffectsatlargerscales.Therelativelysmall amountofresearchdoneon aeroelasticMAVshasfocusedonstaticaeroelasticityandd oesnotadequatelyaddress thedynamicinteractionsbetweenaeroelasticityandight dynamicsfromadesign perspective[ 71 135 159 ]. Thischaptersystematicallyinvestigatestheeffectsofae roelasticityontheight dynamicsofanelasticmicroairvehicle.Thebendingandtor sionalstiffnessofthewing arechosenasthetwomainparametersofinterest,althought heaircraftightspeedand wingelasticaxisarealsoconsidered.Theeffectsonthetri mconditions,ightdynamics, andstructuraldynamicsarecharacterized. Considerationoftheseeffectsmayprovidetheaircraftdes ignerwithanewdesign parametertoaltertheopen-loopbehavioroftheaircraftin thepreliminarydesign process.Thepurposemaybetoavoidundesirableeffectsont heightdynamicsorto takeadvantageofeffectsthatarebenecialtothemission. Itmayalsobedesirableto rapidlytailorormorphthestructureoftheMAVtoalterthe ightdynamicsinabenecial 134

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way.Anunderstandingofthestructure'sinuenceontheig htdynamicsiscriticalto achievethesegoals. 5.1Methodology TheASWINGmodeloftheGenMAVaircraft,introducedinSecti on 3.2 ,isused inthepresentstudy.Thefuselageandtailaremaderigidwhi letheelasticityofthe wingischangedthroughthebendingandtorsionalstiffness parameters EI and GJ respectively.RecallfromSection 2.2.6 thatthebendingstiffness EI relatesthebending momentaboutthechordaxistothedeectionofthewinginthe z-directionandshould notbeconfusedwithbendingaboutanyotheraxis.Allothers tiffnessparametersofthe wing,suchasthebendingcross-stiffness,bending/torsio ncouplingstiffness,extensional stiffness,andshearstiffness,aresetasperfectlyrigidi nthewing. 5.1.1DesignSpaceOverview Theresearchinthischapterisdividedintothevecasessho wninTable 5-1 .In allcases,themagnitudeofthewing'sstiffnessisuniforma crossthewingspan.The rstcaseanalyzesmediumtohighrangesofbendingandtorsi onalstiffnesswhilethe secondcaseanalyzesalowerrangeofstiffnesses.Cases3an d4analyzetheeffects ofaircraftightspeedontheaeroelasticightdynamics.F inally,case5addresses theeffectsofthelocationofthewing'selasticaxis.Aloca tionof 0.28 c indicatesthe experimentallydeterminedpositiongiveninSection 3.2.3.1 (whichliesclosetothe quarter-chordline). Thelowervalueofstiffnessinthesecaseswaschosenthroug htrialanderrorwith thegoalofpreventingunrealisticwingdeformationsandai rcrafttrimconditions.The uppervaluewaschosentobeanorderofmagnitudemorestifft hantheactualight vehicle.5.1.2ModelingProcedure ThemodelswerecreatedusingMatlab c r tocontrolASWING.Thegeneral procedureforeachtestpointisasfollows: 135

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1.Computethestructuralconguration2.GeneratetheASWINGinputle3.GeneratethenecessaryASWINGcommands4.ExecuteASWING5.Acquireandsavetheresults WhenexecutingASWING,theaircraftisrsttrimmedinstrai ghtandlevelightat V 1 =15m/s.Thetrimsolutionforeachcongurationmustbeveri ed,whichisdoneby examiningtheresolvedforcesandmomentsactingontheairc raftandensuringthatthe momentsarenear-zeroandthattheliftisapproximatelyequ altotheweight. Thenamodalanalysisisperformed.WhenrunonaIntel c r Core TM 2Duo2.53GHz processorwith4GBofmemory,theentireprocesstakeslesst han10secondspertest point.Afteracquiringthedata,caremustbetakentoproper lyidentifytheeigenvalues associatedwitheachmodeofight.Asimplisticalgorithmw orkswelltosortthe eigenvaluesformoststandardcongurations.However,for casesinvolvingmore unusualmodalbehaviorthisapproachisnotdependableando ftenreturnserroneous eigenvalueassignments.Insuchcases,theeigenvaluesare examinedmanuallyby inspectionoftheeigenvectorinordertodeterminewhichmo detheycorrespondto. 5.2Case1:MediumtoHighStiffness Thevaluesofbendingandtorsionalstiffnessarevariedind ependentlyfrom 10.0Nm 2 to1.0Nm 2 inthepresentcase.Thedesignspacewasdiscretizedwith20 Table5-1.Overviewoftheruncasesandtheindependentvari ablesineachcase CaseDistribution EI [Nm 2 ] GJ [Nm 2 ] V 1 [m/s]ElasticAxis[%c] 1Uniform1.0-10.01.0-10.015 0.28 2Uniform0.07-1.00.07-1.015 0.28 3Uniformrigidrigid15-35 0.28 4Uniform0.07-1.00.07-1.015-19 0.28 5Uniform0.07-1.00.07-1.0150-0.4 136

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valuesof EI and GJ (foratotalof400congurations)withthespacingchosento maximizeresolutionintheareaswiththelargestchangesin theresponses.Theresult was10valuesof EI and GJ eachforatotalof100congurations.Thedesignspaceis illustratedinFig. 5-1 ,wheretheintersectionoftwolinesrepresentsanindividu alaircraft conguration. 0 2 4 6 8 10 0 2 4 6 8 10 GJ [Nm 2 ]EI [Nm 2 ] Figure5-1.Case1:Designspaceofbendingandtorsionalsti ffness 5.2.1TrimConditions Theangleofattackandelevatordeectionrequiredforeach congurationtobe trimmedareshowninFig. 5-2 .Neitherbendingnortorsionalstiffnesshaveasignicant effectonthesetrimparameters.Theangleofattackchanges lessthan1/10 th ofa degreeacrossthedesignspace,whiletheelevatordeectio ndecreasesslightlyfrom -4.4 to-4.6 asbendingstiffnessdecreasesfrom10to1Nm 2 .Torsionalstiffnessdoes nothaveadiscernibleeffectontheelevatordeection. Theeffectsofstiffnessonthethedeformationoftheaircra ftwingshapeareshown inFig. 5-3 .Themajoreffectoflowering EI istoincreasethewingtipdeectionwhile GJ doesnothaveanoticeableeffect. 137

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1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 6 6.5 7 7.5 8 a [deg] EI [Nm 2 ] GJ [Nm 2 ]a [deg] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 -5 -4.5 -4 -3.5 -3 d e [deg] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] B Figure5-2.Case1:Aircraft and e attrim.A)TrimangleofattackB)Trimelevator deection 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 4 8 12 16 20 z [mm] EI [Nm 2 ] GJ [Nm 2 ]z [mm] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 -0.5 -0.3 -0.1 0.1 0.3 0.5 t [deg] EI [Nm 2 ] GJ [Nm 2 ]t [deg] B Figure5-3.Case1:Wingdeformationsattrim.A)Wingdeect ionat y = b B)Wingtwist at y =0 : 75 b 138

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-40 -35 -30 -25 -20 -15 -10 -5 0 5 0 2 4 6 8 10 12 14 16 18 20 ImagRe Short Period Dutch Roll Roll Phugoid Spiral Figure5-4.Case1:Allpolescorrespondingtotheightmode sforthedesignspace 5.2.2FlightDynamics Figure 5-4 showsthelociofthepolesonthecomplexplane.Overall,the reislittle changeinthelocationofthepolesinresponsetothisrangeo fstiffness.Thegreatest relativepolemovementoccursintheshortperiodmode.5.2.2.1Lateral-directionaldynamics Thestiffaircrafthasaspiraldivergencewherethespiralp oleislocatedat0.068on therealaxis.Thepolemovestoalocationof0.003inthemost exiblecongurationbut doesnotcrosstheimaginaryaxis. AsillustratedinFig. 5-5 ,thetimeconstantoftherollconvergencemodeshows slightchangesinresponsetochangesinstiffness.Asthewi ngbecomesmoreexiblein bending,thetimeconstantdecreasesby0.8%(atboththehig handlowvaluesof GJ ). Asthewingbecomesmoreexibleintorsion,thetimeconstan tdecreasesby1.6%(at boththehighandlowvaluesof EI ). Thebehaviorofthedutchrollmodeshowsminorchangesinres ponsetochanges inbendingstiffness.Thesechangesaredepictedinplotsof thenaturalfrequencyand dampingratioinFig. 5-6 .Athighorlowtorsionalstiffness,thenaturalfrequency increasesby1.5%inresponsetoloweringbendingstiffness whilethedamping 139

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1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0.025 0.026 0.027 0.028 0.029 0.03 T [s] EI [Nm 2 ] GJ [Nm 2 ]T [s] Figure5-5.Case1:Rollconvergencetimeconstantratioincreases8%.Thereisnodiscernibleeffectoftorsio nalstiffnessonthenatural frequencyordampingratio. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1.3 1.35 1.4 1.45 1.5 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 z [-] EI [Nm 2 ] GJ [Nm 2 ]z [-] B Figure5-6.Case1:Dutchrollmodalcharacteristics.A)Nat uralfrequencyB)Damping ratio Aprobablecausefortheeffectsofbendingstiffnessonthel ateral-directionalmodes isduetotheincreaseofdihedralatlowvaluesofstiffness. Dihedralhasawellknown 140

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1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0.13 0.14 0.15 0.16 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0.05 0.1 0.15 0.2 z [-] EI [Nm 2 ] GJ [Nm 2 ]z [-] B Figure5-7.Case1:Phugoidmodalcharacteristics.A)Natur alfrequencyB)Damping ratio effectonthelateral-directionalstabilityderivatives, whichwillaffectthelateral-directional modesofight[ 161 ]. 5.2.2.2Longitudinaldynamics Thephugoidmodeshowsveryminorchangesinresponsetochan gesinbending andtorsionalstiffness.Thenaturalfrequencyanddamping ratioareshowninFig. 5-7 Thenaturalfrequencyanddampingratiochangebylessthan1 %acrossthedesign space. Theshortperiodmodeshowsmoreprominentchangesinfreque ncyanddamping. As EI decreases,thenaturalfrequencyrisesby2.9%whilethedam pingratio decreasesby4.6%.Thereisnosignicanteffectoftorsiona lstiffness. 5.2.3StructuralDynamics Thenaturalfrequencyoftherstsymmetricbendingmode,sh owninFig. 5-9 showsastrongdependenceonthebendingstiffness,decreas ingfrom71Hzinthehigh EI and GJ congurationto22.9Hzinthelow EI ,high GJ conguration.Thedamping ratioalsoshowsastrongdependenceonbendingandtorsiona lstiffness.Whenthe 141

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1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 2.75 3 3.25 3.5 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0.3 0.35 0.4 0.45 0.5 z [-] EI [Nm 2 ] GJ [Nm 2 ]z [-] B Figure5-8.Case1:Shortperiodmodalcharacteristics.A)N aturalfrequencyB) Dampingratio wingiskeptstiffinbendingbutmadeexibleintorsion,the dampingratiorises70%to 0.08.Whenthewingisheldstiffintorsionbutmadeexiblei nbending,thedamping ratiorisesby196%to0.14. 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 z [-] EI [Nm 2 ] GJ [Nm 2 ]z [-] B Figure5-9.Case1:Firstsymmetricbendingmodalcharacter istics.A)Naturalfrequency B)Dampingratio 142

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Thenaturalfrequencyoftherstsymmetrictorsionmodeisn otplottedherebut wasseentodecreasedramaticallyinresponsetoloweringto rsionalstiffness.Evenin themostexibleconguration,thefrequencywas114Hz.The rstsymmetricbending modeisstilltheprimarystructuralmodeofconcernwithits naturalfrequencyof22.9Hz inthesameconguration. Ofparticularimportanceisthefrequencyseparationbetwe enthelowestfrequency structuralmode(rstsymmetricbending)andthehighestfr equencyightmode(short period).Thatseparationdecreasesfrom68.2Hzinthehigh EI and GJ caseto19.8Hz inthelow EI and GJ case. 5.2.4Summary ThebendingandtorsionalstiffnessoftheGenMAVwingwerev ariedfrom10.0to 1.0Nm 2 whilethetrimconditions,ightdynamics,andstructurald ynamicsoftheaircraft wereanalyzed.Ingeneral,boththisrangeofbendingandtor sionalstiffnesshasvery littleeffectontheaircraftdynamics. Thereisaminoreffectonthedutchrollfrequency(whichcha ngedby1.5%)and damping(whichchangedby8%)inresponsetobendingstiffne ss.Theshortperiod modealsoshowednoticeablechangesinresponsetobendings tiffness:thenatural frequencyincreasedby2.9%andthedampingratiodecreased by4.6%asbending stiffnessdecreased. Themostsignicanteffectofthestructuralstiffnessinth israngeisonthestructural modes.Thenaturalfrequencyanddampingofrstsymmetricb endingmodechangeby -68%and196%,respectively,asbendingstiffnessdecrease sfrom10to1Nm 2 5.3Case2:LowtoMediumStiffness Thevaluesofbendingandtorsionalstiffnessarevariedind ependentlyfrom1.0Nm 2 to0.07Nm 2 inthepresentcase.Thediscretizationofthedesignspacew aschosento maximizeresolutionintheareaswiththelargestchangesin theresponses.Theresult was20valuesof EI and GJ eachforatotalof400congurations.Thedesignspace 143

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isillustratedinFig. 5-10 ,wheretheintersectionoftwolinesrepresentsanindividu al aircraftconguration. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 GJ [Nm 2 ]EI [Nm 2 ] Figure5-10.Case2:Designspaceofbendingandtorsionalst iffness 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 4 8 12 16 20 a [deg] EI [Nm 2 ] GJ [Nm 2 ]a [deg] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -20 -16 -12 -8 -4 0 d e [deg] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] B Figure5-11.Case2:Aircraft and e attrim.A)TrimangleofattackB)Trimelevator deection 144

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5.3.1TrimConditions Theconditionsaredeterminedatwhicheachcongurationin thedesignspaceis trimmed.Theassociatedvaluesforangleofattackandeleva tordeectionareshown inFig. 5-11 .Theeffectsaredominatedbychangesinbendingstiffnessw hilechanges intorsionalstiffnesshavenosignicanteffect.Astheben dingstiffnessdecreasesfrom 1.0Nm 2 to0.07Nm 2 ,thetrimangleofattackincreasesfrom6.9 to14.2 (aincreaseof 106%)andtheelevatordeectiondecreasesfrom-4.4 to-13.6 (adecreaseof209%). Itisimportanttonotethatthereisverylittleinteraction betweenbendingandtorsional stiffness,i.e.,theeffectofbendingstiffnessisapproxi matelythesameatallvaluesof torsionalstiffnessandviceversa. Theaeroelasticeffectsonthetrimangleofattackandeleva tordeectionare adirectresultoftheeffectsofstiffnessonthedeformatio noftheaircraft,shown inFig. 5-12 .Themajoreffectoflowering EI istoincreasethewingtipdeectionwhile GJ hasveryminoreffect.Asthewingtipdeectsasaresultofde creasingbending stiffness,thereislossofliftovertheouterspanofthewin g.Tocompensate,theaircraft 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 42 84 126 168 210 z [mm] EI [Nm 2 ] GJ [Nm 2 ]z [mm] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -10 -7 -4 -1 2 5 t [deg] EI [Nm 2 ] GJ [Nm 2 ]t [deg] B Figure5-12.Case2:Wingdeformationsattrim.A)Wingdeec tionat y = b B)Wing twistat y =0 : 75 b 145

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-14 -12 -10 -8 -6 -4 -2 0 0 5 10 15 20 25 30 x [cm]y [cm] x cp at a =5 x cp at a =15 Elastic Axis Figure5-13.Elasticaxisandcenterofpressurealongthewi ng angleofattackmustincrease,whichrequiresamorenegativ eelevatordeection.There isasmalleffectfromlowering EI whichactstodecreasethewingtwist,whichcomes throughaerodynamiccouplingbetweenthebendingandtorsi onofthewing.Asthewing bendingincreases,thewingtwistisaffectedbythechangin gliftforceandthechanging centerofpressurelocation.Thereisnosignicantinterac tionbetween EI and GJ Theminoreffectoflowering GJ onthewingtwistmaybearesultoftherelative locationofthecenterofpressureandtheelasticaxis.Assh owninFig. 5-13 for =5 and =15 ,thecenterofpressureliesveryclosetotheelasticaxis.D ependingonthe angleofattack,thetheoreticalpositionofthecenterofpr essurecouldbeforwardoraft oftheelasticaxis. Therelativepositioningisseenmoreclearlyinaplotofthe averagelocationofthe centerofpressurealongthespanversustheangleofattack, showninFig. 5-14 .Inthe rangeofanglesofattackseenhere,thecenterofpressureli esclosetotheelasticaxis. Thisresultsinaverysmallmomentarmbetweenthenormalfor ceonthewingandthe elasticaxis,resultinginverylittlewingtwistevenatlow valuesoftorsionalstiffness. 146

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5.3.2FlightDynamics Thepolesoftheightmodesforallcongurationsareshowni nFig. 5-15 .Thereis considerablechangeinthebehaviorofeachofthelateral-d irectionalmodesofightin responsetochangesinstiffness.Atverylowvaluesofstiff ness,thechangesbecome morenoticeable,asseenintherollconvergencepoleasitmo vestotherightalong therealaxisandthedutchrollmodeasitmovestotheleftint hecomplexplane.The individualpolesareshowninFig. 5-16 foradditionalclarity. 5.3.2.1Lateral-directionaldynamics Thespiralpoleprogressesfromadivergentbehaviorwithat imeconstantof56.4 secondsinthemoststiffcongurationtoaconvergentbehav iorwithatimeconstant of0.9secondsinthemostexibleconguration.Themodesha pecanbededuced fromthenormalizedeigenvectorcomponentslistedinTable 5-2 .Athighstiffness,the spiraldivergenceischaracterizedbyarollanglewhichis3 %oftheyawanglewhile thewingdeectionsandtwistremainatzero.Decreasingben dingstiffnesswhile holdingtorsionalstiffnessconstantresultsinaspiralco nvergencewheretherollangle 0.2 0.25 0.3 0.35 0.4 0 5 10 15 20 x/ca [deg] span-averaged x cp Elastic Axis Figure5-14.Locationofthespan-averaged x cp andelasticaxisversusangleofattack 147

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is-261%oftheyawangle.Wingtipdeectionsareslightlyas ymmetric.Averyexible congurationin EI and GJ resultsinarollanglethatis-228%oftheyawangle. AsillustratedinFig. 5-17 ,thetimeconstantoftherollconvergencemodeincreases from0.028secondsinthemoststiffcongurationto0.087se condsinthemostexible -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 0 5 10 15 20 25 ImagRe Short Period EI,GJ Dutch Roll EI,GJ RollEI,GJ Phugoid Spiral Figure5-15.Case2:Allpolescorrespondingtotheightmod esforthedesignspace Table5-2.Case2:Normalizedeigenvectorcomponentsofthe spiral convergence/divergence f EI GJ g [Nm 2 ] f 1.0,1.0 gf 1.0,0.07 gf 0.07,1.0 gf 0.07,0.07 g p [deg/s]0.000.002.992.36 r [deg/s]0.02-0.01-1.09-0.97 [deg]0.000.00-0.08-0.06 [deg]0.03-0.02-2.61-2.28 [deg]1.001.001.001.00 z ( y = 0 : 3) [mm]0.000.000.000.00 z ( y = 0 : 15) [mm]0.000.000.000.00 z ( y =0) [mm]0.000.000.000.00 z ( y =0 : 15) [mm]0.000.000.000.00 z ( y =0 : 3) [mm]0.000.000.000.00 ( y = 0 : 3) [deg]0.000.000.000.02 ( y = 0 : 15) [deg]0.000.000.000.00 ( y =0) [deg]0.000.000.000.00 ( y =0 : 15) [deg]0.000.000.000.00 ( y =0 : 3) [deg]0.000.000.00-0.01 148

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-1.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 1 ImagRe Spiral GJ = 1 GJ = 0.07 EI = 1 EI = 0.07 A -0.1 -0.095 -0.09 -0.085 -0.08 -0.075 -0.07 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 ImagRe Phugoid GJ = 1 GJ = 0.07 EI = 1 EI = 0.07 B -18 -16 -14 -12 -10 -8 -6 -4 -2 0 6 7 8 9 10 11 ImagRe Dutch Roll GJ = 1 GJ = 0.07 EI = 1 EI = 0.07 C -10 -9 -8 -7 -6 -5 -4 -3 -2 16 17 18 19 20 21 22 23 24 ImagRe Short Period GJ = 1 GJ = 0.07 EI = 1 EI = 0.07 D -60 -50 -40 -30 -20 -10 0 -1 -0.5 0 0.5 1 ImagRe Roll GJ = 1 GJ = 0.07 EI = 1 EI = 0.07 E Figure5-16.Case2:Polesoftheightmodes.A)Spiralconve rgence/divergencepole B)PhugoidpoleC)DutchrollpoleD)ShortperiodpoleE)Rollconvergencepole 149

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conguration,anincreaseof210%.Themajorityofthiseffe ctisduetobending stiffness. Thenormalizedeigenvectorcomponentsoftherollconverge ncemodearelisted inTable 5-3 .Therigid-bodyportionofthemodeshapestaysapproximate lythesame 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 T [s] EI [Nm 2 ] GJ [Nm 2 ]T [s] Figure5-17.Case2:RollconvergencetimeconstantTable5-3.Case2:Normalizedeigenvectorcomponentsofthe rollconvergence f EI GJ g [Nm 2 ] f 1.0,1.0 gf 1.0,0.07 gf 0.07,1.0 gf 0.07,0.07 g p [deg/s]-35.49-47.93-15.59-11.37 r [deg/s]-1.28-1.53-0.53-0.66 [deg]0.070.080.150.13 [deg]1.001.001.001.00 [deg]0.040.030.030.06 z ( y = 0 : 3) [mm]-0.11-0.29-0.14-0.08 z ( y = 0 : 15) [mm]-0.040.01-0.08-0.05 z ( y =0) [mm]0.000.000.000.00 z ( y =0 : 15) [mm]0.04-0.010.090.05 z ( y =0 : 3) [mm]0.110.300.150.09 ( y = 0 : 3) [deg]0.051.92-0.02-0.01 ( y = 0 : 15) [deg]0.010.340.000.00 ( y =0) [deg]0.000.000.000.00 ( y =0 : 15) [deg]-0.01-0.350.000.00 ( y =0 : 3) [deg]-0.05-1.920.020.01 150

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exceptatthelowvalueofbendingstiffness,wherethecontr ibutionofthesideslipangle doublestobecome13%oftherollangle.Thewingdeectionis slightlyasymmetricand increasesgreatlyasstiffnessdecreases.Largewingdeec tionsareseenatthemedium stiffnessvaluesof EI = GJ =0 : 2 Nm 2 andhighamountsofwingtwistareseeninthe high EI ,low GJ conguration. Thebehaviorofthedutchrollmodeshowslargechangesinres ponsetochanges inbendingstiffness.Thesechangesaredepictedinplotsof thenaturalfrequency anddampingratioinFig. 5-18 andtabulatedaspercentchangesfromthestiffest congurationinTables 5-4 and 5-5 .Thereisaninteractionbetween EI and GJ present: athightorsionalstiffness,thenaturalfrequencyincreas esfrom1.42Hzathigh EI to2.37Hzatlow EI (67%change),whereasatlowtorsionalstiffness,itincrea sesto 2.88Hz(103%change).Athigh GJ ,thedampingratiorisesfrom0.22athigh EI to 0.90atlow EI ,achangeof309%,whereasatlow GJ ,itrisesto0.87. Theeigenvectorcomponentsofthedutchrollmodeareshowni nTable 5-6 .There isastrongdependenceofthemagnitudeandphaseoftherigid -bodycomponentson 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1.4 1.8 2.2 2.6 3 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.31 0.47 0.63 0.79 0.95 z EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-18.Case2:Dutchrollmodalcharacteristics.A)Na turalfrequencyB)Damping ratio 151

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thebendingstiffness.Athighstiffnessvalues,themodesh apeischaracterizedbyyaw andsideslipangleswhichare57%and67%oftherollangle.At lowvaluesofbending stiffness,themodeshapehasnearlythesamecharacteristi csastherollconvergence mode:thecontributionfromyawandsideslipanglesare3%an d19%oftherollangle. Thesecomponentsareslightlyoutofphase,however,withya wleadingrollby33 andsidesliplaggingrollby10 .Thewingshapeshowsanearly-symmetricwingtip deectionwithverylittletwist. Animationsshowingthesedifferencesinthedutchrollmode shapeareshownin Fig. 5-19 Aprobablecausefortheeffectsofbendingstiffnessonthel ateral-directionalmodes isduetotheincreaseofdihedralatlowvaluesofstiffness. Dihedralhasawellknown effectonthelateral-directionalstabilityderivatives, whichwillaffectthelateral-directional modesofight[ 161 ]. 5.3.2.2Longitudinaldynamics Thephugoidmodeshowsaslightresponsetochangesinbendin gstiffnessand noappreciableresponsetochangesintorsionalstiffness. Thenaturalfrequencyand Acompatiblepdfviewerisrequired. Table5-4.Case2:Naturalfrequenciesoftheoscillatoryi ghtmodes EI [Nm 2 ] GJ [Nm 2 ]DutchRollPhugoidShortPeriod 1.01.0 n [ Hz ] 1.420.143.14 1.00.07% 0.00.02.2 0.071.0% 66.9-28.62.9 0.070.07% 102.8-28.63.5 Table5-5.Case2:Dampingratiosoftheoscillatoryightmo des EI [Nm 2 ] GJ [Nm 2 ]DutchRollPhugoidShortPeriod 1.01.0 0.220.090.38 1.00.07% 0.00.07.9 0.071.0% 309.0955.56-53.63 0.070.07% 295.4555.56-55.26 152

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Table5-6.Case2:Normalizedeigenvectorcomponentsofthe dutchrollmode f EI GJ g [Nm 2 ] f 1.0,1.0 gf 1.0,0.07 gf 0.07,1.0 gf 0.07,0.07 g magphasemagphasemagphasemagphase p [deg/s]8.93101.818.95101.8614.87153.8218.06150.68 r [deg/s]5.10-167.425.10-167.530.49-171.050.51-176.28 [deg]0.66-84.250.66-84.580.17-13.500.19-10.15 [deg]1.000.001.000.001.000.001.000.00 [deg]0.5789.910.5789.880.0335.010.0332.93 z ( y = 0 : 3) [mm]0.0145.080.0038.160.12130.290.20123.51 z ( y = 0 : 15) [mm]0.0043.730.0044.720.08127.870.12122.06 z ( y =0) [mm]0.000.000.000.000.000.000.000.00 z ( y =0 : 15) [mm]0.00-121.400.00-120.890.08-50.930.13-56.92 z ( y =0 : 3) [mm]0.01-121.620.01-123.050.13-48.790.20-55.61 ( y = 0 : 3) [deg]0.0076.140.0194.840.01135.750.03-153.75 ( y = 0 : 15) [deg]0.00107.270.00113.640.00135.100.01166.35 ( y =0) [deg]0.00-74.120.00-118.860.00-45.320.00-157.79 ( y =0 : 15) [deg]0.00-16.220.0016.000.00-44.410.01-14.77 ( y =0 : 3) [deg]0.00-104.500.00-17.280.02-43.770.0323.47 A B Figure5-19.Animationofthedutchrollmodewithvarying EI and GJ =1.0.A) EI =1.0 Nm 2 B) EI =0.07Nm 2 153

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dampingratioareshowninFig. 5-20 .Thenaturalfrequencyvariesfrom0.14Hzinvery stiff EI congurationsto0.10Hzinveryexible EI congurations,achangeof-29%. Thedampingratioincreases56%from0.09inverystiffcong urationsto0.14invery exiblecongurations. Normalizedeigenvectorcomponentsofthephugoidmodeares howninTable 5-7 Asthebendingstiffnessisreduced,theinuenceofangleof attackrisesto32%ofthe pitchanglewhiletheinuenceofpitchratedecreasesfrom8 9%to64%ofthepitch angle.Theeffectoftorsionalstiffnessisminor,onlychan gingthewingtwistslightlyin exible EI congurations.Thewingdeformationsremainsymmetric. Thetrendsinnaturalfrequencyanddampingoftheshortperi odmodeareopposite thoseofthephugoidmode,asshowninFig. 5-21 .Themaineffectonbothnatural frequencyanddampingratiocomesfromthebendingstiffnes s.Theeffectof EI on naturalfrequencyissmall,causingittorisefrom3.14Hzat high EI to3.23Hzatlow EI,althoughapeakvalueof3.42Hzisreachedat EI =0 : 18 Nm 2 ,whichcorrespondsto adampingof0.26.Thedampingratiodecreasesfrom0.38athi gh EI to0.18atlow EI achangeof-53%. Thenormalizedeigenvectorcomponentsoftheshortperiodm ode,shown inTable 5-8 ,revealrelativelyminorchangesin and butalargeinuenceofthe structuraldeformations.Loweringtorsionalstiffnessat highvaluesofbendingstiffness resultsinincreasedtwistandwingdeection.Atlowvalues ofbendingstiffness, loweringtorsionalstiffnessresultsinthesametrendbutw ithamuchsmallermagnitude. Loweringbendingstiffnessathightorsionalstiffnessres ultsinverylargeincreasesin wingdeection.Thewingdeformationsremainsymmetric. Animationsoftheshortperioddynamicswithhighandlowlev elsofbending stiffnessarepresentedinFig. 5-22 Ingeneral,theeffectofbendingstiffnessonthelongitudi nalmodesissmallerthan theeffectonthelateral-directionalmodes.Aprobablerea sonisthattheprimaryeffect 154

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.04 0.08 0.12 0.16 0.2 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.04 0.08 0.12 0.16 0.2 z EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-20.Case2:Phugoidmodalcharacteristics.A)Natu ralfrequencyB)Damping ratio Table5-7.Case2:Normalizedeigenvectorcomponentsofthe phugoidmode f EI GJ g [Nm 2 ] f 1.0,1.0 gf 1.0,0.07 gf 0.07,1.0 gf 0.07,0.07 g magphasemagphasemagphasemagphase u [m/s]0.1993.440.1993.600.2585.630.2585.50 q [deg/s]0.8995.300.8995.150.6498.120.6497.87 [deg]0.0689.790.0689.810.3284.910.3384.79 [deg]1.000.001.000.001.000.001.000.00 z ( y = 0 : 3) [mm]0.32-87.760.33-87.992.57-95.462.39-95.49 z ( y = 0 : 15) [mm]0.12-87.740.12-87.741.71-95.421.59-95.46 z ( y =0) [mm]0.000.000.000.000.000.000.000.00 z ( y =0 : 15) [mm]0.12-87.740.12-87.741.71-95.421.59-95.46 z ( y =0 : 3) [mm]0.32-87.760.33-87.992.57-95.462.39-95.49 ( y = 0 : 3) [deg]0.01-85.430.01-23.680.30-95.520.37-95.71 ( y = 0 : 15) [deg]0.00-80.760.00-32.990.08-95.530.09-95.81 ( y =0) [deg]0.00-87.770.0089.620.00-94.870.0082.62 ( y =0 : 15) [deg]0.00-80.770.00-33.220.08-95.530.09-95.79 ( y =0 : 3) [deg]0.01-85.440.01-23.970.29-95.520.37-95.69 155

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 2.5 2.8 3.1 3.4 3.7 4 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 z EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-21.Case2:Shortperiodmodalcharacteristics.A) NaturalfrequencyB) Dampingratio Table5-8.Case2:Normalizedeigenvectorcomponentsofthe shortperiodmode f EI GJ g [Nm 2 ] f 1.0,1.0 gf 1.0,0.07 gf 0.07,1.0 gf 0.07,0.07 g magphasemagphasemagphasemagphase u [m/s]0.03175.640.03173.540.06-175.530.07-174.63 q [deg/s]18.02109.5517.71109.6020.20107.7620.42107.72 [deg]1.000.001.000.001.000.001.000.00 [deg]0.91-2.750.88-4.731.007.511.007.71 z ( y = 0 : 3) [mm]1.47-169.331.90-175.824.85167.434.91173.26 z ( y = 0 : 15) [mm]0.52-169.060.58-172.063.16168.413.19173.75 z ( y =0) [mm]0.000.000.000.000.000.000.000.00 z ( y =0 : 15) [mm]0.52-169.060.58-172.063.17168.433.19173.77 z ( y =0 : 3) [mm]1.47-169.331.90-175.834.86167.454.92173.28 ( y = 0 : 3) [deg]0.05-36.491.54-18.250.56164.500.61148.96 ( y = 0 : 15) [deg]0.01-16.200.29-18.300.15164.420.16148.66 ( y =0) [deg]0.00-169.070.00-5.600.00-179.100.00-19.15 ( y =0 : 15) [deg]0.01-16.240.29-18.330.15164.450.16148.70 ( y =0 : 3) [deg]0.05-36.571.54-18.290.56164.530.61149.03 156

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A B Figure5-22.Animationoftheshortperiodmodewithvarying EI and GJ =1.0.A) EI =1.0Nm 2 B) EI =0.07Nm 2 ofbendingstiffnessisonthelateral-directionalstabili tyderivativesthroughthechanges ineffectivedihedralangle.Torsionalstiffnessismoreli kelytoaffectthewingtwist,which couldhaveaneffectonthelongitudinalstabilityderivati ves.However,substantialeffects oftorsionalstiffnessarenotobservedhereduetotheclose proximityoftheelasticaxis andcenterofpressure.5.3.3StructuralDynamics Thenaturalfrequencyoftherstbendingmode,showninFig. 5-23 ,showsastrong dependenceonthebendingstiffness,decreasingfrom22.9H zinthehigh EI and GJ congurationto10.2Hzinthelow EI ,high GJ conguration.Thedampingratioshows anequallystrongdependenceonbendingandtorsionalstiff ness.Whenthewingiskept stiffinbendingbutmadeexibleintorsion,thedampingrat iorises200%to0.30.When thewingisheldstiffintorsionbutmadeexibleinbending, thedampingratiorisesbya similaramountof207%.Animationsoftherstbendingmodea reshowninFig. 5-24 Thenaturalfrequencyofthersttorsionmode,showninFig. 5-25 ,showsastrong dependenceonbendingstiffnessandtorsionalstiffness,a swellasastronginteraction 157

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betweenthetwo.Athighvaluesofbendingstiffness,decrea singtorsionalstiffness causesa69%decreaseinthenaturalfrequency,butatlowval uesofbendingstiffness, thesamereductionintorsionalstiffnessonlycausesa50%d ecrease.Likewise,at hightorsionalstiffness,decreasingbendingstiffnessre sultsina73.9%dropinnatural frequency,butcausesa56.7%dropwhileatlowvaluesof GJ .Sobeforepredictingthe effectofchangesinbendingstiffness,theleveloftorsion alstiffnessmustbeknown,and vice-versa. Bothbendingandtorsionalstiffnessaffectthedampingrat ioofthersttorsion mode.Athigh EI ,lowering GJ increasesthedampingratiobyupto28.6%.Butatlow EI ,lowering GJ resultsina22.2%dropindamping.Athigh GJ ,lowering EI resultsin a37.0%dropindamping,whereasatlow GJ ,lowering EI resultsina61.1%dropin damping.Theoverallresultsforthedampingratiodisplaya complicatedbehaviorand shouldbeinterpretedcautiously. Ofparticularimportanceisthefrequencyseparationbetwe enthelowestfrequency structuralmode(rstsymmetricbending)andthehighestfr equencyightmode(short 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 z EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-23.Case2:Firstsymmetricbendingmodalcharacte ristics.A)Natural frequencyB)Dampingratio 158

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A B Figure5-24.Animationoftherstbendingmodewithvarying EI and GJ =1.0.A) EI =1.0Nm 2 B) EI =0.07Nm 2 period).Thatseparationdecreasesfrom19.8Hzinthehigh EI and GJ caseto8.0Hz inthelow EI and GJ case.Theminimumfrequencyseparationof6.7Hzoccursat 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 25 50 75 100 125 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.06 0.12 0.18 0.24 0.3 z EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-25.Case2:Firstsymmetrictorsionmodalcharacte ristics.A)Naturalfrequency B)Dampingratio 159

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EI =0 : 1 and GJ =1 : 0 .Thereducedfrequencyseparationcouldhaveaneffectonbo th thelateral-directionalandlongitudinalmodes. Theobservedcross-couplingbetweenbendingandtorsional stiffnessandthe bendingandtorsionalmodescouldcomefromaerodynamiccou plingorfrequency separationeffectsasthebendingandtorsionmodesbecomec loserinfrequency. Anotherfeaturethatisobservedintherstbendingmodebeh aviorisaslight increaseinthenaturalfrequencyatverylowvaluesofbendi ngstiffness.Thisincrease wasdeducedtobetheresultofaninteractionwithunsteadya erodynamicsby examiningtheeffectofchangingthefreestreamvelocity.A sshowninFig. 5-26 ,this effectdiminisheswithdecreasingfreestreamvelocity. 9 9.5 10 10.5 11 11.5 12 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 w n [Hz]EI [Nm 2 ] V =19 m/s V =15 m/s V =11 m/s Figure5-26.Effectofairspeedontherstbendingnaturalf requencyatlow EI Table5-9.Case2:Naturalfrequenciesofthestructuralmod es EI [Nm 2 ] GJ [Nm 2 ]FirstBendingFirstTorsion 1.01.0 n [ Hz ] 22.93113.67 1.00.07% 17.6-69.3 0.071.0% -55.4-73.9 0.070.07% -50.7-86.5 160

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Table5-10.Case2:Dampingratiosofthestructuralmodes EI [Nm 2 ] GJ [Nm 2 ]FirstBendingFirstTorsion 1.01.0 0.150.06 1.00.07% 200.0220.8 0.071.0% 206.6760.4 0.070.07% 206.6724.8 5.3.4Summary ThebendingandtorsionalstiffnessoftheGenMAVwingwerev ariedfrom1.0 to0.07Nm 2 whilethetrimconditions,ightdynamics,andstructurald ynamicsofthe aircraftwereanalyzed.Ingeneral,thebendingstiffnessi sfoundtohaveasignicant aeroelasticeffectwhiletheeffectofthetorsionalstiffn essisminimal. Loweringbendingstiffnessincreasesthetipdeectionoft hewing,causingaloss ofliftandresultinginanincreasedangleofattackandincr easedelevatordeectionfor trim.Changesintorsionalstiffnesshavelittleeffectbec auseofthecloseproximityofthe wingelasticaxisandcenterofpressure. Thelateral-directionalightdynamicsaresignicantlya ffectedbychangesin stiffness.Theprimaryeffectcomesthroughchangesinbend ingstiffnesswhichaffect thewingdihedralangle,causingthedutchrollnaturalfreq uencyanddampingto increase.Thedutchrollmodeshapebecomesverysimilartoa noscillatoryrollmodein veryexiblecongurations.Asbendingstiffnessdecrease s,thespiraldivergencemode becomesaconvergencemodewhilethetimeconstantoftherol lconvergencemode increasessubstantially. Decreasingtorsionalstiffnesshasaslighteffectonthero llconvergencebehavior, causingafasterresponseathighlevelsofbendingstiffnes sandaslowerresponseat lowlevelsofbendingstiffness.Torsionalstiffnessalsoh asaslighteffectonthedutch rollnaturalfrequency. Thelongitudinalightdynamicsareprimarilyaffectedbyc hangesinbending stiffnesswithtorsionalstiffnesshavinglittleeffect.L oweringbendingstiffnesscauses 161

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thephugoidnaturalfrequencytodecreaseandthedampingto increase.Theshort periodnaturalfrequencyisrelativelyunchangedbutdoesd isplayinterestingbehavior duetoalocalmaximumat EI =0.18Nm 2 .Dampingoftheshortperiodmodedecreases asbendingstiffnessdecreases. Theprimaryeffectofloweringstiffnessistolowerthefreq uencyofthestructural modes.Asthesefrequenciesdecrease,thefrequencysepara tionbetweentheightand structuralmodesalsodecreases,contributingtotheeffec tsontheightdynamics. Theeffectsobservedarenonlinearinnature.Atlowvalueso fstiffness,small changesinstiffnessproducesmuchgreatereffectsthanthe samechangesinstiffness producesathighvaluesofstiffness.Itisalsocriticalton otethepresenceofinteraction effectsbetween EI and GJ inthelateral-directionalandstructuralmodes. 5.4Case3:AirspeedEffectsontheRigidAircraft Theeffectofairspeedintherangeof15-35m/sisrststudie dwithanall-rigid congurationtoserveasabaselinebeforestudyingelastic congurations. 5.4.1TrimConditions Thetrimresults,illustratedinFig. 5-27 ,showthattheangleofattackandelevator deectiondecreasesignicantlyasairspeedincreases.Th isdecreaseresultisexpected andduetotheincreasedliftathigherairspeeds.Infact,th eangleofattackrequiredfor trimat35m/sisslightlynegativeandtheelevatordeectio nlikewisechangessignto positivedeection.5.4.2FlightDynamics ThemovementofthepolesontheimaginaryplaneisshowninFi g. 5-28 .Dutchroll, shortperiod,androllalldisplaylargemovement.Thespira lconvergence/divergence polemovesfromtheright-halfplaneintotheleft-halfplan e,asshowninFig. 5-29A endingwithatimeconstantof11.5s.Therollconvergencemo demovesfurthertothe left,asshowninFig. 5-29E ,withitstimeconstantdecreasingfrom0.029sto0.014s. 162

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-1 0 1 2 3 4 5 6 7 8 15 20 25 30 35 a [deg]V [m/s] A -5 -4 -3 -2 -1 0 1 15 20 25 30 35 d e [deg]V [m/s] B Figure5-27.Case3:Trimresults.A)TrimangleofattackB)T rimelevatordeection -80 -70 -60 -50 -40 -30 -20 -10 0 10 0 5 10 15 20 25 30 35 ImagRe V Spiral Phugoid Dutch Roll Short Period Roll Figure5-28.Case3:Allpolescorrespondingtotheightmod es Thefrequencyofthephugoidmode,showninFig. 5-30 ,morethanhalvesfrom0.14Hz to0.06HzwhileitsdampingratioinFig. 5-31 increasesfrom0.09to0.22. Thedutchrollandshortperiodpolesarethemostaffectedby thechangesin airspeed,asseeninFigs. 5-30 and 5-31 .Thedutchrollfrequencyincreasesfrom 1.4Hzto3.3Hzwhileitsdampingratiodecreasesfrom0.20to 0.14.Theshortperiod 163

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-0.15 -0.1 -0.05 0 0.05 0.1 0.15 -1 -0.5 0 0.5 1 ImagRe V = 15 V = 35 A -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ImagRe V = 15 V = 35 B -3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 5 10 15 20 25 ImagRe V = 15 V = 35 C -20 -15 -10 -5 0 10 15 20 25 30 35 40 45 50 ImagRe V = 15 V = 35 D -80 -70 -60 -50 -40 -30 -20 -1 -0.5 0 0.5 1 ImagRe V = 15 V = 35 E Figure5-29.Case3:Polesoftheightmodes.A)Spiralconve rgence/divergencepole B)PhugoidpoleC)DutchrollpoleD)ShortperiodpoleE)Rollconvergencepole 164

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naturalfrequencyincreasesfrom3.0Hzto5.9Hzwhileitsda mpingratioincreasesfrom 0.40to0.49.Bothdutchrollandshortperiodbecomefaster, butdutchrollbecomesless dampedwhiletheshortperiodbecomesmoredamped. 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 15 20 25 30 35 w n [Hz]V [m/s] Dutch Roll Short Period A 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 15 20 25 30 35 w n [Hz]V [m/s] Phugoid B Figure5-30.Case3:Naturalfrequencies.A)Dutchrollands hortperiodmodesB) Phugoidmode 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 15 20 25 30 35 zV [m/s] Phugoid Dutch Roll Short Period Figure5-31.Case3:Dampingratios 165

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5.4.3Summary Increasingtheairspeedoftheaircraftfrom15m/sto35m/sh aspredictableresults. Theangleofattackandelevatordeectionrequiredfortrim reducesignicantlyas airspeedincreases. Thespiraldivergencebecomesaspiralconvergencewhileth erollconvergence becomes52%faster.Thedutchrollmodenaturalfrequencyin creases136%whileits dampingratiodecreases30%. Thephugoidmode'snaturalfrequencydecreasesby57%andit sdampingratio increasesby144%.Theshortperiodmode'snaturalfrequenc yincreasesby97%while itsdampingratiodecreases23%. 5.5Case4:AirspeedEffectsonFlexibleCongurations Theeffectofairspeedonelasticcongurationsisnowstudi edusingtherangesof parametersgiveninTable 5-11 Table5-11.Case4:Rangeoffactors FactorLowHigh Airspeed[m/s]1519 EI [Nm 2 ]0.071.0 GJ [Nm 2 ]0.071.0 5.5.1TrimConditions Thegeneraltrendofchangesinangleofattackandelevatord eectionremain similaratvaryingairspeed.Thesetrendsareillustratedi nFig. 5-32 ,whereeach surfacerepresentsthebehavioroftheexibleconguratio nsataparticularairspeed. Accordingly,eachsurfaceinFig. 5-32 hasasimilarshapebecausetheresponsewith respectto EI and GJ issimilarateachairspeed.Theprimaryeffectofairspeedi sto shiftthewholeresponsesurfacevertically(requiringles sangleofattackandcontrol deectionwithincreasingairspeed,asseenpreviously). 166

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 4 6 8 10 12 14 a [deg] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]a [deg] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -12 -9 -6 -3 d e [deg] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] B Figure5-32.Case4:Trimresults.A)TrimangleofattackB)T rimelevatordeection 5.5.2FlightDynamics Airspeedhasaminoreffectonthetipdeectionandtwistcom paredtothebending andtorsionalstiffness.ThesurfacesinFig. 5-33 areallveryclosetogether(sometimes 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 34 68 102 136 170 D z tip [mm] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]D z tip [mm] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -16 -12 -8 -4 0 t [deg] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]t [deg] B Figure5-33.Case4:Wingdeformationsattrim.A)Wingdeec tionat y = b B)Wing twistat y =0 : 75 b 167

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cuttingthroughoneanother).Theexceptionisthenoticeab lechangeinwingtwistatlow valuesoftorsionalstiffness.Forexample,atlowairspeed ,decreasingtorsionalstiffness toaverylowlevelresultsinaslightincreaseinwingtwist. However,athighairspeed, verylowvaluesoftorsionalstiffnessdecreasethewingtwi st. Thedutchrollmodenaturalfrequencyanddampingratioares howninFig. 5-34 Thereisastrongrelationshipbetweenthefrequencyandthe airspeedathighvalues ofbendingstiffness(increasedairspeedcausesthenatura lfrequencytoincrease).As foundinpreviouscases,torsionalstiffnessdoesnothavea signicanteffectondutch roll.Similarresultsareseenforthedampingratio,althou ghtheeffectofairspeedonthe dampingathighvaluesofstiffnessissmaller. Theresponseofthephugoidmodenaturalfrequencyanddampi ngratioare showninFig. 5-35 andarenearlyidenticaltopreviousresultsinFig. 5-20 .Ingeneral, increasedairspeedreducesthenaturalfrequency,althoug hthiseffectdiminishesatlow GJ .Thereislittleeffectoftheairspeedonthedamping. Theshortperiodmodenaturalfrequencyanddampingratioar eshowninFig. 5-36 anddisplayasimilartrend.Forthenaturalfrequency,thee ffectofairspeedisseenas 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.45 1.74 2.03 2.32 2.61 w n [Hz] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.16 0.32 0.48 0.64 0.8 z 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-34.Case4:Dutchrollnaturalfrequencyanddampin gratio 168

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 w n [Hz] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.04 0.08 0.12 0.16 0.2 z 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-35.Case4:Phugoidnaturalfrequencyanddampingr atio 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3.2 3.4 3.6 3.8 4 w n [Hz] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.16 0.24 0.32 0.4 0.48 z 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-36.Case4:Shortperiodnaturalfrequencyanddamp ingratio ashiftintheresponsesurfacestowardhighernaturalfrequ encies.Increasingairspeed alsoincreasestheresponsesurfaceofthedampingratio. 169

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 12 16 20 24 28 w n [Hz] 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.14 0.28 0.42 0.56 0.7 z 15 17 19 V [m/s] EI [Nm 2 ] GJ [Nm 2 ]z B Figure5-37.Case4:Firstbendingmode5.5.3StructuralDynamics Figure 5-37 showstherstbendingmodenaturalfrequency,whichshowsa small increaseinresponsetoincreasingairspeed.Thedampingra tiogenerallyincreases slightlywithincreasedairspeedandthateffectbecomesmo reprominentatlow GJ 5.5.4Summary Theeffectofairspeedontheresponseofanelasticaircraft closelyfollowsthe trendsseenfortherigidaircraft,withsomevariations. Thetrimangleofattackandelevatordeectionvaryinrespo nsetotheexibilityof theaircraft,asseeninCase2.Theeffectofairspeedistode creasetheangleofattack andelevatordeectionrequiredfortrimforallvehiclesin thedesignspace.Thebending stiffnessremainstheprimarycontributortothewingtipde ection;theairspeedhasa negligibleeffectonthewingtipdeection.However,theai rspeeddoeseffectthewing tiptwistatlowvaluesof GJ ,butthiseffectissmallcomparedtotheeffectofbending stiffnessonthewingtwistthroughtheaerodynamiccouplin g. RecallfromCase2thatoneofthemajorcausesoftheaeroelas ticeffectsonthe ightdynamicswerefromthewingdihedralanglechangingas thebendingstiffness 170

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changes.Becausetheairspeeddoesnothaveaconsiderablee ffectontheeffective wingdihedralangle,thetrendsoftheairspeedeffectsonth eaeroelasticvehiclesfollows closelytothetrendsseenontherigidvehicle.Exceptionse xistatlowvaluesofbending stiffness.Forexample,airspeedhasverylittleeffectont herollconvergencetime constantatlowvaluesof EI .Additionally,theshortperiodmodenaturalfrequencyand dampingundergoanadditionalincreaseatlowvaluesof GJ inresponsetoincreasing airspeed,whereasinCase2 GJ didnothaveasignicanteffectontheshortperiod mode.Thiseffectisduetotheincreasedwingtwistatlowval uesoftorsionalstiffness andhighairspeeds. 5.6Case5:EffectsoftheElasticAxis Incases1-4,theelasticaxiswaslocatedaccordingtotheex perimentalresults giveninSection 3.2.3.2 ,whichisapproximatelyatthe x =0 : 28 c linealongthewing. Resultsshowedverylittlechangeintheightdynamicsinre sponsetochangesin torsionalstiffnesswiththatelasticaxis.Theprobablere asonisthecloselocationof centerofpressurerelativetotheelasticaxis,resultingi nasmallmomentarmbetween theliftforceandtheelasticaxisandverylittleresulting wingtwistdespiteloweringthe torsionalstiffness. Inthissection,thelocationoftheelasticaxisisvariedwh ilethedesignspaceof case2( 0 : 07 EI;GJ 1 : 0 )isre-evaluatedwiththedifferentelasticaxislocations .Two additionallocationsoftheelasticaxisarechosen:onealo ngtheleadingedgeofthe wingandonealongthe x =0 : 4 c line.TheseresultsarecomparedinFigs. 5-38 to 5-51 5.6.1TrimConditions Startingwiththewingdeformation,thewingtipdeections howslittledependence onthetorsionalstiffnessdespitechangesintheelasticax is,asshowninFig. 5-38 Thetwistofthewingatthe3/4spanlocationisshowninFig. 5-39 andrevealsastrong dependenceonthelocationoftheelasticaxis.InFig. 5-39A ,theleadingedgeposition oftheelasticaxisresultsinalargeamountofnegativewing twist(leadingedgedown) 171

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developingastorsionalstiffnessdecreases.Thewingtwis tdecreasesfrom-1.2 to -14.9 as GJ decreasesfrom1.0Nm 2 to0.07Nm 2 at EI = 1.0Nm 2 .Thisdecreaseisa resultofthecenterofpressurelyingwellbehindtheelasti caxislocation,resultingina leading-edge-downmomentonthewing. InFig. 5-39C ,theaftpositionoftheelasticaxisresultsinincreasingw ingtwist from0.5 to8.9 astorsionalstiffnessdecreaseswhilebendingstiffnessi sheldatthe highvalue.Overall,thereislittleinteractionbetween EI and GJ ,thatis,theeffectof decreasingtorsionalstiffnesswhileatahighvalueofbend ingstiffnessisverysimilarto theeffectatalowvalueofbendingstiffness. Theresultingwingtwistfromtheforwardoraftpositionoft heelasticaxishasa strongeffectonthetrimconditions.InFig. 5-40A ,thenegativewingtwistatlowvalues oftorsionalstiffnesswiththeforwardelasticaxisresult sinhighervaluesoftrimangleof attack,upto17.0 .InFig. 5-40C ,theaftlocationoftheelasticaxisresultsindecreasing valuesoftrimangleofattackwithdecreasingtorsionalsti ffness.Thisdecreaseisa resultoftheincreasingwingtwistinthisareaofthedesign space,whichresultsin increasedliftatloweranglesofattack. Therearecorrespondingchangesintheelevatordeectionr equiredattrim,as showninFig. 5-41 .Withtheforwardpositionoftheelasticaxis,asthetriman gle ofattackincreasesatlowvaluesoftorsionalstiffnessthe elevatordeectionalso decreases.Contrastingbehaviorisseenwiththeaftpositi onoftheelasticaxis:as thetrimangleofattackdecreasesintheregionofincreased wingtwisttheelevator deectionrequiredfortrimalsodecreases.5.6.2FlightDynamics Thebehaviorofthedutchrollmodechangesverylittledespi tethechangesin theelasticaxis,asshowninFigs. 5-42 and 5-43 .Therearesmallchangesinthe frequencyanddampingofcongurationswhichareveryexib leinbothbendingand torsion,butthemajorityofthedutchrollbehaviorisgover nedbythechangesinbending 172

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stiffness,nottorsionalstiffness.Thisresultagreeswit hthepreviousconclusionsthatthe aeroelasticeffectsonthedutchrollmodearedrivenbychan gesinthedihedralangleof thewing.Sincethewingtwistdoesnotdirectlyaffectthewi ngdihedral,theeffectsof GJ areminordespitethechangingelasticaxislocation. Theresultsforthephugoidmodeshowastrongerrelationshi pbetweenthetorsional stiffnessandthenaturalfrequencyanddampingratiowhent heelasticaxisisnot closetothecenterofpressure.First,fortheleadingedgep ositionoftheelasticaxis, decreasingthetorsionalstiffnessresultsinadecreasein thenaturalfrequencyof21% fromitsinitialvalueof0.136Hzat EI =1Nm 2 ,asshowninFig. 5-44A .Thiseffectisa similartotheeffectofdecreasingbendingstiffness,whic hresultsina24%decreasein thephugoidnaturalfrequency(at GJ =1Nm 2 ).Theeffectonthedampingratio,shownin Fig. 5-45A issmallbutactstoincreasethedampingby 11%asthetorsionalstiffness decreases. Theaftpositionoftheelasticaxisresultsinmoreextremeb ehaviorofthephugoid naturalfrequencyanddamping,asshowninFigs. 5-44C and 5-45C .Athighbending stiffness,decreasingtorsionalstiffnessresultsina133 %riseinthenaturalfrequency anda94%dropinthedampingratiotothelowvalueof0.006at EI =1.0Nm 2 and GJ =0.07Nm 2 .Thischangeindampingistrendingtowardbecomingnegativ e,which wouldresultinanunstablephugoidmode. Thedifferencesintheresponseofthesetwoelasticaxisloc ationsshowthe stabilizingordestabilizingcontributionoftorsionalst iffnessandwingtwistasitrelates totheelasticaxis.Withtheforwardelasticaxis,decrease dtorsionalstiffnessresults innegativewingtwist,whichactstoreducethelocalangleo fattackonthewingand decreasetheloadonthewing.Withtheaftelasticaxis,redu cingtorsionalstiffness resultsinpositivewingtwist,whichincreasesthelocalan gleofattack,increasingthe loadonthewing,whichfurtherincreasestheangleofattack untilanequilibriumwith thestructuralstiffnessisreached.Thisinteractionisde stabilizingandcausesthe 173

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naturalfrequencytoincreasesharplyandthedampingratio todecreaseandapproacha negativevalue. Similarbehaviorisseenintheshortperiodmode,asshownin Figs. 5-46 and 5-47 First,withtheforwardelasticaxis,decreasingtorsional stiffnessresultsinasmall 0.19Hzriseinnaturalfrequencyascomparedtothe0.25Hzri seinresponseto loweringbendingstiffness.Thedampingratiodisplaysala rgerdecrease,droppingfrom 0.36to0.26as GJ decreasesfrom1.0Nm 2 to0.07Nm 2 at EI =1.0Nm 2 .Thereisa signicantinteractionbetween EI and GJ onthedampingratiowhichcausestorsional stiffnesstohaveanegligibleeffectonthedampingratioat verylowvaluesofbending stiffness. Theaftpositionoftheelasticaxisresultsinextremebehav iorintheshortperiod naturalfrequencyanddampingratio,asshowninFigs. 5-46C and 5-47C .Thenatural frequencyrisesfrom3.1Hzto5.5Hzinanonlinearmanneras GJ decreasesat thehigh EI .Atthelowlevelof EI ,thenaturalfrequencydecreasesfrom3.2Hzto 3.1Hzas GJ decreases.Thechangeintheeffectof GJ fromthehightolowlevels of EI showsthesignicanceoftheinteractionbetween GJ and EI inthiscase.This interactionisalsoseeninthedampingratio,whichincreas esfrom0.39to0.58at EI =1.0Nm 2 butdoesnotsignicantlychangeat EI =0.07Nm 2 Theaftlocationoftheelasticaxishadasimilarnonlineare ffectontheresponseof theshortperiodnaturalfrequencyanddampingtothebendin gandtorsionalstiffness, exceptthatthetrendwasanincreaseindampinginsteadofad ecreaseindampingasin thephugoidmode.5.6.3StructuralDynamics Theeffectoftorsionalstiffnessontherstbendingmodeal sochangesdepending onthelocationoftheelasticaxis,asshowninFigs. 5-48 and 5-49 .Thesechanges aremostlyathighvaluesof EI andthusathighfrequencies.Atlowvaluesof EI thebendingstiffnessdominatesthebehaviorandtorsional stiffnessdoesnothavea 174

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strongeffect.Onenotableresultisthesimilarityoftheef fectoftorsionalstiffnessonthe dampingratiodespitethepositionoftheelasticaxis. Thersttorsionmode,whosenaturalfrequenciesanddampin gratiosareshown inFigs. 5-50 and 5-51 ,showssignicantchangesinresponsetochangesintorsion al stiffnessandtheelasticaxislocation.Foreverylocation oftheelasticaxis,lowering torsionalstiffnessdecreasesthenaturalfrequencyofthe rstsymmetrictorsionmode whiledampingtendstorise.However,theresultsforthedam pingratioarecomplexand shouldbeinterpretedwithcaution.Forexample,withthemi doraftelasticaxislocation, inveryexible EI and GJ congurations,dampingbecomesverysmallandtrends towardbecomingnegative.Anegativedampingcanindicatei nstabilitiessuchaswing utterordivergence. 175

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 42 84 126 168 210 D z tip [mm] EI [Nm 2 ] GJ [Nm 2 ]D z tip [mm] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 42 84 126 168 210 z [mm] EI [Nm 2 ] GJ [Nm 2 ]z [mm] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 42 84 126 168 210 D z tip [mm] EI [Nm 2 ] GJ [Nm 2 ]D z tip [mm] C Figure5-38.Wingtipdeection.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -20 -14 -8 -2 4 10 t [deg] EI [Nm 2 ] GJ [Nm 2 ]t [deg] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -20 -14 -8 -2 4 10 t [deg] EI [Nm 2 ] GJ [Nm 2 ]t [deg] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -20 -14 -8 -2 4 10 t [deg] EI [Nm 2 ] GJ [Nm 2 ]t [deg] C Figure5-39.Wingtwistat y =0 : 75 b .A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 176 176 176

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 a [deg] EI [Nm 2 ] GJ [Nm 2 ]a [deg] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 a [deg] EI [Nm 2 ] GJ [Nm 2 ]a [deg] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 25 a [deg] EI [Nm 2 ] GJ [Nm 2 ]a [deg] C Figure5-40.Aircrafttrim .A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -25 -20 -15 -10 -5 0 d e [deg] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -25 -20 -15 -10 -5 0 d e [deg] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 -25 -20 -15 -10 -5 0 d e [deg] EI [Nm 2 ] GJ [Nm 2 ]d e [deg] C Figure5-41.Aircrafttrim e .A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 177 177 177

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3 3.5 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] C Figure5-42.Dutchrollnaturalfrequency.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.31 0.47 0.63 0.79 0.95 z EI [Nm 2 ] GJ [Nm 2 ]z A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.31 0.47 0.63 0.79 0.95 z EI [Nm 2 ] GJ [Nm 2 ]z B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.31 0.47 0.63 0.79 0.95 z EI [Nm 2 ] GJ [Nm 2 ]z C Figure5-43.Dutchrolldampingratio.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 178 178 178

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.08 0.16 0.24 0.32 0.4 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] C Figure5-44.Phugoidnaturalfrequency.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 z EI [Nm 2 ] GJ [Nm 2 ]z A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 z EI [Nm 2 ] GJ [Nm 2 ]z B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 z EI [Nm 2 ] GJ [Nm 2 ]z C Figure5-45.Phugoiddampingratio.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 179 179 179

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3 3.6 4.2 4.8 5.4 6 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3 3.6 4.2 4.8 5.4 6 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 3 3.6 4.2 4.8 5.4 6 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] C Figure5-46.Shortperiodnaturalfrequency.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.28 0.41 0.54 0.67 0.8 z EI [Nm 2 ] GJ [Nm 2 ]z A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.28 0.41 0.54 0.67 0.8 z EI [Nm 2 ] GJ [Nm 2 ]z B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.15 0.28 0.41 0.54 0.67 0.8 z EI [Nm 2 ] GJ [Nm 2 ]z C Figure5-47.Shortperioddampingratio.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 180 180 180

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 9 13 17 21 25 29 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 9 13 17 21 25 29 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 9 13 17 21 25 29 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] C Figure5-48.Firstsymmetricbendingnaturalfrequency.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 z EI [Nm 2 ] GJ [Nm 2 ]z A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 z EI [Nm 2 ] GJ [Nm 2 ]z B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 z EI [Nm 2 ] GJ [Nm 2 ]z C Figure5-49.Firstsymmetricbendingdampingratio.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 181 181 181

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0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 30 60 90 120 150 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 30 60 90 120 150 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 30 60 90 120 150 w n [Hz] EI [Nm 2 ] GJ [Nm 2 ]w n [Hz] C Figure5-50.Firstsymmetrictorsionnaturalfrequency.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.06 0.12 0.18 0.24 0.3 z EI [Nm 2 ] GJ [Nm 2 ]z A 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.06 0.12 0.18 0.24 0.3 z EI [Nm 2 ] GJ [Nm 2 ]z B 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.06 0.12 0.18 0.24 0.3 z EI [Nm 2 ] GJ [Nm 2 ]z C Figure5-51.Firstsymmetrictorsiondampingratio.A) x EA =0 c B) x EA 0 : 28 c C) x EA =0 : 4 c 182 182 182

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5.6.4Flutter Theelasticaxislocationcanhaveaprofoundaffectontheae roelasticbehavior ofthewingandintroducesthepotentialforbending-torsio nuttertooccurbecauseof verylowvaluesofstructuralstiffness.Thepotentialforb ending-torsionutterisnow investigated.Amedium-stiffnesscongurationiscreated with EI = GJ =1.0Nm 2 Thenaturalfrequenciesanddampingratiosoftherstbendi ngandtorsionmodesare analyzedinresponsetothechangingelasticaxislocationa ndairspeedvariationsfrom 15to30m/s. Thenaturalfrequenciesanddampingratiosversusairspeed fortheleadingedge andexperimentallocationsoftheelasticaxisareshowninF igs. 5-52A to 5-52D .Both oftheseelasticaxislocationshowstabilityinthestructu ralmodesacrosstheairspeed range.However,asshowninFigs. 5-52E and 5-52F ,withtheaftelasticaxislocationthe torsionmodebecomesunstableasanegativedampingratiode velopsaround22m/s. 5.6.5Summary Thelocationoftheelasticaxisisacriticalstructuralfea turethatmustbeconsidered whendesigningormodelingaexiblewing.Thelocationofth eelasticaxisrelativeto thecenterofpressurecanhaveastabilizingordestabilizi ngcontributiontothevehicle dynamicsbecauseofthesignofthewingtwistthatdevelops. Foraforwardelastic axisposition,decreasingtorsionalstiffnessresultsini ncreasinglynegativewingtwist, reducingthelocalangleofattack. Theoppositeeffectoccurswithanaftelasticaxisposition .Asthetorsionalstiffness decreases,thewingtwistbecomesincreasinglypositive,r esultinginhigherlocalangles ofattackandhigherloadsandleadingtoevenmorewingtwist .Thisdestabilizingeffect isseeninthephugoidandrstsymmetrictorsionmodes,whos edampingratiostendto becomeverylowatlowvaluesofstiffness. Itisnoteworthythatthedespitethewiderangeofelasticax islocations,torsional stiffnessdidnothaveaprominenteffectonthedutchrollmo de.Theprimarysourceof 183

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0 5 10 15 20 25 30 16 18 20 22 24 26 28 30 w n [Hz]V [m/s] First Bending First Torsion A -0.2 0 0.2 0.4 0.6 0.8 1 16 18 20 22 24 26 28 30 zV [m/s] First Bending First Torsion B 0 5 10 15 20 25 30 16 18 20 22 24 26 28 30 w n [Hz]V [m/s] First Bending First Torsion C -0.2 0 0.2 0.4 0.6 0.8 1 16 18 20 22 24 26 28 30 zV [m/s] First Bending First Torsion D 0 5 10 15 20 25 30 16 18 20 22 24 26 28 30 w n [Hz]V [m/s] First Bending First Torsion E -0.2 0 0.2 0.4 0.6 0.8 1 16 18 20 22 24 26 28 30 zV [m/s] First Bending First Torsion F Figure5-52.Bending-torsionutter.A) x EA =0 c ,naturalfrequenciesB) x EA =0 c dampingratiosC) x EA 0 : 28 c ,naturalfrequenciesD) x EA 0 : 28 c dampingratiosE) x EA =0 : 4 c ,naturalfrequenciesF) x EA =0 : 4 c ,damping ratios 184

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aeroelasticeffectsonthedutchrollmodeisthebendingsti ffnessofthewing,whose effectscomethroughchangesinthewingbendingandthusthe effectivedihedral angle.Thespiralandrollconvergencebehaviorswerenotan alyzedherebutarealso stronglyaffectedbythechangingwingdihedralwhichdoesn otchangeastheelastic axischanges.Thus,theirbehaviorsmightalsoberelativel yunaffectedbydecreasing torsionalstiffness. Largeinteractionsbetweenbendingandtorsionalstiffnes sareobservedinthe congurationswhichhaveasignicantdependenceontorsio nalstiffness.Because oftheseinteractions,theeffectofchangestoonestiffnes sparametermaynotbe accuratelypredictedwithoutknowingtheleveloftheother stiffnessparameter. 5.7ConcludingRemarks Thedirecteffectofloweringstiffnessistochangetheshap eofthewingattrim, whichleadstochangesinthetrimangleofattackandelevato rdeection.Ifthese aeroelasticeffectsarenotaccountedforinthedesignproc ess,theaircraftmaytrim muchdifferentlythanexpected,leadingtoaninefcienti ghtcondition. Ingeneral,aeroelasticeffectsofwingstiffnessontheig htdynamicsarehighly nonlinearanddependmostlyonthebendingstiffness,notth etorsionalstiffness.It isfoundthatthebendingstiffnessmainlyaffectsthelater al-directionalightmodes throughanincreaseintheeffectivedihedralangleduetoin creasedwingtipdeection. Thebendingstiffnesscanalsoaffectthelongitudinalmode sbecauseofthereduced frequencyseparationbetweenthelongitudinalmodesandth erstbendingmodeas bendingstiffnessdecreases. TheightmodesoftheGenMAVaresignicantlyaffectedbych angesinthe bendingstiffnessofthewing.Largechangesareseeninboth thefrequenciesand dampingratiosoftheoscillatoryightmodes,althoughthe effectsonthelateral-directional modesaregreaterthantheeffectsonthelongitudinalmodes .Thedirectionand magnitudeoftheeffectvariesgreatlybetweenmodes.Forex ample,loweringbending 185

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stiffnesscausesanincreaseindutchrolldampingbutadecr easeinshortperiod damping. Itisimportanttonotethatthenatureofamodemaychangeast hestiffness changes.Thischangewasespeciallynoticeableinthedutch rollmode,which transitionedfromatraditionalmodeshapewithcontributi onsofroll,yaw,andsideslipto averynon-traditionalmodeshapewhichisdominatedbyroll withlittleyaworsideslip. Fromahandlingqualitiesperspective,aircraftdesignist ypicallyfocusedonthe dampingoftheightmodes.However,changesinthemodeshap emayintroducea newconsiderationforthedesigner.Theaircraftmayhavead esirablefrequencyand dampinginacertainmode,buttheunusualmodeshapemaymake thepilotingtask moredifcult.Inaddition,thesecharacteristicsarecrit icaltoconsiderwhendesigning anautopilotthatmustcompensatetheightmodestomaneuve r. Thelocationoftheelasticaxisisanimportantparameterwh enconsideringthe effectsoftorsionalstiffness.IntheGenMAV,theelastica xisliesclosetothecenterof pressure,resultinginasmallmomentarmandrelativelysma llchangesinwingtwist asthetorsionalstiffnesschanges.Whentheelasticaxisis notlocatednearthecenter ofpressure,theeffectofdecreasingtorsionalstiffnessb ecomesmostprominentinthe longitudinalmodesbecauseofitseffectonthewingtwist.A stabilizingaeroelasticeffect occursiftheelasticaxisisforwardofthecenterofpressur e.However,whentheelastic axisliesbehindthecenterofpressure,adestabilizingeff ectoccursandincreasesthe possibilityofbending-torsionutter. Theeffectsofairspeedontheightandstructuraldynamics areimportantto considerinthedesignprocess.However,theeffectsofstif fnessaregenerallylarger thantheeffectsofairspeedinthecasesanalyzedhere. Thepresenceoftheseeffectshighlightstheimportanceofc onsideringelastic andrigidbodydegreesoffreedomtogetherforproperanalys isofexibleaircraft.This problemisespeciallyrelevantforMAVs,whosedesignsmayb einherentlyexible.By 186

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consideringtheseeffectsinthedesignprocess,thestruct uralstiffnessmayserveasan additionaldesignparametertoachieveadesiredightchar acteristic. 187

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CHAPTER6 AEROELASTICITYANDFLIGHTDYNAMICS:NON-UNIFORMWINGSTIF FNESS ThischapterextendstheworkofChapter 5 tonon-uniformdistributionsofstiffness acrossthewingspan.Specically,thisresearchseekstole arntheadvantagesor disadvantagesofcertaindistributionsofwingstiffness. Thereareapotentiallyinnite numberofvaryingdistributionsthatcouldbeconsidered,b utonlyasmallsubsetis analyzedhere. Actualaircraftwingsdonothaveauniformstiffnessdistri bution.Instead,itis commonforthestiffnesstodecreasetowardthewingtips.Fo rconventionalwings,one estimateisthatthestiffnessdecreasesasthechordtothef ourthpower[ 41 ]. Arst-orderdistributionofwingstiffnessisinvestigate drst.Thiswillelucidateany basicdependencyofthevehicledynamicsonanon-uniformwi ngstiffness.Inaddition todecreasingwingstiffness,increasingstiffnesseswill beevaluatedtodetermineany uniqueeffectsthatmightarisefromsuchanon-standardcon guration. Theseresultswillbeadvancedbyinvestigatingthemostin uentialareasofthe wingwhenitcomestoaeroelasticinteractionsbetweenthes tiffnessandtheight dynamics.First,theaeroelasticeffectsofchangingstiff nessatthewingrootwillbe comparedtothewingtip.Next,theamountofthewingspanove rwhichachangein stiffnessisrequiredtorealizeanaeroelasticeffectwill beinvestigated.Finally,the requiredchangeinstiffnessoveraxedwingspantoachieve anaeroelasticeffectwillbe studied. InChapter 5 ,thebendingstiffnessofthewingwasfoundtohaveamorepro minent effectontheightdynamicsthanthetorsionalstiffnessfo rtheexperimentally-determined locationoftheelasticaxis.Asaresult,onlybendingstiff nessisconsideredinthis chapter. 188

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6.1LinearlyVaryingDistribution Alineardistributionwitheitherpositiveornegativeslop eisnowinvestigated.For thesecongurations,thedistributionwascalculatedasaf unctionofthespanlocation y accordingtoEq.( 6–1 ),where EI 0 representstherootstiffnessand EI represents thepercentchangeoverthehalf-span,settoeitherzero(no slope)or 0 : 6 inthecurrent study.Asampledistributionwith EI 0 =1 ; EI = 0 : 5 isshowninFig. 6-1 EI ( y )= 8><>: EI 0 + EI ( EI 0 ) y b : y> 0 EI 0 EI ( EI 0 ) y b : y< 0 (6–1) y [in]EI [Nm 2 ] -12 -8 -4 0 4 8 12 1 0 2 Figure6-1.Samplelinearstiffnessdistributionwith EI 0 =1 ; EI =0 : 5 6.1.1Results ThetrimresultsareshowninFig. 6-2 andrevealverylittledependenceofthetrim angleofattackandelevatordeectionontheslope.Theprim aryeffectisdueto EI 0 and theslopehaslittleeffect.Thisisalsoseeninthewingtipd eectioninFig. 6-3 Figure 6-4 showsthemovementofthepolescorrespondingtotheightmo des onthecomplexplanewiththethreevaluesforslope.Theeffe ctofslopeisverysmall comparedtotheoveralleffectofbendingstiffness.Possib leexceptionstothismaybe thelocationofthepolesassociatedwiththedutchrollmode androllconvergence. Inthemostexiblecase( EI 0 =0.07and EI =-0.6),thedutchrollpolemoves awayfromtherealaxisbyaconsiderableamount.Thiseffect canbeseeninthe naturalfrequencyplottedinFig. 6-5 .Thebendingstiffnessinthiscaseatthewing tipis0.042Nm 2 .Thisrepresentsaveryexiblewingbutitisinterestingth atthewing 189

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7 8 9 10 11 12 13 14 15 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a [deg]EI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 A -14 -12 -10 -8 -6 -4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 d e [deg]EI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 B Figure6-2.Trimresultswithlinearlyvaryingstiffness.A )TrimangleofattackB)Trim elevatordeection 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 D z tip [mm]EI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 Figure6-3.Wingtipdeectionwithlinearlyvaryingstiffn ess tipdeectionforthesame EI 0 and EI doesnotincreasegreatlyoverthe EI =0 baselinedeection.Thisisbecausethewingisstillmorest iffatthewingroot.The effectseeninthedutchrollmodemaybeduetothenonlinearn atureoftheaeroelastic 190

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interaction.Evenasmalldecreaseinstiffnesscanhaveala rgeeffectattheselow valuesofstiffness. Similarbehaviorisseenintherollconvergence.Atthebase line EI 0 =0.07Nm 2 fromthe EI =0totheveryexible EI =-0.6,thetimeconstantalmostdoublesfrom 0.77to0.15secondsasthepoleapproachestheorigin.Thisi salargechangedueto thelowerwing EI causedbythedecreasingslopeofthe EI distribution.Conversely,as thewingstiffnessincreasesinthe EI =0.6case,thetimeconstantalmosthalvesfrom 0.77to0.39secondsasthepolemovesawayfromtheorigin. Theotherightmodeswererelativelyunaffectedinfrequen cyanddamping,as showninFigs. 6-5 and 6-6 Thebendingstiffnessnaturalfrequency,showninFig. 6-7 ,showsalarger dependenceontheslopeathighvaluesoftheroot EI .Atlowvaluesof EI ,theeffectof theslopebecomessmall.Theeffectofslopeonthedampingra tioisalsosmall. 6.1.2Summary Aconsiderableamountofslopewasintroducedtocreatea 60% changeinthetip stiffnessascomparedtotherootstiffness.However,thero otstiffnesswasstillseento dominatetheeffectonthetrimconditions,ightdynamics, andstructuraldynamics. Theonlyexceptionsnotedaretheeffectsofslopeonthedutc hrollandroll convergencebehaviors.Thisismostlikelyduetothenonlin earnatureoftheaeroelastic effectsattheseverylowlevelsofstiffness. Thepresenceofalinearslopemostlyaffectsthestiffnessi ntheouterspan,butthe outerspanstiffnessdoesnotaffecttheightdynamicsasmu chastherootstiffness. ThiswillbeanalyzedfurtherinSection 6.2 191

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-2 -1.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 1 ImagRe EI 0 = 1.0 D EI = 0.6 D EI = 0 D EI = -0.6 A -0.16 -0.14 -0.12 -0.1 -0.08 -0.06 0.6 0.7 0.8 0.9 1 ImagRe EI 0 = 1.0 D EI = 0.6 D EI = 0 D EI = -0.6 B -20 -15 -10 -5 0 5 4 6 8 10 12 14 16 18 ImagRe EI 0 = 1.0 D EI = 0.6 D EI = 0 D EI = -0.6 C -9 -8 -7 -6 -5 -4 -3 -2 -1 17 18 19 20 21 22 ImagRe EI 0 = 1.0 D EI = 0.6 D EI = 0 D EI = -0.6 D -40 -35 -30 -25 -20 -15 -10 -5 0 -1 -0.5 0 0.5 1 ImagRe EI 0 = 1.0 D EI = 0.6 D EI = 0 D EI = -0.6 E Figure6-4.Polesoftheightmodeswithlinearlyvaryingst iffness.A)Spiral convergence/divergencepoleB)PhugoidpoleC)Dutchrollp oleD)Short periodpoleE)Rollconvergencepole 192

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1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w n [Hz]EI 0 [Nm 2 ] Short Period Dutch Roll D EI = 0.6 D EI = 0 D EI = -0.6 A 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w n [Hz]EI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 B Figure6-5.Naturalfrequencieswithlinearlyvaryingstif fness.A)Dutchrollandshort periodmodesB)Phugoidmode 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 zEI 0 [Nm 2 ] Short Period Dutch Roll Phugoid EI = 0.6 EI = 0 EI = -0.6 Figure6-6.Dampingratioswithlinearlyvaryingstiffness 193

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0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 w n [Hz]EI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 A 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 zEI 0 [Nm 2 ] D EI = 0.6 D EI = 0 D EI = -0.6 B Figure6-7.Firstbendingmodewithlinearlyvaryingstiffn ess.A)NaturalfrequencyB) Dampingratio 6.2RootvsTipStiffness Section 6.1 showedthatthewingrootstiffnesshasalargereffectonthe ight dynamicsthanthewingtipstiffness.Nowrootversustipsti ffnesswillbeanalyzed furtherbycreatingstepwisedistributionsofstiffness(r ecallfromSection 3.1.2 how stepwisechangesinstiffnessarepossibleinASWING). Auniformstiffnessof EI =0.8Nm 2 isapplied,withthestiffnessontheinnerorouter 25%ofthespanshiftedby 40%.Thewingisheldrigidintorsion.Thegeneralshapes oftheresultingdistributionsareillustratedinFig. 6-8 Theresultsfromthesefourcongurationsarecomparedagai nstthebaseline, whichhasuniformstiffnesswithnostepwisechange.Additi onally,twocongurationsof thebaselineshiftedby 40%acrosstheentirespanareincluded.Theresultsforthe ightdynamicsaretakenaspercentchangesfromthebaselin econgurationandare tabulatedinTables 6-1 to 6-3 194

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0 2 4 6 8 10 12 Relative shape of EI distributionsSpan [in] Figure6-8.Stepwisechangescreatedintherootandtip EI 6.2.1Results Considerrstthecongurationswithastepwisechangeinst iffnesswhileignoring thecongurationswiththeshiftedbaseline.Thegreatestp ercentchangeoccurswith changesinrootstiffness,especiallyreductionsinrootst iffness.Thisimportanteffectof rootstiffnessholdstrueforthenaturalfrequencyanddamp ingratiooftheoscillatory modesandtimeconstantsofthenon-oscillatorymodes. Thesetrendscanbeexplainedbythedirecteffectsofthecha ngesinstiffnessfrom oneportionofthestructureonanother.Forexample,achang einthestiffnessofthe Table6-1.Changesinnaturalfrequencyresultingfromstep wisechangesin EI on inner/outer25%span Phugoid DutchRoll ShortPeriod FirstBending Baseline n [ Hz ] 0.13 1.51 3.39 11.34 Tip-40%% 0.01 0.04 0.00 0.03 Tip+40%% 0.00 0.03 0.01 0.01 Root-40%% 6.03 3.96 0.82 9.66 Root+40%% 2.09 1.51 1.44 8.37 Baseline-40%% 7.97 5.94 0.31 11.32 Baseline+40%% 2.74 2.04 1.62 13.13 195

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Table6-2.Changesindampingratioresultingfromstepwise changesin EI on inner/outer25%span Phugoid DutchRoll ShortPeriod FirstBending Baseline n [ Hz ] 0.10 0.31 0.28 0.34 Tip-40%% 0.02 0.01 0.02 0.22 Tip+40%% 0.05 0.09 0.01 0.20 Root-40%% 7.35 23.68 21.54 23.93 Root+40%% 2.16 7.97 10.17 13.24 Baseline-40%% 11.12 35.50 24.28 26.96 Baseline+40%% 2.93 11.48 13.15 17.88 Table6-3.Changesinspiralandrollconvergencemoderesul tingfromstepwise changesin EI oninner/outer25%span Spiral Roll Baseline n [ Hz ] 3.61 0.03 Tip-40%% 0.01 0.04 Tip+40%% 0.34 0.06 Root-40%% 40.31 1.63 Root+40%% 36.96 0.07 Baseline-40%% 49.13 3.84 Baseline+40%% 67.26 0.00 structureintheinnerspanstillhasaneffectontheoutersp anstructure.Therestofthe wingwillbesubjecttosomechangeindeectionortwistduet otheexibilityoftheroot. However,changesintheouterspancannotdirectlyaffectth ewingdeformationinthe innerspan.Therecouldbeanindirectaerodynamiccoupling ,butthisindirectcoupling wouldbeweakerthanadirectstructuraleffect.Asaresult, changesintherootstiffness havealargereffectontheightdynamicsthanchangesintip stiffness. Anotherinterestingtrendfromtheseresultsistheproport ionofchangethatis achievedbymodifyingthestiffnessintherootcomparedtos hiftingthewholebaseline bythesameamount.Forexample,theroot-40% EI caseinTable 6-1 showsa6.03% changeinphugoidnaturalfrequencyandthebaseline-40%ca seshowsa7.97% changefromtheuniformconguration.Thechangeinthestif fnessoftheroot25%span achieved76%oftheeffectthatchangingthestiffnessofthe entirewingwouldhave 196

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achieved.Similarly,thephugoiddampingratioresultingf romtheroot-40%caseis66% ofthechangeindampingratioresultingfromshiftingtheen tiredistributionby-40%. Thistrendissurprisinglysimilarforallthemodes,asshow ninTables 6-4 to 6-6 .Across allthemodes,theminimumeffectofloweringthestiffnesso ftheinner25%spanas comparedtoloweringthestiffnessoftheentirewingis42%, themaximumeffectis89% andthemeaneffectis74%.Thisresultdoesnotincludetheef fectontheshortperiod naturalfrequency,whichisconsideredtobeanoutlierduet othesmallmagnitudeofthe effect( O 10 2 ). Table6-4.Comparisonofchangesinrootstiffnesstooveral lstiffness:natural frequencies Phugoid DutchRoll ShortPeriod FirstBending Baseline n [ Hz ] 0.13 1.51 3.39 11.34 Root-40%% 6.03 3.96 0.82 9.66 Baseline-40%% 7.97 5.94 0.31 11.32 Root/Total% 0.76 0.67 2.65 0.85 Table6-5.Comparisonofchangesinrootstiffnesstooveral lstiffness:dampingratios Phugoid DutchRoll ShortPeriod FirstBending Baseline n [ Hz ] 0.10 0.31 0.28 0.34 Root-40%% 7.35 23.68 21.54 23.93 Baseline-40%% 11.12 35.50 24.28 26.96 Root/Total% 0.66 0.67 0.89 0.89 Table6-6.Comparisonofchangesinrootstiffnesstooveral lstiffness:timeconstants Spiral Roll Baseline n [ Hz ] 3.61 0.03 Root-40%% 40.31 1.63 Baseline-40%% 49.13 3.84 Root/Total% 0.82 0.42 6.3AeroelasticSpan AconclusionfromtheSection 6.2 isthatthestiffnessoftheinnerspanofthe winghasagreatereffectontheightdynamicsthanthestiff nessoftheouterspan. 197

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Furthermore,thereissomeportionoftheinnerspanoverwhi chchangesinstiffnesswill resultinobtainingthemajorityoftheeffect(aschosenbyt heengineerorresearcher) thatthesamechangesinstiffnessacrosstheentirespanwou ldhaveresultedin.This innerportionofthespanistermedtheaeroelasticspanandi sthesubjectofthepresent study.Specically,itisdesiredtoquantifythelengthoft heaeroelasticspanwhen deningthemajorityoftheeffecttobe80%oftheeffectofch angingthestiffnessofthe entirespan.Theaeroelasticspanisdenotedas b 80 Ifacertaineffectontheightdynamicsisdesired,changin gthestiffnessofthe entirewingmaybeonewaytoaccomplishit.However,alterin gthestiffnessina percentageoftherootspanmaybeeasierandfasterinsomesi tuations,suchas inight.Onepossibilitymaybetoaccomplishthischangein ightviaanactuator. Conceptually,theactuatorcouldalterthestiffnessovert heaeroelasticspanandachieve adesiredeffectontheightdynamicswithouthavingtoalte rthestiffnessoftheentire span.Ifpossible,theremaybesituationswhereaffectingt heaeroelasticspancouldbe lessdifcultthanalteringthestiffnessoftheentirewing 6.3.1StepwiseDistributionsintheAeroelasticSpan Onlychangesin EI arestudiedhereinbecauseofthesmalleffectof GJ .Two baselinecongurationsareconsidered( EI =0.4Nm 2 or EI =1.0Nm 2 )andchangesof 50% intheaffectedspan( EI =0.5or EI =1.5)areapplied.Airspeedissetto15m/s andthewingisheldrigidintorsion.Thissetupproducescon gurationswitharange of EI from0.2to1.5Nm 2 andanalyzesbothreducedandincreasedstiffnessinthe aeroelasticspan.Foreachconguration,theresultsareno rmalizedbywhattheresults wouldhavebeenif EI hadbeenappliedtotheentirespan. ThetrimresultsareshowninFig. 6-9 .Fortheangleofattack,theaeroelasticspan isapproximately35%ofthespan( b 80 =35% ).Forelevatordeection, b 80 =30% for allfourcongurations.Notethepresenceofnumericalnois einsomeportionsofthe data.Whentheabsoluteeffectisminor,thenormalizedeffe cthasthisappearance 198

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ofnoise.Forexample,considerthe EI =0.4Nm 2 and EI =1.5conguration.The changeinangleofattackfromacongurationwithuniform EI =0.4Nm 2 stiffnessto acongurationwithuniform EI =0.6Nm 2 isonly0.14 (apercentchangeof1.269%). Thepercentchangetoacongurationwith EI =0.6Nm 2 on50%ofthespanis1.179%. However,thepercentchangetoacongurationwith EI =0.6Nm 2 on55%ofthespanis 1.360%.Thesedifferencesaresmallandcouldbecausedbynu mericalround-offerrors. However,whennormalizedbythe1.269%change,theyresulti nnormalizedvaluesof 0.929and1.071,respectively.Thiscausesthescatteredap pearanceofsomeofthe pointsinFig. 6-9A Forthedutchrollmode,showninFig. 6-10 ,40%ofthespanmustbeaffectedin ordertoensurethat80%oftheeffectinfrequencyanddampin gisobtained. Forthephugoidmode,showninFig. 6-11 ,theaeroelasticspanis30%forthe naturalfrequencyand35%forthedampingratio.Thereismor enumericalscatterinthe normalizedpercentchangeindampingratio. Fortheshortperiodmode,showninFig. 6-12 ,only25%ofthespanisneededfor thenaturalfrequencyand30%isneededforthedampingratio .Aninterestingbehavior innaturalfrequencyisobservedinthe EI =0.4Nm 2 EI =0.5case:100%oftheeffect isreachin25%ofthespan.When30%to50%ofthespanisaffect edby EI ,it appearsthereisagreatereffectonthenaturalfrequencyth anifthewholespanhad beenaffectedbythechangeinstiffness.However,theactua lamountofthisdifference (0.0053Hz)isnotsignicant. Therstbendingmode,showninFig. 6-13 ,canalsobeanalyzedfromthe perspectiveoftheaeroelasticspan.Foritsnaturalfreque ncy,80%oftheeffectis reachedin25%ofthespanfortheincreasedstiffnesscases( EI =1.5).Forthe decreasedstiffnesscases,ittakes35%ofthespantoreacht hesameeffect.Thespan neededtoseethemajorityoftheeffectondampingratioisab out30%. 199

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0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Da% span with D EI EI=1.0, D EI=0.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dd e% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 B Figure6-9.Aeroelasticspananalysisoftrim.A)Angleofat tackB)Elevatordeection 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% span with D EI EI=1.0, D EI=0.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 B Figure6-10.Aeroelasticspananalysisofthedutchrollmod e.A)NaturalfrequencyB) Dampingratio 200

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0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% span with D EI EI=1.0, D EI=0.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 B Figure6-11.Aeroelasticspananalysisofthephugoidmode. A)NaturalfrequencyB) Dampingratio 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 B Figure6-12.Aeroelasticspananalysisoftheshortperiodm ode.A)Naturalfrequency B)Dampingratio 201

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0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% span with D EI EI=1.0, D EI=0.5 EI=1.0, D EI=1.5 EI=0.4, D EI=0.5 EI=0.4, D EI=1.5 B Figure6-13.Aeroelasticspananalysisoftherstbendingm ode.A)Naturalfrequency B)Dampingratio 6.3.2Non-StepwiseDistributionsintheAeroelasticSpan Theanalysisoftheaeroelasticspanisexpandedtoincluden on-stepwise distributionsoverthewing.Astepwisechangeinstiffness isaconvenientwaytostudy therelativeimportanceofstiffnessondifferentpartsoft hespanbutmaybehardto implementphysically.Linearandexponentialslopesareim plementedintheaeroelastic spantodetermineiftrendsfromSection 6.3.1 holdtrueforthesecases.Theseslopes areillustratedinFig. 6-14 for EI =0.5(reducedstiffness)and EI =1.5(increased stiffness)intheaffectedspan. Examiningthephugoidmode'snaturalfrequencyinFig. 6-15A ,itisclearthatit takesmuchmoreofthespantoachievethemajorityoftheeffe ctwhenthestiffness changeswithalinearslope.Infact,whenconsideringthe EI =0.5case,changing thestiffnessoftheentirewingusingalinearslopeonlyach ieves70%oftheeffectof changingthestiffnessoftheentirewinguniformly.Thisre sultisobviouslyduetothefact thattheaveragestiffnessacrossthewingisstillhigherwh enalinearslopeispresent thanwithauniformdistribution.Inthe EI =1.5case,usingalinearslopeonlyachieves 202

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0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 EI [Nm 2 ]span [in] Figure6-14.Stepwise,linear,andexponentialslopesover aportionoftheinnerspan 88%oftheeffect,evenwhenchangedovertheentirewing.The exponentialdistribution comesmuchclosertotheeffectivenessofthestepwisedistr ibutionbutstillrequires10% moreaeroelasticspantoachievethesameeffect.Sincethee xponentialdistributionisa closerrepresentationofthestepwisedistributionthanth elinearslopedistribution,these resultsmakephysicalsense. Thesametrendholdsfortheshortperiodandrstbendingmod es,asseen inFigs. 6-16 and 6-17 .Thelinearslopeisverydifferentfromthestepwisecasean d insomecasescannotachieve80%ofthenormalizedeffecteve nwhenexpanded toincludethefullspan.Theexponentialdistributionrequ iresabout10%morespan toachievethesamenormalizedeffect.Forallmodesanalyze d,itisclearthatthe aeroelasticspanisthehighestforthelinearslopecongur ationandlowestforthe stepwiseconguration,withtheexponentialcasebeingver yclosetothestepwisecase. 203

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0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 B Figure6-15.Effectsofslopeintheaeroelasticspan:phugo idanddutchroll.A)Phugoid naturalfrequencyB)Dutchrolldampingratio 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 B Figure6-16.Effectsofslopeintheaeroelasticspan:short period.A)Naturalfrequency B)Dampingratio 204

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0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dw n% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 A 0 0.2 0.4 0.6 0.8 1 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 normalized Dz% Span with D EI Stepwise, D EI=0.5 Linear, D EI=0.5 Exponential, D EI=0.5 Stepwise, D EI=1.5 Linear, D EI=1.5 Exponential, D EI=1.5 B Figure6-17.Effectsofslopeintheaeroelasticspan:rstb endingmode.A)Natural frequencyB)Dampingratio 6.4AeroelasticRootStiffness Evidenceisshownfortheeffectivenessofchangesinrootst iffnessoverchangesin tipstiffness.Inaddition,itisshownthatthereissomeeff ectiveinnerportionofthespan, dubbedtheaeroelasticspan,overwhichchangesinstiffnes scanachievethemajority oftheeffectofthechangeinstiffnessovertheentirewing. Fortherangesofstiffness considered,theaeroelasticspanisfoundtovarybetween25 -40%ofthewing.This amountisstillasignicantportionofthewing,however. Earlier,theideaofusinganactuatortoalterthestiffness intheaeroelasticspan wasconceptualized.Alteringthestiffnessovertheaeroel asticspanmightbeeasier insomesituationsthanalteringthestiffnessovertheenti respan.However,asseen inSection 6.3 ,thisactuatorwouldstillhavetoaffect30%to40%ofthewin g,inmost cases.Ifaffectingthatmuchofthespanistoodifcultorno tpossible,itistheorizedthat thereisasmallerportionofthespanoverwhichanadditiona lchangeinstiffnesscould produceasimilareffect.Forexample,theeffectofreducin gthetotalwingstiffnessby 50%couldbeachievedbya50%reductioninstiffnessoverthe aeroelasticspanoran 205

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evengreaterreductionontheinner10%span.Findingthisae roelasticrootstiffnesson theinner10%spanisthesubjectofthepresentstudy. Onlybendingstiffnessisstudiedandcongurationswithre ducedstiffnessinthe wingrootwillbeconsideredrst.Airspeedissetto15m/s.T wobaselinevaluesof EI areused(0.4and1.0Nm 2 )alongwithtwolevelsofvariation: EI =0.75or0.5Nm 2 .For clarity,whenachangeisappliedtotheentirewingitisrefe rredtoas EI wing .Whenit isappliedtotheinner10%span,itisreferredtoas EI root 6.4.1AeroelasticRootStiffnessforDecreasingStiffness Fortheeffectonthetrim,showninFig. 6-18 EI root needstobe0.55to0.60to meetorexceed80%oftheeffectfrom EI wing =0.75.For EI wing =0.5,a EI root of 0.30to0.35isneededintheroottoachievethemajorityofth eeffect.Thischangeis anadditionaldecreaseof15-20%instiffnessbeyondthe EI root neededtoobtainthe effectof EI wing =0.75. Similarreductionsin EI root areneededtoachievetheeffectonthephugoidand dutchrollmodes,asshowninFigs. 6-19 and 6-20 .Oneexceptionisthedutchroll naturalfrequency,whichrequiresarootstiffnessof25-30 %and60%inthe EI wing =0.5 and0.75,respectively.Fortheshortperiodmode,Fig. 6-21 ,15-20%additional changeisalsosufcient,exceptforthe EI =0.4Nm 2 EI wing =0.5conguration.In thatconguration,a EI root of0.45intherootspanisenoughtoachieve80%ofthe effectof EI wing =0.5.Thiseffectissurprisinglysmallandcorrespondstot heverysmall aeroelasticspanfoundinFig. 6-12A .Fortherstbendingmode,a15–20%additional changewasalsosufcienttoachievethemajoreffect. 206

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0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized DaD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 A 0 0.2 0.4 0.6 0.8 1 1.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 normalized Dd eD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 B Figure6-18.Aeroelasticrootstiffnessfordecreasing EI :trimconditions.A)Angleof attackB)Elevatordeection 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized Dw nD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 A 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized DzD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 B Figure6-19.Aeroelasticrootstiffnessfordecreasing EI :dutchrollmode.A)Natural frequencyB)Dampingratio 207

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0 0.2 0.4 0.6 0.8 1 1.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 normalized Dw nD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 A 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized DzD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 B Figure6-20.Aeroelasticrootstiffnessfordecreasing EI :phugoidmode.A)Natural frequencyB)Dampingratio 0 0.2 0.4 0.6 0.8 1 1.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 normalized Dw nD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 A 0 0.2 0.4 0.6 0.8 1 1.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 normalized DzD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 B Figure6-21.Aeroelasticrootstiffnessfordecreasing EI :shortperiodmode.A)Natural frequencyB)Dampingratio 208

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0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized Dw nD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 A 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 normalized DzD EI root EI=0.4, D EI wing =0.75 EI=0.4, D EI wing =0.50 EI=1.0, D EI wing =0.75 EI=1.0, D EI wing =0.50 B Figure6-22.Aeroelasticrootstiffnessfordecreasing EI :rstbendingmode.A)Natural frequencyB)Dampingratio 6.4.2AeroelasticRootStiffnessforIncreasingStiffness Nowincreasedstiffnessintherootspan(inner10%)isconsi dered.Thesame baseline EI areused(0.4and1.0Nm 2 )withthesamerelativechangeinstiffness ( EI wing =1.25or1.5).TheseresultsareshowninFigs. 6-23 to 6-27 Itisimmediatelyseenthatamuchlargerrelativeincreasei ntherootstiffnessis neededtoobtainthemajorityoftheeffectofincreasingthe stiffnessacrosstheentire wing.Forangleofattackandelevatordeection,with EI wing =1.25,a EI root of2.0or 1.75,respectively,isneededtoachievethemajorportiono ftheeffect.For EI wing =1.5, EI root =3.0wasnotenoughtoachieve80%oftheeffectonangleofatt ack,butdid achievethateffectonelevatordeection. Forthedutchrollmode,showninFig. 6-24 ,the EI =0.4Nm 2 and EI wing =1.25 caseonlyrequires EI root =1.5toachievetheeffectinnaturalfrequency,butrequire s EI root =2.25toachievetheeffectindampingratio.Forthe EI wing =1.5cases, EI root =3willobtainthemajorityoftheeffectonnaturalfrequenc ybutisnothigh enoughtoobtainthemajorityoftheeffectonthedampingrat io. 209

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Forthephugoidmodeandthe EI wing =1.25conguration,a EI root of1.75to2.0 isneeded.Forthe EI wing =1.5conguration,a EI root =3.0willachievethemajority oftheeffectonnaturalfrequencybutnotonthedampingrati o.The EI root neededto obtainthemajorityoftheeffectonthedampingratiowasnot capturedintherangeof EI root considered. Theshortperiodmodewith EI wing =1.25requiresa EI root of1.5to1.75to achievetheeffect.Forthe EI wing =1.5conguration,theeffectonnaturalfrequency canbeachieveslightlyeasierwith EI root =2.5. Therstbendingmode,for EI wing =1.25,a EI root of1.75to2.0isneeded.For EI wing =1.25,a EI root of3.0isneededandisstillnotenoughtogainthemajoreffec t ondampingratio. Evenintheeasiestcases,anadditional50%increaseinstif fnessisrequiredonthe inner10%spantogetthemajorityoftheeffectoftheorigina lstiffnessovertheentire span.Insomecases,doublingthestiffnessintherootissti llnotenoughtoachievethe effectacrosstheentirewing.Thischangeismuchmorethant hechangeneededinthe decreasingstiffnesscase.Thereasonforthehigherchange istheexponentialchange intheightdynamicsduetostiffness.Whendecreasingstif fness,theresponsechanges morethanwhenincreasingstiffness.Thisresultmeansthat lessofachangein EI root willbenecessarytoachievetheeffectof EI wing inthereducedstiffnesscasethanthe increasedstiffnesscase.Thisnonlineareffectcouldhave implicationsforaircraftdesign. Forexample,itmaybebettertodesignawingslightlystiffe rbecauseitwillbeeasierto gettheightdynamicseffectthroughreducingstiffnessin steadofincreasingstiffness. Ingeneral,inthedesignspacecoveredhere,itispossiblet oachieveacertain effectontheightdynamicsbyalteringthestiffnessinthe wingrootratherthanaltering stiffnessovertheentirewing.Thisconclusionbuildsonth eearlierresultswhichshowed thatrootstiffnesshasabiggereffectontheightdynamics thantipstiffness.As expected,thechangeintherootmustbegreaterthanthechan geovertheentirespan 210

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wouldhavebeen.Inthereducedstiffnesscase,a15-20%incr easedreductionin stiffnessintherootisneeded.Consideringthattheinner1 0%spanisonlyasmall portionoftheaeroelasticspansfoundearlier,thisadditi onalreductiondoesnotseem verylarge.However,intheincreasedstiffnesscase,atlea stanadditional50%increase intherootisneeded.Insomecases,therootstiffnessmustb emorethandoubled. 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized DaD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 A 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized Dd eD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 B Figure6-23.Aeroelasticrootstiffnessforincreasing EI :trimconditions.A)Angleof attackB)Elevatordeection 211

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0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized Dw nD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 A 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized DzD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 B Figure6-24.Aeroelasticrootstiffnessforincreasing EI :dutchrollmode.A)Natural frequencyB)Dampingratio 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized Dw nD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 A 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized DzD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 B Figure6-25.Aeroelasticrootstiffnessforincreasing EI :phugoidmode.A)Natural frequencyB)Dampingratio 212

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0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized Dw nD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 A 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized DzD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 B Figure6-26.Aeroelasticrootstiffnessforincreasing EI :shortperiodmode.A)Natural frequencyB)Dampingratio 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized Dw nD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 A 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 normalized DzD EI root EI=0.4, D EI wing =1.25 EI=0.4, D EI wing =1.50 EI=1.0, D EI wing =1.25 EI=1.0, D EI wing =1.50 B Figure6-27.Aeroelasticrootstiffnessforincreasing EI :rstbendingmode.A)Natural frequencyB)Dampingratio 213

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6.5ConcludingRemarks Non-uniformdistributionsofbendingstiffnesswereappli edtotheGenMAVwing.A linearslopevariationinstiffnessshowslittleeffect,an dfurtheranalysisconrmsthatthe stiffnessoftheinnerwingspancontributestheprimaryaer oelasticeffecttothevehicle dynamics. Itisfoundthatthereisaportionoftheinnerspan,termedth eaeroelasticspan, overwhichchangesinstiffnesscanproducethemajority(80 %)oftheeffectofchanges madetotheentirespan.Thisaeroelasticspan,denoted b 80 ,isfoundtobebetween30% and40%ofthetotalspanforthetrimconditionsandeffecton ightmodes.Theresultis thatinsteadofchangingthestiffnessacrosstheentirespa ntoachieveaneffect,avery similareffectcanbeachievedbyapplyingthosechangestot heinner30%-40%ofthe span. Theeffectofslopeinthespanwisevariationofstiffnessin theaeroelasticspan wasfoundtobesignicant.Alinearvariationofstiffnessi ncreasestheaeroelasticspan signicantlywhileanexponentialsloperesultsinanaeroe lasticspanwhichmoreclosely matchestheresultsfromastepwisechange. Itwasfurthertheorizedthatifonlytheinner10%spancanbe affectedbychanges instiffness,thereisanadditionalincreaseordecreasein stiffnesswhichcouldbe appliedtoobtainthemajorityoftheeffectofthechangetot hewholewing.Indeed, itisfoundthatanadditional15%-20%decreaseinstiffness intheinner10%spanis enoughtoachieve80%oftheeffectofchangingstiffnessacr osstheentirespan.For example,adesiredeffectcouldbeachievedbyreducingthes tiffnessofthewingby50% orreducingthestiffnessoftheinner10%spanby70%.Forinc reasingstiffness,a50% increaseormoreisneededtoachieveasimilareffectasstif feningtheentirewing. Overall,theseresultsrevealanadditionaldesignparamet erthatcanbeusedto inuencethedynamicbehavioroftheaircraft.Inadditiont othebaselinewingstiffness, thestiffnessoftheinnerwingspancouldbealteredbythede signertoachieveadesired 214

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effect.Thisabilitymaypresentadvantagestothedesigner sinceachangeinstiffness overasmallerareaofthewingmaybeeasiertoachieve.Furth ermore,itmaybe possibletoinuencechangesinthewingstiffnessinightt hroughapiezoelectric actuator,forexample. 215

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CHAPTER7 AEROELASTICEFFECTSOFWINGBATTENS Abattenisaslenderpieceofrelativelystiffmaterialused tostrengthenastructure. Suchpiecesarefairlycommonindesignsacrossmanycommuni ties.Theyare ubiquitousinnauticalapplicationswhenconstrainingthe shapeofasailortheouter moldlinesofahull.Theyarealsoanimportantcomponentofa irships[ 93 185 ]and spacecraftstructures[ 113 ]. Theconceptofbattensisevenprevalentinbiology[ 119 ].Batshavebonesin theirwingsthatprovidelocalizedstiffnessinamannersim ilartobattens[ 132 ].Insects wingsaremadeofmembranesreinforcedwithblood-lledvei nsthatprovideadditional stiffnesslikeabatten[ 37 ].Theconceptactuallyexistedlongagoinpterosaurs,whic h hadactinobrilsthatwereessentiallybattensandhelpedp rovidestiffnessinlocalized partsoftheirwings[ 31 ]. Battensarealsoanintegralpartofaxed-wingclassofmicr oairvehicles(MAVs). Adistinguishingfeatureofthisclassishighexibilityin thewings.Suchexibilityhas beendemonstratedbypilotstoeasepilotworkloadinmainta ininglevelightdueto thepassivewash-outthatoccursinresponsetoloading[ 69 ].Assuch,thewingsare constructedusingalow-stiffnessmaterialwithinwhichah igh-stiffnessskeletalframe, includingbattens,isembedded.Manyaircraftinthisclass areconstructedusingbattens andsuccessfullyown[ 2 – 4 68 69 71 158 ]. Theroleofwingexibilityisextensivelyanalyzedusingco mputationaluid dynamics[ 150 ].Auidandstructuralsolveriscoupledtogethertostudyt heaerodynamic characteristicsofamembranewinginlowReynoldsnumbero w[ 87 88 ].Amembrane wingwiththreebattensshowsincreasedliftduetopassivei nationaswellasalower effectiveangleofattackduetothetrailingedgedeection .Thepassiveinationalsohas theeffectofdelayingstall.Otherstructuralmodelingofa batten-reinforcedmembrane wingissuccessfullycomparedtoexperimentaldata[ 157 ]. 216

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Thecomputationaleffectsarevalidatedusingwindtunnels .Onestudydemonstrates thepronounceddelayinstallthatoccursforexiblewingsa scomparedtorigidwingsfor aMAV[ 71 ].Anotherdetailedtestingandcharacterizationofamembr anewingshows improvedlongitudinalstabilityandimprovedgustrejecti onoverarigidwing[ 7 ]. Thebattencongurationisgenerallynotchosenthroughari gorousdesign methodology;instead,thebattensareoftenchosenusing adhoc metricsortrial-and-error ighttesting.Someamountoftailoringisintroducedbysel ectingthebattensinorder tobalancetheamountofdeformationagainsttheneedtomain tainadesiredwing shape[ 159 ].Atopologyoptimizationisusedtodesignapatternofstif fnesselements, whicharenotrestrictedtoclassicbattenshapes,toenhanc epropertiessuchasgust rejection[ 156 ]. Thispaperconsiderstherelationshipbetweenbattencong urationsandight dynamics.Inparticular,theeffectofnumberofbattensisd emonstratedonthetrim characteristicsandstaticstabilityaswellasthefrequen cyanddampingoftheight modes.Theresultsagreewithpreviousstudiesthatindicat eexibilityenhancesgust rejectionandlongitudinalstability;however,theaeroel asticeffectsareshowntohavea complicatedrolewhenconsideringtheightperformancesu chthatsomemetricsare improvedwhileothersaredegraded. 7.1Methodology Theightdynamicsareanalyzedforasetofcongurationswi thinadesignspace whichisconstructedtoencompasswingswithvariednumbero fbattens.Each congurationusesthegeometryandmassoftheGenMAVasabas elinewherethe empennageandfuselagearemodeledasrigidelementswhilet hestiffnessofthewing isvariedacrossthedesignspace.Theaircraftistrimmedin straightandlevelightat V =15m/sineachcase. Thebattensinthisdesignspacearestripsthatrunchordwis ealongthewing. Eachstriphasawidthof1.27cmandalengththatextendsalon gtheentirechordatits 217

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b 2 = 30.5cm c = 12.7cm Figure7-1.GenMAVwinggeometrywiththreebattensspanwiselocation.Theselocationsarechosentolieequall yspacedalongthespanas illustratedinFig. 7-1 foracongurationwith3battensoneachwing. Theinitialwingismodeledascarbonberwithvaryingthick nessandathin leading-edgespar.Thissparservesastheattachmentpoint forthebattensand providesadditionalbendingstiffnesssothatthewingcans upportaload.Itismodeled as4-plycarbonberwhiletheremainderofthewingismodele das1-plycarbonber. Theelasticaxisofthewingisplacedalongtheleadingedge. Suchawinghasan approximatebendingstiffnessof EI =0.5Nm 2 andatorsionalstiffnessof GJ =0.1Nm 2 Thisapproachgivesagoodapproximationforthetorsionals tiffnessalongthespan ofabatten-reinforcedwingbutslightlyunrealisticappro ximationforbendingstiffness.In anactualwing,onlyasmallcomponentalongtheleadingedge wouldpossessahigh bendingstiffness.Inthismodel,thebendingstiffnessrem ainsconstantalongthechord. Battensareintroducedintothewingbyalteringthestiffne ssproperties.Thenarrow widthofthebattenprecludesitshavingasignicanteffect onthebendingstiffness; conversely,thetorsionalstiffnesswillbestronglyaffec tedbythebattens.Assuch,the torsionalstiffnessincreasesto1.0Nm 2 wherethebattenslie,whichapproximately correlatestoa4-plycarbonberlayup.Thevariationoftor sionalstiffnessacrossthe semi-spanisdemonstratedinFig. 7-2 forcongurationsrangingfrom0battensto 10battensalongeachwing. 218

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1 2 3 4 5 6 7 8 9 10 0 15 30 0 0.5 1GJ [Nm 2 ] Number of battens per wing y [cm]GJ [Nm 2 ] Figure7-2.Proleoftorsionalstiffnessacrossthehalf-s panforeachbatten conguration 7.2ResultsandDiscussion 7.2.1TrimCharacteristics Theconditionsaredeterminedatwhicheachcongurationin thedesignspace istrimmed.Theassociatedvaluesforangleofattackandele vatordeectionare showninFig. 7-3 .Thesevaluesindicatetherequiredangleofattackincreas eswhile therequiredelevatordeectiondecreasesasthenumberofb attensisdecreased. Specically,thecongurationwith10battenstrimsatanan gleofattackof13.4 and elevatorof-10.7 whilethecongurationwith0battenstrimsatanangleofatt ackof 16.3 andelevatorof-13.5 whichisachangeof17.8%and-20.7%. Thevariationsinangleofattackandelevatordeectionare purelyaresultof aeroelasticeffectssinceeachcongurationoperatesatth esameairspeedwith identicaldistributionsofmassandnominalgeometry.Inth iscase,theaeroelastic effectsmanifestaswingtwist.Theaveragedtwistacrossth ewingvaries,asshown inFig. 7-4 ,from-7.3 foracongurationwith10battensto-11.6 foracongurationwith 0battenswiththenegativesignindicatingleading-edge-d owntwist.Thebattens,while beinggeometricallysmall,changetheoverallstiffnessen oughtocausethissignicant variationintwisttooccur. 219

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0 5 10 15 20 25 0 1 2 3 4 5 6 7 8 9 10 -25 -20 -15 -10 -5 0 a [deg] d e [deg]Number of battens per wing a d e // Figure7-3.Trim and e -12 -10 -8 -6 -4 -2 0 0 1 2 3 4 5 6 7 8 9 10 t avg [deg]Number of battens per wing // Figure7-4.Span-averagedwingtwist Theangleofattackandwingtwistareactuallyquitecoupled .Thewingessentially hasaneffectiveangleofattackresultingfromtheaddition oftheaircraftangleofattack andthewingtwist.Thiseffectiveangleofattack,asshowni nFig. 7-5 ,isnearlyconstant foreachvariationinbattens.Thisnearconstancyimpliest hecongurationswithless stiffnessaretwistingmoreandtheaircraft,includinghor izontaltail,isincreasingthe angleofattacktocompensate.Theresultingorientationha sthewinggeneratingthe sameamountofliftforanyofthecongurationsand,thus,th eelevatordecreasingto reducethemomentitprovidesastheangleofattackisincrea sed. 220

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2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 a + t avg [deg]Number of battens per wing // Figure7-5.Effectiveangleofattackattrim Also,thetrimcharacteristicsaregivenforawingmodeleda sasinglepieceof carbonber.Thisdatapointisgivenas 1 forthenumberofbattensinFigs. 7-3 and 7-4 .Thevaluesforangleofattack,elevatordeectionandtwis tmonotonicallytrend acrossthecongurationstowardsthissolid-wingcongura tionbutdonotnecessarily converge.Considerthatacongurationwith25battenswoul dessentiallybeentirely constructedofcarbonberstrips;however,thesestripswo uldbeconnectedbya membraneasopposedtobeingasinglepieceofcarbonber.Th estiffnessisthusquite differentforacarbon-berwingmadeof25thinpiecesor1la rgepiece. 7.2.2SensitivityofLiftandDrag Theeffectofbattensongustsensitivitycanbesomewhatind icatedbythelift-curve slopeknownas C L andgiveninFig. 7-6A .Thisparameterindicatestheincrease inliftthatresultsfromanincreaseinangleofattackwhich mayoccurfrom,among severalpossibilities,awindgust.Thecongurationswith fewerbattensclearlyhavea lowerlift-curveslopewith C L =4 : 83 whenthewinghas10battensascomparedto C L =4 : 27 whenthewinghas0battens.Thisdecreaseinthelift-curves lopecorrelates withthetrimcharacteristicsinthatthelowerstiffnessas sociatedwithfewerbattens causesthewingtodeformmorewhenitisloadedandthusgener atelesslift.Sucha signicantdecreaseinlift-curveslopeandgustsensitivi tyagreeswithpilotobservations. 221

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4 4.5 5 5.5 6 6.5 0 1 2 3 4 5 6 7 8 9 10 C L aNumber of battens per wing // A 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0 1 2 3 4 5 6 7 8 9 10 C D aNumber of battens per wing // B Figure7-6.Aircraftliftanddrag.A)Aircraftlift-curves lopeB)Aircraftdragdueto changeinangleofattack Thisvariationoflift-curveslopeacrossthedesignspaced irectlycorrelateswith stiffness.Thecongurationswithlessstiffnesswilldefo rminresponsetotheincreased loadingthatresultsfromanincreaseinangleofattack.Thi sbehaviorisassociatedwith lessstiffnessduetofewerbattens,asshowninFig. 7-6A ,orlessstiffnessduetothinner battens[ 159 ]. Thevariationofdragwithangleofattack,knownas C D ,isgiveninFig. 7-6B for thedesignspace.Thisvariationissomewhatcounterintuit iveinthat C L decreasesbut C D increasesasthenumberofbattensisreduced.Thedecreasei nliftisaccompanied byadecreaseininduceddrag;however,theincreaseddeform ationcausesasignicant increaseinparasiticdrag.Theresultingdragisactuallym oresensitivetochangesin angleofattackasthenumberofbattensisreduced.7.2.3StaticStability Eachcongurationinthedesignspaceisstaticallystablea boutthelongitudinalaxis asshownby C m inFig. 7-7 .Thismetricrelatingstaticstabilityisslightlybetterf orthe congurationswithreducednumberofbattens. 222

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-2.85 -2.8 -2.75 -2.7 -2.65 -2.6 -2.55 -2.5 -2.45 0 1 2 3 4 5 6 7 8 9 10 C m aNumber of battens per wing // Figure7-7.Longitudinalstaticstability Theimprovementinstaticstabilityisapproximatedbynoti ngthecontributionsoflift anddragfromthewing.Inthiscase,Eq.( 7–1 )derivestheresultingpitchmomentusing theliftanddragasdecomposedintotheirbody-axisforces. Theliftanddragarefurther decomposedintorst-orderfunctionsoftheangleofattack inEq.( 7–2 ).Thisangleof attackissmallsotheapproximationsinEq.( 7–3 )maybeused. M = x ( L cos( )+ D sin( ))+ z ( D cos( ) L sin( ) (7–1) = x (( L o + L )cos( )+( D o + D )sin( )) + z (( D o + D )cos( ) ( L o + L )sin( )) (7–2) = x (( L o + L )+( D o + D ) )+ z (( D o + D ) ( L o + L ) ) (7–3) Theexpressionfor C m inEq.( 7–4 )resultsbytakingthederivativeofEq.( 7–3 ) andintroducinganon-dimensionalizingdivisor.Thisappr oximationestimatesthat C m shoulddecreaseby0.08asthebattensdecreasefrom10to0wh ichisclosetothe actualdecreaseof0.05computedbythenumericalmodeling. C m = x c ( C L + C D o +2 C D )+ z c ( C D C L o 2 C L ) (7–4) Staticstabilityaboutthedirectionalaxisisappreciably affectedbythenumberof battens.Thevalueof C n ,whichmustbenegativefortheaircrafttobestaticallysta ble, 223

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increasesfrom-0.113to-0.083asthenumberofbattensdecr easesfrom10to0as showninFig. 7-8 0.05 0.06 0.07 0.08 0.09 0.1 0 1 2 3 4 5 6 7 8 9 10 C n bNumber of battens per wing // Figure7-8.Directionalstaticstability Thevariationin C n inFig. 7-8 isattributedtoaeroelasticeffectsonliftanddrag duetothebattens.Thedominantcontributortothistermist heverticaltailwhichis unchangedthroughoutthedesignspace;however,thegeomet ricdihedralgeneratesa smallcontributionwhichdoesindeedvarywithnumberofbat tens.Considertheyaw moment, N ,inEq.( 7–5 )whichresultsfromdecomposingtheliftanddragintobodyaxis forcesandnotingthecontributionsfromtherightandleftw ingswithamomentarmof y Expandingtheseforcesasafnefunctionsofangleofattack andnotingthesensitivities areidenticalfortherightandleftwingsresultsinEq.( 7–6 ).Thisangleofattackissmall sotheapproximationsinEq.( 7–7 )arethusvalid.Theeffectofgeometricdihedralisto differtheangleofattackoneachwingasintroducedinEq.( 7–8 ). 224

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N dihedral = y ( D right cos( right ) L right sin( right ) D left cos( left )+ L left sin( left )) (7–5) = y (( D o + D right )cos( right ) ( L o + L right )sin( right ) ( D o + D left )cos( left )+( L o + L left )sin( left )) (7–6) = y ( D L o )( right left )+ L 2 left 2 right (7–7) = y ( ( D L o )(( + ) ( ))+ L ( ) 2 ( + ) 2 (7–8) Thecontributionofthegeometricdihedraltostaticstabil ityisgiveninEq.( 7–9 ). Thiscontributiondirectlyresultsbytakingthederivativ eofEq.( 7–8 )andconvertinginto anon-dimensionalterm.Itmustbenotedthattheactualvalu eofgeometricdihedral is12 whichresultsfromthenominalvalueof7 plusanother5 ofbendingattrim. Usinganeffectivemomentarmof10cmfromthecenterline,th enumericaldifference betweencongurationswith10battensand0battensisthus 0 : 008 whichcompares quitereasonablywiththedifferenceof : 011 indicatedbyFig. 7-8 C n dihedral =2 y b ( C D C L o 2 C L ) (7–9) Notethatgeometricdihedralisactuallyanegativecontrib utiontostaticstability. Thiscontributionisoftenpositiveformanyaircraft;howe ver,thecontributionisnegative inthiscasebecausetheliftcomponentinthebody-axislong itudinaldirectionislarge enoughtohaveaneffectontheyawmoment.So,theaircraftha salargepositive contributionfromtheverticaltail,whichisconstantfora llcongurations,butan increasinglynegativecontributionfromthegeometricdih edralasthenumberofbattens isdecreased. Thevalueof C l ,whichrelatesstaticstabilityinthedirectionalaxis,is given inFig. 7-9 foreachconguration.Thesevaluesremainnegativetoindi catethatevery congurationisstaticallystablealthoughthevalueof C l increasessteadilyasthe 225

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-0.2 -0.15 -0.1 -0.05 0 0 1 2 3 4 5 6 7 8 9 10 C l bNumber of battens per wing // Figure7-9.Lateralstaticstabilitynumberofbattensisreduced.Thespecicvaluesare-0.113f orthecongurationwith 10battensand-0.083forthecongurationwith0battens. ThevariationobservedinFig. 7-9 canalsobeattributedtoaeroelasticeffectsofthe battenscoupledwiththegeometricdihedral.Considerther ollmomentinEq.( 7–10 ) whichnotesthecontributionsfromeachwing.Expandingthe seforcesasafne functionsofangleofattackandnotingthesensitivitiesar eidenticalfortherightand leftwingsresultsinEq.( 7–11 ).Thisangleofattackissmallsotheapproximations inEq.( 7–7 )arethusvalid.Theeffectofgeometricdihedralistodiffe rtheangleofattack oneachwingasintroducedinEq.( 7–13 ). L dihedral = y ( D left sin( left )+ L left cos( left ) D right sin( right ) L right sin( right )) (7–10) = y (( D o + D left )sin( left )+( L o + L left )cos( left ) ( D o + D right )sin( right ) ( L o + L right )cos( right )) (7–11) = y D 2 left 2 right +( D o + L )( left right ) (7–12) = y D ( ) 2 ( + ) 2 +( D o + L )(( ) ( + )) (7–13) 226

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Thecontributiontostaticstabilityresultsbytakingader ivativeofEq.( 7–13 )with respecttoangleofsideslipandthenconvertingintonon-di mensionalrepresentation. TheresultingcontributionisgiveninEq.( 7–14 ).Thisapproximationcomputesthat C n shouldbe.034greaterfor0battensthanfor10battenswhich agreeswellwithFig. 7-9 thatnotes0.03isthedifferencecomputednumerically. C l dihedral = 2 y b ( C L + C D o +2 C D ) (7–14) 7.2.4ControlEffectiveness Theeffectofbattensonthecontrolpoweroftheaircraftisn egligible,asshown inFig. 7-10 .Theprimarycontrolderivatives C m e ;C l a and C n r areessentially unaffectedbythechangingnumberofbattens.Itisimportan ttonotethattheGenMAV's elevonshavebeenseparatedintoaileronsandanelevatorfo rthisanalysis.Accordingly, theaileronsresideonthehorizontaltailandtheireffecti venessisnotchangedbythe battens. -0.052 -0.051 -0.05 -0.049 -0.048 -0.047 -0.046 0 1 2 3 4 5 6 7 8 9 10 C m d eNumber of battens per wing // A -0.002 -0.0015 -0.001 -0.0005 0 0.0005 0.001 0.0015 0.002 0 1 2 3 4 5 6 7 8 9 10 CoefficientNumber of battens per wing C l d a C n d r // B Figure7-10.Primarycontrolderivatives.A)Elevatorcont rolpowerB)Aileronandrudder controlpower 227

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7.2.5LongitudinalFlightDynamics Thephugoidmodeshowsminorchangesinfrequencyanddampin gacrossthe designspaceasshowninFig. 7-11A .Thefrequencydecreasesslightlyfrom0.114Hz inthe10battencaseto0.110Hzinthe0battencase,a-3.7%ch ange.Thedamping ratioincreasesfrom0.092inthe10battencaseto0.099inth e0battencase,a6.8% increase. Themodeshapeofthephugoidmodeisgivenbytheeigenvector inTable 7-1 andshowssomewhatmorevariationthaneitherthenaturalfr equencyordamping.In particular,theamountofvariationintheangleofattackis noticeablydifferentbetween thecongurations.Thesolidwingshowsaclassicbehaviorw iththeangleofattack beinglessthan9%ofthepitchangle;however,theratioincr easessuchthatthe congurationwith10battenshasanangleofattackbeingnea rly20%ofthepitch angleandthecongurationwith0battenshasitover26%ofth epitchangle.This non-traditionalmodeshaperesultsfromthewingdeforming duetoaeroelasticeffects. Thedeformation,asindicatedbytheeigenvector,isneglig ibleinbendingbuthasatwist atthewingtipsthatis60%ofthepitchangle. Thebattenshavetheoppositeeffectontheshortperiodmode asthephugoidmode andacttoincreasethenaturalfrequencyveryslightlyandd ecreasethedampingasthe numberofbattensisreduced,asshowninFig. 7-11B .Thenaturalfrequencyincreases from3.37Hzto3.39Hzasthebattensdecreasefromtentozero whilethedamping ratiodecreasesfrom0.27to0.25,a8.0%decrease. Thenormalizedeigenvectorcomponentsoftheshortperiodm odeforvariousbatten congurationsarelistedinTable 7-2 .Themodeshapeforallcongurationsretains theclassicbehaviorofbeingpredominatelypitchrateanda ngleofattackwitheach beingnearly90 outofphase.Theaeroelasticeffectsresultinatwistangle thatis roughlytwicethesizeoftheangleofattackorthepitchangl ewhilethebendingremains negligible. 228

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Table7-1.Normalizedeigenvectorcomponentsofthephugoi dmodeforvariousbatten congurations battensperwing:zerotensolid magphasemagphasemagphase u [m/s]0.23790.2250.23290.9940.20192.929 q [deg/s]0.69095.6940.71795.3050.84195.248 [deg]0.26389.5190.19790.1160.08590.648 [deg]1.0000.0001.0000.0001.0000.000 z ( y = 0 : 3) [mm]0.345-90.7450.435-89.9250.595-88.150 z ( y = 0 : 15) [mm]0.299-90.5640.270-89.8020.247-88.122 z ( y =0) [mm]0.000-29.3960.0000.0000.0000.000 z ( y =0 : 15) [mm]0.299-90.5680.270-89.8040.247-88.123 z ( y =0 : 3) [mm]0.345-90.7480.434-89.9270.595-88.151 ( y = 0 : 3) [deg]0.590-90.5410.649-89.7590.168-88.022 ( y = 0 : 15) [deg]0.197-90.4720.142-89.6990.032-87.981 ( y =0) [deg]0.000-99.1830.000-89.6500.000-86.334 ( y =0 : 15) [deg]0.197-90.4770.142-89.7020.032-87.982 ( y =0 : 3) [deg]0.590-90.5460.649-89.7620.168-88.023 Table7-2.Normalizedeigenvectorcomponentsoftheshortp eriodmodeforvarious battencongurations battensperwing:zerotensolid magphasemagphasemagphase u [m/s]0.074-172.8430.060-174.5780.037178.847 q [deg/s]20.809109.61020.326109.50618.952109.237 [deg]1.0000.0001.0000.0001.0000.000 [deg]0.9765.1170.9583.8250.925-0.324 z ( y = 0 : 3) [mm]1.546-170.9811.867-171.3792.601-171.107 z ( y = 0 : 15) [mm]1.084-169.7131.013-170.1581.056-170.795 z ( y =0) [mm]0.000157.1410.0000.0000.0000.000 z ( y =0 : 15) [mm]1.084-169.7211.013-170.1631.056-170.799 z ( y =0 : 3) [mm]1.546-170.9901.867-171.3862.601-171.111 ( y = 0 : 3) [deg]1.966-169.2142.215-169.2940.653-169.024 ( y = 0 : 15) [deg]0.618-168.5790.465-168.5650.123-168.407 ( y =0) [deg]0.000-178.5500.00068.4740.000-96.620 ( y =0 : 15) [deg]0.618-168.5850.465-168.5680.123-168.407 ( y =0 : 3) [deg]1.967-169.2182.215-169.2960.653-169.024 229

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0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0 1 2 3 4 5 6 7 8 9 10 0.07 0.08 0.09 0.1 0.11 0.12 0.13 w n [Hz] zNumber of battens per wing w n z // A 3 3.1 3.2 3.3 3.4 3.5 3.6 0 1 2 3 4 5 6 7 8 9 10 0.2 0.25 0.3 0.35 0.4 w n [Hz] zNumber of battens per wing w n z // B Figure7-11.Naturalfrequencyanddampingofthelongitudi nalmodes.A)Phugoid modeB)Shortperiodmode 7.2.6Lateral-DirectionalFlightDynamics Themodalparametersofthedutchrollmode,showninFig. 7-12 ,showsome changesinresponsetodecreasingbattens.Thenaturalfreq uencydecreasesfrom 1.40Hzto1.37Hzwhilethedampingratioincreasesby8.5%fr om0.31to0.34asthe congurationchangesfrom10battensto0battens. Themodeshapeofthedutchrollmode,showninTable 7-3 ,changesacrossthe congurationsfromatraditionalbehaviortoanovelbehavi or.Inparticular,thephasing 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 0 1 2 3 4 5 6 7 8 9 10 0.2 0.25 0.3 0.35 0.4 w n [Hz] zNumber of battens per wing w n z // Figure7-12.Naturalfrequencyanddampingofthedutchroll mode 230

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relationshipsvarynoticeablybetweentheangleofsidesli pandtherollangleandthe yawangle.Theangleofsideslipis169 outofphasewiththeyawangleforthesolid wingbuttheintroductionofbattensreducesthevaluesucht hattheyvarybyonly 152 forthecongurationwith0battens.Similarly,theangleof sideslipislessoutof phasewiththerollangleforthecongurationwith0battens ascomparedtoanyother congurations.Themagnitudeoftherollangleinthemotion isalsobecomemore pronouncedasthenumberofbattensisreduced.Thephasedif ferencebetweenthe wingtipdeectionandtherollangledecreasesslightlyfro mbeing133 outofphase to107 outofphase.Thephasedifferencebetweenthewingtwistand therollangle increasesslightly.Oneachwingtip,thedeectionandtwis tremainsabout180 outof phasefromeachother. Thetimeconstantofthespiralconvergencedecreasesfrom9 sto6.7sas thenumberofbattensisreducedfrom10to0battens,asshown inFig. 7-13 .The Table7-3.Normalizedeigenvectorcomponentsofthedutchr ollmodeforvariousbatten congurations battensperwing:zerotensolid magphasemagphasemagphase p [deg/s]8.611105.9728.773105.0569.094103.196 r [deg/s]3.191-162.7263.461-163.8223.877-164.243 [deg]0.504-64.8360.507-70.6010.519-78.180 [deg]1.0000.0001.0000.0001.0000.000 [deg]0.37687.4490.39888.1130.42791.430 z ( y = 0 : 3) [mm]0.01972.5080.01966.7000.01346.524 z ( y = 0 : 15) [mm]0.01465.9360.00959.2420.00541.665 z ( y =0) [mm]0.000-103.8310.0000.0000.0000.000 z ( y =0 : 15) [mm]0.015-113.0950.011-117.2380.008-124.190 z ( y =0 : 3) [mm]0.020-108.8290.020-113.2340.020-122.579 ( y = 0 : 3) [deg]0.02870.5530.02263.0320.00347.285 ( y = 0 : 15) [deg]0.00864.3510.00455.1470.00140.748 ( y =0) [deg]0.000-119.0580.000-114.4340.0000.340 ( y =0 : 15) [deg]0.009-113.4030.005-118.1880.001-120.695 ( y =0 : 3) [deg]0.030-108.8570.026-113.8220.005-119.328 231

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normalizedeigenvectorcomponentsarelistedinTable 7-4 .Inthesolidwingconguration, therollcomponentissmall,contributingonly0.08 perdegreeofyawangle.Asthewing becomesmoreexible(reducingbattens),thecontribution fromrollgrowsto0.25 per degreeofyawangle.Atthesametime,theyawrateincreasesi nmagnitude,indicating 6 8 10 12 14 16 18 20 22 0 1 2 3 4 5 6 7 8 9 10 T [s]Number of battens per wing // Figure7-13.TimeconstantofthespiralconvergenceTable7-4.Normalizedeigenvectorcomponentsofthespiral convergencemodefor variousbattencongurations battensperwing:zerotensolid p [deg/s]0.0630.0340.006 r [deg/s]-0.146-0.110-0.049 [deg]-0.007-0.005-0.002 [deg]-0.246-0.178-0.075 [deg]1.0001.0001.000 z ( y = 0 : 3) [mm]0.0000.0000.000 z ( y = 0 : 15) [mm]0.0000.0000.000 z ( y =0) [mm]0.0000.0000.000 z ( y =0 : 15) [mm]0.0000.0000.000 z ( y =0 : 3) [mm]0.0000.0000.000 ( y = 0 : 3) [deg]0.0010.0000.000 ( y = 0 : 15) [deg]0.0000.0000.000 ( y =0) [deg]0.0000.0000.000 ( y =0 : 15) [deg]0.0000.0000.000 ( y =0 : 3) [deg]-0.0010.0000.000 232

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afasterrecoveryfromtheperturbation.Thereisnodeforma tionofthestructureinthe shapeofthespiralconvergence. Thetimeconstantoftherollconvergence,showninFig. 7-14 ,increasesslightly from0.033sinthe10battencaseto0.039sinthe0battencase .Theassociatedmode shapesaregiveninTable 7-5 andindicatethebehaviortransitionsawayfromaclassic pure-rollmotionasthenumberofbattensisreduced.Thesol id-wingcongurationhas anangleofsideslipthatisonly8%oftherollanglewhereast he0battenconguration hasanangleofsideslipthatisover17%oftherollangle.Asm allamountofwing deectionandtwistremainintheresponsebutitisnotsigni cantforanyofthe congurations.7.2.7StructuralDynamics Therst-bendingmode'snaturalfrequencyisaffectedvery littlebythebattens, asshowninFig. 7-15A .Thedampingratioexperiencesastrongereffectthannatur al frequency,however,increasingfrom0.36to0.41from10to0 battens,a13.0%change. Theeffectondampingmaycomethroughaerodynamiccoupling ofbendingandtorsion. Thersttorsionmodenaturalfrequencyseesaslighteffect fromthebattens, decreasingfrom35.2Hzto30.4Hz,a13.6%decrease.Thetren dforthedampingratio 0.03 0.032 0.034 0.036 0.038 0.04 0 1 2 3 4 5 6 7 8 9 10 T [s]Number of battens per wing // Figure7-14.Timeconstantoftherollconvergence 233

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14 15 16 17 18 19 20 0 1 2 3 4 5 6 7 8 9 10 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 w n [Hz] zNumber of battens per wing w n z // A 20 30 40 50 60 70 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 w n [Hz] zNumber of battens per wing w n z // B Figure7-15.Naturalfrequencyanddampingofthestructura lmodes.A)Firstsymmetric bendingmodeB)Firstsymmetrictorsionmode ofthersttorsionmodeshowsnoappreciablechangefrom10t o0battens,asshown inFig. 7-15B Table7-5.Normalizedeigenvectorcomponentsoftherollco nvergencemodeforvarious battencongurations battensperwing:zerotensolid p [deg/s]-25.651-30.552-33.120 r [deg/s]-1.933-1.786-1.386 [deg]0.1730.1480.083 [deg]1.0001.0001.000 [deg]0.0760.0590.042 z ( y = 0 : 3) [mm]-0.123-0.167-0.173 z ( y = 0 : 15) [mm]-0.074-0.073-0.069 z ( y =0) [mm]0.0000.0000.000 z ( y =0 : 15) [mm]0.0750.0750.070 z ( y =0 : 3) [mm]0.1240.1690.176 ( y = 0 : 3) [deg]-0.080-0.062-0.007 ( y = 0 : 15) [deg]-0.031-0.018-0.002 ( y =0) [deg]0.0000.0000.000 ( y =0 : 15) [deg]0.0320.0190.002 ( y =0 : 3) [deg]0.0830.0670.008 234

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7.2.8Interpretation Oneeffectofchangingthenumberofbattensistochangethea veragetorsional stiffnessofthewing,from0.48Nm 2 inthecaseof10battensto0.1Nm 2 inthecaseof 0battens.Astheaveragetorsionalstiffnessdecreases,th eaveragewingtwistnaturally increasesinmagnitude.Thiseffectcanbeseenbycomparing thetwistofthe10batten wingtotheone-battenwing,showninFig. 7-16 .Manyoftheobservedeffectsfollow trendsthatwouldnormallyarisefromdecreasingtorsional stiffnessofasolidwing. However,themagnitudeofthoseeffectsdonotcorrespondto dataobtainedwiththe sameaveragetorsionalstiffnessappliedasauniformdistr ibutioninasolidwing. Forexample,the10battencongurationhasauniformbendin gstiffnessof 0.5Nm 2 ,anaveragetorsionalstiffnessof0.48Nm 2 andtrimsatanangleofattack of13.4 andelevatordeectionof-10.7 .Forasolidwingwithauniform EI =0.5Nm 2 and GJ =0.48Nm 2 stiffnessdistribution,thetrimangleofattackis9.3 andtheelevator deectionis-6.8 adifferenceof-30.6%and36.5%,respectively.Thisdispar ityis illustratedinFig. 7-16 ,wherethetwistofthe10battenwingcanbecomparedto -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -0.3 -0.15 0 0.15 0.3 t [deg]Span [m] one batten ten battens GJ = 0.48 Figure7-16.Twistofbatten-reinforcedwingscomparedtoa wingwithuniformtorsional stiffness 235

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thetwistofthe GJ =0.48Nm 2 wing.Thebatten-reinforcedwingresultsinmoretwist despitethefactthatithasthesameaveragestiffness.This effectisbecausetheexible portionsofthespanbetweenthebattensareabletotwistmor esignicantlythanthe wingwiththeuniformdistribution.Becauseofthisdiffere nce,thebehaviorofanaircraft withabattenedwingcannotbeassumedtofollowthebehavior ofanaircraftwithasolid wingofthesameaveragetorsionalstiffness 7.3Summary Aircraftcongurationswithzerototenbattenswerecreate dbyapplyingstepwise changesinthetorsionalstiffnessofthewing.Resultsshow thatasthenumberof battensdecreasesfromtentozero,thetrimsolutionrequir esapproximately22% moreangleofattackandelevatordeectiontomaintainstra ightandlevelight.This isbecauseoftheincreasinglynegative(leadingedgedown) amountofwingtwist whichreducestheliftonthewing.Theeffectiveangleofatt ackonthewingremains approximatelyconstant. Thelift-curveslope C L decreasesasthenumberofbattensisreducedwhilethe variationindragduetoangleofattack( C D )increasesslightly. Thenumberofbattenshasapronouncedeffectonthestaticst abilityofthe aircraft.Thelateral-directionalstabilityderivatives C l and C n decrease36%and 18%,respectively,asthebattensdecreasefromtentozero. Thisisduetoaeroelastic effectsfromthebattenswhichcomesthroughthegeometricd ihedraleffect.Overthe samechangeinbattens,thelongitudinalstaticstabilityd erivative C m decreases3%, indicatingaslightlymorestableconguration.Thebatten shaveanegligibleeffecton theprimarycontrolderivatives. Theinuenceofbattensontheightdynamicsisprimarilyin thedampingof theightmodes.Decreasingthenumberofbattensactstoinc reasedutchrolland phugoiddampingby7%and9%butdecreaseshortperioddampin gby8%.Thetime 236

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constantofthespiralconvergencemodedecreases26%andth etimeconstantofthe rollconvergenceincreases18%asthenumberofbattensisde creasedfromtentozero. Themodeshapesalsoshowchangesinresponsetodecreasingt henumberof battens.Forexample,thecontributionofangleofattackin thephugoidmodealmost doubles.Therollconvergenceshapeshowsaslightasymmetr ybetweenthewingtip deectionsastheinertiaofthewingresiststherollingmot ion.Changesintherelative phaseofdutchrollmodeshapeareobserved. Thebattenshadaminoreffectontherstbendingmodenatura lfrequencybuta strongereffectitsdampingratio,increasingitby13%asth ebattensdecreasefromten tozero.Conversely,thebattenshadastrongereffectonthe rsttorsionmode'snatural frequencythanitsdampingratio. 7.4ConcludingRemarks Battensareshowntohaveastrong,althoughcomplicated,ef fectontheight performanceofmicroairvehicles.Reducingthenumberofba ttensimprovesgust rejectionandlongitudinalstaticstability;conversely, thisreductionalsorequireslarger anglesfortrimanddegradesstaticstabilityinthelateral anddirectionalaxes.The effectsareequallydiversewhenconsideringightdynamic sinthatthedutch-roll modehasimproveddampingwhiletheshort-periodmodehasde gradeddamping. Furthermore,thenatureofthemodeschangeasthebattencon gurationchangesand theaircraftiesnoticeablydifferent.Thesecharacteris ticsareespeciallycriticaltonote whendesigninganautonomoussystemthatrequiresbothanai rcraftandassociated autopilotthatmustcompensatetheightmodestomaneuver. Assuch,battensshould beconsideredasanotherdegreeoffreedomforaircraftdesi gnthatcanbeintroducedto tailorcertainmetricsandbalancetheeffectsofotherdesi gnparameters. 237

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CHAPTER8 AEROSERVOELASTICDESIGNUSINGWINGSTIFFNESS Aeroservoelasticity(ASE)isamultidisciplinaryscienti cendeavorwhichconsiders theinteractionsbetweenthesteadyandunsteadyaerodynam icforcesactingonthe aircraft,theinertialforces,theelasticstructure,andt heightcontrolsystem.ASE interactionscanaffecttheightperformance,structural integrity,andhandlingqualities oftheaircraft. TheimportanceofconsideringASEinteractionsincreasesi naircraftwithlightweight, exiblestructuresandhigh-gaindigitalightcontrolsys tems.Forexample,in1948a B-36aircraftexperiencedanASEinstabilityinducedbythe autopilot[ 54 ].Thecause wastheplacementofasensorpackageintheaircrafttailwhi chintroducedthebending motionofthefuselageintothefeedbackcontrolsystem.Ano theraeroservoelastic instabilityoccurredonamodiedF-4aircraftduringaside slipmaneuver[ 53 ].The instability,excitedbythecontrolsurfaces,occurredbec auseofanoverlapinfrequency betweenastructuralmodeandthepitchingmotionoftheairc raft.Aeroservoelastic interactionshavebeenobservedinmanyotheraircraft,suc hastheYF-17aircraftand F/A-18aircraft[ 12 173 ]. TraditionalASEanalysesinthedesignprocessarefocusedo nreducingthe undesirableASEeffectsonaircraftloads,controleffecti veness,utter,anddivergence[ 182 ]. However,theseanalysesaretypicallydonelaterinthedesi gnprocesswhenthe structuraldesignissomewhatmatureandthereforefocuson modifyingthecontrollerto avoidtheseeffects. Forexample,alinearquadraticgaussian(LQG)controlalgo rithmisappliedtoan aeroelasticwingmodelwiththegoalofuttersuppression[ 32 ].Thewingislow-aspect ratiowingwithasingleaileron.Twostructuralmodes(rst bendingandrsttorsion) areincludedinthestatespacemodel.Oncefull-statefeedb ackgainsaredesigned,the controllerisabletostabilizethewingandpreventutter. 238

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AcontrollerisdesignedforanASEtransportaircraftwitht heobjectiveofreducing wingbendingusingoutboardaps[ 76 ].Thesensordesignandplacementisfoundtobe acrucialaspectofthefeedbackcontroller. AightcontrolsystemforagenericASEaircraftisdevelope dtoimprovehandling qualities[ 79 ].ALQGcontrollerwithadynamicobserveriscomparedtoali near quadraticregulator(LQR)controllerwithoutputfeedback .Bothcontrollersincrease therobustnessofthesystemandprovidegoodhandlingquali ties. Threedifferentoptimalcontrollersaredevelopedandcomp aredusinganASE modeloftheB-52bomberwhichincludesveexiblemodes[ 11 ].Theperformance objectiveisalleviationofgustloadsontheaircraft.Thec ontrollersareabletoadequately reducetheimpactofthegustontheaircraft'sverticalacce lerationwithreasonable controlsurfacedeections. AsopposedtoASEanalysesdonelaterintheaircraftdesignp rocess,ASEdesign synthesisinvolvesvaryingbothstructuralparametersand controldesignearlyinthe aircraftdesignprocess.Thisapproachisuncommoninaircr aftdesignbuthasthe potentialtorevealimportanttrade-offsbetweenthecontr olsystem,structure,and aerodynamics. OneexampleofASEdesignsynthesisinpreliminaryaircraft designinvolvestheroll rateperformanceofaghteraircraft[ 67 74 111 ].Therollrateisanalyzedoverarange ofCGlocations,aircraftmass,andmomentsofinertia.Open -loopandclosed-loop responsesoftheaircraftareconsidered,wheretheloopisc losedusingan H 1 optimal controller.Thestabilityboundariesacrossthemulti-dim ensionaldesignspacevary dramaticallybetweentheopenandclosed-loopcases. ASEeffectswereparticularlyimportantduringthedevelop mentoftheB-2bomber duetoitsunconventionaldesign[ 23 ].Anintegrateddesignapproachwasused,which consideredtheaircraftplanform,yingqualities,gustlo adalleviation,andcontrollaw synthesis.Thecontrollawsincludedacombinationofclass icalandmoderntechniques. 239

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ASEdesignsynthesisisperformedonahigh-aspectratioUAV [ 60 ].Aircraftdesign parametersincludetheaspectratio,wingarea,wingtwist, andwingwallthickness (whichaffectsthewingstiffness).Theresultsshowthatsy nthesizingthesedesign parametersearlyinthedesignprocessallowsforgreaterim provementsinaircraft performance. InChapters 5 and 6 theaeroelasticeffectsofwingstiffnessontheaircrafti ghtand structuraldynamicswereexplored.Manyprominenteffects ontheightdynamicswere observed,suchaslargechangesinthefrequencyanddamping ofthestructuralmodes aswellassignicantchangestothemodeshapes.Thoseresul tslaidthegroundwork forusingstiffnessasaparameterintheaircraftdesignpro cess. Inthischapter,theaeroservoelasticdesignsynthesispro blemisconsideredby evaluatingtheeffectofchangesinwingstiffness.Twobasi cstructuralparameters ofthewing(bendingandtorsionalstiffness)arevariedwhi leopenandclosed-loop controlresultsareanalyzed.Trackingandmodel-followin gtasksfortheaeroservoelastic GenMAVaircraftareperformedtodeterminewhatdesigntrad e-offsareavailable. 8.1Methodology TheASWINGmodeloftheGenMAVaircraft,introducedinSecti on 3.2 ,isused inthischapter.Thefuselageandtailareheldrigidwhileth estiffnessofthewing ischangedthroughthebendingandtorsionalstiffnesspara meters EI and GJ respectively.Theexperimentallocationoftheelasticaxi satapproximately 0 : 28 c is usedforallcongurations.8.1.1ModelReduction Inordertodevelopandtestcontrolmethodologies,eachair craftmodelisimported intoMatlabfromASWING.Becauseofitsuniquestructuralfo rmulation,theASWING modelfollowsthegeneralizedstatespaceform(alsoknowna sthedescriptorstate 240

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spaceform)giveninEqs.( 8–1 )and( 8–2 ). E x = Ax + Bu (8–1) y = Cx (8–2) Apracticalrequirementforcontroldesignistouseamodelw hichcapturesthe essentialaeroelasticcharacteristicsofthesystembutis alsoofamanageablesize. SinceASWING'sstructuralformulationresultsinapproxim ately2500statesineach model,thesemodelsmustbereduced.Atruncationapproachi susedtoobtaineach reducedordermodel(ROM). AcustomizedversionofASWINGwritesthe E ; A ; and B matricestothediskafter thesystemislinearizedaboutthetrimcondition.Inadditi on,theleft( W i )andright ( V i )eigenvectorscorrespondingtotheeigenvalue i areexportedfromASWINGand importedintoMatlab.Onlytheeigenvectorsandeigenvalue scorrespondingtothe modeschosenbytheuser(usuallytheslowestmodes)areexpo rted.Asaresult, W and V arerectangularmatricesofsize n q ,where n isthenumberofstatesand q is thenumberofmodesretainedinthemodel. Toaccomplishthemodelreduction,analternatestatevecto r z isusedinplaceof x asshowninEq.( 8–3 ). x = Vz (8–3) SubstitutingthisstatevectorintoEqs.( 8–1 )and( 8–2 )andpremultiplyingthestate matricesby W yieldsEqs.( 8–4 )and( 8–5 ). WEV z = WAVz + WBu (8–4) y = CVz (8–5) 241

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NowthesystemofequationsmaybewrittenasshowninEqs.( 8–6 )and( 8–7 ). E y z = A y z + B y u (8–6) y = C y z (8–7) The E y ; A y ; B y ,and C y systemmatricesareofsize q q;q q;q m ,and p q (where m and p arethenumberofsysteminputsandoutputs,respectively). Thesematricesare denedinEqs.( 8–8 )to( 8–10 ). E y = WEV (8–8) A y = WAV (8–9) B y = WB (8–10) Anindicationthatthetransformationwassuccessfulistoc heckthat E y isanidentify matrix.Ifso,thegeneralizedstatespacesystemcanbesimp liedintothestandardform showninEqs.( 8–11 )and( 8–12 ). z = A y z + B y u (8–11) y = C y z (8–12) Asaresultofthistransformation,the A y matrixisacomplex-valueddiagonalmatrix. The B y ; C y matricesarealsocomplex-valued.Anadditionaltransform ationisneededto createreal-valuedsystemmatrices,whichisdonebyconver ting A y intoblockdiagonal form,denoted A .Since ( A )=( A y ) ,atransformationmatrix T maybefoundby solvingEq.( 8–13 ). A = TA y T 1 (8–13) 242

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Thetransformationmatrix T isthenusedtoconvert B y and C y intorealformviaEqs.( 8–14 ) and( 8–15 ). B = TB y (8–14) C = C y T 1 (8–15) Thestatespaceformofthegoverningequationsmaynowbewri ttenintermsofonly real-valuedmatricesasshowninEqs.( 8–16 )and( 8–17 ). z = A z + B u (8–16) y = C z (8–17) Sincethereduced-ordermodelisinmodalform,itcanbeeasi lymanipulatedto removeadditionalmodesortoseparateitintothelongitudi nalandlateral-directional components. Intheresultingmodel,thestructuralmodesaremodiedtoe nforceadampingof 2%.Inaddition,theinuenceofthecontrolinputsonthestr ucturalmodesthroughthe B isampliedbyafactorof10.Thismodicationcreatesadiff erentaircraftthanthe physicalGenMAVbutallowsforamoreinterestingcharacter izationofthestructural inuenceontheASEdynamics.8.1.2ROMValidation AreducedordermodeloftheGenMAVaircraftwithawing EI of0.1Nm 2 and GJ of1.0Nm 2 iscomparedagainstthefullmodelinASWING.TheROMcontain s thetraditionalightmodes,theeightlowest-magnitudeun steadyaeropoles,andthe fourlowest-frequencystructuralmodes(rstsymmetricbe nding,rstanti-symmetric bending,rstsymmetrictorsion,andrstanti-symmetrict orsion,inorder).Thedamping ofthestructuralmodesandthe B matrixintheROMareunmodiedforthiscomparison. TheresultingBodeplotsofthetransferfunctionsfromthei nputstotheEulerangles andratesareshowninFig. 8-1 .TheROMcomparesverywelltothefullmodel,with 243

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-140 -120 -100 -80 -60 -40 -20 0 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] A -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] B -140 -120 -100 -80 -60 -40 -20 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -400 -350 -300 -250 -200 -150 -100 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] C -70 -60 -50 -40 -30 -20 -10 0 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -300 -250 -200 -150 -100 -50 0 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] D -160 -140 -120 -100 -80 -60 -40 -20 0 20 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -100 -80 -60 -40 -20 0 20 40 60 80 100 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] E -80 -70 -60 -50 -40 -30 -20 -10 0.01 0.1 1 10 100 1000 Magnitude [dB]w [Hz] Full ROM -90 -80 -70 -60 -50 -40 -30 -20 -10 0 0.01 0.1 1 10 100 1000 Phase [deg]w [Hz] F Figure8-1.ComparisonoftheROMtothefullmodel( EI =0 : 1 GJ =1 : 0 ).A)Aileronto BankAngleB)AilerontoRollRateC)ElevatortoPitchAngleD )Elevatorto PitchRateE)RuddertoYawAngleF)RuddertoYawRate 244

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theexceptionofa90 phaseshiftandadisparityinthelow-frequencymagnitudeo fthe ruddertoyawangletransferfunction.The90 phaseshiftislikelycausedbyanextra poleortheeliminationofazeroattheorigin,andthediffer enceinslopeisduetoan extrazero.8.1.3DesignSpace Thedesignspaceisusedinthischapteristheoneforverylow valuesofstiffness usedinSection 5.3 .Bendingandtorsionalstiffnessarevariedindependently from 1.0Nm 2 to0.07Nm 2 .Thedesignspacewasdiscretizedwith20valuesof EI and GJ (foratotalof400congurations)withthespacingchosento maximizeresolutionin theareaswiththelargestchangesintheresponses.Thedesi gnspaceisillustrated inFig. 8-2 ,wheretheintersectionoftwolinesrepresentsanindividu alaircraft conguration. 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 GJ [Nm 2 ]EI [Nm 2 ] Figure8-2.Designspaceofbendingandtorsionalstiffness 8.2OpenLoop Theopen-loopresponseofaircraftinthedesignspaceisrs tanalyzed.Both frequencyandtimedomainperformancesareevaluated. 245

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8.2.1FrequencyDomainResults8.2.1.1Longitudinal Themagnitudeofthefrequencyresponsefortwolongitudina ltransferfunctionsis showninFigs. 8-3 and 8-4 .Thefrequencyresponseisplottedasathree-dimensional surface,wheretheverticalaxisrepresentsthemagnitude, therighthorizontalaxis representsthefrequencyrange,andthelefthorizontalaxi srepresentsthebending stiffness.Whenviewedinthismanner,theeffectsoftheben dingstiffnessareeasily apparent.Sincetheeffectsoftorsionalstiffnesswerepre viouslyfoundtobeminor(with theelasticaxisinthe 0 : 28 c location),thetorsionalstiffnessisnotdirectlyplotted Instead,theeffectsofbendingstiffnessareevaluatedacr osstwolinesofconstant GJ in thedesignspace. Startingwiththeelevatortoangleofattacktransferfunct ion,showninFig. 8-3 ,the effectofloweringbendingstiffnessistoslightlyincreas ethemagnitudeoftheresponse inthephugoidandshortperiodfrequencies.Inthehigherst ructuralfrequencies, themagnitudeoftheresponseincreasesandbecomesalmosta shighastheshort 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] A 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] B Figure8-3.Elevatortoangleofattack.A)Varying EI at GJ =0 : 07 Nm 2 B)Varying EI at GJ =1 : 0 Nm 2 246

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periodandphugoidmodes.Thedecreasingfrequencyofthest ructuralresponseas bendingstiffnessdecreasesisalsovisible.Similarbehav iorisseenatthehighandlow levelsoftorsionalstiffness,showninFig. 8-3A andFig. 8-3B ,respectively.Thisresult correspondstotheweakeffectoftorsionalstiffnessonthe response. Theelevatortopitchratetransferfunctionshowsslightly differentresults,shown inFig. 8-4 .Asbendingstiffnessdecreases,themagnitudeofthephugo idresponse decreasesdramaticallywhiletheshortperiodmagnitudein creasesslightly.Atthelowest bendingstiffness,theresponseinthestructuralfrequenc iesisashighastheshort periodresponse.Thistrendisonlytrueforthelowtorsiona lstiffnesscaseinFig. 8-4A however.Athightorsionalstiffness,inFig. 8-4B ,thestructuralresponseincreasesand ishigherthanthephugoidresponsebutstilllowerthanthes hortperiodresponse. 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] A 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] B Figure8-4.Elevatortopitchrate.A)Varying EI at GJ =0 : 07 Nm 2 B)Varying EI at GJ =1 : 0 Nm 2 8.2.1.2Lateral-directional TheailerontobankangletransferfunctionisshowninFig. 8-5 .Thesteady-state responsemagnitudedecreasesslightlyasbendingstiffnes sdecreases,butthelargest changeisinthestructuralresponseinthe10-100Hzrange.A tverylowvaluesof 247

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0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] A 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] B Figure8-5.Ailerontobankangle.A)Varying EI at GJ =0 : 07 Nm 2 B)Varying EI at GJ =1 : 0 Nm 2 bendingstiffness,theresponsemagnitudeatthesefrequen ciesisalmostaslargeas thesteady-stateresponsemagnitude.Thechangesfromlow GJ ,showninFig. 8-5A ,to high GJ ,showninFig. 8-5B ,areminor. Theaeroelasticeffectintheailerontorollratetransferf unction,showninFig. 8-6 isevenmoreprominentthantheailerontobankangletransfe rfunction.Inthestructural frequencyrange,theresponsemagnitudeatlowvaluesofben dingstiffnessismuch higherthanthesteady-stateresponse(whichalsodecrease swithdecreasing EI ). Thereisalsoaneffectonthesteady-stateresponse,whichd ecreasesas EI decreases. Themagnitudeofthetransferfunctionfromruddertoyawrat eisshowninFig. 8-7 Thistransferfunctionshowsasimilarresulttotheaileron transferfunctions:the steady-stateresponsedecreasesslightlywhiletherespon seatthestructuralfrequencies increasesnoticeable.However,thestructuralresponsema gnitudeinthistransfer functionremainslowerthanthesteady-stateresponse. Theeffectonthedecreaseintheresponsemagnitudenearste ady-stateseenin thesetransferfunctionscouldbeduetoalagintheposition ofaveryexiblewingas 248

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comparedtoastiffwingduringamaneuver.Astheaircraftst artstoroll,theexiblewing willexperienceanadditionaldeectionduetotheinertial effectsofthemaneuver.This lagrepresentsadegradationoftheaircraft'sabilitytoro ll. 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] A 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] B Figure8-6.Ailerontorollrate.A)Varying EI at GJ =0 : 07 Nm 2 B)Varying EI at GJ =1 : 0 Nm 2 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] A 0.01 0.1 1 10 100 1000 0 0.2 0.4 0.6 0.8 1 -140 -120 -100 -80 -60 -40 -20 0 20 Magnitude [dB] w [Hz] EI [Nm 2 ]Magnitude [dB] B Figure8-7.Ruddertoyawrate.A)Varying EI at GJ =0 : 07 Nm 2 B)Varying EI at GJ =1 : 0 Nm 2 249

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8.2.2TimeDomainResults8.2.2.1Longitudinal Theopen-looptime-domainresponsesofaircraftinthedesi gnspacearenow analyzed.First,thechangeinangleofattackinresponseto astepinputtotheelevator isshowninFig. 8-8 .Therisetimeremainsverysmallacrossthedesignspace,al though itislowestinthecongurationswithverylowbendingstiff ness,asshowninFig. 8-8A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 Rise Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 30 35 40 45 50 Settling Time [s] B 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 80 100 120 140 160 180 200 220 240 Overshoot [%] C Figure8-8.Open-loopangleofattackresponsetoastepinpu tinelevator.A)Risetime B)SettlingtimeC)Percentovershoot 250

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Athighvaluesof GJ ,therisetimedecreasesapproximately50%from0.06second s to0.03secondsasbendingstiffnessdecreases.Forsystems withverylowtorsional stiffness,thischangeisdiminished. Thechangeinsettlingtimeduetobendingstiffness,showni nFig. 8-8B ,isrelatively minorathighvaluesof GJ .Atlowvaluesof GJ ,thesettlingtimeincreasesbyasmuch as25%. ThepercentovershootoftheresponseisshowninFig. 8-8C .Itrisesdramatically, from80%toover200%,asbendingstiffnessdecreasesandsho wslittledependenceon torsionalstiffness.8.2.2.2Lateral-directional Theresponseofbankangletoanaileronstepinputisshownin Fig. 8-9 .The risetimedecreasesfromapproximately48secondsinthemos tstiffcongurationsto 2.0secondsinthemostexiblecongurations,achangeof-9 6%.Thisdecreaseis accompaniedbyadecreaseinthesettlingtimefrom60second sto3.7secondsoverthe 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 5 10 15 20 25 30 35 40 45 Rise Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0 10 20 30 40 50 60 Settling Time [s] B Figure8-9.Open-loopbankangleresponsetoastepinputina ileron.A)RisetimeB) Settlingtime 251

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samechangeinbendingstiffness,achangeof-94%.Thereisn onoticeableeffectfrom loweringtorsionalstiffness. Asimilareffectisseenintherollrateresponsetoanailero ninput,shown inFig. 8-10 .Inthisresponse,however,therisetimequicklydropstone ar-zeroas thebendingstiffnessstartstodecrease. Similarresultsareseenintheyawrateresponsetoastepinp utinrudder,shown inFig. 8-11 .Therisetimedecreasesfrom48secondsto1.8secondsasthe bending stiffnessdecreases,achangeof-96%.Thesettlingtimedec reasesfrom59secondsto 3.3seconds,achangeof-94%.8.2.3Summary Theopen-loopresponseoftheaircraftshowsastrongdepend enceonthebending stiffnessandaminordependenceonthetorsionalstiffness .Inthelongitudinal frequencyresponses,thephugoidmodeisaffectmorethanth eshortperiodmode. Thereisanoticeabledecreaseinthemagnitudeoftherespon seatsteady-statein 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Rise Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0 10 20 30 40 50 60 Settling Time [s] B Figure8-10.Open-looprollrateresponsetoastepinputina ileron.A)RisetimeB) Settlingtime 252

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thelateral-directionaltransferfunctions,suggestinga decreaseinlateral-directional maneuverability. Therisetimeandsettlingtimeoftheopen-loopresponses,b othlongitudinal andlateral-directional,generallydecreaseasstiffness decreases.Thesedecreases representanimprovementinperformance,whichissomewhat counterintuitive.A possibleexplanationmaylieinthedifferencebetweenther eactionintherigidbody portionoftheaircraft(thefuselageandempennage)versus theexiblepartofthe aircraft(thewing).Asthewingbecomesmoreexible,itpro videslessresistanceto motion(frombothinertialanddragforces).Asenergyisinp uttothesystemthrough astepchangeinthecontrolsurfaces(whichresideontheemp ennage),moreof thisenergygoesintotherigidbodymotion,whichshowsupas theseperformance increases. 8.3ClosedLoop Theclosed-loopperformanceoftheaircraftinthedesignsp aceareevaluatedby designinganoptimalcontrollertotrackareferencecomman d. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 5 10 15 20 25 30 35 40 45 Rise Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0 10 20 30 40 50 60 Settling Time [s] B Figure8-11.Open-loopyawrateresponsetoastepinputinru dder.A)RisetimeB) Settlingtime 253

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8.3.1ControlDesign Toevaluatetheclosed-looptrackingperformance,afeedba ckcontrollermust bederivedforeachsysteminthedesignspace.Alinearquadr aticregulation(LQR) techniqueisusedtondtheoptimalcontroller.TheLQRcont rollerischosenbecauseof itsabilitytogeneratecomparablecontrollersfordiffere ntsystemsinthedesignspaceby usingthesameweightingmatrices. AnLQRtrackingcontrollerisdesignedtoregulateallstate sexceptthedesired trackingstate(forexample,headingangle).Inthecontrol formulation,theactual headingangle( )iscomparedtothedesiredvalue( ref )andthedifference( e )is driventozerobythefeedbackcontroller.Ablockdiagramof thiscontrollerisillustrated inFig. 8-12 ,where E isanelementalarraythatselectsthe outputfrom y 1 s K ff P K E ref e u y x Figure8-12.BlockdiagramofaLQRtrackingcontroller Tocalculatethestatefeedbackgainmatrix K andthefeedforwardgain K ff ,the statespacesystemmustbesupplementedwithanintegraloft heheadingstateinthe statevector,asshowninEq.( 8–18 ).Theoutputvectormustalsobesupplementedwith thesamequantity,showninEq.( 8–19 ). x + = 264 x R 375 (8–18) y + = 264 y R 375 (8–19) 254

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ThenasupplementedstatespacesystemisformedasshowninE qs.( 8–20 )and( 8–21 ). x + = A + x + + B + u + (8–20) y + = C + x + (8–21) Theterms A + B + C + ,and u + aredenedinEq.( 8–22 ).Notethatthecomponents oftheoutputmatrix C correspondingtothedesiredtrackingstate areusedto supplementthestatematrix A .Thisoutput-basedformulationisneededbecause thestatevectorlacksphysicalsignicanceafterthetrans formationinEq.( 8–3 ). A + = 264 [ A ]0 [ C ( )]0 375 B + = 264 [ B ] 0 375 C + = 264 [ C ]0 01 375 u + = 264 u 0 375 (8–22) TheLQRcontrollerndsthestate-feedbackcontrolsignal u = Kx thatminimizes thequadratic,innite-horizoncostfunctionshowninEq.( 8–23 ). J ( u )= Z 1 0 y T Qy + u T Ru dt (8–23) InEq.( 8–23 ), Q and R areweightingmatricesdesignedbythecontrolengineer, typicallythroughtrialanderror,topenalizecertainoutp utsorinputs,respectively. Thechoiceof Q and R affectthecontrolgains K and K ff ofFig. 8-12 .Theweighting matricesareheldconstantforeachaircraftcongurationi nthedesignspace. NotethatimplementinganLQRcontrollerrequiresthatthes ystembestabilizable anddetectable,bothofwhichareimpliedwhenthesystemisc ontrollableand observable,respectively.Thecontrollabilityandobserv abilitygramians,shown inEqs.( 8–24 )and( 8–25 ),areusedtoevaluateeachsystem'scontrollabilityand observability.Recallthatthecontrollabilitygramian W c ispositivedeniteifandonlyif thepair ( A ; B ) iscontrollable.Likewise,theobservabilitygramian W o ispositivedenite ifandonlyifthepair ( C ; B ) isobservable.Usingthismethod,eachsystemacrossthe 255

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designspacewasfoundtobecontrollableandobservable. W c = Z 1 0 e A BB T e A T d (8–24) W o = Z 1 0 e A T C T C e A d (8–25) 8.3.2Longitudinal Fortheoutputvectorofacontrollertrackingapitchanglec ommand,shownin Eq.( 8–26 ),thediagonaltermsofthe Q matrixareshowninEq.( 8–27 ).Fortheinput vectorgiveninEq.( 8–28 ),thediagonaltermsofthe R matrixareshowninEq.( 8–29 ) y = uvwpqrxyz Z T (8–26) diag ( Q )= 00101110001011 e 5 (8–27) u =[ a e r ] (8–28) diag ( R )= 11 e 2 1 (8–29) Asampleofthesystemoutputforacongurationwith EI = GJ =1 : 0 Nm 2 tracking a =5 : 4 commandisshowninFig. 8-13 .Thepitchangleoutputhasarisetimeof0.27 secondsandsettlingtimeof0.42secondswithonlya2%overs hoot.Theerror,plotted inFig. 8-13B ,goestozerooverthesametime. Theresultsfortherisetimeandsettlingtimeofthetrackin gbehavioracrossthe designspaceareshowninFigs. 8-14A and 8-14B .Ingeneral,asthebendingstiffness decreases,theaircrafttakesalongeramountoftimetotrac kthereferencecommand. Therisetimeincreasesto0.32secondsinthemostexibleco ngurations,anincrease of23%,andthesettlingtimedisplayssimilarbehavior,inc reasing37%from0.41 secondsto0.56seconds.Thepercentovershootdecreasesto almostzero,asshown inFig. 8-14C .Themaximumelevatordeectionofeachaircraft'srespons e,shownin Fig. 8-14D ,decreasesfromaround-4.5 to-7.5 inthemostexiblecongurations. 256

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Inanattempttodeterminewhatopen-loopparameteriscausi ngthisperformance decrease,trackingperformancewasobtainedfromarticia lsystemsinwhichthe frequencyanddampingofaparticularmodewasxedatitsval ueassociatedwithastiff aircraft( EI = GJ =1 Nm 2 ).RisetimeresultsforthisanalysisareshowninFig. 8-15 Amongthephugoid,shortperiod,andrstsymmetricbending modes,aeroelastic effectsonthephugoidmodecontributethemosttowardthepe rformancedecrease. Whenthephugoidmodeisheldxedatthefrequencyanddampin gassociatedwiththe stiffaircraft,thedecreaseinperformanceisnotobserved inthedata.Whentheshort periodandrstbendingmodesareheldxed,theperformance worsens,suggestingthat thesemodesmaycontributetoimprovedrisetimeasstiffnes sdecreases. Outofthethreemodes,thephugoidmodeistheonewiththelar gestchangesinthe modeshape.Thecontributionofangleofattacktothephugoi dmodeshaperisesfrom 6%to32%asstiffnessdecreases.Thischangeinmodeshapeco uldbecontributing totheperformancedecrease.Theshortperiodandrstbendi ngmodeshapesremain relativelyconstantasstiffnessdecreases. -5 0 5 10 15 20 25 30 0 0.5 1 1.5 2 OutputTime [s] q [deg] a [deg] q [deg/s] d e [deg] A -1 0 1 2 3 4 5 6 0 0.5 1 1.5 2 Error [deg]Time [s] B Figure8-13.Outputofanaircraftwith EI = GJ =1 : 0 Nm 2 trackinga =5 : 4 command. A)OutputhistoryB)Errorhistory 257

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8.3.3Lateral-Directional Foracontrollertrackingaheadingcommand,thediagonalte rmsofthe Q matrix areshowninEq.( 8–30 )andthediagonaltermsofthe R matrixareshowninEq.( 8–31 ). diag ( Q )= 0010110001101 e 4 (8–30) diag ( R )= 1 e 2 1 e 1 1 e 3 (8–31) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.140 0.160 0.180 0.200 0.220 0.240 0.260 0.280 0.300 Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Time [s] B 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.00 0.50 1.00 1.50 2.00 Time [s] C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] -8.0 -7.5 -7.0 -6.5 -6.0 -5.5 -5.0 -4.5 -4.0 d e [deg] D Figure8-14.Longitudinaltrackingperformance( ref =0 : 1 rad)usingxedweighting matrices.A)RisetimeB)SettlingtimeC)Percentovershoot D)Maximum elevatordeection 258

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Asampleofthesystemoutputforacongurationwith EI = GJ =1 : 0 Nm 2 tracking a =5 : 4 commandisshowninFig. 8-16 .Theyawangleoutputhasarisetimeof0.02 secondsandsettlingtimeof0.04secondswitha3.5%oversho ot.Theerror,plottedin Fig. 8-16B ,goestozerooverthesametime. Theresultsfortrackingperformanceacrossthedesignspac eareshownin Fig. 8-17 .Ingeneral,theperformancedegradesasthebendingstiffn essdecreases andtheeffectoftorsionalstiffnessisminor.Therisetime increasesfrom0.02to0.08 seconds,a300%increase.Thesettlingtimeincreasesby200 %from0.04to0.12 seconds.Theperformanceoftheexibleaircraftislowerth anthestiffaircraftdespite alargeincreaseinthemaximumrudderdeection,from0.08i nthestiffconguration to8.4 inthemostexibleconguration.Theailerondeectionals oincreases,butstill remainssmallcomparedtothemaximumrudderdeection.8.3.4Summary AnLQRtrackingcontrollerwasusedtoevaluatetheeffectso fwingstiffnesson theclosed-looptrackingperformanceoftheGenMAVaircraf t.Bothlongitudinaland lateral-directionaltrackingperformancesdecreaseasst iffnessdecreases,despite 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rise Time [s]EI [Nm 2 ] Original System Stiff Phugoid Stiff Short Period Stiff First Bending Figure8-15.Longitudinaltrackingperformance( ref =0 : 1 rad)whilexingcertain modesattheirstiffvalues 259

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-2 -1 0 1 2 3 4 5 6 0 0.5 1 1.5 2 OutputTime [s] f [deg] y [deg] p [deg/s] r [deg/s] d a [deg] d r [deg] A -1 0 1 2 3 4 5 6 0 0.5 1 1.5 2 Error [deg]Time [s] B Figure8-16.Outputofanaircraftwith EI = GJ =1 : 0 Nm 2 trackinga =5 : 4 command.A)OutputhistoryB)Errorhistory increasedactuationbythecontroller.Thistrendsuggests thattheoptimalLQR controllerforaexibleaircraftcannotbefoundusingweig htingmatricesdesigned forastifferaircraft. Inthelongitudinalperformance,aeroelasticeffectsonth ephugoidmodetendto drivetheperformancedecrease.Theaeroelasticeffectson thephugoidmodeconsistof increasedfrequency,decreaseddamping,andperhapsmosts ignicantly,ashifttoan unusualmodeshapewhichincludesamuchhighercontributio nfromangleofattackand alowercontributionofpitchrate,asseeninTable 5-7 8.4ModelFollowing Theideaofalteringtheaircraftperformancebychangingth ewingstiffness on-demandwasdiscussedinChapter 6 whendiscussingtheconceptofaeroelastic spanandaeroelasticrootstiffness.Ifthedesignerwishes tochangethestiffness in-ight,anactuatorisrequired.Someactuatorsexistwhi chcouldpossiblyaccomplish thischange,suchaspiezoeletricactuatorswhichcouldcha ngetheshapeoftheairfoil. 260

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.000 0.020 0.040 0.060 0.080 0.100 Time [s] A 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Time [s] B 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 d a [deg] C 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EI [Nm 2 ] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 GJ [Nm 2 ] 0.0 5.0 10.0 15.0 20.0 d r [deg] D Figure8-17.Lateral-directionaltrackingperformance( ref =0 : 1 rad)usingxed weightingmatrices.A)RisetimeB)SettlingtimeC)Maximum aileron deectionD)Maximumrudderdeection However,thisapproachhasitslimitations,suchastheadde dweightandcomplexityof theactuator. Insteadofactuallychangingthestiffnessofthewinginig ht,itmaybepossible modifytheaircraftbehaviorviathecontrolsystemtoappro ximatethebehaviorofan aircraftwithadifferentlevelofstiffness.Thisideaof“v irtual”changingthewingstiffness isinvestigatedinthissection. 261

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AnLQR-basedmodel-followingcontrollerisdeveloped.Two possiblecasesare considered:astiffaircraftvirtuallymorphingintoavery exibleaircraftandavery exibleaircraftvirtuallymorphingintoastiffaircraft. Sincetheeffectsoftorsional stiffnesswerepreviouslyfoundtobeminor,onlybendingst iffnessismodiedandthe torsionalstiffnessisheldatthestiffvalueof1.0Nm 2 8.4.1ControlDesign Theloopisclosedonthetargetaircraftrst,followingthe methodusedin Section 8.3.1 .Acommandsignalissenttothetargetaircraftandtheresul tingoutput formsanewreferencesignalforthemodel-followingaircra ft.Thisreferencesignal, whencomparedtotheoutputofthemodel-followingaircraft ,formstheerrorsignal thatisdriventozero.Ablockdiagramofthismodel-followi ngapproachisillustratedin Fig. 8-18 .The Q and R weightingmatricesusedinthecontrolderivation,foreith erthe targetormodel-followingaircraft,aretheweightingmatr icesusedinSection 8.3.2 T 1 s K ff P K E com ref e u y x Figure8-18.Blockdiagramofamodel-followingcontroller usingLQR 8.4.2Results Theresultsforastiffaircrafttrackingtheresponseofae xibleaircrafttoa0.1rad pitchcommandareshowninFig. 8-19 .Thesystemresponsesoftheexibleandstiff aircraftastheyeachtrackthepitchcommandareshownforco mparisonpurposes. Thestiffaircrafttracksthecommandfasterandwithlowero vershootthanthe exibleaircraft.Asthestiffaircrafttrackstheexiblea ircraft'sresponsetothepitch 262

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0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 q [rad]Time [s] Command Stiff Command Flexible Command Stiff Flexible A -30 -25 -20 -15 -10 -5 0 5 10 0 0.2 0.4 0.6 0.8 1 d e [deg]Time [s] Stiff Command Stiff Flexible B 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 error [rad]Time [s] C Figure8-19.Astiffaircraftfollowingtheresponseofaex ibleaircrafttrackinga command.A)SystemresponseB)ElevatordeectionC)Error command,itsresponseslowsappropriatelytobeveryclosei nspeedandovershootto theexibleaircraft'sresponse. ThetimehistoryoftheelevatordeectionisshowninFig. 8-19B .Themaximum elevatordeectionismuchhigherintheresponseofthestif faircrafttrackingthe command,whichisconsistentwiththatresponse'sfasterri setime.Afterthatinitial response,theelevatordeectionofthestiffaircraftasit tracksthetargetexibleaircraft 263

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ismuchmoreoscillatory.Theseoscillationsareduetothef actthatthereferencesignal (theexibleaircraft'sresponsetothecommand)isoscilla tory.Theresultisthattherate ofchangeinelevatordeectionispossiblytoohighdependi ngonwhattypeofratelimits theactualactuatorissubjectto. Thefullhistoryoftheerrorsignalbetweenthestiffaircra ft'sresponseand theexibletargetaircraft'sresponseisshowninFig. 8-19C .Theerroroscillates considerablyintherst10secondsastheaircrafttracksth eoscillatingreference signal.After10seconds,theerrorisverynearzero. Theresponseoftheexibleaircraftasittracksthestiffai rcraft'sresponsetothe pitchcommandisshowninFig. 8-20 .Considerrstthattheresponseoftheexible aircrafttothecommandisalreadyslowerthanthestiffairc raft'sresponsetothe commandduetothelagthattheelasticstructureintroduces totheresponse.When theexibleaircrafttracksthestiffaircraft'sresponset othecommandinsteadofthe commanditself,itintroducesadditionallag,whichisevid entinFig. 8-20A .Aninteresting resultisthatastheexibleaircrafttracksthetargetresp onse,theoscillationsinthe responsearelowerthanifitweretrackingthepitchangleco mmand. Thetimehistoryoftheelevatordeectionisconsistentwit htheresultsinthepitch angleresponse.Astheexibleaircrafttracksthetargetre sponse,themaximumelevator deectionislowerandtheoscillationsarelowerinamplitu de. Thehistoryoftheerrorsignalshowshigherinitialerrorth anthepreviouscase,but theerrorsettlesdownslightlyfaster.8.4.3Summary Amodel-followingcontrollerisapotentiallyusefulwayto achievevirtualchanges toanaircraft'sstiffnessprovidedthatanadequatemodelo ftheaircraftwiththedesired levelofstiffnessexists.However,theseresultsshowthat itismostusefulinthecase ofthestiffaircraftfollowingtheresponseoftheexiblea ircraft.Inthatsituation,the 264

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0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 q [rad]Time [s] Command Stiff Command Flexible Command Flexible Stiff A -6 -5 -4 -3 -2 -1 0 0 0.5 1 1.5 2 d e [deg]Time [s] Flexible Command Flexible Stiff B 0 0.02 0.04 0.06 0.08 0.1 0 5 10 15 20 error [rad]Time [s] C Figure8-20.Aexibleaircraftfollowingtheresponseofas tiffaircrafttrackinga command.A)SystemresponseB)ElevatordeectionC)Error responseofthestiffaircraftdoescloselyapproximatethe responseoftheexible aircraft. Inthecaseoftheexibleaircraftfollowingthestiffaircr aft'sresponse,theresponse isnotabletocloselyfollowthestiffaircraft'sresponse. Infact,theresponsedriftsfurther awayfromthetargetbecausethetargetaircraft'sresponse introducesadditionallaginto thereferencesignal.However,thismethoddoesresultinad ecreaseinoscillationsin 265

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theexibleaircraft'sresponse,sincethereferencesigna lfromthestiffaircraftdoesnot containstrongoscillations. 8.5ConcludingRemarks AeroservoelasticdesignsynthesisfortheGenMAVaircraft wasanalyzed.Open andclosed-loopresultsshowastronginuenceofthestruct uralstiffnessontheight performance.Closed-loopresultsrevealtrackingperform ancedecreasesasthewing becomesmoreexible.Thisperformancedecreasesuggestst hattheoptimalLQR controllerchangesbasedonthestructuralstiffness.Aero elasticeffectsofwingstiffness onthephugoidmodewerefoundtodrivethesedecreasesinlon gitudinaltracking performance. Amodel-followingcontrolschemeisdevelopedtoexploreth eabilitytovirtually changethelevelofwingstiffness.Itisfoundthatthismeth odismostusefulforastiff congurationtrackingtheresponseofamoreexiblecongu ration.Ifanadequateplant modelexistsforthedesiredlevelofwingstiffness,modelfollowingcouldbeausefulway toapproximateitsresponsewithoutphysicallyalteringth estructure. 266

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CHAPTER9 AEROSERVOELASTICGUSTALLEVIATION Agust,whichissimplyachangeinthevelocityoftheairacti ngontheaircraft,can haveimportanteffectsonanaircraft'sstructuralloading ,fatiguelife,safety,stability,and ightcontrol.Formannedaircraft,passengercomfortandc argosafetyinthepresence ofwindgustsareanimportantconcern.ForsmallUAVsandMAV s,sensorstabilization isoftenacriticalissue.Manysensorssuchascamerasandta rgetingdevicesrelyona highlevelofdirectionalstabilitywhichcanbecompromise dbyawindgust. Animportantdistinctionexistsbetweengustsandturbulen ce[ 186 ].Turbulence isoftenconsideredtoberandomspatialuctuationsofthea irvelocitywhichare temporallycontinuous;however,agustisatemporallydisc retechangeinairvelocity inadeterministicmanner.Inotherwords,agustisadiscret eeventthattheaircraft encounterswhereasturbulenceisacontinuousphenomenont hataffectstheaircraft. Thetypicalgoalofgustalleviationistoreducetherootmea nsquare(RMS)valueof somepartoftheaircraft'sresponse(usuallyaload,moment ,oracceleration)[ 82 ].Two basicgustalleviationapproachescanbedistinguished. Therstapproachistopassivelyreducethegustsensitivit yoftheaircraftby alteringtheaircraft'sdesign.Forexample,changingthes izeandplacementofthe empennageortheverticalpositionofthewingcouldalterth eaircraft'ssensitivitytoa gust. Rudimentaryestimationsofanaircraft'spassiveresponse toagustdisturbance wereperformedasearlyasthe1930s[ 26 ].Laterresearchledtospeculationsthat aeroelasticeffectscouldbeusedtopassivelyalleviateth egustbyplacingtheelastic axisnearoraheadoftheleadingedgetomakethewingtwistin suchawaytoreduce theappliedload[ 34 ]. Thebenetofaforwardshearcenterinreducinggustsensiti vity(aneffectcalled wash-out)isdemonstratedwithatwodimensionalpitch-plu ngemodel[ 82 83 ].Results 267

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indicatethatwash-outisusefulinlimitingthepitchingco mponentofthegustresponse, whereaswash-incanlimittheheavingmotion.Theratioofbe ndingtotorsionalnatural frequenciesandthecenterofmasshavestrongeffectsonthe gustresponse. Wash-outisaviablemethodtoreducethegustsensitivityof MAVswithmembrane-type wings[ 159 ].Themembranepassivelyadaptstothegust,sheddingexces sliftand reducingtheoverallforcesactingontheaircraft.Further more,thismethodofgust alleviationallowseachwingtoadapttothegustdifferentl y,alleviatinganyasymmetryin theresponse. Anaeroelasticmodelofaveryexible,high-altitude,long -enduranceaircraftis usedtoshowthatgustsprimarilyexcitetheightmodesandl ow-frequencystructural modes[ 126 ].Afollowonstudycomparestheeffectsofpayloadsizeandg ustamplitude ontheaircraft'sgustsensitivity[ 167 ].Theonsetofstallisfoundtobeanimportant factorinthegustresponsefortheheavieraircraftwithlar gegustamplitudes. Anexperimentalinvestigationndsthatwingexibilityis generallybenecialfor reducingthegustsensitivityoflargeMAVs[ 70 ].Varyingtypesofgustsarecreated duringgroundtestingoftwoMAVs.Themeasuredforcesando w-eldresultsshowa reducedresponseoftheMAVwithaveryexiblewingascompar edtotheMAVwitha stiffwing. Thesecondapproachtogustalleviationistouseafeedbackc ontrolsystemto activelyrejecttheinuenceofthegustontheaircraft'str ajectoryandorientation. Importantconsiderationsforgustrejectionincludesatur ationlimitsoftheactuators, modelingtheeffectsofunsteadyaerodynamics,andperform anceinthefaceof modelingerrors[ 81 184 ]. AgustloadalleviationcontrollerisdesignedforanASEtra nsportaircraftwiththe objectiveofreducingwingbendingusingoutboardaps[ 76 ].Theutilityofthecontroller inreducingthegustisfoundtodependstronglyonthesensor designandplacement. Thecontrollerismoreeffectiveatrejectingthegustwhenm ultiplesensorsareused. 268

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Optimalcontrollersforgustalleviationaredevelopedand comparedusinganASE modeloftheB-52bomberwhichincludesveexiblemodes[ 11 ].Thecontrollersare abletoadequatelyreducetheimpactofthegustontheaircra ft'sverticalacceleration withreasonablecontrolsurfacedeections. Agustloadalleviationcontrollerwasanimportantpartoft hedevelopmentofthe B-2bomber[ 23 ].Thecontrolstrategyinvolvedrapidlypitchingtheaircr afttocontrol thedevelopingangleofattackandwasdevelopedusingacomb inationofclassicaland moderncontroltechniques. Variousgustloadalleviationcontrollersaredevelopedan dcomparedforavery exible,smallUAV[ 59 ].Thetimeresponseofaircraftparameterssuchaspitchang le andpitchrateinthepresenceofthegustareanalyzed.Eachg ustloadalleviation controllershowsanabilitytoimprovetheaircraft'srespo nsetoagustascomparedtoa standardightcontrolscheme. Large,heavyaircrafttypicallyoperateatspeedsmuchgrea terthantheguststhey encounter.Thus,changesinairowdirectionandspeedcaus edbyagustareminor comparedtothevelocityoftheaircraft.However,MAVscane ncountergustswhose amplitudesareonthesameorderastheirightspeed[ 120 ].Thesegustscandrastically changetheairowdirectionrelativetotheairframe,causi nglargechangesintheforces andmomentsactingontheaircraft.Whencombinedwiththelo winertiaofMAVs,these gust-inducedforcesandmomentscouldeasilyupsettheairc raft.Additionally,many intendedightprolesforaMAVinvolveyingthroughanurb anenvironmentwhere buildingsandtreescancreatefrequentgustencountersfor theaircraft. TheeffectsofgustsonMAVsbecomeevenmoreimportantwhenc onsidering theaeroelasticeffectsthatarepossibleinexibleMAVs.A sthewingbecomesmore exible,itcanabsorbsomeenergyofthegust,decreasingth esensitivityoftheoverall vehicletothegust.However,highlevelsofexibilitycoul ddelayacontroller-induced reactiontothegust,whichwouldreducethegustrejectiona bility.Balancingthese 269

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trade-offsinthedesignprocessrequiresunderstandingth eaeroservoelasticresponse oftheaircrafttoagust.Furthermore,theightdynamicsof anaeroelasticaircraftcan changedrasticallyanditisnotclearhowthesechangesinte ractwiththegustsensitivity andgustrejectionproperties. Thischapterwillevaluatechangesinthewingbendingstiff nessasamethodofgust alleviation.Boththeopen-loopgustsensitivityandclose d-loopgustrejectionproperties willbeanalyzed.Understandinghowtheaeroservoelastice ffectsofwingstiffness interactwiththegustalleviationpropertiescanallowthe gustresponsetobeconsidered asanintegralpartofthedesignprocess. 9.1Methodology 9.1.1GustModeling Agustissimplyachangeinoneormorecomponentsofthefrees treamvelocity. Inthisanalysis,onlyhorizontalorverticalgustsarecons idered,asshowninEqs.( 9–1 ) and( 9–2 ),where u 0 and w 0 aretheinitialvaluesand u and w arethegustdisturbances. u = u 0 + u (9–1) w = w 0 + w (9–2) Adiscretegustisdesignedusinga“one-minus-cosine”mode l.Thetemporal distributionofa w gust,forexample,atacertainfrequencyisobtainedfromEq .( 9–3 ), where g isthefrequencyofthegust. w ( t )= 1 2 A w (1 cos( g t )) (9–3) Thetime t variesfromzerototheperiodofthegust,asshowninEq.( 9–4 ). 0 t 1 g (9–4) Theamplitudeofthegust( A w )ischosentoproduceacertainchangeinangleofattack, whichallowsforusefulcomparisonsbetween u and w gustsbecausetheyresultin 270

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thesameangleofattackdisturbance.Amplitudesfor u and w gustsareobtainedfrom Eqs.( 9–5 )and( 9–6 ),where g isthedesiredchangeinangleofattackduetothegust and u 0 w 0 ,and 0 arethevaluesattrim.Theresultinggustispurelytranslat ional;no rotationalcomponentismodeled. A u = w 0 tan( 0 + g ) u 0 (9–5) A w = u 0 tan( 0 + g ) w 0 (9–6) Anexampleofa1HzgustisshowninFig. 9-1 .A u and w gustproleisshown alongwiththeangleofattackhistorywhichresultsfromeit herdisturbance.Notethat g ischosentobenegativeinordertoavoidhighangleofattack regimesinwhich nonlinearaerodynamicswouldnotbecapturedintheASWINGm odel. It'salsoimportanttonotethedifferenceintheamplitudes ofthe u and w gust disturbancesinFig. 9-1 .Ittakesamuchhighermagnitudechangein u velocity( u =10.8 m/s)thanitdoes w velocity( w =0.8m/s)toachievethisrelativelysmallchangein angleofattack.Thechangein u velocityisonthesameorderastheightspeedofthe aircraft,aphenomenonwhichcontributestothechallengin gnatureofgustsforMAVs. -2 0 2 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 Gust ParameterTime [s] d u [m/s] for a g =-3 d w [m/s] for a g =-3 a g [deg] Figure9-1.Examplesof u and w disturbanceswhicheachproducea1Hzgustwith g = 3 271

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Theresultinggustprolehasaspatialdimensionthatisuse fultoconsider.Since thefrequencyofthegustisconsideredrelativetotheairfr ameasitmovesthroughthe gust,thephysicallengthofthegustdependsontheaircraft speed.Forexample,forthe GenMAVaircraftyingat15m/s,a1Hzgusthasalengthof15me ters.A25Hzgust hasalengthof0.6meters,whichisequaltotheGenMAVwingsp an. Adiscretegustisessentiallyalow-frequencyphenomenon[ 151 ].Asthefrequency ofthegusteventsincreases,thegustsstarttobecomeindis tinguishablefromturbulence. Onemethodofdeningthehigh-frequencycutoffpointbetwe engustingandturbulence phenomenonusesthewingadvectiontime[ 11 70 126 ].Inthismethod,achangeinair velocityisconsideredtobeagustifitsperiodislessthant hewingadvectiontime.For theGenMAVaircraftat15m/swithachordlengthof12.7cm,th ewingadvectiontime is8.5 10 3 seconds.Thisadvectiontimeresultsinagustcutofffreque ncyof118Hz, whichisrelativelyhighcomparedtolargeraircraft.9.1.2GenMAVModel GustsareappliedtothelongitudinalmodeloftheGenMAVthr oughthedisturbances u and w asshowninEqs.( 9–1 )and( 9–2 ).Recallthebasicstatespaceformofthe modelshowninEqs.( 9–7 )and( 9–8 ).Thegustdisturbancesareaddedtothestate vector x whichisshowninEq.( 9–9 ).Thepositionandvelocityofthe i th structural nodearerepresentedby z i and z i ,where i =1 ;:::;n and n representsthenumberof structuralnodesmodeledinthesystem. x = Ax + Bu (9–7) y = Cx (9–8) 272

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x = 26666666666666666666666664 u 0 + u w 0 + w q z i z i ... z n z n 37777777777777777777777775 (9–9) Forsimplicity,considerthemultiplication Ax withonlyrigid-bodytermsasshownin Eq.( 9–10 ). Ax = 266666664 a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 377777775 266666664 u 0 + u w 0 + w q 377777775 (9–10) Thecontributionofthegustdisturbancetothestatedynami cscanthusbeseparatedas showninEq.( 9–11 ). Ax = 266666664 a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 377777775 266666664 u 0 w 0 q 377777775 + 266666664 a 11 a 12 a 21 a 22 a 31 a 32 a 41 a 42 377777775 264 u w 375 (9–11) Equation( 9–11 )showsthattheimpactofthegustdisturbanceonthestatedy namicsis capturedbythecolumns1and2ofthestatematrix,whicharet hesamecolumnsthat modeltheimpactof u and w onthestatedynamics. Astatespacemodeloftheaircraftwhichincludesthegustdi sturbancesisshownin Eqs.( 9–12 )and( 9–13 ),wherethematrix F consistsofthersttwocolumnsof A and 273

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thedisturbancevector w isdenedinEq.( 9–14 ). x = Ax + Bu + Fw (9–12) y = Cx (9–13) w = 264 u w 375 (9–14) Forconvenienceintheimplementation,theinputsanddistu rbancescanbe combinedintoanewinputmatrix B andinputvector u ,denedinEqs.( 9–15 ) and( 9–16 ) B = BF (9–15) u = 264 u w 375 (9–16) Thissimplicationresultsinthefamiliarstatespaceform inEqs.( 9–17 )and( 9–18 ). x = Ax + B u (9–17) y = Cx (9–18) Thenalmodelisrestrictedtothelongitudinalstatedynam icswhichinclude phugoidandshortperiodmodes,rstsymmetricandanti-sym metricbendingmodes, andthefourslowestunsteadyaerodynamicpoles.9.1.3ControlDesign Forgustrejectionanalyses,aclosed-loopregulationsyst emisdesignedas depictedinFig. 9-2 Thegainmatrix K isdesignedusingLQRwithasimpliedsynthesismodelthat doesnotincludethegustdisturbances u and w .RecallthatLQRcalculatestheoptimal 274

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P K e ( gust ) w u y x + Figure9-2.Blockdiagramofagustrejectioncontroldesigngainmatrix K tominimizethequadraticcostfunctiongiveninEq.( 9–19 ). J ( u )= Z 1 0 x T Qx + u T Ru dt (9–19) Theweightingmatrices Q and R arechosentobalanceperformanceandactuation andaregiveninEqs.( 9–20 )and( 9–21 ),where q i =1e6foreachofthestructuralnodes. Theseweightingmatricesareheldconstantforeveryplantm odelinthestructural designspace.Thisapproachresultsingainsontherigidbod yvariablesasshownin Fig. 9-3 Q = diag 111 e 51 e 5 q 1 :::q n (9–20) R =0 : 1 (9–21) 9.1.4DesignSpace Themainparameterofinterestisthebendingstiffnessofth ewing,whichisvaried from1.0Nm 2 to0.07Nm 2 inkeepingwiththedesignspaceofChapters 5 and 8 Becauseofthelackofeffectfromtorsionalstiffnessintho sechapters,onlybending stiffnessisanalyzedinthischapter;torsionalstiffness remainsattherelativelyhighlevel of1.0Nm 2 DiscretegustsareevaluatingbycomparingRMSvaluesofthe responseparameter ofinterest.TheRMSvalueisastatisticalmeasureofthemag nitudeofthevariancein theresponseandiscalculatedaccordingtoEq.( 9–22 ),where i variesfrom 0 tothe 275

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-1600 -1400 -1200 -1000 -800 -600 -400 -200 0 200 0 0.2 0.4 0.6 0.8 1 Gain [-]EI [Nm 2 ] u w q q Figure9-3.LQRgainsfortheGenMAVaircraftwithvaryingle velsofstiffness naltime t inintervalsofthetimestep t .Bothhorizontalandverticaldiscretegusts areanalyzed,whereeachgustisdesignedtocreateanangleo fattackdisturbanceof -3 .Thefrequencyofthegustischosentomatchtheaircraft'sp hugoidorshortperiod naturalfrequencies. y rms = r 1 n X y 2 i (9–22) Theeffectsofgustsarealsoanalyzedusingatransferfunct ionapproach.Transfer functionsfroma u or w inputareevaluatedwithaBodemagnitudeplotwhich expressesthemagnitudeofthefrequencyresponsegainfrom thegusttotheoutput variable.Thisapproachallowsthegustresponseoftheoutp utvariabletobeeasily evaluatedacrossarangeoffrequenciesbutblursthedistin ctionbetweenturbulence andgusts.Thetransferfunctioninputhasqualitiesofagus tinthatit'sadeterministic disturbancebutalsoqualitiesofturbulenceinthatit'sac ontinuousinput. 9.2GustSensitivity Theconceptofgustsensitivityrelatesthemagnitudeofthe open-loopaircraft responsetothemagnitudeofthegustdisturbance.Lowgusts ensitivityisdesirable, 276

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wherethegoalistomaintainsufcientairspeedandsuitabl ysmalldeviationsfromthe nominalsideslipandangleofattackvaluesinresponsetoag ust.Thegustsensitivityis analyzedbylookingattransferfunctionsoftheopen-loopa ircraftandRMSvaluesofthe aircraftresponsetoadiscretegust.9.2.1FrequencyResponse Inthegusttoangleofattacktransferfunctions,thesteady stateresponseinangle ofattackmagnitudenaturallyincreasesasstiffnessdecre ases.Thiseffectwasseenin Chapter 5 andisduetoadecreaseinliftasthewingbends.Thiseffectw asmanually removedfromthedatabysubtractingtheincreaseinangleof attackatsteadystate fromtheentirefrequencyrangeforeachlevelof EI .Thiscorrectionmakesthegust sensitivityeffectsatthehigherfrequenciesmoreclearan dcomparableacrosstherange ofstiffnessvalues. Considerahorizontalgust.Thetransferfunctionfrom u totheangleofattackand pitchrateoftheaircraftareshowninFig. 9-4 .Bothtransferfunctionsshowsignicant changesasthestiffnessdecreases. 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 3 6 9 12 15 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 40 80 120 160 200 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-4.Gustsensitivityinresponsetoa u -gust.A)AngleofattackB)Pitchrate 277

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Inthe u to transferfunctioninFig. 9-4A ,thegustsensitivityatthephugoidnatural frequency(near0.1Hz)remainsaboutthesameasstiffnessd ecreases.Intheregion betweenthephugoidandshortperiodmodes,theveryexible aircraft'sangleofattack isanaverageof20%lesssensitiveto u guststhanthestiffaircraft.Attheshortperiod frequencyaround3Hz,thereisatrendtowardlowergustsens itivityformoreexible aircraft,resultingina25%reductionintheresponsemagni tude. Inthe u to q transferfunctioninFig. 9-4B ,thegustsensitivityatthephugoid modefrequencydecreasesby65%asthestiffnessdecreases. Thegustsensitivity inbetweenthephugoidandshortperiodfrequenciesisanave rageof43%lower fortheexibleaircraftthanforthestiffaircraft.Gustse nsitivityattheshortperiod modefrequencydecreasestoaminimumat EI =0.15Nm 2 ,afterwhichitrisesslightly althoughstillremainsbelowthegustsensitivityofthesti ffaircraft.Overall,thevery exiblecongurationhasa39%reductioninthegustrespons eascomparedtothestiff conguration.Abovetheshortperiodnaturalfrequency,th erearesharprisesinthe responseasthegustexcitesthestructuralmodes. Forverticalgusts,thetrendsarequitedifferentfromahor izontalgust.Inthe w to transferfunction,showninFig. 9-5A ,thereisapeakatthephugoidfrequencyfor veryexibleaircraftthatrepresentsanincreaseingustse nsitivityof39%fromthestiff aircraft.Attheshortperiodfrequency,thegustresponser isesgraduallyasthestiffness decreases,reachingapeakthatis29%higherthanforthesti ffaircraft. Inthe w to q transferfunctioninFig. 9-5B ,theresultscanchangedramatically basedonthegustfrequency.Thegustsensitivityoftheexi bleaircraftdecreasesby about47%atthephugoidnaturalfrequencybutincreasesby1 11%attheshortperiod frequency.Forveryexibleaircraft,thestructuralmodes cancauseahighdegreeof sensitivitytogustsaround10Hz.Inbetweenthephugoidand shortperiodfrequencies, thegustsensitivityisrelativelyunchangedfromstiffto exibleaircraft. 278

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0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 3 6 9 12 15 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 40 80 120 160 200 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-5.Gustsensitivityinresponsetoa w -gust.A)AngleofattackB)Pitchrate Itisalsopossibletoevaluatetheimpactofa u or w gustonthewingbendingby examiningthegusttransferfunctiontothewingtipvelocit y( z ),showninFig. 9-6 .For a u gust,onlysmallchangesinwingbendingbetweenstiffande xibleaircraftare 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 1 2 3 Magnitude [(m/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(m/s)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 1 2 3 Magnitude [(m/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(m/s)/(m/s)] B Figure9-6.Responseofwingtipvelocitytothegustdisturb ance.A) u -gustB) w -gust 279

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observed.However,a w gustnearthestructuralfrequenciescanstronglyexciteth e bendingmode. Recallingthattheessenceofalongitudinalgustisachange inangleofattack, gustscanalsobemodeledthroughadisturbanceintheelevat orinputbecauseofits effectontheforcesandmoments.Gusttransferfunctionsfo rthismethodaredepicted inFig. 9-7 .Forthe e to transferfunction,theresultsaresomewhatdifferentthan a u or w gust.Thegustsensitivityincreasesslightlyaroundtheph ugoidandshortperiod modesfortheexibleaircraft.Inthefrequencyrangebetwe enthephugoidandshort periodmodes,thegustsensitivitydecreasesslightly. The e to q transferfunction,showninFig. 9-7B ,showssimilartrendstothe w gust. Thegustsensitivityatthephugoidmodefrequencydecrease sby84%asthestiffness decreases.Attheshortperiodmode,thegustsensitivityin creasesby76%asthewing stiffnessdecreases. 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 3 6 9 12 15 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-7.Gustsensitivityinresponsetoa e -gust.A)AngleofattackB)Pitchrate Overall,thegustresponseseemstobemoresensitiveto w guststhan u gusts.This resultshouldbeinterpretedcautiously,however,because someofthisbehaviorcomes fromthefactthataunitchangein w willproduceahigherchangeinangleofattackthan 280

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aunitchangein u .Intrimata V 1 of15m/s,theGenMAVhasa u velocityof14.88m/s anda w velocityof-1.87m/s.Aunitincreasein u resultsina6%changein whilea unitincreasein w resultsina54%change. Foranyvalueof EI ,theshortperiodmodeismoreeasilydisturbedbyagustthan thephugoidmode.Theshortperiodmodeisprimarilycompose dofoscillationsinangle ofattack.Sincetheessentialeffectofagustistochangeth eangleofattack,thismode iseasilydisturbedbythegust,causinglargeresponsesina ngleofattackandpitchrate. Incontrast,angleofattackremainsnearlyconstantinatyp icalphugoidmotion,making thismodeharderforagusttodisturb. Thereisadistinctdifferencebetweenthegustsensitivity atthephugoidandshort periodnaturalfrequenciesinresponsetoaverticalgust.F orthepitchrateresponse,the gustdecreasesas EI decreaseswhenthegustisnearthephugoidfrequency.Atthe shortperiodfrequency,however,theaircraftbecomesmuch moresensitivetoagustas EI decreases.Thedampingofthesemodesmaybeeffectingthisb ehavior.Asstiffness decreases,thedampingofthephugoidmodeincreasesby56%w hilethedampingof theshortperiodmodedecreasesby54%.Thehigherdampingof thephugoidmode coulddecreasethegustsensitivityatitsfrequencywhilet helowerdampingoftheshort periodmodecouldincreasethegustsensitivityatitsfrequ ency. Totesttheeffectsofmodaldamping,theaircraftmodelsare modiedtoholdthe dampingofthephugoidandshortperiodmodesconstantassti ffnessdecreases.The resultsfora w gustareshowninFig. 9-8 .Whilethegustsensitivityatthephugoid frequencyislargelyunchanged,thetrendsattheshortperi odarereversedfromthose seeninFig. 9-5 Percentchangesfromhightolow EI inFig. 9-8 arecomparedtoFig. 9-5 in Table 9-1 .Thesepercentchangesshowthattheincreaseingustsensit ivityattheshort periodfrequency,foreitherthe or q response,comesmostlyfromthedecreased dampingoftheshortperiodmode.Withoutthatdecreaseinda mping,theexibleaircraft 281

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0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 3 6 9 12 15 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 40 80 120 160 200 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-8.Gustsensitivitytoa w gustwithconstantphugoidandshortperioddamping. A)AngleofattackB)Pitchrate displayatrendtowardlowergustsensitivityinsteadofhig hergustsensitivity.The changeinmodaldampingdidnothaveaneffectonthegustsens itivityatthephugoid frequency.Table9-1.Percentchangesfromhightolow EI inresponsetoa w gust g responsetrue constant phugoid 39%39% phugoid q -47%-47% shortperiod 29%-16% shortperiod q 111%-11% Theeffectsofdampingwerealsoevaluatedwitha u gust,asshowninTable 9-2 Thedecreaseinshortperioddampingwithdecreasing EI actstoincreasethegust sensitivityoftheaircraftrelativetotheresultswithcon stantdamping.However,even withthedecreaseddamping,thegustsensitivityoftheexi bleaircraftatthisfrequency isreducedcomparedtothestiffaircraft.Thephugoiddampi nghasnomajoreffecton thegustsensitivity. 282

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Table9-2.Percentchangesfromhightolow EI inresponsetoa u gust g responsetrue constant phugoid 3%3% phugoid q -65%-65% shortperiod -25%-36% shortperiod q -39%-70% Anotherinuenceonthegustsensitivitycouldbefromaeroe lasticeffectsonthe phugoidandshortperiodmodeshapesasstiffnessdecreases .Forthephugoidmode, asthebendingstiffnessdecreasesthecontributionofangl eofattackandpitchrateto themodeshapeincreaseby27%and25%,respectively.Thisun usualmodeshape couldinuencehowthatmoderespondstoagust.Fortheshort periodmodeatlow valuesofstiffness,themodeshapeexperiencesonlyminorc hangesin and q AnecdotalevidencefromMAVpilotsseemstoindicatethatin creasedexibility lowerstheaircraft'ssensitivitytoagust[ 69 71 ].Ingeneral,theresultsfromahorizontal gustinFig. 9-4 agreewiththeseobservations.However,theresultsforave rticalgustin Fig. 9-5B bothagreeanddisagreewithpilots'comments,dependingon thefrequencyof thegust.Twofactorsthatmightcontributetothisdifferen cehavetodowithwhichtypes ofgustsaremoreprobableandhowapilotremotelyobservesa gust. First,itislikelythatMAVpilotsencounterhorizontalgus tsmoreoftenthanvertical gusts.Pilotstypicallychoosetoyinwide,openeldswhil eavoidingightnear obstaclessuchasbuildingsortreelines.Verticalgustste ndtobeproducedbythe windpassingovertheseobstacles,producingdownwashandv orticesontheleeward side.Anopeneldhasnoobstaclestocreatetheseverticalg ustsandthegustsmay thereforebeprimarilyhorizontalinnature.Thermalupdra ftscouldimposeverticalgusts onMAVsinanopeneld,butthegroundandatmosphericcondit ionsmustbecorrect. Thisreasoningcouldindicatethatpilotreportsmaybebias edtowardhorizontalgusts andthusagreewiththeresultsforhorizontalgustsensitiv itypresentedhere. 283

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Unfortunately,itisdifculttoempiricallyknowwhetherh orizontalgustsaremore commonthanverticalgusts.Thevastmajorityofexistingkn owledgeaboutgustsin theatmosphericboundarylayerarerelevantmainlytostruc turesthatareordersof magnitudelargerthanmicroairvehicles.Workhasbeendone tocharacterizethelevels ofturbulencethataMAVmightencounterbutnotthenatureof thediscreteguststhat aremostcommon[ 179 ]. Second,aremotepilotreliesonvisualcuestodetectdistur bancestotheaircraft. Aphugoidmotionischaracterizedbysignicantchangesina ttitudeandaltitudewhile ashortperiodmotionischaracterizedbysmall,fastchange sinangleofattack.Asa result,aphugoidmotioniseasiertoseethanashortperiodm otionandthusthepilot reportsmaybebiasedtowardtheeffectsoflowfrequencygus ts. 9.2.2TimeResponse Thegustsensitivityofaircraftwithvaryingwingstiffnes sisalsoanalyzedinthe timedomainbyapplyingadiscretegustandanalyzingtheRMS valuesoftheresponse. Recallthatbothgusttypes( u and w )aredesignedtoproducethesameamountof changeinangleofattack. Thepitchrateresponsefora u gustatthephugoidfrequencyisshowninFig. 9-9 Theresponseoftheexibleaircrafthasasimilarsettlingt imebutlowerRMSvaluethan thestiffaircraft.Thestiffaircraft'sresponsehasaRMSv alueof0.057rad/swhereasthe exibleaircraft'sresponsehasaRMSvalueof0.043rad/s,a 24%decrease.Thelower RMSvalueindicatesthatthegustdisturbanceaffectedthe exibleaircraftlessthanthe stiffaircraft,whichisclearlyseeninthetimehistory.Th econclusionthattheexible aircraftislesssensitivetogustsatthisfrequencyfollow stheconclusiondrawnfromthe frequencyresponsedatainFig. 9-4B TheRMSvaluesofthepitchrateresponsefor u and w gustsnearthephugoidand shortperiodfrequenciesareplottedagainstthewingstiff nessinFig. 9-10 .Wheneither a u or w gustencounterstheaircraftnearthephugoidmodefrequenc y,theRMSvalues 284

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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 Pitch Rate [rad/s]Time [s] Stiff Flexible Figure9-9.Pitchrateresponsetoa u gustatthephugoidmode'snaturalfrequency conrmadecreaseingustsensitivityforexibleaircraft. Thedecreaseingustsensitivity ismuchgreaterfora u gustthana w gust(-77%RMSvs-23%RMS,respectively). Thesetrendsagreewiththedecreasinggustsensitivityfor exibleaircraftseenaround thephugoidnaturalfrequencyinFigs. 9-4B and 9-5B Foragustwhichoccursaroundtheshortperiodnaturalfrequ ency,theRMSvalues forthesegustsshowdifferingtrends.TheRMSvaluedecreas esforthe u gustbut increasesforthe w gust.ThisresultfollowsthedifferingtrendsseeninFigs. 9-4B and 9-5B attheshortperiodnaturalfrequency. 9.2.3AerodynamicDerivatives Staticstabilityderivativesindicatetheinitialtendenc yofanaircraftasitresponds toadisturbance.Assuch,theyaresometimesusedasindicat orsofanaircraft'sgust sensitivity.Inparticular,the C L C m ,and C m u derivativesareusedtogaugethe longitudinalgustsensitivityofanaircraft[ 7 169 ].Someamountofgustinsensitivity resultsfromtheaircraftdesignprocesswhichisfocusedon ndinggoodaerodynamic stabilityderivativesforuncontrolledightinsmoothair Therelativechangeofthesederivativeswithrespecttocha ngesinwingbending stiffnessisshowninFig. 9-11 .The C L derivativedecreasesasbendingstiffness 285

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.2 0.4 0.6 0.8 1 RMS [rad/s]EI [Nm 2 ] d u w g =phugoid d w w g =phugoid d u w g =short period d w w g =short period Figure9-10.GustsensitivityRMSvaluesforthepitchrater esponse decreases,indicatingthattheaircraftproduceslesslift inresponsetoachangeinangle ofattack.Decreasing C L isinterpretedasareductioningustsensitivityforlowerv alues ofbendingstiffness.Themagnitudeofthe C m derivativeincreaseswithdecreasing stiffness,indicatingastrongerrestoringmomentandared uctioningustsensitivity. Thepitchingmomentduetoforwardspeed( C m u )istypicallynegativeduetothe aerodynamiccentermovingaftasairspeedincreases.Howev er,forveryexibleaircraft -4 -2 0 2 4 6 8 0 0.2 0.4 0.6 0.8 1 Coefficient [-]EI [Nm 2 ] C L a C m a C m u Figure9-11.Aerodynamicderivativesrelevanttogustsens itivity 286

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congurations,thevalueof C m u becomespositive,indicatingatendencytopitchup asforwardspeedincreases.Thisbehaviorisrelevantfor u gusts,sinceapositivegust wouldincreasetheforwardspeed.Theincreaseofthisderiv ativesuggestsincreasing sensitivityto u gustsasstiffnessdecreases,whichdisagreeswiththeresu ltsobserved inFig. 9-4 Thetrendsfromtheaerodynamicderivativesdonotcapturet hesamelevelof detailthatthefrequencyandtimeresponsemethodscapture .Thederivativesshow agreementwithtransferfunctionresultsinsomeareasofth efrequencyspectrumbut notinothers.Theaerodynamicderivativesdonotshowanytr endsthatcorrespondto thehighergustsensitivityathighfrequencies. 9.3GustRejection Gustrejectionisamethodofgustalleviationinwhichthego alistoreducethe inuenceofagustontheaircraftresponsebyusingafeedbac kcontrolsystem.The gustrejectionpropertiesoftheaircraftinthedesignspac earejudgedbyexamining transferfunctionsoftheclosed-loopaircraft,wherealow er-magnituderesponse indicatesastrongergustrejection.RMSvaluesofthetimer esponsearealsoevaluated. 9.3.1FrequencyResponse Transferfunctionsfora u gustareshowninFig. 9-12 ,whereahighlevelofgust rejectionisindicatedbyalowmagnituderesponse.Forthea ngleofattackresponsein Fig. 9-12A ,thecontrollerisabletomitigatemostofthechangeinangl eofattackdue tothegust.Inthehighfrequencyrange,thegustrejectioni mprovesformediumvalues ofstiffnessandthendecreasessharplyatverylowvaluesof stiffness.Fromthestiff aircrafttothemostexible,theaveragegustrejectiongoe sdownby36%. InthepitchratetransferfunctioninFig. 9-12B ,thecontrollerisalsoableto rejectthegustconsistentlyacrossthefrequencyspectrum andstiffnessrange.The responsestillrisessharplyinthefrequencyrangeofthest ructuralmodes.However,the magnitudeofthisresponseisstilllowerthanthemagnitude oftheopen-loopresponse 287

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0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 40 80 120 160 200 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-12.Gustrejectioninresponsetoa u -gust.A)AngleofattackB)Pitchrate showninFig. 9-4B ,indicatingthatthecontrollerhassomeabilitytomitigat ethegust evenwhenitisnearinfrequencytothestructuralmodes. Theabilityofthecontrollertorejectthe u gustonthepitchrateresponseatlow frequenciesdecreasesslightlyaswingstiffnessdecrease s.However,inthestructural frequencyrange,gustrejectionimprovesatverylowvalues ofstiffness. Inthe w to transferfunctionshowninFig. 9-13A ,thegustrejectionproperties varyacrossthestiffnessandfrequencyranges.Forstiffan dexibleaircraft,the controllerachievesbettergustrejectionatlowfrequenci esthanathighfrequencies. Forveryexibleaircraft,thegustrejectiondecreasesbya bout29%fromlowtohigh frequency. Inthe w to q transferfunctioninFig. 9-13B ,thegustrejectionresultsaresimilarto thosefromthe u gustinFig. 9-12B .Thelevelofgustrejectionremainssomewhatsteady asstiffnessdecreases,exceptforaminimumthatisreached around EI =0 : 2 Nm 2 Thecontroller'sgustrejectionhasaneffectonthewingtip velocity( z ),shownin Fig. 9-14 .Acrossthedesignspace,thewingtipvelocityexperiences verylittleeffect 288

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0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 -5 0 5 10 15 20 Magnitude [(deg)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 40 80 120 160 200 Magnitude [(deg/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(deg/s)/(m/s)] B Figure9-13.Gustrejectioninresponsetoa w -gust.A)AngleofattackB)Pitchrate fromthegust,exceptatthestructuralmodefrequency.When comparedtoFig. 9-6 ,the closed-loopwingtipvelocitiesareloweronaveragebecaus eofthecontroller. 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 1 2 3 Magnitude [(m/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(m/s)/(m/s)] A 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 0 1 2 3 Magnitude [(m/s)/(m/s)] w [Hz] EI [Nm 2 ]Magnitude [(m/s)/(m/s)] B Figure9-14.Responseofwingtipvelocitytothegustdistur bance.A) u -gustB) w -gust 289

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Overall,stiffnesshasaminoreffectontheabilityoftheco ntrollertorejectthegust. Effectsofstiffnessaremostnoticeablewhenagustexcites thestructuralmodes,a situationwhichishardforthecontrollertonegate.9.3.2TimeResponse TheclosedloopRMSvaluesversuswingstiffnessareshownin Fig. 9-15 .The trendsinRMSvaluesfortheclosed-loopresponsesgenerall yagreewiththetrends fromthefrequencyresponses.Ingeneral, u gustshaveahigherRMSvalueacrossthe stiffnessrange,whichisnoteworthybecausethediscrete u and w gustsaredesigned toproducethesameangleofattackdisturbance.AhigherRMS valuemayindicatethat thecontrollerisnotbeabletorejecta u gustaswellasa w gust. Dependingonthestiffnessofthewing,thegustrejectionpr opertiesmayincrease ordecrease.AlloftheresponsesreachamaximuminRMSvalue saround EI = 0 : 2 Nm 2 ,indicatingaminimuminthegustrejectionabilityoftheco ntrolleratthis point.Inthemostexiblecongurations,theRMSvaluesare lowerthanthestiff congurations,indicatingimprovedgustrejection. 0 0.02 0.04 0.06 0.08 0.1 0 0.2 0.4 0.6 0.8 1 RMS [rad/s]EI [Nm 2 ] d u w g =phugoid d w w g =phugoid d u w g =short period d w w g =short period Figure9-15.GustrejectionRMSvaluesforthepitchrateres ponse 290

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9.4Summary Theeffectsofagustonthelongitudinaldynamicsofanaeros ervoelasticaircraft dependonthedirectionandfrequencyofthegust.Afrequenc y-domainapproachto analyzingthegustresponsecanrevealsomeoftheseintrica ciesandagreeswellwith traditionalRMSresults. Forahorizontalgust,theexibleaircraftislesssensitiv einangleofattackandpitch ratethanthestiffaircraft.Thisresultholdstrueforgust satanyfrequency. Foraverticalgust,theexibleaircraftislesssensitivei npitchratetothegust whenthegustfrequencyisnearthephugoidmode.Withagustf requencynearthe shortperiodmode,theaircraft'spitchrateismuchmoresen sitivetothegust.The angleofattackresponseismoresensitivetothegustatboth phugoidandshortperiod frequencies. Attheshortperiodfrequency,theaircraft'sgustsensitiv ityisobservedtobe stronglyinuencedbythechangesinthemodaldampingassoc iatedwithincreasing wingexibility.Whenthegustsoccuratthephugoidfrequen cy,theresponsewasnot dependentonthemodaldamping.Instead,changesinthephug oidmodeshapedueto theincreasedwingexibilitymayberesponsibleforthecha ngesingustsensitivity. ForabasicLQRcontrollerusingconstantweightingmatrice sanddesigned withoutforeknowledgeofthegustcharacteristics,thegus trejectionpropertiesare veryacceptableatalllevelsofstiffness. Usingthestaticstabilityderivativestogaugethegustsen sitivityisoflimited usefulnessforaeroelasticaircraft.Thederivativesagre edwithresultsinsomeareasof thefrequencyspectrumbutdisagreedwithotherareas. 9.5ConcludingRemarks Awing'sbendingstiffnesscanhavealargeeffectthegustal leviationproperties oftheaircraft.Bothgustsensitivityandgustrejectionca nincreaseordecreasebased onthetypeandfrequencyofthegust.Theaeroelasticeffect songustsensitivitycome 291

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mainlythroughchangesinthedampingandshapeofthetradit ionalightmodes. Tailoringthesemodescouldreduceoraccentuatethechange singustsensitivityas desiredbytheaircraftdesigner. Ingeneral,wingexibilityhasonlyaminoreffectonthegus trejectionproperties oftheaircraft.Theimplicationisthatduringanaeroservo elasticdesigneffortforgust alleviation,thestructuraldesignfocuscouldbeongustse nsitivitywhileknowingthatthe impactongustrejectionwillbeminor. Foreknowledgeoftheaeroservoelasticinteractionsbetwe enthetraditionalight modes,structuralmodes,andgustalleviationpropertiesc anallowtheaircraftdesign processtoproduceanaircraftthatisspecicallytailored forcertainqualitiesofgust alleviationintheintendedightenvironment. 292

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CHAPTER10 CONCLUSIONS 10.1ResearchSummary Thisresearchconnectsseveralaspectsofaeroservoelasti citytotheaircraftdesign processforaxed-wing,exiblemicroairvehicle.Theprim arydesignparameters consideredarethebendingandtorsionalstiffnessofthewi ng,althoughthelocationof thewing'selasticaxisisalsoconsidered. Ingeneral,theaeroelasticeffectsofwingstiffnessonthe ightdynamicsare highlynonlinear.Foranelasticaxislocationwhichisclos ethethecenterofpressure, theeffectsofchangingtorsionalstiffnessareminorandth elevelofbendingstiffness dominatestheaeroelasticinteractions.Themaineffectof uniformlyloweringbending stiffnessistochangetheshapeofthewingattrim,whichlea dstochangesinthetrim angleofattackandelevatordeection.Iftheseaeroelasti ceffectsarenotaccountedfor inthedesignprocess,theaircraftmaytrimmuchdifferentl ythanexpected,leadingtoan inefcientightcondition. TheightmodesoftheGenMAVaresignicantlyaffectedbych angesinthe bendingstiffnessofthewing.Largechangesareseeninboth thefrequenciesand dampingratiosoftheoscillatoryightmodes.Thebendings tiffnessmainlyaffectsthe lateral-directionalmodesofightthroughanincreaseint heeffectivedihedralangledue toincreasedwingtipdeection.Thedirectionandmagnitud eoftheeffectvariesgreatly betweenmodes.Forexample,loweringbendingstiffnesscau sesanincreaseindutch rolldampingbutadecreaseinshortperioddamping.Reducin gthebendingstiffness canalsoaffectthelongitudinalmodesbecauseofthereduce dfrequencyseparation withtherstbendingmode,whosenaturalfrequencydecreas esasbendingstiffness decreases. Thebendingstiffnesscanalsohaveaprominenteffectonthe shapeoftheight modes.Non-traditionalmodeshapesresultingfromdecreas edbendingstiffness 293

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areobservedinthedutchrollmodeandphugoidmode.Fromaha ndlingqualities perspective,aircraftdesignistypicallyfocusedontheda mpingoftheightmodes. However,changesinthemodeshapemayintroduceanewconsid erationfortheaircraft designer. Elasticaxislocationswhicharefurtherawayfromthecente rofpressureresultin strongereffectsfromchangesintorsionalstiffness.Decr easingtorsionalstiffnesshas astrongereffectonthelongitudinalmodesthanthelateral directionalmodes,which comesthroughchangesintheangleofattackandpitchingmom entonthewing.An elasticaxislocationforwardofthecenterofpressureresu ltsinwash-outwhereas anelasticaxislocationaftofthecenterofpressureresult sinwash-in.Incertain congurationsandightconditions,thewash-incanresult inutterordivergence. Whenapplyingnon-uniformdistributionsofstiffnesstoth ewing,itisfoundthat thereisaportionoftheinnerspanoverwhichchangesinstif fnesscanproducethe majorityoftheeffectofchangesmadetotheentirespan.Thi slengthistermedthe aeroelasticspanandisfoundtobebetween30%and40%ofthet otalspanforthetrim conditionsandeffectonightmodes.Itwasfurthertheoriz edthatifonlytheinner10% spancanbeaffectedbychangesinstiffness,thereisanaddi tionalincreaseordecrease instiffnesswhichcouldbeappliedtoobtainthemajorityof theeffectofthechangeto thewholewing.Indeed,itisfoundthatasmalladditionalde creaseinstiffnessinthe innerspanisenoughtoachievethemajorityoftheeffectofc hangingstiffnessacross theentirespan. Battensarefoundtohaveastrongandcomplexeffectonthei ghtperformance ofmicroairvehicles.Reducingthenumberofbattensimprov eslongitudinalstatic stability;conversely,thisreductionalsorequireslarge ranglesfortrimanddegrades staticstabilityinthelateralanddirectionalaxes.Theef fectsareequallydiversewhen consideringtheightdynamics.Thedutch-rollmodehasimp roveddampingwhilethe short-periodmodehasdegradeddamping(forafewernumbero fbattens).Furthermore, 294

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thenatureofthemodeschangeasthebattencongurationcha ngesandtheaircraft iesnoticeablydifferent. Thedynamicaeroelasticpropertiesofamembranewingareex perimentally investigatedinlowReynoldsnumberairows.Thedynamicde rivatives,specically thepitch,lift,anddragdampingderivativesaremeasuredi nresponsetodifferent levelsofmembranepre-tensionanddynamicchangesinpitch angleorangleofattack. Anoticeabledifferenceofliftanddragbetweentherigidan dexiblewingsinstatic anddynamicconditionsisobserved.Thelift-to-dragratio wasalsoinuencedbythe dynamicparameters.Substantialchangesinthepitchingmo mentcoefcientswere observedinthepresenceofdynamicchangesinangleofattac korpitchangle. Aeroservoelasticdesignsynthesisusingwingstiffnessas themainparameter isdeveloped.Openandclosed-loopresultsshowastrongin uenceofthestructural stiffnessontheightperformance.TheperformanceoftheL QRcontrollerchanges considerablyasthewingstiffnesschanges.Amodel-follow ingcontrolschemeis developedtoexploretheabilitytovirtuallychangethewin gstiffness.Whileitiswas foundthatthestiffaircraftcouldapproximatetheightre sponseoftheexibleaircraft, thereversewasnotfeasibleinthisexample. Thebendingstiffnessofthewingcanhavealargeeffecttheg ustalleviation propertiesoftheaircraft.Bothgustsensitivityandgustr ejectioncanincreaseor decreasebasedonthetypeandfrequencyofthegust.Theaero elasticeffectsmainly comethroughchangesinthedampingandshapeofthetraditio nalightmodes. Tailoringthesemodescouldreduceoraccentuatethechange singustsensitivityas desiredbytheaircraftdesigner.Foreknowledgeoftheaero servoelasticinteractions betweenthetraditionalightmodes,structuralmodes,and gustalleviationproperties canallowtheaircraftdesignprocesstoproduceanaircraft thatisspecicallytailoredfor certainqualitiesofgustalleviationintheintendedight environment. 295

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Thepresenceoftheseeffectshighlightstheimportanceofc onsideringelasticand rigidbodydegreesoffreedomtogetherforproperdesignana lysisofexibleaircraft. ThisproblemisespeciallyrelevantforMAVs,whosedesigns maybeinherentlyexible. Byconsideringtheseeffectsinthedesignprocess,thestru cturalstiffnessmayserveas anadditionaldesignparametertoachieveadesiredightch aracteristic. 10.2FutureResearch Therearenumerousopportunitiesforfurtherresearch.For example,thefuselage andempennagewereheldrigidthroughoutthisanalysis.Ane lasticfuselagehasa knowneffectonthelongitudinalbehavioroftheaircraftan danelastictailwouldstrongly affecttheaeroservoelasticpropertiesduetothecontrols urfaceplacement.Bothof theseresearchdirectionswouldbelogicalextensionsofth isresearch.Additionally,only oneairframetypeandgeometrywasstudiedhere.Itwouldbei nterestingtonotehow theseeffectschangewithaircraftgeometryparameterssuc hasthesizeandplacement ofthewing,tail,orcontrolsurfaces. Apossibilitythathasbeensuggestedinthisresearchisthe modicationofthe wingstiffnessduringighttoquicklyachievethedesireda eroelasticeffect.Changing wingstiffnessisawell-knownmeansofachievingaircraftm orphing[ 183 ].Thisresearch showsthatthestiffnessdoesnotneedtobemodiedoverthee ntirespantoachievean aeroservoelasticeffect,whichmaymakethisgoalofwingmo rphingeasiertoachieve. Futureresearchcouldattempttoimplementthisideaingrou ndtestingbyusingan actuatororsmartmaterialtochangethethicknessofthewin gneartheroot,which wouldalterthestiffnessandtheaerodynamicsofthatporti onofthewing.Another avenuemaybetoanalyzeahingedwinganddeterminewhathing estiffnessisneeded toapproximateacertainlevelofwingstiffness.Inight,a nactuatorcouldchangethe stiffnessofthehingetomimicachangeintheentirewingsti ffness. Thesechangeswouldresultindynamicaeroelasticeffectso ntheightdynamics andstabilitypropertieswhichwouldcreatenewcomplexiti esfortheightcontrolsystem. 296

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Additionally,sensingthesedeformationsinightwouldre quireinnovationsinsensor technology.Properlyusingthesesensormeasurementsinth econtrolschemewould presentmanynewresearchopportunities[ 1 ]. Thechangesintheaeroelasticinteractionsduetothewing' selasticaxislocation presentsomepossibilitiesforfutureresearch.First,thi sworkcouldbeextendedina closed-loopsystemtodeterminetheaeroservoelasticeffe cts.Secondly,composite materialsallowforthecreationofphysicalwingswiththes amesizeandshapebut differentelasticaxes.Experimentalanalysescouldbecon ductedonawingwitha forwardoraftlocationsoftheelasticaxistofurtherchara cterizetheireffects.Awing couldalsobeconstructedtoplacetheelasticaxisaheadoft heleadingedge,which wouldamplifythewash-outeffects. Animportantparameterinaircraftdesignisthelocationof thecenterofgravityand theresultingstaticmargin.Itwouldbeusefultodetermine howthisdesignparameter interactswiththewingstiffnessandelasticaxislocation Futureresearchinthegustinganalysiscouldincludetheae roelasticresponse oftheaircraftloadsasthewingstiffnesschanges.Theclos ed-loopworkcouldbe expandedtodeterminehowthestiffnessaffectstheability oftheaircrafttoaccomplish acertaintaskinthepresenceofgusts.Also,havingamodelo fhowagustaffectsthe dynamicsallowsforthepossibilityofincludingthatinfor mationinthecontrolsynthesisto improvegustrejection. Wash-outhasarecognizedeffectontheaircraft'sgustsens itivity[ 8 82 159 ].The amountofwash-outisrelatedinparttothelocationoftheel asticaxis.Futureresearch shouldconsiderlocationsofelasticaxispositionwhichpr oducevaryingamountsof wash-outinordertofurtherquantifyitseffectontheaeros ervoelasticgustalleviation. ItisrecognizedthatthesmallsizeofMAVscreatesnewpossi bilitiesforaircraft maneuvering.Indeed,theconnedspacesinwhichMAVsmight operatenecessitates highlevelsofmaneuverability.Somelooktobio-inspirati ontoovercomethese 297

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challenges[ 120 ],whichwouldimplyaexibleaircraftstructureandthein uenceof aeroservoelasticinteractions.However,itisnotwellund erstoodhowtheseinteractions wouldaffectaMAV'smaneuverability.Oneapproachmaybeto implementrobust trajectoryoptimizationtechniquestosimulatemaneuvers andanalyzetheircharacteristics[ 61 ]. Itmaybepossibletodesigncontrollerstoexploittheseeff ects,which,combined withthepossibilitiesofalteringthewingstiffnessinig ht,couldintroducenew possibilitiesforaircraftmaneuvering.Additionally,ma neuveringinthepresenceof winggustspresentsnewchallengesfortheaeroservoelasti cdesignofMAVs.Any suchresearchwouldneedtoincorporateamoredetailedanal ysisoftheunsteadyand possiblynonlinearaerodynamicowsaroundaexible,mane uveringvehicle.More fundamentally,itisnotwellunderstoodhowthetraditiona ldenitionsandmeasuresof maneuverabilityandagilityforlargeaircraft[ 14 ]applytoamicroairvehicle. 298

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APPENDIX:EXTENDEDTHEORY A.1DyadicProduct Thedyadicproduct,alsocalledtheouterproduct,isaspeci alcaseofthetensor productontwovectorsofthesamedimension.Forexample,tw ovectors a and b are denedinEq.( A–1 ). a = 266664 a 1 a 2 a 3 377775 b = 266664 b 1 b 2 b 3 377775 (A–1) Thedyadicproduct,denotedwiththe n symbol,isshowninEq.( A–2 ): a n b = 266664 a 1 a 2 a 3 377775 b 1 b 2 b 3 = 266664 a 1 b 1 a 1 b 2 b 1 b 3 a 2 b 1 a 2 b 2 a 3 b 3 a 3 b 1 a 3 b 2 a 3 b 3 377775 (A–2) TwoimportantpropertiesofthedyadicproductarenotedinE q.( A–3 )andEq.( A–4 ). a ( b c )=( a n b ) c (A–3) ( a b ) c = a ( b n c ) (A–4) A.2RateofChangeTransportTheorem Bothvectorsandscalarquantitiesareindependentofthere ferenceframefrom whichtheyareobserved.Therateofchangeofascalarisalso independentofthe referenceframe,buttherateofchangeofavectordependson thereferencefromfrom whichthemotionisbeingobserved.Therefore,thetimerate ofchangeofavectormust alwaysbeaccompaniedbythereferenceframeinwhichitwaso bserved.Forexample, A d dt a istherateofchangeof a in F A andisnotequalto B d dt a Thetransporttheoremisusedtocomputetherateofchangeof anarbitrary vectorinframe F A whenobservationsofthevectoraremadeinframe F B .Itisgiven inEq.( A–5 )where b isthearbitraryvectorand A B istheangularvelocityvectorof 299

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referenceframe F B asviewedbyanobserverinreferenceframe F A .Itcanbestated as:therateofchangeofavector b asviewedbyanobserverin F A isequaltothesum oftherateofchangeof b asviewedbyanobserverin F A andthecrossproductof A B withthevector b .Theonlyassumptioninthederivationisthatthereexistsa pointthatis commontobothreferenceframes[ 137 ]. A d b dt = B d b dt + A B b (A–5) Itisimportanttonotethat A B = B A .Furthermore,iftherearemultiplereference frames F A ; F B ; F C ,then A C = A B + B C andthetransporttheoremcanbeappliedas normaltoobtaintherateofchangeofavectorin F A givenobservationsofthevectorin F C .Thisisknownastheangularvelocityadditiontheorem. A.3Lagrange'sEnergyEquations ThegeneralformofLagrange'sequationforasystemis: d dt @ L @ q i @ L @q i = Q i (A–6) where Q i representsthenon-conservativeforcesactingonthesyste mand q i isthe generalizedcoordinate.TheLagrangian L isthedifferenceofthekineticenergy T and thepotentialenergy V ofthesystem: L = T V Fornon-conservativeforcesthatareafunctionofthegener alizedcoordinate q i Rayleigh'sdissipativefunctionmaybeusedtorewriteLagr ange'sequation.Anysuch forcemaybewritten: Q i = n X j =1 c ij ( q;t )_ q i (A–7) wherethe c ij arethedampingcoefcients.Rayleigh'sdissipativefunct ionisdenedas: F = 1 2 n X i =1 n X j =1 c ij q i q j (A–8) 300

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Afterwhich,thefollowingcanbeshown: @F d @ q i = n X j =1 c q i j q j = Q q i (A–9) NowthegeneralformofLarange'sequationinEq.( A–6 )canberewritteninthe dissipativeform: d dt @ L @ q i @ L @q i + @F d @ q i = Q 0i (A–10) where Q 0i accountsfortheremainingforcesactingonthesystem. 301

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BIOGRAPHICALSKETCH AsachildgrowingupinYarmouth,NovaScotia,Judson'savia tion-relatedinterests includedbuildingmodelairplanesandawaitingtheannualY armouthAirShow.Atthe ageof13,heexperiencedhisrstightinaglider.Thisexpe riencewouldleadhimto earnhisprivategliderpilot'slicenseattheageof17andhi sprivatepilot'slicensefor poweredaircraftattheageof18. Judson'senthusiasmformilitaryserviceandloveofaviati onledhimtopursuea careerintheUnitedStatesAirForce.Afterayearofcollege atNorwichUniversityin Vermont,heearnedanappointmenttotheAirForceAcademyan djoinedtheclass of2004.AttheAcademy,henaturallychosetostudyaeronaut icalengineering.His undergraduatefocuswasonaircraftdesign,aerodynamics, andighttesttechniques. AftergraduationfromtheAirForceAcademy,heattendedthe AirForceInstituteof Technology(AFIT)asa2ndLt.AtAFIT,hepursuedhismaster' sdegreeinaeronautical engineeringwithafocusoncomputationaluiddynamics(CF D)andconventional weaponseffects.Aftergraduating,heacceptedanassignme ntwiththeAirForceSEEK EAGLEOfceatEglinAirForceBase,Florida.Hereheapplied hisCFDknowledgeto theproblemsofstoreseparationonvariousplatformssucha stheF-16Falcon,B-52 Stratofortress,andMQ-9Reaper.FromtheSEEKEAGLEOfceh etransferredtothe AirForceResearchLaboratory,MunitionsDirectorateatEg lin,wherehebroadenedhis technicalbackgroundbyconductingexperimentalresearch inalow-speedwindtunnel onsmall,micro,andnanoUAVs. JudsonattendedtheUniversityofFloridawhereheearnedhi sPh.D.underDr. RickLindasamemberoftheFlightControlLab.Histechnical interestsincludeight dynamicsandcontrol,moderndesignofexperiments,experi mentalaerodynamics, aeroelasticity,andcomputationaluiddynamics. 318