NASA Technical Reports Server (NTRS) 20180000681: Development of an Integrated Nonlinear Aeroservoelastic Flight Dynamic Model of the NASA Generic Transport Model

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by  NASA Technical Reports Server (NTRS)

Development of an Integrated Nonlinear Aeroservoelastic
Flight Dynamic Model of the NASA Generic Transport Model
NhanNguyen∗
NASAAmesResearchCenter,MoffettField,CA94035
EricTing†
NASAAmesResearchCenter,MoffettField,CA94035
Thispaperdescribesarecentdevelopmentofanintegratedfullycoupledaeroservoelasticflightdynamic
modeloftheNASAGenericTransportModel(GTM).Theintegratedmodelcouplesnonlinearflightdynamics
toanonlinearaeroelasticmodeloftheGTM.Thenonlinearityincludesthecouplingoftherigid-bodyaircraft
states in the partial derivatives of the aeroelastic angle of attack. Aeroservoelastic modeling of the control
surfaceswhicharemodeledbytheVariableCamberContinuousTrailingEdgeFlapisalsoconducted. The
R. T. Jones’ method is implemented to approximate unsteady aerodynamics. Simulations of the GTM are
conductedwithsimulatedcontinuousanddiscretegustloads.
I. Introduction
Theaircraftindustryhasbeenrespondingtotheneedforenergy-efficientaircraftbyemployinglight-weightma-
terialsforaircraftstructuresandincorporatingmoreenergy-efficientaircraftengines. Reducingairframeoperational
emptyweight(OEW)usingadvancedcompositematerialsisoneofthemajorconsiderationsforimprovingenergyef-
ficiency. Modernlight-weightmaterialscanprovidelessstructuralrigiditywhilemaintainingsufficientload-carrying
capacity. As structural flexibility increases, aeroelastic interactions with aerodynamic forces and moments can alter
aircraftaerodynamicsandflightdynamicssignificantly,therebypotentiallydegradingaerodynamicefficiency,stability
andcontrol.
Thegeneralmotionofarigid-bodyaircraftisunconstrainedinthree-dimensionalspaceastheaircraftpossesses
allsixdegreesoffreedomintranslationandrotation. Thismotionishighlyinfluencedbyalltheaerodynamicforces
andmomentsaswellasthepropulsiveandgravityforces. Themotioncanexhibitstabilitywhichenablestheaircraft
toreturntoitsequilibriumortrimstate,orinstabilityifthemotiondivergeswhenitissubjectedtoadisturbance. The
generalequationsofmotionarenonlinear,eventhoughtheaerodynamiccharacteristicsoftheaircraftaremodeledas
linear. Itisimportanttorecognizethataflightdynamicmodelisonlyamathematicalrepresentationofapproximate
dynamicsofanaircraftinflight. Manyfactorscancauseaflightdynamicmodeltodeviatefromanobservedmodel
ofanaircraft. Someofthesefactorsare:
• Nonlinearaerodynamics-Manyassumptionsareusuallybuiltinthederivationofaerodynamicforceandmo-
ment parameters such as coefficients and derivatives. A linear approximation is usually employed in most
aerodynamic models of an aircraft. In reality, aerodynamic coefficients are not always linear and can exhibit
nonlinearity at high angle of attack and sideslip angle. The linear aerodynamic approximation can provide a
reasonable prediction of aircraft flight dynamics. However, in certain flight regimes, such an approximation
maynolongerbevalid.
• Aeroelasticity - Aircraft is an elastic body which experiences stresses and strains under applied aerodynamic,
propulsive, and gravity forces and moments. Elastic deformation of an aircraft results in changes in aerody-
namiccharacteristics. Therefore,aeroelasticityisasignificanteffectthatcontributestoaircraftflightdynamics.
∗NASAAmesResearchCenter,ResearchScientist,AIAAAssociateFellow,[email protected]
†NASAAmesResearchCenter,ResearchEngineer,[email protected]
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Arigid-bodyflightdynamicmodelisusuallyaninitialaircraftmodelthatananalystdevelopstoprovideapre-
liminaryunderstandingofaircraftdynamics. Whenastructuraldesigniscarriedout,theeffectofaeroelasticity
mustbeincorporatedintotherigid-bodyflightdynamicmodelinordertoproperlypredictappliedloads.
• Atmosphericdisturbances-Atmosphericturbulence, windgust, andlocalizedweatherphenomenacansignif-
icantly affect aircraft dynamics. The angle of attack depends not only on the aircraft velocity vector but also
thewindvelocityvector. Changesinthewindvelocityvectorcanalsoaffectaircraftaccelerationandapplied
loadingwhichcancauseaeroelasticresponses.
The notion of a rigid-body aircraft is idealized. When aircraft structures are designed to maintain their intended
aerodynamic shapes in-flight without significant deformation under aerodynamic loading, aircraft is thought of as
being a rigid body and its flight characteristics are described by a six degrees of freedom flight dynamic model. As
aircraftstructuresbecomeincreasinglymoreflexible,theinfluenceofaeroelasticitybecomesmorepronounced.Flight
control under aeroservoelastic interactions can be challenging. The mishap of the NASA Helios aircraft illustrates
the complex aeroservoelasticity of flexible flight vehicles. Flight dynamics of flexible flight vehicles are intimately
coupledwithstructuraldynamicsoftheaircraft.
This paper describes the development of an aeroservoelastic (ASE) flight dynamic model of the NASA Generic
Transport Model (GTM).1 The GTM represents a notional single-aisle, mid-size, 200-passenger transport aircraft
genericallyapproximatingaBoeing757,asshowninFig. 1. TheGTMhadbeenextensivelytestedinthe14-foot–by-
22-footwindtunnelatNASALangleyResearchCenter. Thus,windtunneltestdataareavailablethatcanbeusedto
validatecomputationalmodels. TheGTMmodelalsohasbeenusedextensiveinflightcontrolresearch.
Figure1. NASAGenericTransportModel
The aircraft has a mid-cruise weight of 210,000 lbs for a typical operating load (gear up, flap up) that includes
cargo, fuel, and passengers. Fuel weighs about 50,000 lbs for a range of about 3,000 nautical miles. At the design
cruiseconditionofMach0.797at36,000ft,thedesignliftcoefficientis0.51.
UndertheAdvancedAirVehiclesProgramoftheNASAAeronauticsResearchMissionDirectorate,theAdvanced
AirTransportTechnology(AATT)Projectisconductingmultidisciplinaryresearchtoinvestigateadvancedconcepts
andtechnologiesforfutureaircraftsystems. ANASAstudyentitled“ElasticallyShapedFutureAirVehicleConcept”
was conducted in 20102,3 to examine new concepts that can enable active control of wing aeroelasticity to achieve
dragreduction. Thisstudyshowedthathighlyflexiblewingaerodynamicsurfacescanbeelasticallyshapedin-flight
byactivecontrolofwingtwistandbendingdeflectioninordertooptimizethelocalanglesofattackofwingsections
toimproveaerodynamicefficiencythroughdragreductionduringcruiseandenhanceliftperformanceduringtake-off
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andlanding. OneconceptresultingfromthisstudyistheVariableCamberContinuousTrailingEdgeFlap(VCCTEF)
developedinitiallybyNASA.2 InitialstudyresultsindicatethattheVCCTEFsystemmayofferapotentialpay-offin
dragreductionthatcouldprovidefuelsavings.
NASAandBoeinghavejointlydevelopedtheVCCTEFfurtherunderaresearchprogramfrom2012to2014.5,6,16
This research program was built upon the initial development of the VCCTEF system for the NASA GTM in 2010.
The resulting VCCTEF system developed under this program employs light-weight Shape Memory Alloy (SMA)
technologyforactuationandthreeseparatechordwisesegmentsshapedtoprovideavariablecambertotheflap. This
camberedflaphasgreaterpotentialfordragreductionascomparedtoaconventionalstraight, plainflap. Theflapis
alsomadeupofindividual2-footspanwisesections,whichenabledifferentflapsettingsateachflapspanwiseposition.
This results in the ability to actively control the wing twist shape as a function of span, resulting in a change to the
wingtwisttoestablishthebestlift-to-dragratioL/Datanyaircraftgrossweightormissionsegment. Wingtwiston
traditional commercial transport designs is dictated by the aeroelastic deflection of a fixed “jig twist” shape applied
at manufacture. The design of this jig twist is set for one cruise configuration, usually for a 50% fuel loading or
mid-pointonthegrossweightschedule. TheVCCTEFoffersdifferentwingtwistsettings,hencedifferentspanwise
loadings, for each gross weight condition and also different settings for climb, cruise and descent, a major factor in
obtaining best L/D conditions. The second feature of VCCTEF is a continuous trailing edge flap. The individual
2-footspanwiseflapsectionsareconnectedwithanelastomertransitionmaterial,soastoproducenogapsinbetween
thespanwisesections. Thiscontinuoustrailingedgeflapcanpotentiallyhelpreduceviscousdragandairframenoise.
TwowindtunnelexperimentswereconductedfortheflexibleGTMwingattheUniversityofWashingtonAeronautical
Laboratoryin2013and2014.7,8 TheexperimentalresultsconfirmtheaerodynamicbenefitsoftheVCCTEF.
TheVCCTEFisamulti-functionalflapsystemenvisionedtobe:
• Awingshapingcontroldevicetotwisttheflexiblewingandchangethespan-loaddistributiontoreducecruise
dragthroughouttheflightenvelope,
• Ahigh-liftdevicefortake-off,climb-out,let-downandfinalapproachbyusingthefullspancamberedflap,
• Afullspanrollcontroleffectorinlieuoftraditionalaileronsusingtheaftsectionofthecamberedflap,and
• Anaeroservoelastic(ASE)controldevicetocompensateforreducedfluttermarginsofflexiblewingsandpro-
videloadalleviationcontrol.
TheVCCTEFisdividedinto14sectionsattachedtotheouterwingand3sectionsattachedtotheinnerwing,asshown
in Fig. 2.5 Each 24-inch section has three chordwise cambered flap segments that can be individually commanded.
Thesecamberedflapsarejoinedtothenextsectionbyaflexibleandsupportedmaterial(showninblue)installedwith
thesameshapeasthecamberandthusprovidingcontinuoustrailingedgeflapsthroughoutthewingspanwithnodrag
producinggaps.
Amajorgoaloftheprogramistodevelopalight-weightflapcontrolsystemthathasasignificantweightadvantage
as compared to current flap screw-jack actuators. Hydraulic, electric and Shape Memory Alloy (SMA) torque rod
actuation were evaluated with the result that the SMA actuation has the best weight advantage. Moreover, the use
of hinge line actuation eliminates the large and heavy externally mounted actuators, and permits all actuators to be
interiortothewingandflapmoldlines,thuscontributingtotheoveralldragreductiongoal.
Figure3showsaschematicrepresentationofanoutboardwingflapsectionhavingthreecamberedflapsegments.5
SMAactuatorsdrivethefirstandsecondcamberedflapsegmentsandafasteractingelectricactuatordrivesthethird
cambered flap segment. SMA actuators can deliver large hinge moments, but generally move at a slow rate. The
outboard wing flap uses the full-span third cambered segment as a roll command effector and as a control device
for suppressing aeroelastic wing structural dynamic modes, both requiring high rates which can be met by electric
actuators.
Usingthecamberpositioning,afull-span,high-liftconfigurationcanbeactivatedthathasnodragproducinggaps
andalowflapnoisesignature. ThisisshowninFig. 4. Tofurtheraugmentlift,aslottedFowlerflapconfigurationis
formedbyanairpassagebetweenthewingandtheinnerflapthatservestoimproveairflowovertheflapandkeepthe
flowattached. Thisairpassageappearsonlywhentheflapsareextendedinthehighliftconfiguration.
Inthehigh-liftconfiguration,theouterwingflapusesthethirdcamberedsegmentforrollcontrol,asshowninFig.
5.Thisprovidesrollingmomentthatisequivalenttoaileroncontrol.Itissomewhatsimilartodeflectingtheaileronsin
adrooppositiontoactasflaps,acommonprocedureusedontacticalaircraftandonsometransportaircraft. Thehigh-
lift configuration distributes the required flap hinge moment throughout the span of the wing while using actuation
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componentsthatarealllocatedinteriortothewingandflap.ThiscanbeachievedbytheuseofSMAhingelinetorque
rods,sizedtomeetthehingemomentrequirementsateachspanwiselocationonthewing.
Figure2. WingConfiguredwiththeVariableCamberContinuousTrailingEdgeFlap
Figure3. VariableCamberFlapControlUsesShapeMemoryAlloyTorqueRodandElectricDriveActuation
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Figure4. CruiseandHighLiftVCCTEFConfigurations
Figure5. Three-SegmentVariableCamberFlap
Figure6. GTMwithwithVariableCamberContinuousTrailingEdgeFlap
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Figure6illustratestheGTMequippedwiththeVCCTEFforwingshapingcontrol. Byactivelyshapingthewing
aerodynamicsurfaceusingtheVCCTEF,optimalaerodynamicperformancecouldpotentiallyberealizedatanypoint
intheflightenvelope,therebyenablingamissionadaptivecapability. Itisakeyenablingfeatureoftheresearcharea
PerformanceAdaptiveAeroelasticWing(PAAW)intheAATTproject. Theterm“performanceadaptiveaeroelastic”
distinguishesitselffromthefamiliarterm“missionadaptive”inthattheeffectofaeroelasticityonaerodynamicperfor-
mancemustbefullyaccountedforasisthecaseformoderntransportdesign.TheVCCTEFreliesontwomechanisms
toimproveaerodynamicperformance:1)wingtwistoptimizationforflexiblewingdesign,and2)variablecamberand
continuoustrailingedgeforimprovedaerodynamics. Thistechnologycouldenablemodernhigh-aspectratioflexible
wingaircraftwithsignificantflexibilitytoadaptivelychangewingshapesin-flighttoachievecruisedragoptimization,
whileatthesametimesatisfyingoperationalconstraintssuchasstructuralloadlimitations,fluttermargins,gustand
maneuverloadresponses,andothersbyactiveaeroservoelasticcontrols.
Toassesstheeffectivenessofwingshapingcontrolformoderntransportaircraft,theGTMwingismodeledwith
a high degree of flexibility, similar to estimated flexibility distributions on state-of-the-art passenger aircraft wings.
Thewingbendingstiffnessistailoredtoachievea10%wingtipdeflectionat1-gflightconditions, whichresultsin
abendingstiffnessabouthalfthatofolder-generationtransportwings,whilethetorsionalstiffnessisaboutthesame.
This10%wingtipdeflectionisaboutthesameasthatofamoderncompositehigh-aspect-ratiowingdesigninmodern
transportaircraftsuchastheBoeing787.
II. FlightDynamicsofRigidAircraft
The development of a flight dynamic model of the GTM requires mass and inertia properties, and stability and
control (S&C) derivatives. The mass and inertia properties of the GTM are based on a Boeing’s report and are
modified to account for the reduced weight of the flexible wings. The S&C derivatives are estimated using three
differentconceptualaerodynamicvortex-latticecodesalongwithanalyticalcalculation. Thethreeaerodynamiccodes
areVORLAX,AVL,andVSPAERO.Theresultsshowreasonableagreementamongthefoursetsofestimates.Because
VORLAXhastheaeroelasticcapabilityaswellastransonicandboundarylayercorrections,17 VORLAXresultsare
selectedfortheflightdynamicmodel.
AstaticaeroelastictrimfortheflexiblewingGTMisdeveloped. Thetrimsolutioncalculatestheangleofattack,
enginethrust,andelevatordeflectionsforvariousdeformedGTMconfigurationsatdifferentfuelweight,altitude,and
airspeed.Thefuelweightismodeledasanaddedweighttothewingweightwhichaffectsthestaticdeflectionshapeof
thewingsat1-gcruiseconditions. Afinite-elementmodelisdevelopedtocomputethestaticwingdeflections. Once
thewingdeflectionshapeiscomputed,theS&Cderivativesareevaluatedforthedeformedaircraft.
Thenonlinear6-degree-of-freedomflightdynamicequationsofmotionintheaircraftbody-fixedreferenceframe
aregivenby
p˙ =RV (1)
Φ˙ =Tω (2)
d(mV)
+ω˜mV=F (3)
dt
d(Iω)
+ω˜Iω =M (4)
dt
(cid:104) (cid:105)(cid:62) (cid:104) (cid:105)(cid:62) (cid:104) (cid:105)(cid:62)
wherep= x y h isthepositionvector, V= u v w isthevelocityvector, Φ = φ θ ψ is
(cid:104) (cid:105)(cid:62)
theEuleranglevector,andω = p q r istheangularratevector.
Theangularratematrixω˜ isgivenby
 
0 −r q
ω˜ = r 0 −p  (5)
 
−q p 0
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TheinertiamatrixoftheaircraftabouttheaircraftrollaxisX,pitchaxisY,andyawaxisZisgivenby
 
I¯ −I¯ −I¯
XX XY XZ
I= −I¯ I¯ −I¯  (6)
 XY YY YZ 
−I¯ −I¯ I¯
XZ YZ ZZ
RandTaretherotationmatricesgivenby
 
cosθcosψ −cosφsinψ+sinφsinθcosψ sinφsinψ+cosφsinθcosψ
R= cosθsinψ cosφcosψ+sinφsinθsinψ −sinφcosψ+cosφsinθsinψ  (7)
 
sinθ −sinφcosθ −cosφcosθ
 
1 sinφtanθ cosφtanθ
T= 0 cosφ −sinφ  (8)
 
0 sinφsecθ cosφsecθ
Theforcevectorisgivenby
 
X+T−mgsinθ
F= Y+mgcosθsinφ 1 (9)
 
Z+mgcosθcosφ
whereX =C q Sistheaxialforce,Y =C q Sisthesideforce,Z=C q Sisthenormalforce,andT istheengine
X ∞ Y ∞ Z ∞
thrust. TheforcecoefficientsC andC arerelatedtotheliftanddragcoefficientsas
X Z
C =C sinα−C cosα (10)
X L D
C =−C cosα−C sinα (11)
Z L D
Themomentvectorisgivenby
 
l
M= m+Tz  (12)
 e 
n
where l is the rolling moment, m is the pitching moment about the aircraft center of gravity (CG), n is the yawing
momentabouttheaircraftCG,andz astheenginethrustoffsetfromtheaircraftCG.
e
III. InertialandAeroelasticForcesandMomentsofFlexibleWings
ConsideranairfoilsectionontheleftwingasshowninFig. 7undergoingbendingandtorsionaldeflections. Let
(x,y,z)betheundeformedcoordinatesofpointQonawingairfoilsectioninthereferenceframeDdefinedbyunit
vectors(d ,d ,d ). Letp =xd beapositionvectoralongtheelasticaxis. Then, pointQisdefinedbyaposition
1 2 3 0 1
vectorp=p +qwhereq=yd +zd definespointQinthey−zplanefromtheelasticaxis.
0 2 3
Figure7. LeftWingReferenceFrameofWinginCombinedBending-Torsion
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Let Θ be a torsional twist angle about the x-axis, positive nose-down. LetW andV be flapwise and chordwise
bendingdeflectionsofpointQ,respectively. LetU betheaxialdisplacementofpointQ.Then,thedisplacementand
rotationvectorsduetotheelasticdeformationcanbeexpressedas
r=Ud +Vd +Wd (13)
1 2 3
φ =Θd −W d +V d (14)
1 x 2 x 3
wherethesubscriptsxandt denotethepartialderivativesofΘ,W,andV.
Let(x ,y ,z )bethedeformedcoordinatesofpointQontheairfoilintheleftwingreferenceframeDandp =
1 1 1 1
x d +y d +z d beitspositionvector. Then,thecoordinates(x ,y ,z )arecomputedas10
1 1 1 2 1 3 1 1 1
p =p+r+φ×q (15)
1
where
   
x x+U−yV −zW
1 x x
 y = y+V−zΘ  (16)
 1   
z z+W+yΘ
1
A. InertialForcesandMoment
LetV=ub +vb +wb andω = pb +qb +rb betheaircrafttranslationalandrotationalvelocityvectorsatthe
1 2 3 1 2 3
aircraftcenterofgravity(CG)where(b ,b ,b )aretheunitvectorsintheaircraftbody-fixedreferenceframeB.Let
1 2 3
r =−x b −y b −z b bethepositionvectorofpointQintheaircraftbody-fixedreferenceframeBrelativetothe
a a 1 a 2 a 3
aircraftCGsuchthatx ispositivewhenpointQisaftoftheaircraftCG,y ispositivewhenpointQistowardtheleft
a a
wingfromtheaircraftCG,andz ispositivewhenpointQisabovetheaircraftCG.ThevelocityatpointQduetothe
a
aircraftvelocityandangularvelocityinthereferenceframeDisthencomputedas
v =V+ω×r =(ub +vb +wb )+(pb +qb +rb )×(−x b −y b −z b )
Q a 1 2 3 1 2 3 a 1 a 2 a 3
=(u+ry −qz )b +(v−rx +pz )b +(w+qx −py )b =xd +yd +zd (17)
a a 1 a a 2 a a 3 t 1 t 2 t 3
where
   
x −(u+ry −qz )sinΛcosΓ−(v−rx +pz )cosΛcosΓ−(w+qx −py )sinΓ
t a a a a a a
 y = −(u+ry −qz )cosΛ+(v−rx +pz )sinΛ  (18)
 t   a a a a 
z (u+ry −qz )sinΛsinΓ+(v−rx +pz )cosΛsinΓ−(w+qx −py )cosΓ
t a a a a a a
Thetransformationbetween(b ,b ,b )and(d ,d ,d )isgivenby
1 2 3 1 2 3
    
b −sinΛcosΓ −cosΛ sinΛsinΓ d
1 1
 b = −cosΛcosΓ sinΛ cosΛsinΓ  d  (19)
 2    2 
b −sinΓ 0 −cosΓ d
3 3
    
d −sinΛcosΓ −cosΛcosΓ −sinΓ b
1 1
 d = −cosΛ sinΛ 0  b  (20)
 2    2 
d sinΛsinΓ cosΛsinΓ0 −cosΓ b
3 3
The local velocity at point Q due to aircraft rigid-body dynamics and aeroelastic deflections in the left wing
referenceframeDisobtainedas11,12
∂∆p
v=v + +ω×∆p=v d +v d +v d (21)
Q x 1 y 2 z 3
∂t
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where∆p=p −pand
1
   
v x +U −yV −zW −ω (V−zΘ)+ω (W+yΘ)
x t t xt xt z y
 v = y +V −zΘ +ω (U−yV −zW )−ω (W+yΘ)  (22)
 y   t t t z x x x 
v z +W +yΘ −ω (U−yV −zW )+ω (V−zΘ)
z t t t y x x x
   
ω −psinΛcosΓ−qcosΛcosΓ−rsinΓ
x
 ω = −pcosΛ+qsinΛ  (23)
 y   
ω psinΛsinΓ+qcosΛsinΓ−rcosΓ
z
Thekineticenergyisformedby
T = 1(cid:90) ρv.vdA= 1(cid:90) ρ(cid:0)v2+v2+v2(cid:1)dA (24)
2 2 x y z
We use the method of separation of variables to express the displacements asU(x,t)=Φ (x)q (t), V(x,t)=
u u
Φ (x)q (t),W(x,t)=Φ (x)q (t),Θ(x,t)=Φ (x)q (t). Then,thevirtualworkquantitiesduetothegeneralized
v v w w θ θ
coordinatesq (t),q (t),q (t),andq (t)arecomputedintermsofthevirtualdisplacementsas12
u v w θ
(cid:20)d (cid:18)∂T (cid:19) ∂T (cid:21) (cid:90) (cid:20)dv (cid:21)
−fiδU = − δq = ρ x −v (ω +V )+v (ω −W ) δUdA (25)
x dt ∂q˙ ∂q u dt y z xt z y xt
u u
(cid:20)d (cid:18)∂T (cid:19) ∂T (cid:21) (cid:90) (cid:20)dv (cid:21)
−fiδV = − δq = ρ y +v (ω +V )−v (ω +Θ) δVdA
y dt ∂q˙ ∂q v dt x z xt z x t
v v
(cid:90) (cid:26)d[v (−y−V+zΘ)+v (U−yV −zW )] (cid:27)
x y x x
+ ρ +v (yω +yV )−v (yω −yW ) δV dA (26)
y z xt z y xt x
dt
(cid:20)d (cid:18)∂T (cid:19) ∂T (cid:21) (cid:90) (cid:20)dv (cid:21)
z
−f δW = − δq = ρ −v (ω −W )+v (ω +Θ) δWdA
z w x y xt y x t
dt ∂q˙ ∂q dt
w w
(cid:90) (cid:26)d[v (−z−W−yΘ)+v (U−yV −zW )] (cid:27)
x z x x
+ ρ +v (zω +zV )−v (zω −zW ) δW dA (27)
y z xt z y xt x
dt
(cid:20)d (cid:18)∂T (cid:19) ∂T (cid:21) (cid:90) (cid:26)d[v (−z−W−yΘ)] d[v (y+V−zΘ)]
y z
−m δΘ= − δq = ρ +
x dt ∂q˙ ∂q θ dt dt
θ θ
(cid:9)
−v (yω +zω +zV −yW )+v (yω +yΘ)+v (zω +zΘ) δΘdA (28)
x y z xt xt y x t z x t
(cid:82) (cid:82)
Let ydA=Ae where A= dA is the mass area and e is the offset of the CG of a wing section from the
cg cg
elasticaxis,positiveiftheCGisaftoftheelasticaxis. WedefineI =(cid:82)(cid:0)y2+z2(cid:1)dA,I =(cid:82)z2dA,andI =(cid:82)y2dA.
xx yy zz
(cid:82) (cid:82)
Furthermore, We assume zdA≈0 and I =− yzdA≈0. Integrating the integrals that contain δV and δW by
yz x x
parts,weobtainthelinearcontributionsoftheaeroelasticdeflectionstotheinertialforcesandmomentas12
fi=ρA(cid:2)−x +yω −zω +(cid:0)ω2+ω2(cid:1)U+(ω˙ −ω ω )V−(ω˙ +ω ω )W+2ωV −2ω W +yV
x tt t z t y y z z x y y x z z t y t t xt
+zW −U ]+ρAe (cid:2)−(ω˙ +ω ω )Θ−(cid:0)ω2+ω2(cid:1)V −2ω Θ +V (cid:3) (29)
t xt tt cg y x z y z x y t xtt
fi=ρA(cid:2)−y −xω +zω −(ω˙ +ω ω )U+(cid:0)ω2+ω2(cid:1)V+(ω˙ −ω ω )W−2ωU +2ω W +zΘ −xV
y tt t z t x z x y x z x y z z t x t t t t xt
∂
−V ]+ρAe [(ω˙ −ω ω )Θ+(ω˙ +ω ω )V +2ω Θ +2ωV ]+ [ρA(y U−x V+yU −xV)]
tt cg x y z z x y x x t z xt tt tt t t t t
∂x
+ ∂ (cid:8)ρAe (cid:2)−x +yω −zω +(cid:0)ω2+ω2(cid:1)U+(ω˙ −ω ω )V−(ω˙ +ω ω )W−y V +2ωV
∂x cg tt t z t y y z z x y y x z tt x z t
−2ω W +zW −U ](cid:9)+ ∂ (cid:8)ρI (cid:2)−(ω˙ +ω ω )Θ−(cid:0)ω2+ω2(cid:1)V −2ω Θ +V (cid:3)(cid:9) (30)
y t t xt tt ∂x zz y x z y z x y t xtt
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fi=ρA(cid:2)−z +xω −yω +(ω˙ −ω ω )U−(ω˙ +ω ω )V+(cid:0)ω2+ω2(cid:1)W+2ω U −2ω V −yΘ −xW
z tt t y t x y x z x y z x y y t x t t t t xt
−W ]+ρAe (cid:2)(cid:0)ω2+ω2(cid:1)Θ−(ω˙ −ω ω )V −2ω V −Θ (cid:3)+ ∂ [ρA(z U−x W+zU −xW)]
tt cg x y y x z x y xt tt ∂x tt tt t t t t
+ ∂ [ρAe (−x Θ−z V −xΘ −zV )]+ ∂ (cid:8)ρI (cid:2)−(ω˙ −ω ω )Θ−(cid:0)ω2+ω2(cid:1)W −2ω Θ +W (cid:3)(cid:9) (31)
∂x cg tt tt x t t t xt ∂x yy z x y y z x z t xtt
mi =ρA(−z V+y W−zV +yW)+ρAe [−z +xω −yω +(ω˙ −ω ω )U−(ω˙ +ω ω )V
x tt tt t t t t cg tt t y t x y x z x y z
+(cid:0)ω2+ω2(cid:1)W+y Θ+2ω U −2ω V −xW −W (cid:3)−ρI Θ
x y tt y t x t t xt tt xx tt
+ρI (cid:2)(cid:0)ω2+ω2(cid:1)Θ−(ω˙ +ω ω )W −2ωW (cid:3)+ρI (cid:2)(cid:0)ω2+ω2(cid:1)Θ−(ω˙ −ω ω )V −2ω V (cid:3) (32)
yy x z z x y x z xt zz x y y x z x y xt
In addition to the inertial forces and pitching moment acting on a wing section, the contributions of half of the
fuselage mass and inertias and the engine mass to the inertial forces and pitching moment without the rigid-body
aircraftinertialforcecouplingaregivenby
(cid:18) (cid:19)
1
∆fi=δ(x) − m U +δ(x−x )(−m U ) (33)
x 2 f tt e e tt
(cid:20) (cid:18) (cid:19)(cid:21)
1 ∂ 1
∆fi=δ(x) − m V + I V +δ(x−x )[−m V −m z Θ ] (34)
y 2 f tt ∂x 2 f,zz xtt e e tt e e tt
(cid:20) (cid:18) (cid:19)(cid:21)
1 1 ∂ 1
∆ff =δ(x) − m W + m y Θ + I W +δ(x−x )(−m W +m y Θ ) (35)
z 2 f tt 2 f f tt ∂x 2 f,yy xtt e e tt e e tt
(cid:18) (cid:19)
∆mi =δ(x) 1m y W −1I Θ +δ(x−x )(cid:2)m y W −m z V −m (cid:0)y2+z2(cid:1)Θ (cid:3) (36)
x 2 f f tt 2 f,xx tt e e e tt e e tt e e e tt
wherem ,I ,I ,andI arethemassandinertiasofthefuselage;m isthemassoftheengine;y istheoffset
f f,xx f,yy f,zz e f
of the fuselage CG from the elastic axis, positive if the fuselage CG is forward of the elastic axis; (x ,y ,z ) is the
e e e
coordinateoftheengineCGintheleftwingreferenceframeD,positiveiftheengineCGisbelowandforwardofthe
elasticaxis;andδ(x−a)istheDiracdeltafunctionwhichisdefinedas
(cid:90)
δ(x−a)f(x)dx= f(a) (37)
B. AeroelasticForcesandMoment
Inordertocomputetheaeroelasticforcesandmoments,thevelocitymustbetransformedfromtheleftwingreference
frameDtotheairfoillocalcoordinatereferenceframedefinedby(µ,η,ξ)asfollows:
        
v 1 0 0 1 V 0 1 0 W v v +v V +vW
µ x x x x y x z x
 v = 0 1 Θ  −V 1 0  0 1 0  v ≈ −v V +v +v Θ  (38)
 η    x   y   x x y z 
v 0 −Θ 1 0 0 1 −W 0 1 v −v W −v Θ+v
ξ x z x x y z
Thelocalaeroelasticangleofattackontheairfoilsectionduetothevelocitycomponentsv andv intheleftwing
η ξ
referenceframeD,asshowninFig. 7,iscomputedas12
v v¯ +∆v v v¯ ∆v
ξ ξ ξ ξ ξ η
α = = = − (39)
c v v¯ +∆v v¯ v¯2
η η η η η
where
   
v x +U −yV −zW −ω (V−zΘ)+ω (W+yΘ)
x t t xt xt z y
 v = y +V −zΘ +ω (U−yV −zW )−ω (W+yΘ)  (40)
 y   t t t z x x x 
v z +W +yΘ −ω (U−yV −zW )+ω (V−zΘ)
z t t t y x x x
v¯ =z (41)
ξ t
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