NASA Technical Reports Server (NTRS) 20170000931: Multi-Objective Flight Control for Drag Minimization and Load Alleviation of High-Aspect Ratio Flexible Wing Aircraft

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by  NASA Technical Reports Server (NTRS)

Multi-Objective Flight Control for Drag Minimization and
Load Alleviation of High-Aspect Ratio Flexible Wing Aircraft
NhanNguyen∗
NASAAmesResearchCenter,MoffettField,CA94035
EricTing†
NASAAmesResearchCenter,MoffettField,CA94035
DanielChaparro‡
StingerGhaffarianTechnologies,Inc.,MoffettField,CA94035
MichaelDrew§
StingerGhaffarianTechnologies,Inc.,MoffettField,CA94035
SeanSwei¶
NASAAmesResearchCenter,MoffettField,CA94035
Asaircraftwingsbecomemuchmoreflexibleduetotheuseoflight-weightcompositesmaterial,adverse
aerodynamicsatoff-designperformancecanresultfromchangesinwingshapesduetoaeroelasticdeflections.
Increased drag, hence increased fuel burn, is a potential consequence. Without means for aeroelastic com-
pensation,thebenefitofweightreductionfromtheuseoflight-weightmaterialcouldbeoffsetbylessoptimal
aerodynamic performance at off-design flight conditions. Performance Adaptive Aeroelastic Wing (PAAW)
technologycanpotentiallyaddressthesetechnicalchallengesforfutureflexiblewingtransports. PAAWtech-
nologyleveragesmulti-disciplinarysolutionstomaximizetheaerodynamicperformancepayoffoffutureadap-
tivewingdesign,whileaddressingsimultaneouslyoperationalconstraintsthatcanpreventtheoptimalaero-
dynamic performance from being realized. These operational constraints include reduced flutter margins,
increasedairframeresponsestogustandmaneuverloads,pilothandlingqualities,andridequalities. Allof
theseconstraintswhileseekingtheoptimalaerodynamicperformancepresentthemselvesasamulti-objective
flightcontrolproblem.Thepaperpresentsamulti-objectiveflightcontrolapproachbasedonadrag-cognizant
optimalcontrolmethod.Aconceptofvirtualcontrol,whichwaspreviouslyintroduced,isimplementedtoad-
dressthepair-wiseflapmotionconstraintsimposedbytheelastomermaterial.Thismethodisshowntobeable
tosatisfytheconstraints.Real-timedragminimizationcontrolisconsideredtobeanimportantconsideration
forPAAWtechnology. Dragminimizationcontrolhasmanytechnicalchallengessuchassensingandcontrol.
Aninitialoutlineofareal-timedragminimizationcontrolhasalreadybeendevelopedandwillbefurtherin-
vestigatedinthefuture.Asimulationstudyofamulti-objectiveflightcontrolforaflightpathanglecommand
withaeroelasticmodesuppressionanddragminimizationdemonstratestheeffectivenessoftheproposedso-
lution. In-flightstructuralloadsarealsoanimportantconsideration. Aswingflexibilityincreases,maneuver
loadandgustloadresponsescanbesignificantandthereforecanposesafetyandflightcontrolconcerns. In
thispaper,wewillextendthemulti-objectiveflightcontrolframeworktoincludeloadalleviationcontrol.The
studywillfocusinitiallyonmaneuverloadminimizationcontrol,andthensubsequentlywilladdressgustload
alleviationcontrolinfuturework.
I. Introduction
Aircraft are typically designed to maintain efficient structural design for safe load-carrying capacity. Modern
engineered materials such as composites have begun to appear in new airframe constructions for weight reduction
∗TechnicalGroupLeadandResearchScientist,AssociateFellowAIAA,IntelligentSystemsDivision,[email protected]
†ResearchEngineer,IntelligentSystemsDivision,[email protected]
‡ResearchEngineer,StingerGhaffarianTechnologiesInc.,NASAAmesResearchCenter,[email protected]
§ResearchEngineer,StingerGhaffarianTechnologiesInc.,NASAAmesResearchCenter,[email protected]
¶ResearchScientist,IntelligentSystemsDivision,[email protected]
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purposes. A typical modern transport aircraft wing can provide less structural stiffness while maintaining the same
load-carrying capacity as an older aluminum wing construction. There is a realization that next-generation aircraft
conceptscanbedevelopedtotakeadvantageofthestructuralflexibilityaffordedbymodernengineeredmaterialsto
improvetheaerodynamicperformance.
Asaircraftwingsbecomemoreflexible,adverseaerodynamicsatoff-designperformancecanresultfromchanges
in wing shapes due to aeroelastic deflections. Increased drag, hence increased fuel burn, is one such potential con-
sequence. Withoutmeansforaeroelasticcompensation,thebenefitofweightreductionfromtheuseoflight-weight
materialcouldbeoffsetbylessoptimalaerodynamicperformanceatoff-designflightconditions. PerformanceAdap-
tiveAeroelasticWing(PAAW)technologycanpotentiallyaddressthesetechnicalchallengesforfutureflexiblewing
transports.PAAWtechnologyleveragesmulti-disciplinarysolutionstomaximizetheaerodynamicperformancepayoff
offutureadaptivewingdesign,whileaddressingsimultaneouslyoperationalconstraintsthatcanpreventtheoptimal
aerodynamic performance from being realized. These operational constraints include reduced flutter margins, in-
creasedairframeresponsestogustandmaneuverloads,anddegradedpilothandlingqualitiesaswellasridequalities.
Alloftheseconstraintswhileseekingtheoptimalaerodynamicperformancepresentthemselvesasamulti-objective
flightcontrolproblem.
A multi-objective flight control framework has been developed to address these operational constraints and the
efficiencygoalsimultaneouslyinordertoarriveatoptimalsolutionsthatcanprovidegoodcompromisebetweenthe
efficiencygoalandoperationalconstraints. Theseoptimalsolutionstakeadvantageofamulti-functionalflightcontrol
systemcalledtheVariableCamberContinuousTrailingEdgeFlap(VCCTEF)conceptinthisstudy. TheVCCTEFis
apossiblecandidatePAAWconceptdevelopedundertheNASAAdvancedAirTransportTechnology(AATT)project.
TheVCCTEFconceptwasoriginally developedbyaNASAInnovationFundstudy entitled “ElasticallyShaped
Future Air Vehicle Concept” in 2010.1,2 This study examined new concepts that can enable active control of wing
aeroelasticitytoachievedragreduction. Theresultsshowedthatahighlyflexiblewingcouldbeelasticallyshapedin-
flightbyactivecontrolofwingtwistandverticaldeflectioninordertooptimizethelocalanglesofattacktoimprove
the aerodynamic efficiency. The VCCTEF concept provides spanwise load tailoring via a continuous trailing edge
formedbymultiplespanwiseflapsectionswhicharejoinedtogetherbyelastomertransitionsectionsasshowninFig.
1. The spanwise load tailoring allows optimal lift distributions to be achieved throughout a given flight envelope,
thereby enabling a mission-adaptive performance. A secondary benefit of the continuous trailing edge flap is the
potentialnoisereductionduringtake-offandlandingwhentheflapisconfiguredforhigh-liftoperations. Incontrast,
conventionalflapswithgapsproduceasignificantnoisesourceathigh-liftconditions.
Figure1. GTMwithwithVariableCamberContinuousTrailingEdgeFlap
The VCCTEF also provides chordwise pressure shaping via a variable camber flap having three chordwise seg-
mentsasshowninFig. 2. Thesethreechordwiseflapsegmentscanbeindividuallycommandedoractuatedinunison
whenaflapdeflectioncommandisgiven. Byvaryingthedeflectionsoftheindividualchordwiseflapsegments,any
camber surface can be created to achieve a desired aerodynamic performance. In general, a cambered flap is more
efficientinproducingliftthanastraightconventionalflapbyachievingthesameliftatalowerdrag. Thechordwise
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pressureshapingmodifiesthepressuredistributiononawingsurfacetoachieveadragreductionortoreducetheshock
formationonthewing’suppersurface,thusallowingahighercruisespeedorreducingthetransonicdragrise.
Initial study results indicate that the VCCTEF system can offer a potential pay-off in drag reduction that could
translate into significant fuel savings. In order to realize the potential benefit of drag reduction by aeroelastic wing
shaping control while meeting all other performance requirements and operational constraints, an integrated multi-
disciplinaryapproachisdevelopedtoincorporateaflightcontrolsystemdesignintoamodelingenvironmenttomaxi-
mizethesystembenefitsofPAAWtechnology. Figure1illustratestheVCCTEFconceptinstalledonaflexiblewing
GTM.
Figure2. Three-SegmentVariableCamberFlap
Figure3. GTMWingConfiguredwiththeVariableCamberContinuousTrailingEdgeFlap
Subsequently, Boeing Research & Technology conducted a joint study with NASA in 2012 to develop a system
design of the VCCTEF3,4 as shown in Fig. 3. This study was built upon the development of the original VCCTEF
systemfortheNASAGenericTransportModel(GTM)whichisessentiallybasedontheBoeing757airframe. The
VCCTEF design includes 14 spanwise sections attached to the outer wing and 3 spanwise sections attached to the
inner wing, as shown in Fig. 3.4 Each 24-inch section has three chordwise cambered flap segments that can be
individually commanded. These cambered flaps are joined to the next section by a flexible and supported material
(showninblue)installedwiththesameshapeasthecamberandthusprovidingcontinuousflapsthroughoutthewing
span with no drag producing gaps. The VCCTEF design results in the ability to control the wing twist shape along
thewingspan,therebyeffectivelyproducingachangetothewingtwisttoestablishthebestlift-to-dragratio(L/D)at
anyaircraftgrossweightormissionsegment. Thedesignwingtwistonatraditionalcommercialtransportisdictated
by the aeroelastic deflection of a fixed “jig twist” shape applied during manufacturing. The design of this jig twist
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is set for only one cruise configuration, usually for a 50% fuel loading or mid-point on the gross weight schedule.
TheVCCTEFoffersdifferentwingtwistsettings,hencedifferentspanwiseloadings,foreachgrossweightcondition
and also different settings for climb, cruise and descent. This ability is a major factor in obtaining best L/D flight
conditions.
Theflexibilityofmoderntransportwingscancauseareductioninafluttermarginwhichcancompromiseaircraft
stability.Inaddition,aflexiblewingismoreresponsivetogustormaneuverloadswhichcanleadtostructuralissuesas
wellascompromisedrideandhandlingqualities. Inapreviousstudy,aflutteranalysiswasconductedtoexaminethe
effectofincreasedflexibilityoftheGTMwing.5 ThebaselinewingstiffnessoftheGTMisreducedby50%. Table
1 shows the flutter speed prediction at 35,000 ft for the stiff wing GTM with the baseline stiffness and the flexible
wingGTMwiththereducedstiffness. ThecriticalfluttermodefortheflexibleGTMwingisthefirstanti-symmetric
bendingmodewhichfluttersatMach0.938atanaltitudeof35,000ft.
SymmetricMode Anti-SymmetricMode
StiffWingGTMFlutterMach@35Kft 1.358 1.310
StiffWingGTMFlutterFrequency@35Kft,Hz 4.31 3.87
FlexibleWingGTMFlutterMach@35Kft 0.938 0.925
FlexibleWingGTMFlutterFrequency@35Kft,Hz 6.94 2.85
Table1. FlutterSpeedPrediction
TheFAA(FederalAviationAdministration)requiresaircraftcertificationtodemonstrateafluttermarginofatleast
15%abovethedivespeedwhichisnormallydeterminedfromflighttesting. ForamaximumoperatingMach0.8,the
divespeedmaybeestimatedtobeabout20%overthemaximumoperatingMach,or0.96. Thus,theflutterclearance
forthisnotionalaircraftwouldrequireaminimumflutterspeedofMach1.10at35,000ft. ThestiffwingGTMmeets
thisflutterclearancebuttheflexiblewingGTMdoesnot. Toincreasetheflutterspeed,activefluttersuppressionisan
option. Currently,activefluttersuppressionhasnotbeencertifiedfortransportaircraft,butthissituationmaychange
as the FAA has begun to investigate certification requirements for active flutter suppression control for commercial
transportaircraft.
Toaddresstheadaptiveaeroelasticwingshapingcontrolobjective, amultidisciplinarydesignanalysisoptimiza-
tion(MDAO)frameworkmustbeconsideredbyincorporatingtheaerodynamicperformancepredictiontogetherwith
aeroelasticity, flutter suppression control, and maneuver load alleviation control. The objective of the MDAO is to
identify a desired wing flexibility that would provide the best overall aerodynamic, structural, aeroelasticity, and
control benefits. For example, the wing stiffness could be optimized to maximize the aerodynamic performance at
off-design cruise conditions with the least amount of control effort to maintain a flutter margin and maneuver load
constraints. TheMDAOwouldincludethecontrolsynthesisdirectlyintheMDAOprocess.
Flightcontroldesignofconventionalaircrafthasalongheritagewiththesingle-axisflightcontrolphilosophyfor
pitch control by the elevator, roll control by the aileron, and yaw control by the rudder. The VCCTEF represents a
newclassofmulti-functionalflightcontrolforfuturePAAWtechnology. Thispotentiallycouldopenupanewflight
control paradigm. In the presence of multi-functional flight control surfaces such as the VCCTEF, a conventional
flight control task such as a pitch command could be designed in conjunction with other flight control requirements
particularlyforflexiblewingaircraft.Theseadditionalrequirementsmayinclude:1)dragminimization,2)aeroelastic
modesuppression,3)maneuverloadalleviation,and4)gustloadalleviation.Inaddressingtheserequirements,aflight
controlsystemcouldprovideimprovementsinpilothandlingqualitiesandpassengerridecomfort.
To address all of these flight control objectives simultaneously can be a challenge in a flight control design. A
multi-objective optimization framework can be developed to address the needs for satisfying multiple, competing
flightcontrolrequirements. Thispaperwillpresentamulti-objectiveflightcontrolapproachtoaddresssomeofthese
multi-disciplinaryinteractionsinaflexiblewingaircraftemployingPAAWtechnology.Amulti-objectiveflightcontrol
system has been previously developed to simultaneously gain aerodynamic efficiency and maintain traditional pilot
command-trackingtasksforguidanceandnavigation.6,7 Thisstudyextendsthepreviouslydevelopedmulti-objective
optimal control design to include gust and maneuver load alleviation objective in conjunction with the aeroelastic
modesuppressionanddragminimizationobjectives.
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II. AdaptiveAeroelasticWingShapingControlAnalysis
A. DragMinimizationControl
Aeroelasticdeflectionscanaffectaircraftaerodynamics. Asanaircraftcruises,fuelisburnedandthewingloadingis
reduced,therebycausingtheliftdistribution,hencethewingshape,tochange.Thechangeinthewingshapecancause
adragpenaltysincethewingshapenolongerretainsitsoptimaldesignshape. Thisparticularlycanbeanimportant
issueforlight-weightairframes. Thus,aircraftwithflexiblewingstructurescanpotentiallybecomelessfuel-efficient
ifthereisnomechanismtocompensateforaeroelasticdeflections.
Theelasticangleofattackcontributestotheaircraftliftanddrag. Aircraftwingsareusuallydesignedforoptimal
aerodynamicsatasinglepointintheflightenvelope. Atypicaldesignpointforatransportaircraftwingisusuallyat
themid-pointofatypicalcruiseprofilewherethefuelishalfspent. Asthefuelisburnedoff,thewingshapechanges.
The resulting static aeroelastic deflection alters the elastic angle of attack which in turn causes the lift distribution
to be non-optimal. The net effect is a drag penalty at off-design flight conditions. In modern transport aircraft with
flexiblehigh-aspectratiowings,specialcontroldevicesareusedtore-optimizethewingtwisttocompensateforthe
aeroelasticeffectonaerodynamics.
Thestaticaeroelasticequationsforbendingandtorsionaregivenby
∂2W ∂2 (cid:18) ∂2W(cid:19) ∂m¯
ρA + EI =l¯cosΛ−ρAgcosΛ+ cosΛsinΛ (1)
∂t2 ∂x2 yy ∂x2 ∂x
∂2Θ ∂ (cid:18) ∂Θ(cid:19)
ρI − GJ =m¯cos2Λ−ρAge cos2Λ (2)
xx ∂t2 ∂x ∂x cg
whereW(x,t) is the bending deflection, Θ(x,t) is the torsional twist, x is the coordinate of the elastic axis inclined
from the aircraft pitch axis y¯ by the sweep angle Λ as shown in Fig. 4, l¯(x,t) is the unsteady lift force, m¯(x,t) is
thepitchingmomentinthestreamwisedirectionalongtheelasticaxis,ρAgisthewingweightdistribution,e isthe
cg
offsetofthecenterofmassfromtheelasticaxis(positivewhenthecenterofmassliesforwardoftheelasticcenter),
andthemassandelasticpropertiesρA,ρI ,EI andGJarealldefinedwithrespecttotheelasticaxis.
xx yy
Figure4. SweptWingCoordinates
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Usingtheaerodynamicstriptheory,theunsteadysectionalliftcoefficientc (x,t)isgivenby
L
l¯ (cid:18) ∂W e cosΛ∂Θ 1 ∂W(cid:19)
c
c = =c (α+ξ−α )+C(k)c δ+C(k)c ΘcosΛ− sinΛ+ −
L q c Lα 0 Lδ Lα ∂x V ∂t V ∂t
∞ ∞ ∞
πc (cid:18) ∂Θ e cosΛ∂2Θ 1 ∂2W(cid:19)
m
+ cosΛ + − (3)
2V ∂t V ∂t2 V ∂t2
∞ ∞ ∞
whereq isthedynamicpressure,c (x)istheliftcurveslope,α(t)istheangleofattack,ξ(x)isthejigshapetwist,
∞ Lα (cid:104)
α (x)isthezero-liftangleofattack,c(x)isthewingchordinthestreamwisedirection,c (x)= c (x) c (x)
0 L L L
δ δ1 δ2
(cid:105)(cid:62) (cid:104) (cid:105)(cid:62)
··· cLδn(x) isavectoroftheliftcontrolderivatives,δ(t)= δ1(t) δ2(t) ··· δn (t) isavectorofthe
flap deflections, e (x) is the offset of the elastic center from the three-quarter-chord point (positive when the elastic
c
center lies forward of the three-quarter-chord point), ρ is the air density, andC(k) is Theodorsen’s function of the
∞
reducedfrequencyparameterk= ωc fortheelasticwing.
2V∞
Thestreamwiseunsteadysectionalpitchingmomentcoefficientc (x,t)isgivenby
m
m¯ e(cid:2) (cid:3)
c = =c +C(k)c δ+ c (α+ξ−α )+C(k)c δ
m q c2 mac mδ c Lα 0 Lδ
∞
e (cid:18) ∂W e cosΛ∂Θ 1 ∂W(cid:19) πe (cid:18)e cosΛ∂2Θ 1 ∂2W(cid:19)
c m m
+C(k) c ΘcosΛ− sinΛ+ − − −
c Lα ∂x V ∂t V ∂t 2V V ∂t2 V ∂t2
∞ ∞ ∞ ∞ ∞
πe cosΛ∂Θ πc2cosΛ∂2Θ
c
− − (4)
2V ∂t 64V2 ∂t2
∞ ∞
(cid:104)
wherec (x)isthepitchingmomentcoefficientabouttheaerodynamiccenter,c (x)= c (x) c (x) ···
mac mδ mδ1 mδ2
(cid:105)(cid:62)
c (x) isavectorofthepitchingmomentcontrolderivatives,e(x)istheoffsetoftheaerodynamiccenterfrom
m
δn
theelasticcenter(positivewhentheaerodynamiccenterliesforwardoftheelasticcenter),ande (x)istheoffsetof
m
theelasticcenterfromthemid-chordpoint(positivewhentheelasticcenterliesforwardofthemid-chordpoint).
ThecruiseconditiondictatesthatthetotalliftLontheaircraftisequaltotheweightW oftheaircraft. So,
L=W (5)
The wing lift typically constitutes at least 90% of the total lift of a typical transport aircraft. For a given design
grossweight,airspeed,andaltitude,adesignaircraftliftcoefficientiscomputedas
(cid:115)
W
C∗= (6)
L q S
∞
whereSisthewingreferencearea.
Forsteadystateaerodynamics,thegeneralizedrigidaerodynamicforcevectoriscomputedas
´
(cid:34) (cid:35)
LΦ(cid:62)(x)l dx
Q = ´0 r =Q +Q α+Q δ (7)
r LΨ(cid:62)(x)m dx 0 α δ
0 r
(cid:104) (cid:105)(cid:62) (cid:104) (cid:105)(cid:62)
whereΘ(x)= Θ (x) Θ (x) ··· Θ (x) andΨ(x)= Ψ (x) Ψ (x) ··· Ψ (x) arevectorsofthe
1 2 N 1 2 N
bendingmodeshapefunctionsandtorsionalmodeshapefunctions,respectively,and
(cid:2) (cid:3)
l = c (α+ξ−α )+c δ q ccosΛ−ρAgcosΛ
r Lα 0 Lδ ∞
+ ∂ (cid:110)c +c δ+e(cid:2)c (α+ξ−α )+c δ(cid:3)(cid:111)q c2cosΛsinΛ (8)
∂x mac mδ c Lα 0 Lδ ∞
m =(cid:110)c +c δ+e(cid:2)c (α+ξ−α )+c δ(cid:3)(cid:111)q c2cos2Λ−ρAge cos2Λ (9)
r mac mδ c Lα 0 Lδ ∞ cg
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Thegeneralizeddisplacementisthenexpressedas
(cid:18) K (cid:19)−1
a
q= K+q (Q +Q α+Q δ) (10)
∞q 0 α δ
∞
(cid:104)
whereK isthestructuralstiffnessmatrix,Kaistheaerodynamicstiffnessmatrix,andq= w1 w2 ··· wN θ1
(cid:105)(cid:62)
θ ··· θ isavectorofthegeneralizedcoordinateswhichformthedisplacementsolutions
2 N
N
W(x)=∑Φ (x)w (11)
i i
i=1
N
Θ(x)=∑Ψ (x)θ (12)
i i
i=1
Sincethegeneralizedrigidaerodynamicforceisafunctionoftheangleofattackα andflapdeflectionvectorδ,
thewingbendingandtorsionaldeflectionsarealsofunctionsofα andδ. Thisiswrittenby
W(x)=W (x)+W (x)α+W (x)δ (13)
0 α δ
Θ(x)=Θ (x)+Θ (x)α+Θ (x)δ (14)
0 α δ
The local aeroelastic angle of attack on a swept wing is influenced by the bending and torsional deflections ac-
cordingto
cL ∂
α =α+ξ−α + δ δ+[Θ (x)+Θ (x)α+Θ (x)δ]cosΛ− [W (x)+W(x)α+W (x)δ]sinΛ (15)
c 0 c 0 α δ ∂x 0 δ
Lα
Atthedesigncondition,theoptimalliftdistributioncanbeachievedbyproperlyselectingthejigtwistξ(x). This
optimalliftdistributionisexpressedas
1
Γ= V cc (16)
∞ L
2
Theaerodynamicdragcoefficientatthedesignconditionisexpressedinaparabolicdragpolaras
C∗2
C∗ =C∗ + L (17)
D D0 πARε∗
whereC∗ isparasiticdragwhichincludestheskinfrictiondragandtransonicwavedrag,ARisthewingaspectratio,
and0<Dε0∗<1isthespanefficiencyfactor.
Thejigtwistreferstothegeometrictwistthatisbuiltintothewingconstructionprocessduringmanufacturing. A
jig-shapewingisgenerallyofaplanarplanform. Asthewingdeflectsunderloadingatthedesigncruisecondition,the
jigtwistwouldcompensateforthelocalaeroelasticangleofattacktoachievetheoptimalliftdistribution. Whenthe
aircraftoperatesatoff-designflightconditions,thedesignliftdistributioncannolongerbemaintained. Consequently,
alossinthespanefficiencyfactorresultsandcausesanincreaseintheinduceddrag. Atanoff-designliftcoefficient
C >C∗,anincrementalliftcoefficient∆C isrequiredtotrimtheaircraftatanincrementalangleofattack∆α,then
L L L
thechangetheliftdistributionwithoutaeroelasticcompensationbywingshapingcontrolisgivenby
(cid:20) (cid:21)
1 ∂W (x)
∆Γ= V cc ∆α Θ (x)cosΛ− α sinΛ (18)
2 ∞ Lα α ∂x
Thechangeintheliftdistributionresultsinareductioninthespanefficiencyfactorwhichaffectstheinduceddrag.
Theaerodynamicdragcanbeexpressedas
(C∗+∆C )2
C =C∗ +∆C + L L (19)
D D0 D0 πARε
whereε <ε∗fornon-optimalliftdistribution.
The parasitic drag usually is a convex function of the angle of attack. So, if the parasitic dragC∗ at the design
D0
conditionisaminimum,then∆C >0.
D0
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Withaeroelasticcompensation,theliftdistributioncanbeoptimizedwithapositiveflapdeflectionδ toachievethe
sameincrementalliftcoefficient∆C withareducedincrementalangleofattack∆α∗. Thisoptimizedliftdistribution
L
isexpressedas
(cid:20) (cid:21) (cid:20) (cid:21)
1 ∂W (x) 1 1 ∂W (x)
∆Γ∗= V cc ∆α∗ Θ (x)cosΛ− α sinΛ + V cc δ+ V cc δ Θ (x)cosΛ− δ sinΛ (20)
2 ∞ Lα α ∂x 2 ∞ Lδ 2 ∞ Lδ δ ∂x
Assuming the optimal span efficiency factor ε∗ is achieved, then the aerodynamic drag due to aeroelastic wing
shapingcontrolisexpressedas
(C∗+∆C )2
C =C∗ +∆C∗ + L L (21)
D D0 D0 πARε∗
Adragreductionisthenobtainedas
(cid:18)1 1 (cid:19)(C∗+∆C )2
∆C =∆C −∆C∗ + − L L (22)
D D0 D0 ε ε∗ πAR
Itcanbeseenthatthedragreductioncomesfromboththeinduceddragandtheparasiticdragsince∆C∗ <∆C
because∆α∗<∆α.Thus,aeroelasticwingshapingcontrolcanofferapotentialdragreductionbenefitforfuDe0lsavinDg0s
intransportaircraft.
Ingeneral, thedragminimizationcontrolcanbesolvedbyacoupledaerodynamic-structuraloptimization. This
optimization can be performed off-line to establish a schedule of flap settings based on the aircraft gross weight
and flight conditions. Aerodynamic-structural optimization studies as well as a wind tunnel experiment have been
conductedtoassessthedragreductionbenefitoftheVCCTEF.8–10Thesestudiesshowadragreductionbenefitranging
from 1% to 6%. Figure 5 shows a typical solution of the aerodynamic-structural optimization of the spanwise lift
distributionoftheflexiblewingGTMwiththeVCCTEFatanoff-designcruisecondition.8
Figure5. OptimizedSpanwiseLiftDistributionofFlexibleWingGTMat30%OverDesignCL
Inpractice,variancesintheaircraftgrossweightandflightconditionscanresultindifferentperformancecharac-
teristicsthantheoptimalsolutionsobtainedfromtheoff-lineaerodynamic-structuraloptimization.Thus,oneapproach
in multi-objective flight control is to incorporate a real-time drag minimization control to compensate for operating
variances. Thisreal-timeminimizationcontrolstrategyhasbeendevelopedrecentlyandwillbevalidatedinawind
tunnel experiment.11 The real-time drag minimization would require real-time drag and aeroelastic deflection mea-
surements for estimating the aerodynamic characteristics of the flexible wing. This information is then used for the
on-lineoptimization.
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B. LoadAlleviationControl
As aircraft wings become more flexible, maneuver and gust loads can result in potentially large load excursions.
Dynamic responses due to a rapid flight maneuver such as a sudden pull-up or sharp roll can cause wing bending
momentstoincreaserapidly.Asaresult,forhighaspectratioflexiblewingaircraft,flightmaneuversmustbeexecuted
in a manner so as to maintain flight loads to within the allowable limits. Multi-functional distributed flight control
surfacessuchastheVCCTEFcanbedeployedinthesesituationstoachievetherequiredmaneuverwhilereducingthe
maneuverloads. Themaneuverloadalleviationthuscouldbeincorporatedinthedesignofaflexiblewingtoreduce
thewingstructuralweight.
As an example, consider the Truss-Braced Wing (TBW) aircraft equipped with a VCCTEF system as shown in
Fig. 6. Thereare10flapsections,eachwithtwochordwisecamberedsegments.12
Figure6. Truss-BracedWingAircraftwithVCCTEF
A2.5-gpull-upmaneuveriscommanded.Figure7showstheflapwisewingbendingmomentwhichisamaximum
at the truss juncture.12 The truss is designed to reduce the wing root bending moment as can be seen from the plot
andeffectivelytransfersthebendingmomenttothetrussjuncture. WithouttheVCCTEFdeployment,themaximum
bendingmomentoccursatthetrussjuncture. TheVCCTEFisthendeployedtoalleviatethewingbendingmomentat
thetrussjuncture. Theoptimalloadalleviationoccurswhenboththewingrootbendingmomentandbendingmoment
atthetrussjunctureareequal. ByoptimizationoftheVCCTEF,thewingbendingmomentisreducedby37%.12
Figure7. WingBendingMomentfor2.5-gPull-UpManeuver
Figure8showstheliftdistributionactingonthewingwithandwithouttheVCCTEFdeployment.12 Toreducethe
wingbendingmoment,theVCCTEFincreasesthewingliftinboardanddecreasesthewingliftoutboardofthetruss
juncture. TheVCCTEFeffectivelyre-optimizesthespanwiseliftdistributionfromanearlyellipticalliftdistribution
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foroptimalaerodynamicperformanceatcruisetoaliftdistributioncorrespondingtoaminimumwingbendingmo-
ment. Thus,itcanbeseenthatdragminimizationcontrolandmaneuverloadalleviationcontrolproduceanopposing
effect. Therefore,maneuverloadalleviationcontrolcannotbeachievedsimultaneouslywithdragminimizationcon-
trol.Inmulti-objectivecontrol,thisopposingeffectiscreatedbyconflictingobjectives.Thereexistmultiplecandidate
optimalcontrolsolutionsthatlieonaParetooptimalsurfacethatprovidesasetofcompromisedsolutionsofamulti-
objectiveoptimizationproblem. Themulti-objectiveoptimalcontrolthusprovidesatrade-offactiontominimizeone
objectivewhilenotsignificantlydegradingtheotherobjective.
Figure8. WingLiftDistributionforLoadAlleviationControlusingVCCTEF
Fortheunsteadyliftandpitchingmomentexpressionsintheprevioussection,wecanexpressthemingeneralas
∂Θ ∂2Θ ∂W ∂W ∂2W
l=l +l +l Θ+l +l +l +l +l +l δ+l δ˙+l δ¨ (23)
0 α Θ Θt ∂t Θtt ∂t2 Wx ∂x Wt ∂t Wtt ∂t2 δ δ˙ δ¨
∂Θ ∂2Θ ∂W ∂W ∂2W
m=m +m α+m Θ+m +m +m +m +m +m δ+m δ˙+m δ¨ (24)
0 α Θ Θt ∂t Θtt ∂t2 Wx ∂x Wt ∂t Wtt ∂t2 δ δ˙ δ¨
wherel(x,t)andm(x,t)aretheexpressionsintherighthandsidesofEqs. (1)and(2).
Theweakformexpressionoftheaeroelasticequationsusingafinite-elementmethodorotherequivalentdiscretiza-
tionisgivenby
Mq¨+Kq=Q +Q α+Q q+Q q˙+Q q¨+Q δ+Q δ˙+Q δ¨ (25)
0 α q q˙ q¨ δ δ˙ δ¨
where M is the structural mass matrix, K is the complex stiffness matrix which accounts for structural damping, Q
isthe generalizedforce, andthe subscripts denotethe partialderivatives ofthe generalizedforcewith respectto the
subscriptvariables.
Thewingflapwisebendingmomentandtorsionalpitchingmomentarecomputedas
M =GJ∂Θ =GJ∑N Ψ(cid:48)(x)θ
x ∂x i i
i=1
(cid:104) (cid:105) (cid:16) (cid:17)
=GJΨ(cid:48)(x) 0 I K−1 −Mq¨+Q0+Qαα+Qqq+Qq˙q˙+Qq¨q¨+Qδδ+Qδ˙δ˙+Qδ¨δ¨ (26)
M =EI ∂2W =EI ∑N Φ(cid:48)(cid:48)(x)w
y yy ∂x2 yy i i
i=1
(cid:104) (cid:105) (cid:16) (cid:17)
=EIΦ(cid:48)(cid:48)(x) I 0 K−1 −Mq¨+Qr+Qqq+Qq˙q˙+Qq¨q¨+Qδδ+Qδ˙δ˙+Qδ¨δ¨ (27)
Thebendingmomentatacriticallocationonthewingcanbeevaluatedfromtheexpressionaboveandexpressed
ingeneralas
M =M +M α+M q+M q˙+M q¨+M δ+M δ˙+M δ¨ (28)
c 0 α q q˙ q¨ δ q˙ q¨
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