Table Of ContentAeroelastic Uncertainty Quantification Studies Using the S4T
Wind Tunnel Model
MelikeNikbay∗
NationalInstituteofAerospace,Hampton,Virginia,23666,USA
IstanbulTechnicalUniversity,Maslak,Istanbul,34469,Turkey
and
JenniferHeeg†
NASALangleyResearchCenter,Hampton,Virginia,23681,USA
ThispaperoriginatesfromthejointeffortsofanaeroelasticstudyteamintheAppliedVehicleTechnol-
ogy Panel from NATO Science and Technology Organization, with the Task Group number AVT-191, titled
“ApplicationofSensitivityAnalysisandUncertaintyQuantificationtoMilitaryVehicleDesign.”Wepresent
aeroelasticuncertaintyquantificationstudiesusingtheSemiSpanSupersonicTransportwindtunnelmodelat
theNASALangleyResearchCenter.Theaeroelasticstudyteamdecidedtreatbothstructuralandaerodynamic
inputparametersasuncertainandrepresentthemassamplesdrawnfromstatisticaldistributions,propagat-
ingthemthroughaeroelasticanalysisframeworks.Uncertaintyquantificationprocessesrequiremanyfunction
evaluationstoassestheimpactofvariationsinnumerousparametersonthevehiclecharacteristics,rapidlyin-
creasingthecomputationaltimerequirementrelativetothatrequiredtoassessasystemdeterministically.The
increasedcomputationaltimeisparticularlyprohibitiveifhigh-fidelityanalysesareemployed. Asaremedy,
theIstanbulTechnicalUniversityteamemployedanEulersolverinanaeroelasticanalysisframework, and
implementedreducedordermodelingwithPolynomialChaosExpansionandProperOrthogonalDecomposi-
tiontoperformtheuncertaintypropagation. TheNASAteamchosetoreducetheprohibitivecomputational
timebyemployinglinearsolutionprocesses.TheNASAteamalsofocusedondetermininginputsampledistri-
butions.
Nomenclature
LHS LatinHypercubeSampling
MCS MonteCarloSampling
PCE PolynomialChaosExpansion
POD ProperOrthogonalDecomposition
ROM Reducedordermodel
I. Introduction
Aeroelasticity,asamultidisciplinaryresearchfield,investigatesthebehaviorofanelasticstructureintheairstream
andtheinteractionsbetweeninertial,aerodynamic,andstructuralforces. Withtheuseofadvancedtechnologies
inaircraftdesignandanalysis,theneedforpredictingsuchaerostructuralinteractionsbyusinghigh-fidelitycompu-
tational models has increased. Most analysis procedures involving high-fidelity modeling consider the system and
solution as deterministic. However, as computational models have advanced, parametric sensitivities have become
more intrinsic and more hidden. Recent work has demonstrated solution sensitivity to small changes in geometry,
freestream conditions and computational parameter specifications. This sensitivity leads to considering these simu-
lations from an uncertainty standpoint. As another motivation in experiments, input parameters are only controlled
to within some precision level and the input and output parameters are only known to within measurement errors.
Comparisonsbetweencomputationsandexperimentsshouldconsidertheseuncertainties.
∗Currently,SeniorResearchEngineeratNationalInstituteofAerospace,locatedatNASALangleyResearchCenter,AeronauticalSystems
AnalysisBranch&AssociateProfessor,FacultyofAeronauticsandAstronautics,AIAAAssociateFellow
†SeniorResearchEngineer,AeroelasticityBranch,AIAAAssociateFellow
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In the current paper, we endeavor to recast our aeroelastic simulation inputs and outputs to treat the analyses as
stochastic processes. Two different approaches and philosophical viewpoints are utilized to quantify the effects of
uncertaintiesinaeroelasticsystems. Themethods,modelingandresultsareproducedfromresearcheffortsofteams
fromNASAandIstanbulTechnicalUniversity(ITU).TheNASAworkutilizeslinearaerodynamicmodeling,whilethe
ITUworkutilizeshigherfidelityaerodynamicmodelingandareducedordermodel(ROM)approachforpropagating
theuncertainparameters.
Aflexiblevehicleconfiguration,theNASASemispanSupersonicTransport(S4T)windtunnelmodel,waschosen
asthebasisforuncertaintystudiesandpropagation. Thecurrentworkutilizesinformation,analyticalmodelsanddata
reportedbythe(S4T)projectteamin2012ataspecialsessionatthe53rdAIAA/ASME/ASCE/AHS/ASCStructures,
StructuralDynamicsandMaterialsConference. AprogramoverviewpaperwasauthoredbySilva36 thatdocuments
thewindtunnelmodel,fourwindtunneltestsconductedbetween2007and2010,andreferenceslaboratorytestsand
analyses. Theproductsfromthoseeffortsareutilizedasthestartingpointforthecurrentuncertaintyanalyses;these
aspectswillbefurtheridentifiedinthebodyofthispaper.
Thecurrentpaperfocusesondefiningstatisticaldistributionsofinputparameters, propagatingthemthroughthe
twoanalysisprocesses,andexaminingthestatisticaldistributionsofoutputparameters. Theanalysisapproacheswill
bedescribedfirst,followedbyadescriptionofthedevelopmentofanexampleinputparameterdistributiontorepresent
uncertainty associated with a structural parameter. Example results from each of the analysis methods will then be
presented,followedbydiscussionandconcludingremarks.
II. AnalysisApproaches
A. UncertaintyPropagationUsingLinearAnalyses
Themethods,modelingandresultsgeneratedbytheNASAteamwereproducedfromlinearaerodynamicandstruc-
tural computations. The models used as the baseline cases came from the project team that designed, tested and
analyzedtheconfiguration. Themajorstepstakeninperformingtheuncertaintypropagationswere:
• Defineinputvariation
• Generaterandominputsamples
• Generateinputfileforeachinputsample
• Performsimulationforeachinputsample
• Extractoutputparametersfromeachsimulation
• Performstatisticalanalysesoftheevaluationparameters
ThreetypesofsimulationswereperformedintheNASAanalyses: staticaeroelasticanalyses,normalmodesanal-
ysesandfluttersolutions.EachoftheanalyseswereperformedusingstandardanalysisoptionsinMSCNASTRAN.18
Inputandoutputparameterscreeningwasperformedusingstaticaeroelasticanalysis. Normalmodesanalyseswere
performedtoshowthesensitivityofmodalfrequenciestothestructuralvariations. Flutteranalyseswereperformedto
showthesensitivitytomodalparametersinadditiontosensitivitytothemorefundamentalinputparameters.
B. ReducedOrderModelingUsingEulerSimulations
TheITUteamperformedstaticaeroelasticandflutteranalysesoftheS4TwingbyusingthePolynomialChaosExpan-
sion(PCE)22–25 andProperOrthogonalDecomposition(POD)27,29–31,33,34 basedReducedOrderModels,26,32 which
weregeneratedbyusinganEulerflowbasedaeroelasticsolverinZEUS19 software. TheaeroelasticsolverofZEUS
employsmodeshapesobtainedfromtheNASTRANmodalsolver.TheROMresultswerecomparedtotheinitialcom-
putationalresultsofthehigh-fidelitymodels.TheseROMscouldpredicttheresponsesinasmallinputdatarange,and
inthiscaseboththePCEandPODmethodsgavesatisfactoryresultswhiletheiraccuracieswerethoroughlydiscussed
inthepresentwork. Secondly, boththePCEandPODmethodswereusedtoconstructaROM,whichwasaccurate
whenlargevariationsofinputparameterswereconsidered. TheflutterbehavioroftheS4Twingcouldaccuratelybe
predictedovertheentireflightregimeforMachnumbersrangingfrom0.60to1.20bythePCEandPODbasedROMs.
ThiswillbedemonstratedinthispaperthroughdetailedcomparisonsoftheflutterresultsoftheROMstotheinitial
high-fidelitycomputationalanalyses.
UncertaintydistributionsofstructuralandaerodynamicinputparameterswerethenpropagatedusingtheROMs.
Outputparametersfromthereducedorderstaticaeroelasticandflutteranalyseswereexaminedtoassesstheinfluences
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oftheinputparametersthatwerevaried.TheinputsamplingwasperformedusingMonteCarloSimulation(MCS)and
LatinHypercubeSampling(LHS)techniques,whicharefrequentlyusedtopropagateuncertaintiesinaprobabilistic
framework. The input parameters that were treated as uncertain in these analyses were angle of attack, Young’s
modulusanddynamicpressure. Theinfluenceofthefirsttwowereassessedforboththestaticaeroelasticandflutter
analyses. Thedynamicpressureuncertaintywasonlyevaluatedforthestaticaeroelasticcases. Theoutputparameters
thatwereexaminedwereflutterfrequency,flutterspeed,flutterdynamicpressure,liftcoefficient(C ),dragcoefficient
L
(C ),pitchingmomentcoefficient(C ),andwingtiptwistangle.
D m
1. PolynomialChaosExpansion(PCE)Methodology
NonintrusivePCEdefinesaROMbasisintermsoforthogonalpolynomials.Inthepresentstudy,Hermitepolynomials
are used as the orthogonal polynomials since the input parameters are assumed to vary with respect to Gaussian
distribution. ThedefinitionofthefirstfewHermitepolynomialsaregivenbelowintermsofthestandardvariable,ξ:
H (ξ)=1 (1)
o
H (ξ)=ξ (2)
1
H (ξ)=ξ2−1 (3)
2
H (ξ)=ξ3−3ξ (4)
3
H (ξ)=ξ4−6ξ2+3 (5)
4
H (ξ)=ξ5−10ξ3+15ξ (6)
5
H (ξ)=ξ6−15ξ4+45ξ2−15 (7)
6
Thechaoscoefficientscanbecalculatedbyusingthefollowingrelationforageneralphysicalsystem:
u H H H . . H a
u12 H1200 H1211 H1222. . H12pp a12
. = . . . . . . (8)
. . . . . . .
u H H H . . H a
n n0 n1 n2 np p+1
where u (i=1 to n) is an output response that is known from the analyses and a (j =1 to p+1) shows chaos
i j
coefficients to be determined. The order of the approximation is denoted by p and n shows the dimension. In this
matrix system, the parameters H indicate the multiplication of H and H functions defined in Eq. (4) to (10).
ij i j
However, in the two-dimensional case, H is the function of the first standard random variable, ξ, while H is the
j i
function of the second standard random variable, η. Since three input parameters are defined in the present work,
H =H(ξ)·H (η)·H (γ) (where γ is also a standard random variable) multiplications should be computed. The
ijk i j k
ROMcanbeconstructedbyusingthechaoscoefficients. Inordertocalculatethechaoscoefficients,linearregression
canbeapplied. Consideringthestandardnormalvariables ξ, η andγ, agenerallinearregressionmodel,U, canbe
definedinclosedformasbelow:
U =Haˆ+ε (9)
whereεdenotestheerrorbetweenrealvaluesandapproximatevalues.Theregressioncoefficients,aˆ,canbecomputed
byusingEq. (10):
aˆ=(HTH)−1HTU (10)
Afterdeterminingtheregressioncoefficients,theROMtorepresentthereferenceinitialdesigncanbeconstructedfor
thetwoinputvariablecasebyusingtherelationbelow:
U (ξ∗,η∗)=a H (ξ∗,η∗)+a H (ξ∗,η∗)+...+a H (ξ∗,η∗) (11)
j 1 j0 2 j1 p+1 jp
whereξ∗andη∗arethevaluesofthestandardrandomvariablesforthecasethatneedstobecomputed.
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2. ProperOrthogonalDecomposition(POD)Approach
TheProperOrthogonalDecomposition(POD)methodisusedtogenerateaROM.Thesystemisdefinedasalinear
combinationofindependentorthogonalbasisfunctions. Theorthogonalfunctions,calledthe“PODbasisfunctions,”
areformulatedsuchthattheeffectofthefirstmodeisthehighestandthesecondmodeislesseffectiveandsoon. A
physicalmodelwithnoutputscanbeexpressedasalinearcombinationgivenasbelow:
k
[y] = ∑a [Φ ] (12)
(n×1) j j (n×1)
j=1
whereyisanoutputresponsevectorforasnapshotoftheinputvariables,a arePODcoefficients,Φ arePODbasis
j j
vectors and k is the order of ROM (k<n). In this study, the “method of snapshots”35 is used for creating ROMs.
A snapshot matrix T, is an n×m matrix (m is number of snapshots) and can be constructed by using the output
parametersofthesystem:
(1) (2) (m)
y y . y
1 1 1
. . . .
[T] =
(n×m)
. . . .
(1) (m)
y . . y
n n (n×m)
Anyphysicalsystemgivenwithasnapshotmatrixcanberepresentedbyasurrogateresponsefunctionwhichsuper-
posesthebasicmodesofthemodel. Thecorrelationmatrixisdefinedintermsofthesnapshotmatrixandthenumber
ofPODbasisvectors:
1
[C] = [T] •[T]T (13)
(n×n) n (n×m) (m×n)
The POD basis vectors and coefficients are defined through the eigenvalues and eigenvectors of the correlation ma-
trix[C]Φ=ΛΦ. SingularValueDecomposition(SVD)facilitatestheconstructionofthePODbasedROMs. Inthis
method,thefactorizationofarealmatrix,T isgivenbelow:
[T] =[U] [Σ] [V]T (14)
n×m (n×n) (n×m) (m×m)
where U, the columns of which consist of the eigenvectors of [T][T]T, is the left singular vector of T and these
eigenvectorsarePODbasisvectors.V,thecolumnsofwhichconsistoftheeigenvectorsof[T]T[T],istherightsingular
vectorofT. [Σ]isadiagonalmatrixrepresentingthesquarerootsofPODeigenvalues,λ. ThePODcoefficientsare
i
calculatedbyprojectingthesnapshotsonthePODbasis.
a =[T ]T •[Φ ] (15)
j j (1×n) j (n×1)
The number of POD basis vectors is equal to the number of snapshots m (and also POD coefficients), however, the
number of retained POD basis vectors (POD modes) k is determined by considering the captured energy from the
originalmodel. Thetotalcapturedenergyisdefinedas:
k
E =∑λ (16)
total i
i=1
where λ is the vector of normalized eigenvalues of the correlation matrix,C, and k is the required number of POD
modestoreachthespecifiedcapturedenergylevel. Inthepresentwork,thecapturedenergylevelistakenas99.9%
inallapplications.
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III. DevelopmentofStatisticalDistributionsofInputParameters
Development of statistical distributions to represent input parameter variations were performed by the NASA
team. The information required to formulate the distributions of structural parameters is discussed in this section.
Themodelhardware,testingandfiniteelementmodel(FEM)tuningprocesswerethekeyelementsindevelopingthe
inputparameterdistributions. Thevariationsforotherinputvariablesthatareutilizedinthecurrentwork—angleof
attack, dynamic pressure, Mach number and modal frequencies—are not discussed in detail here; the processes and
assumptionsfordevelopingtheseadditionalinputparametervariationsaredetailedintheNATO-STO-AVT-191Task
GroupReport.17
A. HardwareCharacterizationTesting
TheS4Twingisacomplexaeroelasticsemispanwindtunnelmodel,whichisdesignedforaeroelastic/aeroservoelastic
analysis.20 Thefuselageconsistsofagraphite-epoxyflexiblebeamattachedtoanaluminumC-channelrigidbeam.
Wingboxconstructionconsistsofanupperandlowerskin,eachmadefromfiberglass/epoxyfacesheetssandwiching
apaperhoneycombcore. Figure1showstheconfiguration.
Figure1. S4Tconfiguration.
Extensivewinggeometrymeasurement,stiffnesstestingandmodalsurveytesting(groundvibrationtesting,GVT)
were performed. These results were used to improve the aerodynamic surface definition and the structural model.
Theflexiblefuselagebeamelementmaterialpropertieswereusedasprincipaltuningparametersduringmodelupdate
procedures. Figure2showstheresultsfromtuningtheanalyticalmodeltomatchthemeasureddeflections. Therange
ofthemodificationsusedinperformingthistuningserveasthebasisforthematerialpropertydistributionsusedinthe
uncertaintypropagationstudies.
The GVT results37contain measured values for the frequencies and damping values associated with the first six
modes.TheanalyticalandmeasuredfrequenciesanddescriptionsofthemodeshapesareshowninTable1.38 Compar-
isonsoftheanalyticalandmeasuredfrequenciesgiveindicationsofthesuccesslevelofthestiffnesstuningexercise.
Thescatteramongthemeasuredvaluesforeachmodalfrequencyalsoprovidesanindicationofthevariationassociated
withmeasuringthefrequencies. Measureddampingvalues,reportedinreference37,containedsignificantscatter.
B. StructuralModel
The FEM utilized in this study for the baseline analysis cases is the unmodified model from the S4T team that is
detailed by Sanetrik.38 The FEM elements, emphasizing the fuselage, are shown in Figures 3, 4 and 5. Figure 5 in
particularshowstheflexiblefuselagebeammaterialpropertysegmentsthatwillbemodifiedaspartoftheuncertainty
propagationstudies.Thenumbersgiveninthefigurecorrespondtothedesignationsusedformaterialpropertycharac-
terizationsets. Associatedwitheachmaterialpropertysetisavaluefordensity,Young’smodulus,theshearmodulus
andPoissonratio.Inthecurrentstudies,theonlymaterialpropertytobetreatedasanuncertainvariableistheYoung’s
modulus.
ThefuselagebeamelementsshowninFigure5wereconsideredtheprimarycandidatesforparametricvariationin
thisstudy,basedonthetuningprocessesoftheS4Tteam. Theinputvariablerangeswerebasedonthevaluesusedby
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Figure2. FiniteelementmodeltuningbyS4Ttestteam;DeflectionindegreesvsFlow-wise(chordwise)station.
Figure3. Finiteelementstructuralmodel.
theS4TtestteamtotunetheFEM.Duringthisprocedure,theS4Tteamtunedthesixbeammaterialsinthebaseline
structuralmodel,correlatingwithmeasureddeflectiondata,asshowninFigure2. Inthecurrentstudy,thesesamesix
material properties were evaluated, using minimum, nominal and maximum values employed in the original tuning
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Table1. Measuredandanalyticalmodalfrequencies,Hz.
Modalfrequencies,Hz
Mode Description Analyticalnominalvalue Measuredvalues
1 Pitchaboutforwardnodalmount 6.29 6.395 6.375 6.249
2 Pitchaboutbalance 8.43 8.089 7.935 7.838
3 Fuselage1stbending 10.07 10.323 10.312 10.059
4 Wing1stbending 11.4 11.635 - 11.781
5 - 12.88 12.528 12.585 12.384
6 - 14.25 16.571 15.066 15.603
7 - 15.28 - - -
8 - 17.6 - - -
Figure4. Finiteelementmodel:Connectionpointsfromfuselagetowing,nodalmountpointsandfuselagebeamelements.
Figure5. Beamelements’materialpropertynumbers.
process. The values chosen by the S4T team to use in the final version of the FEM were designated as the nominal
values.
ThreeinputdistributionsanddrawnsamplesetscharacterizingtheuncertaintyoftheYoung’smodulusweregen-
erated: 1)anormaldistributionsampledwithLHS,500samplesdrawnwithoutreplacement;2)anormaldistribution
sampledwithLHS,25regionswith25resamplingswithreplacement(625totalsamples);and3)auniformdistribution
sampledwithLHS,25regionswith25resamplingswithreplacement(625totalsamples). Formaterial21,whichwill
be shown subsequently to be of greatest interest, the Gaussian distributions from which samples were drawn had a
meanvalueof6.390e+06psiandstandarddeviationof0.945e+06psi. Thedetailedstatisticsforallthreedistributions
forthismaterialpropertyaregiveninTable2.
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IV. LinearAnalyses
Asmallsubsetofthelinearanalysisresultsgeneratedinthisstudyarediscussedandpresentedinthecurrentpaper.
Additionalresultsandcomparisonswithdifferentuncertaintyquantificationapproachesaredetailedinreference17.
Inthecurrentsection,anexampleofparameterscreeningusingstaticaeroelasticanalysisispresented,alongwithan
examplecaseofpropagatingadistributionofaninputparameterthroughanormalmodesanalysis. Finally,example
results are shown for propagating both uniform and normal distributions of an input parameter through the flutter
analysisprocess.
A. StaticAeroelasticAnalysesforParameterScreening
Material property screening studies used the flexible-to-rigid ratio of the lift coefficient due to the horizontal tail
deflection as one of the evaluation parameters. This particular output variable was chosen due to several factors: 1)
this parameter was shown to be linear with respect to dynamic pressure; 2) the S4T program’s main research focus
wasonaeroservoelasticity, whichemphasizesthecriticalityofcontrolderivativesand3)mostcontrollawdesigners
associatedwiththeprogramchosethehorizontaltailastheiractuationsurfaceforperformingaeroservoelastictasks.
The screening analysis results, shown in Figure 6, indicate that the Young’s modulus associated with the material
property21hasthegreatestinfluence. Thisisshownbytherelativeslopesofeachofthelines; theslopeassociated
withmaterialproperty21isthesteepest.
Figure 6. Static aeroelastic analysis parameter screening; Flexible-to-rigid ratio variation of CLHT due to variation of
flexiblefuselagebeamYoung’smoduli.
Thus, the Young’s modulus of material 21 was chosen as the structural model parameter to vary and propagate
throughstaticanddynamicaeroelasticsimulations. Thismaterialpropertyisassociatedwithbeamelementscompris-
ingthemodel’sfuselageintheregionnearthemid-chordwingroot,asshowninFigure5.
B. NormalModesAnalysis
The second major type of analysis that was performed using linear methods was normal modes analysis. For this
analysis,theinputparameteridentifiedasthemostsignificantusingthescreeninganalysiswasvaried—theYoung’s
modulus of material property 21. Any variation in dynamic pressure, Mach number, angle of attack or structural
dampingwouldhavenoimpactontheoutputparametersofanormalmodesanalysis. Theoutputparametersthatare
examinedherearethefrequenciesofthefirst(lowestvalue)30naturalmodes. Thethreeinputdistributionsshownin
Table2fortheYoung’smodulusvariationwereeachusedasinputstothisprocess.
Histogramsofthefirst30naturalfrequenciesareshowninthe30subplotsofFigure7forthesecondinputcase
asnumberedinTable2. Noticethatfortheinputdistribution,whichisGaussian,someofthemodalfrequencyoutput
distributionsappearGaussianandsomeappearskewed(i.e.,someoftheseoutputdistributionsarelikelybetterfitby
beta distribution than by Gaussian distributions). There are also a few frequencies where the influence of the input
variation is so small that the histogram shows quantization levels associated with the significant digits of the output
data file (i.e., the first two subplots in the third row, associated with modes 13 and 14). The results for the second
inputcasearepresentedinFigure8. InFigure8,themeanvaluesforthefirst8modalfrequenciesareshownbythe
horizontallines. Thevariationsinthesimulationoutputsarecapturedbytheheightsofthebluerectangles,indicating
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Table2. Statisticalanalysisofinputdistributionsamples,Young’smodulusvaried.
Case Input Sampling Mean Standard Skewness Kurtosis Coefficient
distribution Method deviation of
type variation
1 Gaussian LHS500regions, 6.39E+06 9.45E+05 2.39E-02 2.96E+00 1.48E-01
noreplacement
2 Gaussian LHS25regions 6.39E+06 9.49E+05 -5.91E-02 3.04E+00 1.49E-01
25resamplings,
withreplacement
3 Uniform LHS25regions 6.39E+06 1.09E+06 -4.20E-03 -2.90E-01 1.71E-01
25resamplings,
withreplacement
Figure7. Histogramsofthefirst30modalfrequencies;ResultsfromLHS(25,25)samplesfromGaussiandistributionof
theYoung’smodulusofmaterial21.
the +/- 3 sigma bounds of the frequencies. The figure shows that only the second and third modes are significantly
modifiedbythevariationsappliedtotheYoung’smodulus.
C. FlutterAnalysis
Flutteranalyseswereconductedforeightbaselinecases,andforinputparametervariationsofeachofthoseeightcases.
The baseline cases correspond to the subsonic Mach numbers where the S4T team generated computational results.
Threeuncertaintydistributionswerepropagatedforeachconditionforfiveinputparametersets: structuraldamping,
Young’smodulus,Machnumberandnaturalfrequencies. Eachinputdistributionwaspropagatedindependently. The
complete results are given in reference 17. In the current section of this paper, example results are shown for two
distributions representing the damping variation. The Gaussian distribution of damping was generated with a mean
value of 0.02, and a standard deviation of 0.006; the uniform damping distribution was generated with a minimum
valueof0andamaximumvalueof0.04.
The doublet lattice method was employed to generate the aerodynamics and the PKNL (p-k nonlinear) flutter
solutionofMSC/NASTRANwasemployed. Foreachofthesebaselinecases,thestructuraldampingwassettozero,
and the first thirty flexible modes were included in the structural definition, corresponding to including all modes
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Figure8. VariationofYoung’smodulusformaterial21,GaussianLHS(25,25);Influenceonthefirst8modalfrequencies.
below100Hz. TheflutteranalyseswereperformedechoingthebaselineprocessperformedbytheS4Tteam,keeping
the velocity constant and varying the density. Although flutter results are shown here at transonic Mach numbers
(Mach 0.80 and 0.95), this should not be interpreted as an endorsement to utilize linear methods in this range. The
resultsarepossiblyrelevantatMach0.8forlowanglesofattack,butthismustbeestablishedbycomparisonswiththe
experimentaldataorhigherfidelitysimulations.
Inlinearanalysis, theflutterconditionisdefinedasthedynamicpressurethatcausesamodetohaveadamping
valueofzero.Thestructuraldampingappearsinthefluttersolutionequationsasanadditiveterm.Oncetheaeroelastic
modaldampinghasbeencalculated,changesduetothestructuraldampingcanbeapproximatedbyusingthestructural
dampingasthedefinitionofneutralstability,ratherthanusingthevalueof0. Thisisnotanexactcalculationdueto
thepresenceofnonlinearitiesintroducedbythegeneralizedaerodynamicforces,whicharecalculatedasfunctionsof
reducedfrequencyandthusarefunctionsofstructuraldamping. Theseinfluencesaresmallandareneglectedhere.
The derivative of the aeroelastic damping with respect to dynamic pressure is referred to as the steepness of the
flutter crossing. A steeper flutter crossing corresponds to a case that is more sensitive to aerodynamic damping (or
aerodynamicenergyinput)andlesssensitivetostructuraldamping. ThehigherMachnumbercasesaremoresensitive
toaerodynamicloadingthanthoseatlowerMachnumbers. Conversely, thelowerMachnumbercases—withflatter
flutter crossings—are more sensitive to structural damping than the higher Mach number cases. The result of the
difference in the flutter crossing steepness at different Mach numbers is illustrated in Figure 9, where the flutter
dynamicpressurevariationisobservedtobeconsiderablylargeratMach0.6thanatthehigherMachnumbers.
WenotethatforaGaussianinputdistributionofthedamping,theoutputflutterdynamicpressureisalsoGaussian.
Thedampingvariationflutterresultsareplottedwiththebaselineanalysisresults(shownbytheredcircles)inFigure9.
Inthefigure,theoutputhistogramsareshownontheirsides. Thegreenbargraphsshowthehistogramsoftheflutter
dynamicpressuresduetoauniforminputvariationofthedamping. Theheightofthehistogram(intermsofflutter
dynamic pressure) is greatest for the lowest Mach number indicating that the lower Mach number flutter onset is
moresensitivetouncertaintyindampingthanthehigherMachnumbers. Thisreflectsthesteeperfluttercrossingas
discussedabove. Theuniforminputdistributionsresultinuniformoutputdistributions. Thebluebargraphsshowsthe
histogramsoftheflutterdynamicpressuresduetoGaussianinputvariationofthedamping. Incomparingtheresults
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