single dof flutter calculations

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j jATIONALADVISORYCOMMITTEE ORAERONAUT ICS TECHN I CALNOTE2396 SINGLE -DEGREE -OF-FREEDOM-FLUTTER CALCULATIONSFOR

AWINGIN SUBSONIC OTENTIALFLOW ANDCOMPARISON WITHANEXPERIMENT

I yHarryL .Runyan LangleyAeronauticalLaboratoryLangleyField Va.

N A C A WashingtonJuly1951

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NATIONALADVISORYCOMMITTEEFORAERONAUTICS

TECHNICALNOTE2396

SINGLE-DEGREE-OF-FREEDOM-FLUTTERCALCULATIONSFORAWINGINSUBSONICPOTENTIALFLOWAND

COMPARISONWITHANEXPERIMENTByHarryL .Runyan

SUMMARY

Astudy ofsingle-degree-of-freedompitchingoscillationsofawinghasbeen presented.hisstudyincludestheeffectsofMach numberandstructuraldampingandisprimarily anextensiono farecentpaperbySmilginwhichincompressible flowwasconsidered.heactualexistenceofsingle-degree-of-freedomflutterwasdemonstratedbysomelow-speed'testsofawing,pivotedashortdistanceahead oftheleadingedgewithageometricaspectratioof5.87.ngeneral,goodagreementwasfoundbetweenexperimentalandcalculatedresultsf o rhighvaluesofaninertiaparametercorrespondingtohighaltitudes,butdifferencesexistf o rlowvaluesoftheinertiaparameter.heeffectofaspectratio hasnotbeenconsideredinthecalculationsandcouldhavean appreciableinfluenceontheoscillation.

INTRODUCTION

Thepossibilityoftheexistenceofsingle-degree-of-freedomoscillatoryinstabilityorflutterinincompressibleflow,bothpotentialandseparated,hasbeenknown f o rsometime.searlyas1929Glauert(reference1)notedthepossiblelossofdampingofapitchingwinginincompressible flowwhich mightlead toanoscillatory instabilitythatmaybereferredtoassingle-degree-of-freedom flutter.n1937Possiomadesimilarobservationsf o rsupersonic flow(reference2)andin19U6thisstudywaselaboratedonbyGarrickandRubinow(reference3) ,whoobservedthat,undercertainconditions,asingle-degree-of-freedomoscillationispossibleinincompressible flow.ubsequently,Smilg(referencek)madecalculationsshowingtherangesofaxis-of-rotationlocationandaninertiaparameterwhichcouldleadtoanoscillatoryinstabilityin pitchoryawf o rtheincompressiblecase.

Untilrecentlywhateverinterestwasshowninsingle-degree-of-freedomflutterwaslargelyacademicbecausetherangesofparameters


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involveddidnotappearpractical. However,withcurrentairplanesandmissilesdesigned f o rhighspeedsandhighaltitudes,thesubjectbecomesa morepracticalone,f o rundertheseconditionsundampedoscillationsofevenverysmallamplitudemaybecomeimportant. Inaddition,calcu-lationsofsingle-degree-of-freedomflutter mayrepresentauseful,easilyobtainedlimitf o rcasesofcoupledflutterinvolvingotherdegreesoffreedom.

Thispaperconsidersspecificallythetypeofsingle-degree-of-freedomflutterassociatedwiththepitchingofanairfoilaboutvariouslocationsoftheaxisofrotation. ItextendstheworkofreferencehtoincludetheeffectsofMachnumberupto =0.7nddiscussesth eeffectofstructuraldampingf o ronelocationoftheaxisofrotation.Theresultsofanexperimentalinvestigationwhichconfirmstheexistenceofsingle-degree-of-freedomflutterar ecomparedwithth etheoreticalvalues. Thecalculationswerebasedontwo-dimensionalaerodynamic-forcecoefficientsandinvolvedasingledegreeo ffreedom. Theeffectofaspectratioandthecoexistenceofotherdegreesoffreed omwouldmodifyth eresultstoalargeextent.

SYMBOLS

aondimensionaldistancef r o mmidchordtoaxiso frotation,basedonhalf-chord,positiverearward

balf-chordcpringconstantCaoefficientoftorsionalrigidityperunitlengthdampingcoefficientFandGunctionsof o roscillatingplanefl owgatructuraldampingcoefficientIaomentofinertiaaboutaxisofrotationperunitlengthIaaut-of-phase(imaginary)componentofmomentonairfoilaboutaxisofrotationperunitlengthkeducedfrequency( b c / v )


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N A C A N396m M MaMrMrM 1M2M3 M uZ M

massMachnumberaerodynamicmomentperunitlengthrealparto faerodynamicmoment,forincompressibleandcompressibleflow,respectively

aerodynamic fluttercoefficients( s e e reference )

Haan-phase( r e a l )componento f momentonairfoilaboutaxiso f rotationperunitlength

vluttervelocityxisplacementangulardisplacementaboutaxiso f rotation,positiveinstallingdirectionpluiddensity

ircularfrequencyatflutterMaaturalcircularfrequency

ANALYSISIntroductoryConsiderations

Beforethespecificexampleo fsingle-degree-of-freedompitchingflutter i sdiscussed,i t maybeadvantageousfirst t oreviewtheconcepto f asingle-degree-of-freedom vibratingsystemandthen t oshowtherelation o fthisexample t o anaerodynamicsystem.helineardifferen-tialequationfor afreesystemconsisting o fa mass,aspringhavingaspringconstant,and a viscousdamperhavinga coefficient s

mx dx cx=o 1 )

Themotionrepresented bythisequationi s damped i f s apositivequantity,a si sordinarilythec a s e . I f houldbenegative,


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themotionisundamped,aconditionofdynamicinstabilityexists,and,if szero,harmonicoscillationscorrespondingtoaborderlineconditionbetweendampedandundampedmotionmayexist.

Fo rasystemsuchasanaircraftwingoscillatinginasteadyairstream,thesametypeofequationwouldapplyas f o rthemass-spring-dampersystempreviously mentioned.owever,thecoefficients,,and fequation( l )willnow haveaddedcomponentsassociatedwiththeaerodynamics.heequationf o rawingoscillatinginpitchinasteadytwo-dimensionalairstreamis

Iaa + . (1+iga)Caa= Ma(a,d,,... ( 2 )

whereaepresentsthecomplexaerodynamicmoment,whichisa functioninpartofamplitude,velocityc ,accelerationd reducedfrequencybak = , locationofaxisofrotation,Machnumber,andsweepangle.Equation( 2 )iscomplexandmaybeseparatedintotwocomponents,oneassociatedwiththedampingofthesystem(sometimescalledtheimaginarypart)andtheotherassociatedwiththeflutter f requencyandvelocity(sometimescalledtherealpart).

Equationfor PitchingOscillations, =0Fromreference6 ,thevaluesofthedampingequationandthefrequencyequationf o rtwo-dimensionalincompressiblefloware:ampingequation:

Laa ;-G -)f*-M H -*;& )'- 3 ) Frequencyequation:

Rce - (WMMf - ( i\2G /l _\2F Jak2 npb1m = hEquation( 3 )isequivalenttothevanishingofthedampingcoefficientdfequation(1)andthusrepresentsaborderlineconditionbetweendampedandundampedoscillations. Th e flutterfrequencyandvelocitymaythenbedetermined fro mequation h)


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Equation( 3 )cannotbesolvedexplicitly r incethefunctionsFnd retranscendental functionsof.hereareseveralmethodsofsolvingthisequation;aconvenientone,giveninreference7 ,istoassumevaluesof/kndsolvef o rthestructuraldampingcoefficient.Thetypeofstructuraldampingforcecommonlyusedin fluttercalculationisinphasewiththevelocitybutproportionaltotheamplitude.fthedampingcoefficientisplottedagainst/k,thevalueof/ko ranygivendampingcoefficientmaybedetermined.henthevalueof/kthatsatisfiestheimaginaryordampingpartofthemomentequationhasbeendetermined,,the frequencyofoscillationandthevelocitymaybedeterminedfr omtherealpartofthemomentequation,equation( I 4 ) .Equation( I 4 )maybeputindifferent f ormasfollows:

5 )where

s. 2 11 r I

Ifthetorsionalrestraintoaszero,equation( 5 )reducestoI , - H r 6 )npb

sothat,ifthevalueoftheinertiaparametero/npb^xceedsthevalueofro rthegivenaxis-of-rotationlocation,theoscillationcanexistatallairspeedsabovezerospeed.he frequencyisthenadirectfunctionofthevelocityasdefinedbythe followingequation:

where/ksthevalueoftheflutter-speedparameterassociatedwithth eborderlineconditionbetweendampedandundampedoscillationf o rthegivenaxisofrotation.

Equationf or PitchingOscillationIncludingMachNumberInordertoconsidertheeffectofMachnumber,theresultsof

reference5maybeused.hemethodofcomputationisth esameas


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describedintheprecedingsection;however,th eaerodynamicmomentahasbeenredefinedtoincludetheeffectofMachnumber.Thedamping(imaginary)component(seereferences5and6)is

Laa= K\2 x a+ 2 l 2%-f f + aM 2 + z - Z 2andthefrequency(real)equationis

+ M2=0 ( 8 )

Wwhere

Mr k k 2LM3- Ml+Z3-2Z1Theaerodynamiccoefficients] _ 2,o,L,- J _ ,2, Z O,andKarefunctionsonlyofreducedf requency ndMachnumber(seereference5 > )

ANALYTICALRESULTS

Thepurposeofthissectionistoshowtheresultsofsomecalcu-lationsmadetodeterminetheeffectofsomeoftheindependentvariablesonthe flutterspeedandflutterfrequency.In figure1 ,theflutter-speed parameter/ b o oasplottedagainst

theinertiaparametero/npbUo r threeMachnumbers, =0 , =0 . | p ,and =0.7.heregiontotherightandaboveagivencurveistheunstableregion,whiletheregiontotheleftandbelowisthestableregion. Increasingaltitudeisequivalenttoincreasingvaluesoftheinertiaparameter. (Notethelargechangeinscalebetweenfigs.1(a)to1(f).)

Asanillustrationofthemeaning ofthecurvesoffigure1 ,theM =0aseoffigure1(a)( a=-1.0)isdiscussed. Ifthe valueoftheinertiaparameterisbelow5 > ? 1(theasymptote),theconfigurationwillbestable.sthealtitudeisincreased,theinertiaparameterwillincreaseand,ifitisequalto571,thevelocityatwhichanunstable


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oscillationcouldoccurwouldbeinfinite. slightincreaseintheinertiaparameterwouldnowhaveaverygreateffectinreducingthecriticalvelocity.o rverylargevaluesoftheinertiaparameter,th ecurveisasymptotictoavalueof/ h o oahichisequaltothereducedvelocity/ b c a(thati s ,/k),whichf o rthiscaseis2^.72.

TheeffectofMachnumberisnowexamined.irst,andmostimportant,alargereductioninthestableregion istobenoted.o rexample,infigure1(a),theupperlimitofthestableregionf o r =0f o rth einertiaparametero/npb 1 ^s71andthislimitisreducedto137at =0.7.

Anothereffectisthat,f o ragivenspeed,thefrequency ofoscillationwouldincrease f o ranincreaseinMachnumber.o rinstance,infigure1(a),thefrequencyofoscillationwouldbeincreasedbyafactorof2withanincreaseinMachnumber from 0to0.7.

Infigure2 ,thefrequencyratio( a > / o o a)2isplottedagainsttheinertiaparametera/npbU.hiscurvehasthesameverticalasymptotef o rtheinertiaparameterasf o rthecorrespondingreduced-velocitycurve(fig.1)andtheunstableregionisagaintotherightand aboveth ecurve.heinertiaparameterincreasesasthealtitudeincreases.tlo wvaluesoftheinertiaparametertheconfigurationisstable. Thefrequencyofoscillationisinfiniteattheasymptoticvalueoftheinertiaparameteranddecreasesrapidlyastheinertiaparameterisincreasedfurther.o rverylargevaluesoftheinertiaparameter,thecurveisasymptotictothenaturalfrequencyoafthesystem.nfigure3, theminimumasymptoticvalueoftheinertiaparametero/npb^atwhichtheoscillationcouldbeginisplottedagainstMachnumberf o rvariouspositionsoftheaxisofrotation.nimportanteffecttobenotedisthat,asthedistanceoftheaxisofrotationfromtheliftingsurfaceisincreased,theeffectofMachnumberbecomesincreasinglygreater.

Infigureh, thevalueo freducedvelocity/ks plottedagainstlocationofaxisofrotation o rthreeMach numbers.heareainsideth ecurvef oragiven Machnumberisth eunstableregion.helowerbranchofeachcurveisasymptoticto =-0.5 (quarterchord),buttheupperbranchhasa maximumdependingontheMachnumber.o r =0correspondingtotheresultsofreferenceh, andf o r =0.5hemaximumvalueof ppearstobeapproximately-5.5; f o r =0.7hemaximumvalueof sapproximately-7 .

Itshouldbenotedthat,f o ranairplaneormissilehavingacomparativelyshorttaillength(correspondingtothevaluesof nthispaper),anoscillationinvolvinga yawingmotionoftheverticaltailorapitchingmotionofthehorizontaltailmay beaninstabilityofthetypeconsideredinthispaper.sapointofinterest,valuesofthe


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inertiaparameter f o rusualaircraftconfigurationswhentheverticaltailisconsideredastheliftingsurfacevaryfrom2,000to20,000atsealevelandwould beincreasedbyafactorof10 f o r60,000feet.Sincetheinertiaofan aircraftisusuallylargerabouttheverticalaxisthanaboutthehorizontalaxis,it appearsthatthistypeofanalysismightbemoreapplicabletotheyawingmotion. Itshouldbenotedthatthecalculationsarebasedontwo-dimensionalaerodynamiccoefficientsandtheeffectofaspectratio,especiallyifatailsurfaceiscon-sidered,canbeappreciable.

Infigure5 ,theeffectofstructuraldampingisshownf o ranaxis-of-rotationlocation =- 1 . 2 1 ;nd =0 .heflutter-speedparameterv / b j oasplottedagainsttheinertiaparameterf o rseveralvaluesofstructuraldampingcoefficienta. Itisapparentthatasmallamountofstructuraldampinghasaverygreateffectontheflutterspeed,especiallyatthelow-densityorhigh-altitudeportionofthefigure.IForinstance,atavalueo f =18,000,avalueofa=0.01raisesnpb^thefluttervelocity byafactorof3abovethezero-dampingcurve,andavalueo fa =0.02raisesthe fluttervelocitybyafactorof .However,structuraldampingdidnotinfluenceth eminimum(asymptotic)valueoftheinertiaparameterat whichtheoscillationcouldbegin.

APPARATUSANDTESTPROCEDURE

ThetestswereconductedintheLangley U.^-footflutterresearchtunnelatlo wspeeds(0.O6


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high-amplitudeoscillations.hisvariationofdampingwithamplitudecanaccountf o rthefactpreviouslymentionedthatthemodel wouldstartoscillatingatslightlylowerairspeedifthemodel weredisturbedwiththeleverthanifitwerelefttotheinherentairturbulenceofthetunnel.

EXPERIMENTALRESULTS

Theexperimentalresultsareplottedin figurewheretheordinateistheflutter-speedcoefficient/ b c oaandtheabscissaistheinertiaparameterg/npb^.heoreticalcurvesf o r fourdifferentvaluesofdampingaregiven,andtheexperimentalcurveisshown.

Thevaluesoftheexperimentalcurveatthehigh-altitude(low-density)rangeareincloseagreementwiththetheoreticalcurve f o rga =0.008.ro manexaminationoftherecords,itappearsthatadampingcoefficientof.015>ga>0.008asobtained;amoreexactdeterminationwasno tpossiblebecauseofthedependenceofthedampingontheamplitudeofoscillation.

Theimportant factstobenotedare,first,thatasingle-degree-of-freedomoscillationwasobtainedand,second,thatthetrendinthelower-densityregionwasofthesameorderofmagnitudeasthatofthetheoreticalcurveswithdamping.hereasonf o rth ediscrepancyatthehigher-density partoftheplotisnotknown;asimilarphenomenon hasbeenfo undinothercasesf o rthemoreconventionaltypeofflutterinvolvingmorethanonedegreeoffreedom.

Fromobservationsofth etests,itappearsthatthesingle-degree-of-freedom oscillationdiscussedinthispaper isamildtypeo fflutter,ascontrastedtothemoredestructivetypeofflutterusuallyassociatedwithcoupledflutter. Thistypeo finstability mightbecomeofimportanceinairplanestabilityconsiderations,andthepossibleapplicationtophenomenasuchassnakingshouldnotbeoverlooked. It mustberealizedthatthree-dimensionaleffectsmayexercisesomemodificationoftheseresults.

CONCLUSIONS

Astudyofsingle-degree-of-freedompitchingoscillationso fawinghasbeen presented.hisstudyincludestheeffectsofMachnumberandstructuraldampingandisprimarilyanextensionofarecentpaper bySmilg,inwhichincompressibleflowwasconsidered.heactualexistenceofsingle-degree-of-freedomflutterwasdemonstratedbysomelow-speed


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testsofawing,pivotedashortdistanceaheadoftheleadingedgewithageometricaspectratioof5.87.

Thefollowingconclusionsmaybedrawn:1 .Theexistenceofsingle-degree-of-freedompitchingoscillations

hasbeenexperimentallydemonstrated.2 .Theexperimentaldataareincloseagreementwiththetheoretical

valuesf o rhighvaluesoftheinertiaparameter.tlowvaluesoftheinertiaparameter,theexperimentaldataareinpoor'agreementwiththetheory.

3 .Structuraldampingaasanappreciableeffectonthisinstability andincreasestheflutterspeed.

I 4 .TheanalyticalresultsshowthatanincreaseofMachnumberreducestherangeofvaluesofaninertiaparameter f o rwhich aconfigu-rationwould bestable. Theresultsarebasedontwo-dimensionalcoef-ficientsanditispossiblethataspectratiocouldhaveagreateffect.5 .Theflutterseemstobeofamild variety,inthatit wouldno tnecessarilycausestructuralfailure,butthepossibleapplicationtophenomenasuchassnakingf o raircrafthavingashorttaillengthshould

notbeoverlooked.

LangleyAeronauticalLaboratoryNational AdvisoryCommittee f o rAeronauticsLangley Field,Va.,April10 ,1 9 ? 1


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REFERENCES

1 .Glauert,H.:heForceandMomentonanOscillatingAerofoil.R .&M.No .1 2 1 * 2 ,BritishA.R.C.,1929.

2 .Possio,C:'Azioneaerodinamicasulprofilooscillanteallevelocitultrasonore.cta,Pont.Acad.Sei.,vol.I.,no .11,1937,PP.93-106.

3 .Garrick,I .E.,andRubinow,S .I . :lutterandOscillatingAir-ForceCalculations f o ranAirfoilinaTwo-DimensionalSupersonicFlow.NACARep.8 1 * 6 ,1 9 l * 6 . (FormerlyNACATNll8.)

1 * .Smilg,Benjamin:heInstabilityofPitchingOscillationso fanAirfoilinSubsonicIncompressiblePotentialFlow.our.Aero.Sei.,vol.16,no .11,Nov.1 9 1 * 9 pp.691-696.

. Garrick,I .E.:ending-TorsionFlutterCalculationsModifiedbySubsonicCompressibilityCorrections.ACARep.836,1 9 l * 6 .(FormerlyNACATN 1 0 3 1 * . )

6 .Theodorsen,Theodore,andGarrick,I .E. :echanism ofFlutter-ATheoreticalandExperimentalInvestigationoftheFlutterProblem.NACARep.685,1 9 l * 0 .

7 .Smilg,Benjamin,andWasserman,LeeS.:pplicationofThree-DimensionalFlutterTheorytoAircraftStructures.CTR No .1 * 7 9 8 ,MaterielDiv.,ArmyAir Corps,July 9 ,1 9 i * 2 .


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