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NATIONALADVISORYCOMMITTEEFOR AERONAUTICS ITECHNICAL NOTE 3780

INCOMPRESSIBLE FLUTTER CHARAC(LTXISTJCS OF

REPRESENTATIVE AIRCRAFT WINGS

By c. E wilts

CaltfordaIhMmte ofTechnology

.

Washington

A@ 1957

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https://ntrs.nasa.gov/search.jsp?R=19930084503 2020-03-19T06:32:20+00:00Z

G Iw!lmAL AmT8aRY cmMIl!mE

mccwRwsIBIE cHAmmRmT132a m’

mmmmmmm AIRcRAm

WC. H. Wilts

The presentreportgivesthe resultsof a detdled studyof theflutterchamctaristlcsof fourrepresentativeaircraftwings. Thisstudywas made usingthe electric_ cauputerat the CaliforniaIMtitute of Technology.Duringthe courseof this investigationeightimportantpsmmeters of eachwing were variedsmd, in addition,theeffectsof mass, Inez-Ma,pitchingspring,and locationof a concentratedmsas were investigatedfcm all fourwings - severalsweepbacksagles.

The introductionof thisreportdiscussesin general-&mnt3theflutt~ characteristicsof airplanes. The secondsectioncontainsadiscussionof the electric-analogprinciplesthatmade a studyof thism- feasible. The M sectioncontainsa discussionof the aero-

1

I-c - s-c-- m~ti- - for simpMf’ymlgthe flutter-1s of a wing. The fourthsectiongivesInfcmmationrelatingtothe errorsintroduced~ the finite-differenceapprmcimationsto continu-1

I ous aeroelasticqskms. ti addition,data are givenpertainingto thefluttercharacteristicsof a swept-wing~-tunnel model end the resultsof computationsbased on two assumptionsregardingaer@mmi c forcesona sweptwing. ~ fifthsectionlistsall pertinentdata relatingto thefourrepresentativeaircr= wings and the sixthsectioncontainsthecomputedfluttercharacteristicsof the fourwings.

ImRomcmxv

.

Flutteris a phemmenon which is observedin the transie?rtorunforcedresponseof an aerodynamic~tem. Mathematicallyspealdng,it iS observedin the sotitionof the _eneous dlffe!rentialeqpationdescribingthe behmior of an airplanein flightthroughstillnonturbulentair. An airplanewing which is consideredto be a continuousbeem13kor platelikestructurehas an infinitenwiberof ~ees of freedan,andthe characteristicegpationwhichdescribesthe transientresponsehasen infinitenuniberof roots. ~erience has shownthat onlythe rootsoflowermagnitude(fiequenq)exhibitthe problemof instabili~ or fl.utter.

- - -. —..—-—— —. .. . ——— . —-

2

It is thisfactwhichmaims Itcomputerwhtch repmments onlyorusing afew normalmodes in

m m 37ao

.

possibleto pl%diCt~tter us- an analog -the lowerfkeqpenq males of the structureei.tkwdigitalcm analogcauputation.

,

The exp~ In the transient responseof a linearWtem are therootsof the characteristicegpation. Sincethe characteristicegpationinvolvesrealparameters,the rootsare real or occuras cauplexcoqlugatepairs. The latterrootsare the cmes of Interesthere. The realpart ofa co@ugate pair Is the reciprocal.of the time ccmstantIn the transientresponseand the (posttlve) imghary part is the frequencyof oscillation.This is ilkutrated In figure1. Mathematicaldescriptionof the tran-sienta Is

,. ~le(*b)t , ~(.-~)t

orintems ofrealfunctlons

y = Ae@cos(at + @

If the realpsrt of the pair of roots u Is negativethe “transient”dies out and the root is saidto be stdble. E the real.part is positivethe transimt growsm~tialdy unttl limltedw nmllmarities oraestructlon,- the root Is Satdto flutter. !chetermlnologyi snotstrictlycorrect,but it Is canmonpracticeto referto the aponents ofthe transientresponseas flutterroots,sincethey are numericallyequal.to the rootsof the characteristicequatlcm. Thrm@out thtsreportsuchtmninobgy ull.1be used.

-% of flutterrootsmew be measuredby two dimensionlessnutn-bers ~ and g,whichdifferfraneachather ~afactorof2. Thefoalueris generemy usedW control-systemengineers;the MLtter,wflu- sn&lysts. - ~ c~ ba defin&l~ the-eqpat16nterm in the transientresponsegiveneszkkr

y = A#cos(mt + $4)= lie-%ltcosITTl-~a~t

for the p&tlculac

1+@1U@+g

}

Fluttercomputationsare usuaUy centeredaroundregionswherethe value

of g lieslntherauge -0.2<g <0.2. In such casesthe factorj’differsfranunityby MS than 0.5 percent. Fom thisreasonIt is custan-arytocnKLt this factortithe trigonmYtrictermgtwing the followingepproxlmatlon:

..- . — —. .— - -— -- ------

luclim 3780 3

.

I

Thispracticetill be followedin thisraport. For dsqing which issmall,an qpmxlmate rule of thwib16 thatthe den.pingfactm g isneerlyegpalto the per unit decrementper cycledividedw YC. ~ per-centdecrementper cycle 8 is used,thereresultsthe convenient~roxlmation

6g“—

100X

The flutterrootsof en airplanesre complexfunctionsof allgeometrical,stmctural, and inertialpropertiesof the &rfmme as wellea of the airspeedand alr de.nei*. with all otherpropertiesheldconstant,the .@westSLmpeed at which the flutterroot *bits neutralstdbill@ is calledthe flutterspeed. If g is plottedas a functionof veloci~, the ebscissa(speed)at which the curvefirstcrossestheeXiS g ‘ O IS the flutterspeed. In this stu@ such curveswere usedto determinethe flutterspeed,but such curves=e used in thisreportonlyto illustratethe behmior of someunusualflutterroots. A tdbu-I.ationof flutterspeedsdoes not alwqysgive a goodpictureof the fluttercharacteristics. Anexa@.ei sshuuni nfigure2,w here the dampingoftwo rootsla shown. One rootbeccmesunstdbleat a speedof dxxt300milesper hour and the other,at a speedof dmut 6(M miks per hour.E a parametervariationincreasesthe da@ng g of both rootsby 0.03,one flutterspeed-is raisedto 350 milesper hour,a 17-percmt increase,@ the o- is raisedto 603 milesper hour,a O.5-percentincrease.A furtherincreasein g of 0.02will raisethe secod flutterspeed0.4 percent,to 605lllih?Sper hour,while the firstrootwill now exhibitno flutter. It shouldbe emphasizedthat eventhougha designspeedof,SW, 500 milesper hour hea bem surpassed,the systemmay stillberegardedas uneatisfactoa-y.A systemso closeto flutterat a speedof%0 milesper hourmight actuallyflutterbecauseof weight(fuel)vari-ationsor minordifferencesin stiffnessresultingfrm variationswithinthe manufacturingtolerances.I&m the standpointof thisreport,allthreeof the sets of rootsdiscusseddbuvewill be reg=ded aa havingessentiallythe same “fl.utter-characteristics,“ eventhoughthey exhibitradicallydifferenttheoreticalflutterspeeds. Eq@aeis is givento thispointbecauseremarksto be made laterin thisreportq be misunderstoodwithouta clearcomeption of thisviewpofi

This investigationwas conductedat the CeJdf’ornlaIzietituteof!&bnology * the sponsorshipbnd with the financialassistanceof theNationalAdvisoryCamuitteefor Aeronautics.

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mm m 3780

SYMMIS .

hauchora

half chordat root.

half chordat tlp

symbolic repremntatlonof circulatoryccmponentof liftforceduetoangle at’attack

M.ft Coefficient

wingstationfrcmroot,tn. .

Young‘0 llmdlllusof elastlcl~

eqyivslentbeam flexmal rigidity,(lb)(sq In. )

aQerlmmtal; used as a subscript

fluttarfrequancy,Cps

nomalmodefrequencyofcantilevered englneana nacella,cps

flutbsrfrequencyfor Continuousstructure

shearmodulus

equivalentbeam torsionalrigidi~, (lb)(sqin.)

~t.—

damp~- factorof a dmped sinusoid, e 2 Cosd

verticaldeflection,positiveduun,in.

nment of inertiaper unit length,lb-sec2

torshlal Stiffness

increase In stiffness,percent

radiusof gyration,in.

Senlispl?ulofWing

. .—. —- .- -.. — --—— -. -

mm m 3780 5

..

.

‘.

%?

t

v

%

Vf

Vn

w

%

.

twistlrlg ManeIlt

positivenosedtmut elastic~, m

axisper unit len@l of wing,

mass psr tit length,lhsec2/sq In.

mass of concentrated mass

fuselagemass

total ulng mass, lb-sec2/in.

total wing mass outside of fuselage

lmpea mass

Mft fcmceper unit lengthof wing,positivenosedown,lb/in.

Ia@ace transformationvartable

xc Pres~e b=a m ~ cqonent of w1ocI*,(1/2)Pvn2,ll+q in.

time,eec

alrstreem velocl~, in.jsec

flutter-i~ of aiqlane with barewing

alrstreemveloci~ at which flutteroccurs,In./8ec

ccqponentof airstreemvelocityv cos A, ilL/SeC

reference veloci@c, in./see

peqpediculac

flutterVelocityfor con~ Wing

distancemeasuredalongwing

distancefrcm

distancefrom

distancefRmb - xl, In.

mld.chordaft to elasticaxis,M.

to elastlc ads,

~*

three-fourths

aft to elasticads, ~

chordforwardto elastic

+ (b/2),In.

axis,

-.— —— - —- —.. — —.—. .-— -.

m m 37&

diwtancefixxnelasticaxis aft to centerof mass, in.

~ Varla-

cell sizefor finite-differencestructure

absohrtepitch angle*out elssticaxis,posi.tlvenoseup,radians

percemtdeCre9nentper cycle

per Unit criticaldam@ng

slopeof elasticaxis or roll *out horizontalaxisnormaltoelasticaxis,positivetlp down,radians

sweepbackangleof elasticaxis,deg

Sdr densi*, lb-sac%?

wing twistinggradient,

realpart ofpair of roots

~ fr=T==Y, radtal+x!c

-a ~k~ fre~, radiems/sec

The use of electrical analogsfor the solutionof aeroelastlcproblems has been &Lscussedin detailin ref~ence 1. The purposeof

. the presentsectionIs to summarizethe principlesbriefly. For purposesof flutteranalysis,the structuralsys%emIs assumedto be LLnesz,anda Uneer ehctrical networkIs constructedwhose electricalbehaviorapproximatesthe @namlc behewiorof the linearizedstructure.For thispurpose,capacitorsare ordtmxily used to representconcentratedorlumpedinertiaproperties,Inductorsare used to representlumpedflex-ibill@ properties,emd trensfomersare used to representthe geometricalpropertiesof the structure(refs.1 * 2). h such electricalanalogs,voltagesthrou@ut the networkrepresentvelocitiesin the structuredcurrentsrepresentfOrces. Elec&onlc egpimt is used to producecur-rentswhichd.qpedon voltagesin the electricalsystemin the samemannerIn which aerodynamicforces@X$nllupon the velocitiesof the Slrfoll.

w m 3780 7

KJ

Theccammite electricalstmcture canbe regardedas en electricalInoaelo ftheeircr mtithesememannerthat a wind-tunnelmdel Wuuldberegarded asas tructuralmodel. The advantageof tihib~ llesin the relativeeasewith which ane can alterthe propertiesof thendel,thusperformingflutter“c_tationsn with greatrapidi&. It shouldbeemphasizedthatthe normalmodes of the structureare not used as toolsor elementsin the analysis. !Qw _iS consists} in fact,in obsmthe behsdor of an electric@.model of an aimraft in flight.

Thatbehaviorwhich is most readilyobservedis the transientresponseto a sudden disturbance.Thismethodis thereforesimilarto the testingtechniqpewhich is acdimrily used for wlr.d-tunnelmodels. An advantageof the electricalmethodis that InK& pulses~ be used,so that sep-rationof two or more nearlyunstdblem sldghtlyunstable* of oscil-lationis more readilyaccanplished.Basicrecordeddata comsistof theIOGri=c decrementof the responseand the fregpencyof oscillationFlutterspeed- fregpamy for my configurationare orUnarily foundbycauputingthe dsmping g -10•fregpency f for specificVal.&sof veloc-i~ end in~olating to M the freqpencyand speedat which g is z-.

APPRoxlMvlmnam SIMPLIIYINUFU!!!THl

MrwtUal R@resentation

AmIxs13

.

For dynamicanalysisd airplanewings of largeaspectratio,it iscustauaryto &eat the wing es a bemnlikestmcture in both verticalbendingand tOrSiO1l. It is usuallyassumedfor sinplici~ that an elasticaxis exists. lbranunswept wing,this isastralght linewhichundergoesno verticaldisplacementwhen the wing is subdectedto a pure torgpepsw-allelto this axis and alongwhichno twistinggraaiexltexistswhen mrti-calloads =espplledanywkre al.ongthislhe. Fbranunswept wingofconventionalconstruction,this sbqplificaticmis usuallygpiteaccumte.Foraswept winganelastictis Vbe defined asastrdght lineuhichassumes a constant slope uver its entire length when a twisting moment isapplied parallelto this lineandwhichhas no twistinggredlentwhenverticallaadsare applied-here alongthis llne. For aspectratios~eater than 5 or 6 - forconventionalwingconstruction,a linecanbefoundon the structurewhich satisfiesthisdefinitionreasonablywellexceptn+r the root. It is not uncaamonto find an equivalentelasticaxis at dmut the 35 m=40 percentchti, a line locat~ aft of the leadingedge a distanceegpslto 35 or 40 perc~t of the localchord.

The assumptionof an elasticaxis involvesthe tacitassumptionthatchordwiseb- of the ulng is negligible.It follom, then,thatthemotionof the wing at any spanwisecoordinatecan be describedby twocoordinates,the verticaldisplacementof sanepoint on the chord,end the

8 mm m 3780

=@e of twist of the chord. IY wingmotionis descrtbedIn termsofvertical.motionof the elastlcaxis and twlstlngmotion~out this line,-&m thesemotionsare not coupledthroughthe actionof elasticforcesIntheuing exceptinthe root regionfora sweptwing.

The root regionof a sweptwing is necessarilya relativelycaupll-catedstructure.However,for aeroelasticproblemssa egylvalentsl@.estructure-canbe foundWhich is ca@.&ely satisfactoryfor wings of Lx?geaspectratio. This canbe demonstratedw the folluwlngreasoning. ~outersectionsof a wing exhibitdefinitebemllke Prop-ies, but in theregionof the root cons-~le warpingof the wing surfacemust takeplace. The aerodpamlcforcesnearthe root of the wing are thereforenot adequatelydescribed~ striptheory. In addltton,the inertiaeffectsof this secticmare nat readilycuqputed. However,the effectsof the aeradynsmicforceson the root sectionare insignificantfor ordi-q -tier Cauputations. Thishas beem demonstratedmsny timeswiththe anaQ cauputerw remuvingthe -odynmdc forceson the ~oardcell of the flnlte4ifferences~ture. W Inertiaforcessre alsoInsignificant caupared with the ehstlc forcestranatulttedw the rootsection,snd it Is thmsdbre possibh to replacethis sectionfor purposesof -is ~ a set of “influencecoefficients”rel.atlngtransmittedforcesto relativedisplacementof = outersectionof the wing relativeto the fuselage. It has been fouudthat - sane casestheseinfluencecoefficientsresedblecoefYlcientsfor a simplebesm extendingstraightintothe fuselagesad attaddng thereb saue simplewqy. Thewingstructuralaxisthen consistsof a shortsecttm whichM be perpen&Lc-Utothe fuselagecenterline aduhich issi@yattachedto a6wept-back elasticSXLSwhich extendsto tlw wing tip.

Methcd.sfor aetemllnng the eqptvalentstructureare outsidethescopeof thisrepro-t.Sincethis structureverlesgreatlywith the par-ticularwing constructionused, It was necessazyto choosea siqplethough~ical root structurefor this study. That chosenis illustrated infigure 3 where Ihe ebstlc axes are shown ~ dotted llnes. !& break htheelastlc ~lsassumad tobe attheedge of- fuselage,and theaxLs insidethe fuselageis asmnuadto be strai@t endperpe-culsr tothe airplanecenterline. T!hewingisassmnad to be pinnedatthesldeof the fuselage. Conse~tly, all twistingmauentIs removedat thispointSzd it is ti necessaryto * any assuluptionsregaralngtwistingmm -U the fuselage. ~ ?716UMtVInsidethe fuselageis,ho’m?w=,Importantfor -trlc motion. _ the past 6 yeS3S, exkn-sive f’lut- ccmputatlms have beenmade with the electric-analogcomputerfor Caumercial.endInllltary*craft as well as for wins-tunuelnloaelsincludingthosedescribedin references3 and4. In allcasesixrvesti-gated,Ithasbeen fuundthat relativelylargevariationsb root condi-tionshewe a negligibleeffecton the fluttercharacteristics(in thesame describedh the Introductla). observedchangesIn daqing wereusuaUyin the range O<lAgl< 0.05,whichhaSvez7 amal.leffecton

—.. —. --—— - -- ..—

flutterspeedunlessthe curveof g againstvelocl~ is very flat,nearza’ovalllesof g. kedbSS tO S~> both _h’iC d antiSynUUetriCmotion of the airplane must be permittedsincethe fluttercharacteristicsfor the two Qpes d motionmsy be quitedifferent.

Fuselage stifhess and *18 prqertdes usually Me SUCh VShleSthat an assqption of a rigidfuselagefor ~lc motionalterstheflutterCharactaisticsLLttle. For fighterplanes,the errorintrduceais ne@igible. For largebuibers,the chsngein flutter6peedmay beappreciable,but it does not alterthe trendsto be observedupon vaci-ation of wing properties. It has thereforebeen assumalin this studythat the airplanefuselageis rigid. Tail-surfacefkxibill~ does notsignificantlyaffectwing flutter~oblems. A rigidtail surfacewithsufficientareato prmlde satisfactorystaticstdbili~ has thereforebeen assumed.

For all.the flutterccmputatimsgivenin this report,the aero--C forcestie been si@lfied by two importantassmqptions:

(1) ~ air fbw iS incaupressibla.

(2) If the atrfoilis aivlaeainto stripspeqpdcular to theelasticaxis,then the forceson each stiipcsn be caqputedas a fimctionof the normalcomponentof the airstresmVeloci* and themotionof thatstripindependently0# the motionof adjacentstrips.

The firstasmqption is not reqlireaw analogmethodsh general,but its use greatlyincreasesthe rapidi~ withwhichdata canbe obtained.Sincethe purposeof the studyis not to obtainspecificaccurateflutterspeedsbut to studytreds in fluttercharacteristics,this assmqptiondoes not seemunreasonable.With regazdto the use of stripthecmy,twoassumptions- oftenfoundin the literature.b usingthe “airstresmmethod”the wing is dividedinto stripsp=allel to the airstremn,andthe forcessndmanents on each striparec~ted astbough theulng werenotswept tithe air flow EUmutthesectionwre atwo —almensicnlalincan-pressiblefluw. The aerodynamiccoefficientsw be takento be the sameasthose foran unsweptwing Ormsybe modifiedwafactm COSA. In~lying the “ncxrmal-cmpnentmeth&, ” the wing is di%tdedinto stripsPerpticuk to the elastic axis. ~ ~c f=c- - ~tsare cauputed as though the effective air veloci* were the normal caupo-nent vcos A,titheforces d@_ onlyontbe motiomof theindivldualstripaM not upon themotionof adjacentstrips(exceptthat sane smalltermsmay be Inchd.edwhich are proportionalto the twistinggradi=t andthereforedep*t upon the motionof the neareststrips). A criticaldiscussionof the two alternativesis givenin reference5. Thisrefer-encerecommendsuse of the nonual-ccnnponentmethod.

. .. .. —.- .--— .— —— -—— ——— - — - ..—— --- —-- —— —.

10 m m 3780

B&ore adaptingthe secotiass~tion, em effortwas made to findBCU C~~tiOD fi- eXQ~ r-tO. Beference 3 contains~erl-marlA1.flutterspeeds for a WiI&tUIUEl dial wing wI.th sweepbackangleequalto dbout35°. ~s angleis sufficient to give an e@precidbledifference in results obtained with the various assuqptione mentioned*uve. !& section entitled “Mnlte-Difference Errors” In the presentreport contains the results of caqputations Which show that the normal-cauponentmethodgivesresultswhich =e as satisfactoryas thosegivenww-~ua”

Eqpations for aetexminn - ~c f=c- m -g -= =egivenin reference5. In the egpations gl- therej several terms arefoundwhosetheoreticaldustiflcation Is not well establlshd. These* (~oqped in SX4CW brackets a P. M of ref. 5) me f- to henegligibleeffecton senrplefluttercomputations.It seemsreasonable,therefore,to anittheseternsfrcm computationsi.molvedIn the presenttrendstudy. With theseaulssionsandwith obviouschangesto conformto the symbolfiand notationused In the present report, the equations are:

p= Pl+p2+p3

%=

PI =

p2 =

P~ =

%=

Ml =

%“

9’

%“

1%2(~+6wA)+(tz+e tan A)+%

- - —.. —.

IuuMm 3780 11

1

r

v.

Theta.msaregrouped in the orderShounfa conmnience lnestabllshlng- circULts. Thel.asttermti Pli8notfouna inthecorresponaingeqpationof reference5. l!blstermIs removed(mathamatlcally)byInsertionof an e- but oppositeterm In ~ - a slmll.srtezm In ~.

It is addedto Pl herebecausethe circuitswhichgaerate the term -

a+e tanA &dsoprovidethetemn(x@n) (&+6 tanA), the lastpart

of which is not foundin referace 5. As-Is pointedoutbelow,this tamhas a negligibleeffectso that its inclusiomIs of no importance,W itis idicated in the expressionfw PI for the sdlceof ccnupleteness.

It shouldbe eqphaslzedthat the dynamicpressure ~ is based on Vn,

where Vn is the vM.oci~ caqonent nozmalto the elastlcads. The

coordinates a and f3 =e bothmeasuredin elastlc-axiscoordinates.The synibollsmC(bp/Vn) is used to representthe TheOdorsenOr Wagner

functiatl.A shortdi6CUSSi~ of the interjjretationof thts E@lO~Crepresentation can be foundin reference6.

All tams foundShovecan be _sented by siqpleanalog&cultsWith the Sxceptlmlof ~ and 1$. &sminatlon of eqtiions 6-7 of

reference5showsthateachtem ‘in ~ and ~ Issimilarto (if not

equalto) a tam foundin the specialbrackets. Sincethe latt~ tamshavebeen anitted,thereseemsto be no logicalreasonfor retaining ~

and ~. Inaamch as theirinclusiongreatlycauplicatesthe analogcir-

cuits,thesetermswere also aaltted.

h addition to the fdnitetierence apprcmdmationsand thosecon-tainedin the assumptionsof inccqpressibleflow @ stripthe-, threeotheraercdynsmlcapproximationsshuuldbe numtioned. The firstof theseis the failureto modifyaer~c forcesat the wing tip. The de~Inthegrowth ofllft forcesas describedwthe Wagneror Theodorsenfunctionsfor two-dimensionalflow cannotapplynear the tip. &deed,boththe delayinlift and the magnitudeof thelif’tmustgoto zeroatthe tip. The emtentof the errorintroduceddep- upon the @ortanceof tlp forcesin fluttercauputations.EMofar as their“lmcatlonisconcerned,theseforcesare quiteimportant,but, becauseof wing t~er,the magnitudeof the totalforceper unit lengthU.mlnlshesne= the ttp.Sincewings of considerabletaper=e involnd in this IIrvestigatlon,Itis to be expectedthat the errorwill be relativelysmall. The seedapproximatlcmis failureto canputeaerodynamicforcesproperlyat theroot ofasw@wlng. As 6tated~lAer, the err= introducedby this

. . .. .. . ____ .— ---- . .._ . . .. .—. --— — - — -.— — . .

.- --- ——. —.. .. . . ..

12 I?Mmm 3780

appradmtion is negligible, sincethe aerdyudc forcefor a largesectionof the wing root canbe cxnittedentirelywithoutan appreci~lechsmgeh flutterspeed. The H sQprmdlUation10 introd.uceaby the “

necessi~ of caqputingthe Wagnerfunction(orthe T&mdorsen ftmction)electrlcaUy. ThisfunctionIs creed usingnetworksshownin refer-ence 1 with an errorno greaterthan 2 percentoverthe frequencyrangecm time intervalof *est.

FlmLte-DifferenceStructures

No practicalmethcdshem been devisedfor representinggeneralContimlcnlsBtinctureswiti contimmus e~ctrical systems. The electric-analogcaqputerutilisesLwqpedelectricalelementswhich can, In prln-clple,be used onlyto constructanalogsfor l.mqpedmechanicalsystems.Huuw=, as pointedout In referencesZ d 2, it Is possibleto repre-sentthe dynmic charactqrlsticsof bemulikestructures~ a luqpedstructurebasedupon flnlte-differenceepprculmationsto p-lal dlffer-enttalegpaticms. It Is comenlent to callthis Imqed systema flnlte-Mfference 8tructure,Whetherit is a mechanicalmodel or en electricalanalog. !l?hesereferencesuutllnethe process~ which Inertiati stiff-nesspropertiesd aeroaynamlcforcesare smragea or replacea~ SingleconcentratedInertlas,springs,or forceein the finite-differencestructure.

It shouldbe remarked,at thispotnt,that the assumptionof a flnlte-differencestructureInsuresa finitenumberof flutterroatsor Qonen-tial functiomsin the transientresponse,whereasthe continuousstructurehas, in princtple,en inflnltenuuiber.Sincethe Ugher freqpencyrootshsxehigh dsnrping,itis onlythe lowerfke~rootsthatere ofinterest. Two or threeof thesemay, however,show essentiallyzerodsqing simultaneouslyat a givenvelacl~, and it is sanetimesnecessaryto determinethe characterlsttcsof severalflutterroots. Thereisobviouslya bwer llmitto the mmiberof cellsthatmust be used to obtahsatisfactoryaccur~, simceeach cell aiidsroughlytwo rootsto thesystem.

!Chereis Ilttleinfonuatton In the literaturewhichpertalasto theaccurq with which such Structures representthe contimmus ~tem.Reference7 glvw dataforstatic-deflectionsad normalaode cMracter-istlcsof certainfinlte4ifferencestructuresbut no Informatia &mutELccurq of flutterCunputetlons. It Is the ~ose of this sectiontosummarizework at the AnalysisLaboratoryof the CaliforniaInstituteofTechnologyWhlChwas carriedout to aetermlnefinite-differenceerrorsin fluttercauputatlonsfor severalspeclflcstructures.

I

.— —- .... .- -.

.

Using eqpatlms f= aerdynmuic forcesbasedon tuo4dmensionalstriptheoryand Mnem Incunpresslblefluldflaw, several“exactsolutions”hawebeen obtainedfor flutt~ problems. Sme of thesesrefoundin references8 snd9. Thesesolutlonsarq exactIn the sensethatno furtherplqmicslor mathematicalsimplificationsare Involvedaud theonly -Ors axe introducedby round-off~ illevaluatingtranscen-datal.functionsand Infiniteseries. Solutionof thesesamePmM.ems~ use of finite-differencespproxtmationsto partialdifferentialegpa-tfonsprovidesthe most practical~ of estimatingfinitetierenceerrorsfor otherconfigurationsfor which exactsolutlonsm not obtain-able. It is true that,in dd. CSSeSmentioned~me, the airfoilhasbeen assumedto have unifom spsnwlsepropertiesand that in most prao-tical.casesthe airfoil@s a significanttaper. On the otherhind,reference7 containsa studyof the flnitedifferenceerrors111thedeflectioncharacteristicsand normal+nodepropertiesof both unifonu- ~ered beams. Lllhlsstuayshol?ednounusualdifferencesin theseprop~les~ ~ so It Is ~ti -t W resultsobtaimxlfdr flutterof unlfonnairfoilsare Qpical of resultsthatwuuldbe obtainedforflutterof *erea sirfotls.

Althoughmuch of the work reportedh this sectionwas not done inthe presentIuvestigatlon,it Is inclndd here she most of it does notappe= In ~ readilyavalldblepublication.

UnifcnmAirfoilwithPinnedEnds

A uniformbeam with plnn4 endstill supportmUy sinusoidalmodesin both b- ad torsion. Flutter* sre alsoof sinusoidals@eand it is thereforepossibleto reduce the flutterproblemto sn eigen-valueproblemwhich canbe solvedwith a M@ degreeof numericalaccuracy.The finite-ez%mce analogsfm a pinned-pinnedbeam llkewlsewillsupportonly Sinusoidalmoaes. It Is possiblethereforeto get exactsolutionsfor the finite-differenceqggmxhations to the couMmousairfoil.

I

!t!heairfoil ChOSeIl for thiS analysisIS describedin tdbleI. For~ c~ -j ~ fitter EWea ~ fre~ were foundto bevf = 692milesper hour and ff = 12.72C@JSEJper secofi,respectively.

tiis of ~ ~~-er-e Smf=m w carriedout using eight-,four-,and two-celldivisionsbetweenthe phned ends. Resultsare @min t~le II and figure4. Fa thispartlcuUm caseit Is necessarytouse more than four celJsif flutterspeedIs to be obtainedwith errorlessthan 2 percent. ~ use of synme~ conditionsat the centerof thebemn, ltisnecesssry touseozilyUthisnmdmro fcellswltb anelectrlcanalogccnputer. Thus,useoftwO analOgcellsgLves atheoret-Ical errorof *out 2.2 percent,and four analogcellswould give anerrorof 0m% 0.6 percent.

—.. - -—- — - -—. — .. —-— -

lk

TMfonn CantileverWingWith ConcentratedMass

m m 3780

AneJyticaldetermination of the flutter speedof a centikver wingismuchmore difficult thsn that forabeem uithplqned ends. However,otherinvestigatorshem obtainedaccuratemnuerlcalsolutionsfor a fewc~gurations. Themost @ortant of theseIs describedin reference9.This caseis of importancefor two reascms: 1% involvesseveralspenwlsepoaitloneof a largeeccentricccmcentrat~mass whichhas a greateffectupon the flutterspeed;d, for sauepositions,at leasttwo canpletelydifferentflutterrootscan be f-.

TdbleIIIpresentsthe pbyslcalcharacteristicsof the atrfoilanalyzedIn reference9. In thisref~, theflutterspeedand flutterfreqpencywere cauputedfor sevenmass locations,data for which arereproducedin tdileIV. Sincethe locationof a concentratedmass mqy beiqportantin flutteranalysis,d sinceallpointson a flnlte~f ferencebesmmenot equellysultekd.eas enattaclmentpoint for a concentratedmass, it was believedthat a cmparison of the abovedatawith fi.nite-M.fferencesolutionswas quiteiqportent.Unfortunately,similaraccuratesolutionsfor a finite-differencestructure=e not readilyobtained,soit was nacessaryto use the electric-analogcauputerto obtainthesesolutions. The resultingcmp~(m thereforecorrtalnsboth flnite-differaaceend analog-cauputererrors. PrackYuswork has indicatedthatthe latterare probablynot greaterthan 1 percentif the ‘J!heodorsenfunctionis representedaccurately.

In this analysis,two sllghtlydifferentbesm analogswere used.In both,the beemllh(trausformm

prqp-ies -e represented~ a systemof levers),but in one gr~ the lumpedforceswere appliedat the

junctionsof the leversand h the secti groupthe forceswere eppliedat the mi@@nte of the levers. The enal~ of the secondgroupwas once~ ti give a betterapproximationsinceit resemblesthe Russellbeam_ d16CuSsedin reference7. Recentlmvestlgatiomhas shownthat thisbeliefis withoutfoundation,and the secondanalogis nuu preferredonlyes a matterof convenl~e for sweptbackwings shce it providesthe wingslopedirectlyat the face stationswhere it is neededfor canputationof aerodynamicforces. h both casesthe ca?rtilewerconditionat therootwea pruwLdedby a half cell at the root,- the forcesnearestthetip were eppli~ one half cellfhomthe tip. !T5usthe firstgr~ tmulvedan Integralnuniberof celb, and the seccmdgroupinvolved.a half Integral(Integerplus one-half)nuuiberof cells. Five caseswere imvestlgated;

2, ~, 4, 5$ -6 ce~. SinceIt was shownin the prd.ous section

that lesstllsnbcellswasof

resultsof 4, ~, ad.6 ce~

no interest for presentpurposes,onlythe

are presentedIn thisreport. .

-—. - ----- .. . . .

c

luumm 3780

InvlewofItwas elmectea

the SilqpMcltyof the fluttercumms shownin reference9,thatdata wouldbetaken at onlyafewspanwise mass bca-

tions . i&ew3r, it was soonfoundthat the fluttercharacteristicsweremuchmore cmpllcatedthan anticipated,and dak were takemat A mass@atIons in the 6-ce~ case. The fluttercharacteristicsof the wingwith variSKlelocatiauof the ccmcentratedmass =e sketchedin fig-ure 5(a). As the concentratedmass is _ outwardfra the root,theflutterspeeddropssM@My. At a distanceshout16 percentof the totalspan fkanthe root aminlnmm is reached,andbeyondthe 25-percat posi-tionthe flutterspeedrisesv- r@dly. At the 50-percentpositionthe flutterspeedfor this root has beccmeeqpalto the flutterspeed.ofa caupl.etelydifferentroot. The flutterspeedfor this secod root dropswith Increasingspanwlsepositim of themass makingIt @ossible todeterminewltlrtheanalogcanputerthespeed for the originalrootbeyondthe 30-percentpositiom. The flutterSpd for the secotiroot reachesaminlmm with themass at the 45-percentposition,then risesto a veryhighvalue as the mass la maved towardthe 75-percentposition. A flutterrootWhich Is prob&Elythe secotiis obsti for mass posltlms near thetip, the I.uwestflutterspeedoccurringwithmass at the tip. It wasalEoobservedthatdivergenceof the wing occurredWhenmer the flutterspeedexceededdxmrt5,000hches per secod. Becauseof divergence,itwas not possibleto measurewith accuracyfluttcwspeedswhich exceededdivergencespeedby more than dbout50 p-at. As a result,flutterspeedswithmass ne= the 75-p=cent span couldnot be measured.

~ flutter-acteristtcs for the 4-, ~ -, and 6-cellstructures

are shownin tdblsV d figure5(b). Data for the sevenpositionsana-lyzedin reference9 are alsoplottedin the figure. ~ection of thesecurvesshowsthatmeuy more accuratemmerical solutionsm required.todeterminethe finitetierence mnmrs for allmass positions. In spiteof the Inadqyate nmerical data,an attea@ was made to draw a smoothcurvethin@ w - Po~ts * m refaence 9. b doingms m

~- d6-ce~and.og datawere usedas aguideti &tez@.nhgthe Xe

of the curve. TMs curve,shownIn fi~e 5(a),has alreadybeen dis-cussed. It is realizedthat a slgnMicant errorof as much as 2 or 3 per-cent~ existin this curvefor sunemass positions,but therewas noothermethodfor obtainingestimatederrarsfor the finite-differencestructures.With the *stanMng that the caqparlsondata~ be Inerrorb saueregions,-e 6 wasprepareds- the percentageerrorin flutterspeedfor the ~ious analogsas functionsof the mass location.

For ~- and 6-ceKL stmctures w averageerrorsare ~out 2 percent.

It canbe readilyseenthat,althougha &cell.analoggivesvery satis-fact~ requltsfor the bare wing (masspositionO), it is necess~ touse more than4 cellsIf errorslessthan 5 percentareregpiredat *messlocattons.A furtherdiscussionof thts Izwestlgattonwillbe foundin reference10.

—— ..— ——— —- ——— — --— . .—. — - --

-—- ---- _____ .. ___ . . .

1.6 Hmm 3780

As a result of this analysts} it was decided that all flutter can-

putations made in this trd study would be made using + cells to

represent one-half of the airplane wing.

13cp*ntal Correlation

Wb&lmnnel testshem been made of many mdel structures.It Isdifficult,houwer, to M uncl.assifteddeta in whichthe structureisccm@etely end accuratelydescribed. b the courseof this iuvestlgation,two caseswere feud in uhlch a correl&lonbetweenexperlmsntalandcauputedckracteristicscouldbe atteqted. The firstof theseis the@form unmqpt cantileverwing tiscussedIn the precedingsectlom. Thefluttersp~ ti fieqpencyobservdiin a wind tunnelare repomtedInreference9 ad a ccmpanionreport,reference IL. These data are sumna—rlzd in tdbleIV,which also conti the caqputadvaluesof reference9.A betterundersta?dlngof the correlationis obtainedU the ~erimentaldata areplottedwith the assumedanalyticsolution.BUCh a Ccxuparlson.

-s t33The correhhion for this caseseems

FlutterSpeedof a Swept-wing Model

Reference3 givesresultsofwind-tunnelteststo detemlnetheflutterspeedof a modelwing with sweepbackangleeqpalto 34.5°. Thisulng had two concentratedmassesattachedat ~tely the 30- end&)-paCent spanpositions. lh an effortto caqparethe airstresmdnormal-canponentaa@ynadcs for ftittermawputatlons,an electrical-- constructedfor thiswing. I’orauysweepbackangle,itlstobe expectd thatthe two methodswill give flutterspeedsdiffering~ a

factorof ~oximatdy (COSA)l/2,*S the ~ c Coefflci.ents=modifiad by the factor COSA intheairstream method)ln which casethe two methodsshouldgive shllar results. The prlncl.paldifficul..

tton of the propertiesof the concentratedmassesencouutereawas aeterminaon the wing, sincereference3 doesnotgivecanpleteInformationaboutthesemassesend theirgeawtrical.location. The best datathat couldbedaihc~ frau thisreport=e givenIn =le VI. Sincethe massesareallnadulth the adrstresmbut are repres=ted in elaetlc-sxlscoordlnates,a productof Inertiabetweenroll end pitch exists. Sinceno such infor-mationwas erailsibk,the productof inerttawas anittedfra canputations,and the rollJQgInertiadbouta chordlinewas assumedto be one-halfaslargeas the pitchinginertiadbautthe elastlcas. It Is bellevedthattheseapproximationsand simpllftcationswill effectthe resultsbylessthan 1 perced.

A _tiS~ of observedand -ted characterstics Is givenhteibleVII. The firstthreenormal+nodefrequenciesshow satisfactory

_____ - .. --—

3G

m

●

,

NAC!Am 3780 17

agreement,with differencesof 1, 5, and3 percent,respectively.Theflutt& speedcuqputedwith eitherrepresentation@ aerodynamicforcesis lowerthanthe Wins-turmelvalue. In the case of the airstremumethod,the discrepancyis 19 percent,or, if the aerodynmniccoefficients=emodified,U. percent. Usingthencmlal-cuqpmlentmethod,theaiscrqgmlcyis 12percent. Flutterfreqmncy is in error~out 20 percentin allcases. Althoughthe observeddifferencesare relativelylargeIn allcases,it 16 cikclndedthatreference 5 is satisfactory

cEARAcT.mmm32s

the mnnal-cmponent method-rec&&d.ed infor thismcdel.

Plan formsd stiffnessand inertiadatawere chosen* surv@ngthe vurtous fighter,bti~, and transportplanesdevelopedIn recentyears. Four representativeairplmes were chosen,two figM6rs end twolargebcmibers.Smallerattackbaibersend transports~-not 2ncluded.becauseof lackof time. !J!heairplane13choEenarel16tsMlEw In allrespectsto aqy particularset of four airplanes,but ~ do tie stif&ness - Inertiapropertieswhichresemblefour spec”iflcalrbr~. Planform,sueepbackangle,elastic-axislocation,and conceqtrated+uassloca-tionswere,however,chosenmore arbitrarilyso that-@is reportcouldremainunclassified.The fourbasicplan formsare shuqnIn figure3.!12ubasic fighterA has a bare unsweptwing with span.of’dknztX inches,taperratioof 2.o, and aspectratio6. The basic fighter-~%asa wingsweepbackangleof 30°, a span of aboutkOO inches,and 8 taperrattoof 2.0. Thetwo basihwin@ have the SSme len@hmeasufed’ale@ theelasticaxisand the same chordsmeasuredperpendl- to the elasticaxis.

The basicbaiberA has an unsweptulng with span of dxmt1,700*s, tsperratio of 2.5, end aspectratio 12. It has a concen-tratedmass representingan enginenacelleat the O.46-spanposltlonwithcenterofmassdmut one-halfchordf~ of theelasticaxis.The basicbc@berBhasawlng sweepbacksngleof 30°, aspanofdmut1,X Inchas,and a tqperratioof 2.4. It alsohas a concentratedmassrepresentingan enginenacelleat the ssmerelativepositionas forbanberA. Thetwobasic wingslmvethesmue len@hmeasmed a-elasticas and approximatelyeqpalchordswhenmaasured~allel tothe alrstresm.

.

,

Mass per unit length,pitch inertiap= unit len@h, bendingrlgldl@,and torsional.rigiai* m drawnas smoothClmvesqprmimating thecharacteristicsof SCILUe-iCd. @Odernaircraft. ~ describedin refer-ence2, thesedatamust be collectedor l.wnpedoverdistancescorrespadingto the cell lengthof the analogfinite~erence structure. The assumed.curvesandthelmqped valuessreshowninfigure7. Thelufqpedvalues

—.-— —— —.. .— --- .. .. —.- —.— -.——-1

M m ml 3’780

are also listedin tdit.es VIII to XIj which give all pertinent *act~-tattcs Of the basic ~lanes ..

U ~-t PEWMIeter6Of the baSiC airplaneWingswere vexiedin an effort to find simllsrfeaturesIn the fluttercharacterlstd.csofthe variouswings. N qpantitlesvaried- the extentof theirvari-ation Is sumnaxIzad as follows:

Quantl~ varied. MinhlmvEaue Msamlmlvalle

wlngmass aensi~, per unit basic . . . .Wing pitch inertia} per unit basic . . .Bendingrigidi@, permit basic . . . . .Kbrsionalrigidity,per Unitbasic . . . .Center-ofaasslocatlon,percentchord . .Elastic-exLslocation,percat chord . . .Chord,perunlt basic.... . . . . . .ab~x,~====.==== ● =

0.5 2.00.5 2.0

0.67 1.50.67 1.5

25 6030 50

0.67 1.50 45

With tha exceptionof sueepbackangle,these~titles were v=ledone at a time frcaztheirbasicvalue. Huwever,for all fourbasic air-planes,sane or all of the parameterswere vemleilfor two or threevaluesof sweepbaclcangle. It is reallzedthat the ~we variationsdo notconstitutea canprehenslvesurvey. However,to a considerable~ent the --es ~ ~~ ~e~ @ b seveti -atiom are additiveif thevariationsare smalland aremade shulteneously. Anotherlimitationisthatthe fluttercharacteristics are effected ~ the spanwise ~iationIn the firstsevengpantitiesLLstea. The two fightersa?latwo bdbersconstitutefour chengesin the spsnwise=ation of thesequantitiesbut unfortunatelyare casesin which four or five of them are varied,

simultaneously. CM&m quautlties uhlch were thought to have second-adereffects were not cons-cd. Azmng these exe altitude (representedbyratioof air densityto wingmass),fuselagemass endpitchingInartla,and tail configuration.!l?hlsties not Implythat fluttervelocityisindependentof altitude,but with vezyminorvariationsthe flutterveloci~ vartesinverselyas the sqpereroot of the alr densl~. Sea-lewel~ densi~ was used t&ou@out this study.

It is ~ob~le thatbaibersof the PM form and size studiedtillbe flownwithoutengineson the wimg. Consequently,the basic casesofinterestaxe thoseIn which a concentratedmass Is lacatedthere. On theotherhard, it is of sane Interestto ccaqp=ethe characteristicsof thebsre wing as well as thoseof a wing with concentratedmass. Bothbaib.ers A ~ B were studiedwith bae wing es well as with mncentratedmass in the basicpositia on the wing.

.For purposeof reference, it is necessary to assign a mmiber to

deslgnde each ptiicuhw case. The group discussed dbuve ccxqpxdses

.——

.

EAcAm 37m 19

175 cases. The assiet of casenuuibersis shownin Wle XII. ThistEiblet3howsmost rmiu the variouscasesthatwere studied.

Concenh=atedmasseson fighterwingsusuallyconsistof fuel tanks,bdbS, OS SilUilSr BtORS . It is ~ssible, therefore,to selectasinglevalueformass d Inertiawhich canbe regEmdedas @plcaL Forcertainpositions, maqy values for mass and inertia-e chosen,althoughin mat casesthe mniberof valueswas restrictedw the time msllziblefor callputations. For referencepurposes,the basicmass for fighterplaneswas -itiwrily chosento be one-qpdxm of the mass of the entirewing (halfof themass of one side),the pitchingradiusof gyrationwasseteqpal to30inches, tithe rollredius ofgyratiauwasassmned to.be 1.5inchesor less. Specific&ta for the tuu figlrtersw Mated intaKleXIII.

Concentratedmassesfor baiber~lsnes em usual@ enginenacelles,with a mass which c= be predicteduithlna factorof 2. Mrertheless,it Is of sane interestto studythe effectof vlxriousmass valuesinthesecasesalEo. BaaLcmass value for both bcmberswas assumedto be15 pound-secondssqwxcedpa inch,which corresptmdsto a weightof nearly6,000 pOUldS. Pitchingradiusof gyrationwas assmed to be 35 inches.Basicmass posltlon was assumed to be at the O.&span posttionand60 InchesIn flmntof the elasticexls. Thesedata ewe alsotdbulatedin table

11)2)3)4)5)

XIII.

concentiatedaass~acteristics -ied in this studyare:

MassPitchinginertiaaboutcenterSpamise locationChordwise locationPitchingflexibility

assignmentof casenumbersisthe study. Althoughspecificspauwisepossible to choose chordwise Positions

of mass

more difficultfor thisphase ofposltlonswere chosen,it was notbeforti. The chordwlseposl-

&ns were chosenas the data-wereobtained. fi saue casesmore &n20 positions-e used for a givenspenwiseloctilon. Conse~tly, onecasenuniberwas assignedto all chordwisevariationsat a givenBpanwlselocation. A s~ of all variationswith the correspcmdlngcasenum-bers iS @ven in -b ~.

pit- flextbili@ of the concentratedMS was vari~ in six casasinvolvlngboth babers . In all cases,the chordwlselocatlonof the cen-ter of -S was basic (tdbleXIII). Ih threecasesthemass was In basicspanwisepositionand In threecasesthemass was at the tip. Casenumbersme givenIn tdbleXIV.

- .— . . ...— — —— —.- -.-— — .- — ———-—— . . .

.... . -.

20 mm m 3780

!cRmlmIN FmmmcHmAmmm!Em

Reference@antitles and (lrqphicalFresentatlon

Remllts of thestuay oftrenasi?l flutter Characterlstlcs Which arellstedintablexvaregimln Dlllespe rhourtii nperunltvalueaofa referencespeed. The referencevel.ocl~chosenis

k Ivo.~~.sec.~~

Obviouslya flutterspeedof 1.5muld notrepresenta realistic valuesince this would mrreqpond to supersonic speed ulth a Mach nuuiber ofabout1.1. Howev=, sti a numberstillhas usefulslgnlflcamefortwo reasons: (1)A majorpurposeof this studyis to establlshtrcdsand to determinewhat cmfigurationstend to be more or lesssteiblethanothers,aM (2) a changein stiffnessis eqdwslkmt to a changeh veloc-i~, so that a structurewith one-halfthe st

vSs of another,but other-

wise unchanged,would exhZbita flutterspeed E timesas greatasthatofthe other,avalueeqpal tol.060r 6C%)milesperhourln the case .givendbuve.

All geauetrical,structural,-1A•inertiagpantltlessre givenhper unitvalues. For exaqple,distancesme measuredin units of theairplanesemlspanandmasses,in termsof a basicvalue. For conversionto speclflcmechanicalunits,the refkmencequantitieswill be fmud inme 3J ~ch s- * p- f-, f@re 7,whichgivesInertiaPerunitlen@h sndriglditydata for thewlngs, tit&bles VIIIto XI, whichlist all pertinalt~eZ’iSttCS of the fourbasic dr@anes . ~densim of alr at sea levelwas used tWc@mut thesemnqputati-. Thevalue chosenis:

P = I..Ik6(10-7)~-sec2 ti.4

In presentingresultsgraphically,flutterspeedstie, In general,been reducedto dimensionlessvaluesby usingas the vel.ocl@unit theflutterspeedof the baste conflgn?atlon.For example,whenplottlngazrtlsynnnetrlcflutterspeedas a functionof ulng mass &msl& far aparticuh wingsuchaa thatof fight- B with A = 45°,theflutterspeedsharebeen dividedw the antisymetrlcflutterspeedof fighterB,A = b~, with baaicwing =S. -tic - Srrtisynunetrlcresultsareboth presented,ra~ than choosingthe cm whichgiveslowestflutterspeed. -e suchresultsare presentedIn the Ssmef’iwre,~trlcresultsare generallyindlcat~ by solidllnes,snd~ iiotteillines. Opeclficnumericalvaluesfor theflutter fregyncles iEmefou@ in tdh XV.

antis+tilc results>flutter spe&s ~ .

.

.-— . ------- —

mUAm 3780

Massandlhertia

In most practicalconfigurations,

21

Wd.ations

the noimal mda of tihrationwithlowest freqptiwyis ~Wab-~e~~~ c~~the firstwingbendingmode. In the dmence of a largeconcentratedmassOnthewlng, apr edmdnant torsional motion is usually observed in thetbiraor fourth nloae. Simplefluttercan oftenbe predictedwith engl-neeri.ngaccuracyusingonlythesetwo*as the normalcoordinatesofthe structure.When a largeconcentratedmass Is involved,the situationis much more ccmplex. Tuo Ormoretorston modesas wellastwo or morebendingmodesbeccmeiqmrtant in fIuttercaqputations,and severalflut-ter roots~ be observedwhichprdwdmnt ly involvevariousones ofthesemodes. For eccentricmassesit becmes, In fact,~ssible tospeakof bendingand torsionmodes sincemany * will involvebothlargebendingand torsiondisplacements.

In thosecasesIn which flutterInvolms a be@ing mode md a higherfregpencytorsionrode, it canbe sdd that a structural.changewhichseparatesthe frequenciesof these* ordinarilyratsesthe flutterspeed,_ a changewhich* the frequenciesmore -ly egyallowersthe fl.ut_&mspeed. It will be observedbelowthatthis ~a~~is not aluEqmm. A changein mass densi~ withoutchangeIn pitchinginertiahas greatesteffecton firstbendingfregpencyevenIn caseswith~ge sweepback. Consequently,increaseIn wingmass densitiywouldbe~ected to giveau increasein flutterspeed_ &crease in mass den-sity,a decreasein flutterspeed. Changesin pitchinginertiawouldnormallybe qpectsd to hen an oppositeeffect. 8uchvariationsweremade for threefighterconfigurations,fourbae-uing banbercmftgwa—ttons,and fhe bcmiberC~ tlonswith conce)rln-atedmass. The massdensityandpitcblngLnertiawere separately changed by factors of 2.0and O.~,maktng a total of 48 configurationsin additionto the 12 basiccases. Referencecasenumbersare givenIn tAM.eXII.

T&bul.atSond flutterspeedand frequencyfcm each casewiXL befoundin tablexv. I& results=e also shownin figure8. & mentionedem?lier,theflutterspeedshavebeen reducedto Mmensionlessvaluesbyusingas the velaci~ unitthe flu- speedof the basicwing for eachb=ic c~tl~. The trd predictedZiboveewe foundin most cases.M the caseof bare-wingfightersthe effectIs v- systematic.TheEverageof all casesis givenas follon:

Mass value Plteh Inertiavalue AveragechangeIn vf,percent

o.~ 1.0 -122.0 1.0 61.0 ●51.0 2.0 :

The effect,thoughunifozm,Is quitesmall.

.. .. ---- . .-.. — — —— —--- - —— —-——- -

22 m m 3700

The resultsfor lxmibersslmwmuch less consistency.For casesbothwith - withoutconcezrtrat#imasses,the effectof wlng+uassdensl~~atibn is upredicts31e. Heemlyhalf of thecasesshowtreMs whichsre opposite to that predicted abuvw. !l!haaddition of a.conmmtratedmass at the 0.46-spanpositionremraed the * in severalcases. Ontheother hind,chengein vhgpltchlng inertiadldshow a systematictr~ for allbcsdbercases. On the ewerege,a changeIn pitchingInertiaby a factorof two changedthe flutterspeedabout7 percent.

The folMwing conclusionscanbe drawn:

(1) A changeof wing pitchinginartiashowsa s@xmatic trendfor allwings,althoughthe effectis small.

(2)A changeof wing mass showsa deflnlte trd for @pical

M-$ although the effect is small.

(3) - Of wing mass for @plcal -ge bcmderswith orWithoutmmentratea.massesShomlno Systematictrd.

Wlffness Variatlms

It has been pointedout (e.g.,ref. 1, p. 783)thatwhen incc@press-iblefluldflow is assmned,a changeof stiffnessIs equivalentto a-e of velocityinsofaras transientresponseof an atrfoilIs con-Cti. Ccmsegyently,It canbe saidthat a uniformincreasein stiffnesswll.lralsethe flutterspe~bythe serootof the factor~whlchstiffnessis increased. b nmst ai@anes, it is foundthat the Increasein torsionalrigidl.~is primerllyresponsiblefor the increasein flutterspeed@ tlmt,im general,a changeIn bendingrigidityovcmratherwldalimitsdoesnot changethe flutterspeedsignificantly.

As shown in tdbleXIX, 12 configurationswere stdied to sqpportthis Conclusion. Mnce bothbendingrigidi@fend torsionalrigiditywereseparate~ changed~ factorsof 0.67 - LX, -e are a total of48 case nmbers assigned to this group. The results of this study arelist~ in tdble XV and presented gr@ically in figure 9. For easeofccmparlsml,flutterspeedsare conmrted to Mmensionlessvalues,dfluttercharacterlsticafor changesin bendingand torsionalrigidityareplottedSide~ sib. h gemmal, It was foundthat chsngein torsionalrigldltiyby a factorof 3/2 or 2/3 Incrwkd m decreasedthe flutterspeed~20peroent andtbat asimllar changetib_rigl&L& hadanegld-gibleeffectwon the flutterspee& Amongthe 12 confl@ratlcmsStudied,the followingeXEeptimlsto this trd m noted:

.

(1) Bmber A, A = 0°: In the anttsymetrtccaseboth bendingandtorsionalrigidd.@had roughlyeqyaleffects,ftitterspeedchsn@ng

.

*lo percentfor the rigiditychangegivendbuve.

(2)BariberB, A = 45°: _tric casesame as case (1) ebuve.

--- .-

mm m 3780 23

.

(3) Bdber A, concentratedmassat 0.46 @an, A = 0°: IU theantisymetriccase,torsionalrlgldl~ had a x percent greater effect

(50-percent change In flutter speed) and b7

rigtdt~ had a neg-tive effect(m-percent ehsngein flutterspeed .

(h) Bdber A, concentmtedmass, A = 30°: b thesymmetriccase,thetrendwasnormalOn@ forincreasein torsicmalrlgidi~anddecreaseinbendingrigidt~.

!l!heseexceptionsdo not constitutea majordeviation,- the trendIs consideredmu estEibllshed.

M Stiffness Variations

It Is not to be ~ected that the sme effectwill be obs~ iftorsioualrigidi@ is changedat ~ous stationsalongthe wing. Inthe dbsenceof a cmcentratedtipmess, ~ efYecton flutterspeedmustvanishfor stationsnear the tip, - presmd)ly the Mgest effecttillbe observedfor stationsneax the fuselage. Becauseof the greateasewith whichthesedata couldbe obtti, the &fect of localstiffnessvariatiauwas ob~ for sewereLconfigurate-.

The analogccqputer reqgires luqhg or axmraging of Inertia -stiffhess properties. Consequently$ it Is possible to determine readilyonly the effect of a stiffness -iatlon which must be assumed to exkduver the cdzlre length “of a cell In the finite-difference structure. Thebasicdata consistthereforeof step curves. To obtainem ~tevaluefar the per unit -e in flu- speedper unit changeIn sttffhessper unit lengthat any point alongthe wing it is necess~ to draw aSmoothcurm?passl.ngthroughthis curm?such thatthe areasunderthetwo curvesme ~oximately equal. It is beldevedmore sutixibletopr8s~tth8st8p~tiutthe re8dera0~~ hls applicationreqplres. The configurationsstudiedare listedbelow

[

1) Fi@ta A, A = 0°, _tiiC - entiqmetri c, case 102) RL@rkerB, A = @, _tric and sntiztrlc, case323) BanberA, A = 0°, b=e wing, symetazLc,case93h) ~b~ ~+ A = 0°, concentratedmass at O.~ span,_tric,

Resultsof thisB- =e presentedIn figure10. !lhedbsclssaof amum is the spanwlsestationat whichthe bendingor torsionalrigidi~variationis made. The ordinateis the per unit changein flntterspeedper unit changein stiffnessper unit lengthalongthe wing. E, forexample,the stiffnessIs increased j percentavera distance w alomga wing of semispan 1 betweenthe stations d - (w/2) ad d + (w/2),then the ~te of the.smothed curveat the dbsclssa d/Z whenmulti-pliedby ~(w/2) will give the approximatepercentchangein flutterspeed.

..—.— — —.—— ... —.--— - -- — —.— —— —-—— .

24 mm m 3780

-- for the two fi@t~ show~eat simi~~ both fw --Cand antisymuetz.dcconditions.The greatesteffectis obtti ~ dwmgingtorsionalrigidi~ near themidspsnposition,sli@tly outboardfor~Atisli@tly~_fm~ B. !Cheeffectof amending- -=~ -e ~ f- to be - at all stations. In most casesamall negativeeffectwas observed,the flutterspeeddroppingslightlyas the bendingrigidl~ uas hcresaed.

BaiberA withoutconcentratedmass showeda simi~ trendwith thefollowingeurceptions: “

(1) Msdnrum@P~ was obtainedby cbnging torsionalrigidi~near the rwt of the wing (o.2543peJlposition).

(2) Increasein bindingrigidi~ was observedto decreasethe flutterspeed~ an muountwhich wasatolo times greater thanthat for fighter JL

AMItion of a concentrated mass at spanwise station 0.46has a greateffecton this dKmcteristic. TheInasachosenis tgpicalfor an air-plane en@ne, - is sufficitilylargeso thatthe wing is to a certainextentpinnedat thispoM for the particularflutterroot Irfvolved.

.

Consegpently,stiffnesschangesiriboardof the enginehave a negligibleeffect,and changesoutboscdhem an effectvery similarto that observed -fw a bsre wing of reducedlength.

It shouldbe remarkedat thispointthatthe resultdiscussedmoveis not to be regardedas a tremdfor all cmfigurations. Hhen the fllltlxmis pr~i~ = outer- bu-torsicm titter thenthis remd.tiS tobe expected. Experiencehas shown,howev-, that occasionallyan inner-psneltmsion mode is Imolved in flutter,and changein torslmml.rigid-i~ outboardof the nacellehas no significanteffect. It is unfortunatethat such a comfiguratiau was not investigated for this repro-t.

Cente-of-s Iocation

h location of the wing center of mass has a greateffectuponfitter speedof an airplanewing. The g~ trendis that flutterspeedincreasesas the centerof mass muves faward. It is not gener-aldytrue thatthe centerof mass is at a constantchordlscationat allspanwisestations. However,for purposesof stu@lng the trends,it isnecesseryto assumesauebasicpositionfor the centerof mass. Past~erience has shuwnthat a center-of+nasslocatiannearthe elasticaxis(~ s~t~ ~) iS ba retistic - @Pical. For tlxLsreasomthe basicpositionof the centerof mass was assumedto be the elasticSxisor 40 percentchord. Variatia in center-ofaasslocationwasbetweenthe 25- d 60-percent-chordpoints. Thirteenconfigurations-e studied,the variouscenter-of+nasslocationscmslng 53 casesMsted in Wle XII.

4G

b

w m 3780 25

The resultsae llstedIn tableXV and are showngraphicallyhaimensimd.essfolmlin figureU.. The genEEraltrendIs that the flutterspeedIncreasesas the centerof mass -6 forwazdand decreasesas thecenterof mass moves E&t, excqt for Center-of+llasslocationsfar btithe elasticaxis. For positionsnesr the elastic~, the flutt= speedc&K~m 3 percentfor a shiftin centerof mass egpalto 1 percent

. Fbr the extremeaft posltlons(60percentchord)most ofthe curvesbecaueqpiteflat,and in dxxrtfour casesthe flutterspeedhasstarted torisesllghtlyas thecater ofmassismuwdfm%her aft.Ontheotherhand, the curvesbecanevery steqpfor center-ofaassloca-tionsfcuwardof the elastlcaxis. ti most casesthe increasein flutterspeedwas so greatthatdata couldnot be obtalnadfor the 25--3205-percent-* locationsbec~e the flutterspeedgreatlymeededthe alvergencespeed. The eweragepercentagechangeIn flutterspeedforashlf% incemter ofmassegpal tolpercent of thechorddeperdsupon locatlahof the centerof mass as indicatedbelow

Center-of4nasslocation,percentchord . . . . . . . . 40 ~ 60Changein flutterspeed,percent. . . . . . . . . . . 3.1 1.7 0.8

One unusualcasewas noted. The resultsfor fight= B, A = k5°, infIgureU.(a)show an unusualbehswlorfor - center-ofaaas locationin the antisymmetmiccase. A studyof the frequencyof oscillationforeachpositiontendsto swort the conclusionthat two diffkmentflutterrootsare involwed..In any casethe resultsaxe SnaMMus -couldbear furtherInvestlgatia.

FighterA, A = 0°, ShOWSanotherunusualcharacteristicIn theantisymetriccase. One flutterroot disappearsas the centerof massis moved foti of the 46-percent-chordlocation. Thisresult,shownIn fIguren(a), Is more eastlyunderstoodby referenceto figure12where the curvesof g agatnst v are plottedfor this configuration.A similarcase shownin figuren(b) has two readilyobservableflutterroots,one with low Ed the otherulth high flutterfregpency. Datafor both caaes=e givenin figuren(b). Itistrue that ~the onewith lowerflutterspeedis of practicalinterest,but for purposesofstudyingtrendsboth are egpallyimportant.A plot of g against vfor this case is also shownin figureM.

Elastic-AxisIocation

A main c~onent of the ~c pressures on an airfoil isegplvalent to a force qplled at the qyxrter chord. Consequently, theelastic-axislocationrelativeto the gparterchord&kndnes thenatureof the coupllngbetwecmaerodynamicforcesand the structure.If elastic-axislocattonalonewere changed,both centerof pressureticenterof mass would changewith respectto the assumedstructuralaxis.

. ..— . .. —- —-— -—. ———- —- .-— ..— — ----

26 m m 3780

.

horbrto separa.tetheeffects duetothese tuochanges,the centerof.

mais was movd with the elasticaxis In the conflgumtionsdiscussedhere.AmMJnmic co@lng in which the Cenlxmof pressure(gparter-) is .

forwardof the elastic~ ~ heme a destehillzingInflmnce whilecenterof pressureaft of the elasticexisgeneralJyhas a st~ilizing&fact.

E1.Rstic-edsIacationsbetweenthe 30 ad, x percentchordwereUseainthe ticasesllsteain tehleXEL Resultsa?eglvenlnteblexvand figure13. Ic all casesthe expectetitrendwas observed. For anelasticads near the 40 percentchord,the flutterspeedchanged3.2percenton the merage for a shiftin ehstic -s egpalto 1 percentOftbechora. !lhiseffec tisnotline=uver awiderange, however;thsflutterspeedincreasesmore rapidlyas the qumter chordis approachedsad decreaseslnoreK1.owlyas the elasticEucLsIs moved aft. lbr anelasticaxts at the ~ percentchord,the Correspm changein flutterspeedWRS only 1.8 percent.

ChordVariations

Achange lnchord.ofawingls usuallyaccanpauled~signif’icsnt-es Illmass, Inertia,and stiffnessas WELL as changesIn otherCharacterlstlcs. b an effortto assessthe effectof ~c Pr-- .suresalone,=at@ns were made In uhlchmass, inertia,and stiffhesswere held constantW3111sthe Choralengthwas changed. Iacetionof theelastic~s was maintainedat a constantper tit M stationso thatthe distancebetweengyarterchordand elasticaxis changedIn proportiontothechenge inthechordl.ength. Sincethemagnitudeof the aerodynamicforceincreaseswith _ lengthand sincethe predminantlyaestaillzingleg of the TheodorsenfunctionIncreaseswith chordlength,it Is to beQected thatthe flutterspeedwilJ decreaseas the chomdlengthIsincreased.

Four configurationswere studiedIn which the chordM@h was~ w factorsof 0.67 EIM L50. The eightcasesand the configura-tions ere llsted in txibleXtI. Fluttercharacteristicsare givenintxibleXV and figureU. The resultsare ?xsllarIaibquniform. on theewerage,a 7-percentchangein flutterspeedresultsfrau a lo-percent-~-, malMr chordsgkbg a higherflutterspeed.

Sweepback

The d’feetof sweepbackupon flutterspeeddependsqpa _ factors.In conventionalwing design,the root structure-Ies greatlywith swaep-back engle,emd the egpi.valentelastlcaxis~ show consitih varia-

.

tion in position. For wings of largesweepbackangleeml low aspectratio,

. ..-—

lrAcATN37ao 27

the conceptof an elasticaxismay not be usti h describingstructuralproperties.mall anotherpoint of view the problemis emn more pa@e%ingsincethereIs not generalagreementdxnrtthe natureof the aadymmlcforceson a sweptwing. In the sectionof this reportentitled%nlte-DlfferenceErrors,” the resultsof threemethodsof cauputationwere ccm-pfm3d with resultsof win&tunnel testsof a modelwing whichwas swept-Imck %.50. !Cwumethodswere foundto give simi~ results,whichweresimificantlwbetterthan thoseof the third. Althoughthe agremt wasnoi entirely-satisfactory,it was decidd toremmended in reference~. For the presentassumptionswere thereforemade:

(1)Aer@mmi c forcesare as discussed‘,@nlte-DifferenceErrors.”

-e * ~c forcesi?mstigation,the follawing

In the sectionentitl.cd

(2)!Lbachievea sweepbackangle,the wing is rotakd dbouta =i-cal ads throughthe intersectionof the unsweptelsstlcaxis and theside of the fuselage. The tlp is, however,terminatedparallelto theairstream,so that onlythe spanmeasuredalongthe elastlcsxls is~~ ~ -*

(3)s~~~ ProPertf* of the Win8 are ~ed by swe~back.

(4)~caerofuSofthefie@eis muvedaft asthe sweepbackangleis Increasedso that it coincidesroughlywith the centerof pres-sureof the wing.

(5)NO modlflcatlonsweremde for aer@mmlc forces at the tip.

The fivebasic configurationsewe shownIn t&bleXII, which givesreferencenumbemsfor the 17 cases. !lMeresultsaxe gfvenin lxibleXVand figure1.5.Fluttercharacteristicsof the two fightersshow areasonablecorrelation,azd, in general,a decreaseIn flutterspeedforsweepbackanglesotherthan zero. However,the baibersdonot showacorrelationwith the fightersw with eachother. It is signlficaatthata stistazrtialchangein f1* speedwith sweepbackanglewas obsemed.~ one case,flutterspeedincreasedmore than 60 pert-t for a 45° sweep-back,while othercasesshoweda 30-percentdecreasefm sweepbackangleof &bout25°.

In additionto the casesdbuve,it is possibleto crossplotthev=lation of flutterspeedwith sweepbackanglefw the followingp~-etervarlatlom of bauberB: Wingmass &msi@, wing pitchihginertia,bendingrigldi~, torsionalrlgldi~, and center-of-masslocation. Mostof thesewe plotteUh _ ti. It is in~sting to note that thegeneraltmczdfor bauberB is to a greatextentixdep*t of thesev-ations.

—.—. — .—— —. - .—. —.-— -. -

28 MMA m 37%0

Concentrated-MassPitchingFlexibility

The engtneson present- banbers=e smetimes mountedIn nacelbs Fon ~1.onssanedistancebelowthe wing. Becauseof the inherentflexl-blli~ in such a structureand its fasteningto the wing,the dynamiccharacteristicsof the engineare altered. Becauseof the ~ ofthe structure,it is possibleto write two sets of egpationsf= thenacelle,one Involvingpitching,vertical,and fwe and aftmotionaMthe otherimvolvinglateral,rolling,andyaulngmcrtiti.Thesesets areuncoupledexcqptthroughinteracticmsWith the wing. The Characterlstlcsrepresented~ the eqyatlonsInvolxlngpitchh8ve a greatereffectonflutterdamderlstlcs, or, statedh mlotha?w> - -mim of arigid~lon for lat~al motia has not ordinarilybeen observedto intro-duce greatdlffties h fluttercharacteristics.!Ehtsassw@ionbeccmeslessvalid for wingswith largesweepback. On the * H,a significantvariation~ be observedas the pitchingfle%lbiMtlesarevaried. For pitchingmotionit is usuallyqpiteaccurateto assumeaneffectivecenterof rotationat sanepoint in frontof andbelow theelasticaxis. Unlessa specificcase Is to be considered,however,itia just as satisfactoryto asswnethis centerof rotationat the elasticaxis,sincevariationIn the loctilonof thispointhas only a second-ordereffect. Consequently,In this studythe centerof rotattonforpitchingmotionwas establishedat the elasticaxis and the ~lon wass8smed rigidfor lateralmotion.

SIX casesshownIn tale = were imestl.gated:

I1)BcmiberA, A = @, mass at O.~ span,case 1762)Bdber A, A = @, mass at tip, case 1793) BenderB, A = @, mass at O.M span,case 177~]k~~ A= 00,mass attlp, case 180

A = 3@, mass at 0.46 span,case 1786) ~er B: A = 300,mass at tip, case ml

lh all of thesecasesthe chordwlsepositionof the mass was basic,60inches f~ofthe elasticsxis.

In presentingthe results,an effort has beenmade to put the datain Mmellsimibess form. lhs)thef lutt~speedisglven aaaperunltvalue of the flutter speed with rigidconnection.Thisbasicflutterspeedcanbe foundin tableXIV. The fkdblll@ is convenlqntlymeas-ured~ the nonnal+nodevibrationfi~ Ofthnacelleu ltllthewinghela rigidIn pitch. Eowever,insteadofuslng the valueof frequencyincycles persecoma, this frequencyIsmeasurea lnperunltvalueofthe flutterfregpmcy with rigidconnection.Valuesfor the flutterfke-gpencyulth rigidconnectioncan alsobe fourdin tdbleXIV. Therearetwo fregyenciesof the nacellewhichmightbe regemdedas significant.Oneofthese lsthecantlkfh equency inwhich thewinglsheldrlgid

EAcAm 3780 . 29

In both pitch - plunge. *er, for kge Ixxib-, the ulng has suchgreatflexibillw In verticalb- that great= s~cance mightbeattachedto the fregpencywhen pitchingmotionis constrainedend ticalmotionis completelyunrestrained.Eecme of the locationchosenforthe basicmass, the differencein thesefregpenciesis a factorof 2, thefreqpencywith verticalmotionunrestrainedbeinghigher. For presenta-tion of data,&La hl~ value of freqyencywas chosen,becausein thosecases-e a “tuning effectma observedthemexdmumeffectoccurredwhen thisfre~ was egpalto the flu- frequencyfor the basicrigidmass. One exceptionto this is obsti in the dimussion below.

Results are plottedIn figure17. Mine of the twelvecasesshow a~t decreaseIn flutterspeedas the rigidityis reducedfranan Infinitevalye. Sevenof thesecasesshow a mintmmnflutterspeedWhen the nacellefreqyezq Is ~ly equalto the rigidflutterfrequency.Thisdecrease-es between7 end37 percentwithen 8wragevalueof18percent.Tm casesshowa decreasein flutterspeed,but no tudngeffect.l%emaxhm rateof decreaseoccurs,in fact,whenthenacellefieqyencyIs fa belowthe rigidflutierflwquenq. Ib both casestheflut- speeddropsto en eqn@otic valuedboutah-tenths of the basicvalue.

ThreeaFthetwelve cases showan Increase in flutterspeedas the-’b IS reducedfrom en infinite value. In twe cases Increase takesplace in the regionwherenacellef%eqpencyis roughlyequalto the flut-ter fregpency. In both casesthe flutterspeedincreases-e than~percent. !lQlelastenwMMus caseshowsa resonanceor tuningeffect.It is anmabus for two”reasons:(1)The flutterspeedrisesto a sharppeak dboutlo percentdbuvebasicvalue,and (2)this occurswhen nacellefreqpencyis twiceas -at as the flutterfreqyency. It shouldbe pointedout that,for thisrlgldltqr,the flutt= frequency10 equalto the nacellefrequenq with wing attachMnt COIIStrdlledin bendingM Well ~ pttch.

Effect of a ConcentratedMass

~ aircraft structures ham engines, stores, or externalfueltanksattachedtothewlngti such a~ll&~actdynsmicaKQrasconcentratedmasses. Ithaslong been lumwnthat the I.ocationoct’sucha mass has a significanteffecton flutter. Unfortunately,otheraero-mc @s= s~c~ prob- do not p-t locatlonof suchamass so thatmsximumf~ speedla obtained. Ontheother hana,withinthe restricticmsimposedw otherconsiderations,it Is oftenpossibleto @prove flutterchsracteristtcssignificantlyw properchoiceof masslocation.

This Irmstigationhas includeda detailedexaminationof the effectof a concentratedmass on the fluttercharacteristics‘ofseveralconfig-urationsof the fourbasic ~lane wings. Fre~ studyof this

—... .—— ——— .— .— ——. --— -

30 “ mm m 3780

effectsharedsuch Interesting end unusualeffects that the scope of theinvestigation was eqa beyord that ~ x~os~= ~ re~lt@3data exe so volumlmus that it is difficult to present them effectively. -In particular, It Is Impracticalto constructa tdblewhich givesall &the data obtaind, ad so graphicalpresentatiait3reqdnd. Two methodshm’ebeen adoptd In thisrep-. M a givenspauwiselocationof themess,the flutterspeedcanbe plottedes a functionof the cho?.Wiselocatlon. Thishas been done for alJ.casesiwestlgatcd. Sincethe con-c~ated mass is dined with the alrs&eem, it is most convwlent incaseswl.thsweepbackto mum the mass paralld to the airstresmratherthanperpendicularto the elastlcaxis. Where sufficientdata are Wall-~le, thesecurvescanbe smmemized In a s~le diagremIn which linesofconstant flutterspeedreshown onadralng ofthawl.ngplen form.~ the concentratedmass locatedanywhereon such a ccdour llne,theflutterspeedwill be the same. llheresultis essexrtiaUya topographicmqQ of the flitter-speedsurface,where eachpointon the plan formrepresentsa possiblelocationfor the concentratedmass.

$mal Htitles arise with both methodsof presentation. ~main source of difficulty -lles h the fact that s0w3ral.Important flutterroots exist for a * with concentrated mass. I?orcertain I.ocatlons ofthe mess} one root till show lowest flutter speed, tile for otherloca-tlausanotherrootuil.l-e the luwestflutterspead. WncetheanalogcqputerIs essential.lyan electricalmodel,it Is~ @possibleto=aueofthese flutterspeadsif anotherroothasaflutterspeadfarbelowthefirst. It is possible,therefore,to fiM with certaintyonlythoseptiions of a givenflutter-rootsurfacewhich liebeneathall&her flutw-rod surfaces. For one ComfiguattanEltuaied,four suchdistinctsurfaceswere positivelyidentifiedand it was not possibletoestablishthat surfacesappearingat widelyseparatedregionswere orwere not related. h most casesthe rootswere differentiated~ obtainingessentiallymarginalstabilityfor two aistlnctrootsalongthe LLnewhere the twu surfacesIntersect. It canbe reedllyappreciatedthat_pointsere reqplredto establishthe flutter-speedcontours,particularlywhere s~ intersectings-aces are involved. It was, In fact,@pos-siblein the time avallabl.eto obtdn sufficientdata to estdbldshallinimrestingfeaturesaboutthesecontours. However,it is bellevadthataKl iqportentfeehmes are showncorrectlyin the figurespresentedhere.

The curveswhich showflutterspeedas a functionof chordwlsepositionat a fixedspanfrequentlyshow intersectionsbetwee!ndifferentflutterroots. b identl~ theseroots,it 1S usefulto know theflutterfkequ~ associatedwith eachroot. !Blesln@est ~ to presentthese-a Is to showthe valueof fregpencyat a few selectd pointsalongeach curve. WhererootsIntersectandboth frequencieswere meas-ur~, both valuesere shown. Tn sane caseswhereactualfrequencieswere .notmeasured,low,medium,or M@ freqpencywe shown.

.- . —-

EAcAml 3780 31

,

For ccmvenlence,the concentratedmass was al- placedat thecenterof a ftnite-differencecell.excerptIn threecaseswhere additionalinformationwas obtained~ placingit ~ betweencells. It isconvenientto identi~ theselocations~ the celdnumberas has beendone in tableXIV,Which assignsa casenudberto each cafiguratlon.It must be remembered,hawmmr, thatthe celldivisionsare slightlydlf-ferentfor baibera?d &L@tar airplanes,and thereforethe spmmrfsesta-tion for a givencellnuuiberwill be different. The locationof thesestationsin tams of unit span is giventn tableXIII. & the figures,the spanwisepositionis correctlygivenas a fractioncm per unit valueof the wing Semispen.

Thesize of the concartratedmass and itspitchingend rolling. Incwtiaabo affectthe flutterqpeed. Mnce past ~erlence has shownthat rolldngoinertiahas a mall effect,a few casesme chos~ forfurtkm LnwwMgatlon of themagnitudeof this effect. For a concentrdxdmass locatedin the wing it is reasonableto assumea radiusof ~ationwhichls aamallfractiono ftheEweragehalfchord. For a mass suspendedbelowthe wing, it is unlikelythatthe distaucewill exceedhalf of theeweregehalf chord. Tm valuesfor radiusof ~ation were chosen,egpalto 0.1 and 0.5timesthemerage chordforthefighterplanes. h allcasesconsideredtherewasno signiflcmtdifferencein fluttercharacter-istics_ theroUing inertiawasvariedfrcanzeroto themaximumvalue.!l!hemiatlon was,in fact,so inslgulficantthatnoneof thedataispresentedin thisreport.Iu-tfolb mitmqybedssumedthattherolMng Znertiaof theconcentratedmasshas anyvaluebetwe- thedbom?Mnllts. Sincethemassof theconcentratedmassandItspitchingInertiahavea greatereffect,it is necess~ to considervzmiationsin theseqpantitieain severaltypicalcases. ~ basicvaluesformassandpitchinginertta(orradiusof gy%ationk) hme beendiscussedIn thesectirmentitled‘hacteristics ofFourRepresentativeAlrcrsft”andere@m in tsbleXIII. !& relationsof thesevaluesaremmmexizdin tlm.eXIV.

lhemostlogical~ to givetheresultsIs to pres=t firsttheflutterchm%cteristhsforthebasicmasson eachp~- dr@ane.Fiveairplsneswerechosen:

[

{

1 FighterA, A=@Z F&hter A, A= 4503) BcmberA, A = 0°4] BouibaB, A=@5) BaIiberB, A=30°

Sinceit was dlfflcul.tto choose a Qpical radiusof ggration for a masson a fighterplane,threevaluesme used. -se -us are6, U,and 30 Inches, as EIhownin Ixible XIV. Ffgures18(a)and 18(b)showtheeffectof chordwlselocatiunof the mass at ftve spanwlsepositionsfor

..—. .. —— ——-. . .—— —— -- —— —- .—— . .. --— -——

32 HACA m 3780

fighter AwLth A=OO..

Resultsfor all threeradiiof ~atlon in pitchare plottedon the same sheetusingdlffkment-ols for eachvalue.Circlesare used for the smaUest ved.ue,k = 6 incherii; triangks are .usedfm k=mlnches; asolidllnewlthnosynibo lslsus edfortheintermediatevalue. Abscissasforallcurvesarechordwlsedistancehxm theelastic-s measuredas per unitvalue of the wing SeELSpan.ShELlarma for ftghterA, A = 45°,areshownin figures18(c)-18(d);sixspsawlsestationswere used in thts case. One surprisingfeaturecanbe notedIn all of thesefigures: The characteristicsarerelativelyidepemlent of the pitdblngInertia,eventhoughthisInertiaisvzmledfranaveqlsrge value(k=minches) to nearlyzero(k= 6 inches).~s doesnotman thatat anyparticularpointtheflutterspeedsareIdentical,but theaverallshepesofthecumms showrezlmrklmlesllllllJEirl@.

Althoughthesefigures(fl@. I-8(a)to 18(k))gtvea goodPi*eof thefluttercheracterlstlcs,it is easierto int~et the resultsIfall data are caublnedto constructfluhtercontoursas diBcussedearlder.such Contcnnxlfor the minimumandmaximwnvaluesofkareshoun Inf@res V(a) ~ 19(h) ● -e f@ms SUPPOrtthe followingconclusions:

(1) A chordwlsepositionaf’tof the elastlc-S Is almostal,Unaeslrable.

(2) The 30- to X-percent-spanpositionand the tlp locationareS==* wlaeairsble.

(3)A positionf~ of thee~tic80-percent-spalpositionwill, In general,Speea.

SxlSana naarthe 70-togreatlyincreasethe flutter

Since the U not pamlt a caupletestudyof the Characteristicsfa ~ B, datawere obtainedonlyfor the casesof 0° SM 45° sweep-backwith mass at the tlp. Canpszisonof the resultsshownh fl -

ture 18(e)with the correspa data for fighterA in figures18 a),18(b),18(c),~ 18(d)showsthat for thl.slocatlonthemedoes not seemto be ~ significantdifferenceIn characteristics.Whetherit is safeto _olate thisresultto othermass locationscannotbe said atthis time.

Sincethe studyof fighter A showedthat the pitchradiusof gyrationhad a smalleffect,and sincethe pitchradiusof gyrationof a bcniber_ ~ re~ttve~ ~~ M-, it m ticiM to use only one valuein the S- of bcmberairpbnes. However,the practiceof usingone_ or * as on a s~k WM, = well.as the differentsizesof engines,givesa possiblevariationin masswhichmightwell =ceeda factorof two. All bcaiberdatawere therefme obtainedwith both basicmass and half’basicmass. Far ease of comparisonof the two sets of

.

—.. .. -

5G

.

mlOAm 3780 33

data,@ey are plottedside~ side in the fir

~ti- for bcniberA,A = 0°, are carktned in figures18(f)end 18 g). “Resultsfom bcaiberB,A = 0° are givenin figures18(h)and 18(i),- the case of bcaiberB,A=#, ls summarizedin f-s 18($)and 18(k). A@.n It is possibleto simpli~ interpretationof thesefiguresby ccmblningthe resultsintoflutter-speedcontours. However,it canbe seen thatdata f= basicmass and halfbasicmass are ~ slmt-, - so such CO*S havebeenPreP~~ X for ~ cueS ~ti b=ic DMMW. ~ flutter-speedcontoursare shownIn figures19(1)to 19(n). A studyof thesefiguresshowssanedeviationsfhauthe resultsfor fighterairplanes. The followingcon-clusionscanbe drawn:

(1) A positionaft of the elasticaxis is almostal- ~airdble.

(2) The tip regionis generallyundesirableas a locationfor themass.

(3)Wim f- eXcSPtl~, m positionforuardof theelasticaxisamlbetueentherootandtheW-pexrc@rb-spanpositiontillgl.veflutterspeed eqyal to or greater than the bare-wing flutter speed.

(4) WWWer, there are, in most cases,no practicallocationswhichgive any SlgnlflcsntimprovementIn fluttercharacteristics.Two caseswill be notd in which the speedmightbe Increased@ percent. Theothersare restrictedto a lo- or 20-percentimprovement.

Sincefighterplanesshowedremarkablevariationin fluttercharac-teristicswith mass position,It was believednecess~ to exmuinetheeffectof changesIn the size (mass)of the concentratedmass. ThiswasfIrststudiedat two spanwlseposttlonsfor ftghterA, A = OO. Thepositionsare the tip and station5 (0.~ span). Fluttercluwacterlsticsas functionsof chorduisepositionweremesauredfor severalvaluesofmass. The threevaluesof radiusof gyrationgivenh t&bleXtIIwerewuEd6fWorlll.mv&s excqptthat for very smallvaluesof mass *

. Eouemr, becauseof the slmilarim of results,data me presentedonlyfa the maximumvalue (k=30inches) sndmlninmm*(k= 6 ~). Casenumbers=e llstedin tableXIV.

The resultsfor tlp locationshownin figure20(a)show a veryinteresting.progressionin characteristicsas the mass Is reducedtozero. Most strildngis the factthatno sigulflcantchangetakesplacewhen the mass is variedfrantwicebasicvalueto halfbasicvalue. Evemwtthmass reducedto 8 percentofbasicvalue,thethreeflutterrootsfor~t~c ~i~ @ * * flutterro~ f~ -Mc ~iOMcan sttll be identified, though their chsxacteristics are ~ this thesomewhat EL1.tcuwd. Similar data for the mass at O.~ span Errepresented

tn figure 20(b). ELgure20(a)concludedillustratesgraphicallythedangerin extrapolatingresults. For amass at the O.10 chordwise

.-— .. —-. —— .. . -—.—— — .——-—. —— .

34 mm m 3780

posttlon and symmetric motion, a mass eqpal to 0.008 basic mass increases -flutter speed to 1.07. Dodbling the mass Increases It to 1.M. Agahdodbllng it will increase the sped to saue unlmown value greater than1.40. hrever, if the mass is again increased w a factor of dbout 2.8,

.

the flutter speed (of anotherroot)Will have dmppea to l.m SgalI1.

Fluttercharacteristicswere -O measuredfor basic and halfbasicmass at thewingttpwltbsweepbackangleof45°. ~s casewas chosenbecausethea&isymmeta’lccharacteristicfor a 3CWnch radiusof gyration(fig.M(a)) showeda ~ unusualcharacteristic. - 21(8)showsthat reductionot the mass ~ a factorof two eliminatesthe mmmlousbehavior,but in all utherrespectsgivesresultswhich ~e essentiallythe same as the basicmass. RLgure21(b)givessimilardata for themass at the t).~ spanpositionulth A = 45°. Againthe resultsforbasic IU8SS d half basicperhqpsunwiseto attemptcationthat the essentialof flgure~would notbe*18 were Increasedor

-S - not S~C811t~ diffeZWlt . It iSw gm= statement,but thereis everyindl-fea~s shownIn the fl.utier-speed contoursprofoundlyalteredIf eithermass or pitchingdecreasedm a factoras largeas 2.0.

Accuracyand Probable&TOl?S

beenthat

A brief statement dbout the expected accuraq of these resultshaspurposelydeferreduntilthe d of this report. It la bellevedthis discussionwill be more meabgfu 1 afterthe readerhas observed

the natureaud scopeof tbe data obtsdned. Theanalog canputerts notcomposed of perfectelectricalelements. For exalqplx!,the ixlallctorsusedin this studyhsxe loss characteristicscorrespondingto a dsmplngfactorof dbout g = 0.01. Trallsfamlersalso~ slgnlficsatlosses. Theelectricalanalogof the alrplmes studiedin this reporthad am electri-cal danpingcorrespmdingto a structuraldmnpingbetween g = 0.02 dg = 0.03. !l!hlsIs not greatly different fran the damping to be found Inconvmtional aircraft constructla, so no corrections were made for thisinternal as@ng.

No general statement can be made about the effect of remdan ccanputerermxs. &ue gtve rise F Iytoanemor intheaamplng factor ofcheroots, lllwhlch the g - v) Clime la Shttted verti~. other ~Egive rise basically to an error in velocl~, in which the curve Is prima-rily Shlftea horlltontally. Slncethe sl.opeof thecurveofg againstvIs ~ no mesus comtant, it becanes @ossible to give a Specific figurefor accuraq of flutter speed. Ih unusual csaes, where roots are of the~eshown in figure 12, flxttermsy bepredictedwheninfactitwlllnot occurfcm this root at all. Thisdistinction,whichmust be mademathemstlcally,is of no importancein practic-al.cases. An sdrplanewhichshuusa flutterdaqplngof g = 0.01 and is therefae theoreticallystdble -is not to be regardedas srwmore satlsfac~ or usefulthan onewhich

.-

HAcATN37ao

alma a damping g = -0.01 ad would thereforeManufaduring tolerances azd the safe~ factorswill not permit use of an ticraft unlessit is

35

Xetically flyqart.requiredIn aircraftmoderatelystablefor

a signifi&ut vemLation in all.structural parmneters. It-ls bellevedthat,exclusiveof errorsintroduced~ the finltedifferencestructurea?laappraxtmlXtionsin * ~c theory,the resultsobtainedinthis studyhsm a probdbleerrorin da@ng factorof ebowt g = tO.02- a probebleerrorin flutterspeedof dmut 2 percent,WhicheverIseppl.icdblein the lightof the dbovediscussion.However,trendsobtainedby variationof structuralparmeters are cwsidemiblymore accuratethenthiswmzld i@y, since~ errorwouldp~sist with roughlythe samevalue in all caseshvulwlng suchpammeter variations.

For cases1 to ml, it is possibleto constructcurvesof ge v ~-~hutbe=-a pertof this report.

curvesserveto shuwthe steepnesswith uhfchthe rootspassthroughflutteraud give saue indicationof the accur~ of the flutterspeed. This situationdoes not existfor ce8es181 to 289. l&comentrated+nassvartaticme,cauputatlcmswere carriedout in such a~ that ~ flutterspeedsend frequencywere obtained. Consegpently,it Is not possibleto dellneateareaswhich are “sefe”from the standpointof fI.Utter.It is known,for exen@e, thatwhere a longpendentlobeisobserved,as for case 235 In figure18(d),the ~tem is harem unetebleev~ withinthis lobe. Thereare Edmilsrregionselsewhere,f=-la cue 212,which is eatisymmetric(fig.20(b)),wherethe systemis barelyunstdblewithinan ellliptical-shapedbcnudary. Similarly,case21h is barelystableIn this regionand yet no flutterroot ia evenshownsincethe system.does not beccuneactuallyunstdbleat arw point.

Theseremarks=e notmade to showthe fluttercurvesto be value-less,but to cautionthe readeramdnst maldu Inferencesnot containedin tiereportend not legitimate~~orted

suMMhRYoF~

~ the datapresentedhere.

certain trendsin fluttercharacteristicsfor typical.modern elr-craftseemto be indicated~ this studyof the Incaupresstblefluttercharacteristicsof Sdrcraftwings. tiscmecases afewded.ations arefOulla.It.is probeMe that if more extremeaircraftdesignswere con-sidered,evenmm?e wouldbe observed. Nevertheless,this Sumluu?ygserveas a useful@de.

(1) & the folhwing tablesre llstedthe averagechangesin flutterspeedfor a l-percentchangein eachparsmeter,the changebeingmadefrdm the basicvalueexceptfor center-of+naessad elastic-axislocatlon,in which casesseverallocationsare assumed.

.-— —.— — .. . -. — —.—. — .—. .—

..—. .—--- .

36 I?ACAm 3780

- In parsmeterA~e chamgein

flutterspeed,percent

-ease wingmass 1 percent olhcreasewing pitchingInertia1 percent -.1-ease bendingrigldi~ 1 percent oIncreeaetorsionalrlgidl~ 1 percent .5Increase*chord lpercemt -.7calterofmassfomerdl percentofchoR-afrau-40-percentlmation 3.0X-percentl.ocatim 1.760-percentlocation .8

mastic &xlsfommrd 1 percentof chordfzwl-40-percentloctilon 3.2x-percent location 1.8

.

(2)m-ad cham3ein torsionalrlgiditiis most effectivein

the wing, torsional rigldi.~eitheri?iboardor outboti of the mass till~ ~ ~~ ~ee% *- on the type of flutterexlstlng.!lheeffectof sueepbsckwas not observedto hwe a systematiceffect.Pitchbg flexlblli@ of the concezrtratehuass~ort has a definiteinfluenceon flutterspeed. In msny casesa tuningeffectwas observed,with a IQ- to 40-percentdecreaseIn flutterspeed. This effectwas not=- observed;in S- C~8S, the ~ speedwas stgnlflcantlytncreased.

(3)per~s ~ ~st ~~at~ r-its will be foundin the effectsof a concentrate&massbcatlon. Forawlde rangeofmassandradlus-of-gyrationvaluesthe resultswere very ~tematic. For fighter-@peplanesIt was foundthat:

(a)Aft chordwlsepositionsera usuallyundesirxible.

(b)The 30- to 50-p&rcent-spanend tip locationsare gm~Udes-le.

(c)A forwaxdlocatiomnear the 70- to &)-percent-spanpositlcmwill, in general, greatly increase flutter speed..

For bcmiber-~e planes these results =e saue!whatmodlf’led:

(a) Aft chordwise positions axe usualdy Undeslrsble.

(b) ~ tip location Is genemlly Undeslrdble.

mMm!m 3780 37

(c)With few exceptions,any locationforwardof & elastic=sendbetweenroot and O.gO-spanpositionIs satisfactory,althoughflutterspeedis r=ely greaterthanbare-ulngflutterspeedby ~ siguificsntSmount.

Thereis perhapsno need to rmk that theseresultscanbe alteredby introductionof a flexibill@ in the concentrate&masssupport.

Californialimtituteof TaChuology,Pasadena,CUf ., Ju 6, 1955.

.—. —.. —- -— -— . . ——. —----- --

38 HAcAm 3780

1. MacNeal,R. H., Mc&mn, G. D., endWilts,C. H.: The SokutlonofAeroelasticProblemsby Means of ElectricalAnalogles. Jour.Aero.Sci., ml. 18, m. 12, Dec. Wl, pp. 777”78g.

2. Benscoterj St&My U., and Heal, RlcherdH.: IntroductiontoElectrical.-CircuitAnalogiesfor Be- Analysis. NACA m q85, 1952.

3. SeuaU, JohnL.: ~ erimentalazdkldJttCd. hvesti@tion Of Flutterof a NonuniformSweptbackCantilswerWing With Two Conc~atedWetghts. lWdX W ulEOga, 1951.

4. ~elson,HerbertC., and Taassonl, JohnE.: lhrp~tal Emestigationof theEffectsof Sweepbackon theFlutK of a UniformCantileverWingWitha Vari*ly IacatedConcentratedMass. NACARM ~, *9.

5. m, J. G., ~, H. J., ~ tic% 1.~=: *u@ of ~ectgof sweepa theFlutta of Cantlti Wlnge. mm m 2121,lg50.

6. Scanlan, RobertH., ti Ros=baum, Robert: Introductionto the Studyof Alrcraf%Vibrationand Flutter. The MacmillanCo., 1951.

7. McCann,G. D., andBrahsm,H. S.: AStudyof the Accuracyof InmpedPsmmeter and AnalogCauputerRepresentationsof CantileveredBeamsUhderConditionsof StaticStressd I@muic Vibrations.Rep.Ho. 3, ContractAF 18(600)-6@,OffIceof Scl.Res.andC.I.T., 1955.

8. Golend,MxrW.n,andLuke,Y. L.: The Flutterof a UniformWing With!ELpWeights. Jour.@pi. uch. , vol. 15,no. 1,W. 1948,pp. 1.3-20.

9. Rmyan, HerrYL.,andWatkins,~les E.: Flutterof aUnlformWingWithanArbitrarilyPlacedMass Accordingto a Dlfferentlal-EquationAnalysisend a CcuuparisonWith Exp~t. HA(2ARep. 966,195L(8Uper8ede8HMA !lW1.&8,1949.)

10. wilts, c. E : Finite DifferenceErrorsin the Flutter~eed of hUniformWing With an lu%itrarllyPlacedMess. kalysls Mb. Rep.,C.I.T., June 19.

Il. Runyan,HarryL., and Sewall,JohnL.: Experlimmtel~vestigationof the EE’fectsof ConcentratedWeightson FlutterCharacteristicsof a StraightCantileverWing. HA(X m 1594,19M.

.

39

CW A PINNED-P- AIRFOIL

.

yf&-w&*b” &””-””””. . . . . . . .

ifaimiper&it) l&th, m, lb-6ec2/sq&ertia per unit length,1, lb-sec2

Bendingrigidl~, EI, lh-ti.2. .

!hxwionalrigidi&, (%T,lb-in.2.x~, in. . . . . . . . . . . . .

x3,1n. . . . . . . . . . . . .

Alrdensity,p, lb-see/%# . .

.

.

.

.

.

. .

in:. .. .. .. .. .. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. .

. .

. .

. .

. .

. .

. .

. .

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.I

. .

. .

. .

. .

. .

. .

. .

. .

. .

.●

✎

✎

✎

✎

✎

✎

✎

.

.

.

.

.

.

.

.

. . . 288

. . . 48

. . 0.010355.176

“1:4L (lCP). 6.87(1$). . . 9.6

. . . -9.6

0.0845(104)

P--PINNED AIRFOIL

Vfs f’f).

Nhmberof cells ~fpm Aff/fcamph CPB

m 692 0 U.7 o

8 688 .Cx% 12.6 .008

h 677 .022 12.3 .032

2 644 .069 I.1.o .L34

..—. ..— —— — -.— ——- ..-

—..—.

40 MACATN 3780

!cABIEIII.—l?EYsIcALmmmmmmx aE’muFm4cAmIrAmm

Halfchord,b, tn....... . . . . .Spa?l,la in. . . . . . . . . . . . . .

Mass per unit lemgth,m, l&sec2/sq in.PitchingInertiaper unit length,lh-sec2

Flexuralrigidl~, EI, (lb)(sq in. ) . . .

*StOlld@@i@, GJ, (lb)(sq in.) . .Elastic-axis position, X0, in. . . . . .

center Of mass position, q, in. ● ● ● ●

. . .

. . .

. . .

. . .

. . .

. . .

. . .

. . .

Mass of concentratedmass,~, lb-sec2/in. . .

Pitch Inertiaof concentratedmass, lbsec2/in.Centerof mass of concenbatedmass,(~)c, in.

Alraensl%Y, P,h!lecWn.& . . . . . . . . .

. . . . . . . ..0 b

. . . . . . . . . . b8

. . . . . . l.~(la

. . . . . . 8.cx)(I.04) -

. . . . . .

1%

1.407 10%. . . . . . 0.692. . . . . . . .. . . . . . . . 0:156

. . . . . . 8.23(10-3). . . . . . . o.ti~. . . . . . . -3.*

. . . . . . 1.1.55(lo-7)

-—-. .

6G NAOATN3780 41

TABIJ$lv.—mmmm!ALANDExPmmmw rmm=~=m

cAmImmRmuI!rEco~ MASS

[Data taken frcanreferences9 and U]

Masslocation,per Ullltspan

o.3.67●=9:%

.354

.625

.938;g -

1.(MO

calculated

=7

II

0.400 25.27.-. ----- -----226 .397 W*Z--- ----- -------- 1 ----- I -----

277

II

.M8 28.*c3&9 .631 g.:;

:~251 24:87--- I-----I

-----205 .360 ~.60

228221

~

260---261al231218

Experimental

perunit(a)

0.401.W.388.41.O.451J@

----.459.442●w.384

ff ,cpB

22.119.117.4ti.315.5

bti.3-26.8

[1ad

21.821.621.4

%0 per unit WIQcl@ is d in./Becor ~ @l.

b!l!his~erhnental rectrd seem to shownearlyslmultaneowdlver-geme d flutter at * frequencies.

cC!alcul.ateddivergence speed is dmut ~ mph. However,a flntterspeedcan stillbe calculatedmathematically.

aMvergencewas observedexQerlmentally.

.

-.. —..————- . . . —-—— ..—- .-.

42 mcA TN3780

.

.

—

NAcAm 3780

mm VI.- PHrsJEALCHAmcmm!mcs cmSwEwMcKmm

~t. taken frm reference3. Measof wing is for portion outboard ofroot restretnt. More detailed infonuation will be found In refer-ence 3. Ikd%Lfor concentrated masses are not given explicitly inreference 3 andnut be regar&d as only epprete ~ -

Wing Chez’acterlstics:

%pEUl,ti.. . . . . . . . . . . . . . . ● . . . . . . . . . 48.3%oOthalfti, ~,ti. . . . . . . . . . . . . . . . . . 5.2%tpwchoml, >, in.... . . . . . . . . . . . . ...2.36Wingtotal ma8s,~,lh-sec2/in. . . . . . . . . . . . . . .O.00784

!hnnelfluiddensi&, p, lh-t3ec2/in.4. . . . . . . . . . 3.40 (10-7)Sweepbacken@e,A,~ . . . . . . . . . . . . . . . . ...34.5

Concentrated+nassclmactertstics: moard Chrtboerd

Mas6,~,lh-sec2/in . . . . . . . . . . . . ...0.00806 0. ook52Pitch Inertiaaboutelastlcreds,lh-sec2/in. . . 0.~12 o.olg2Per unit spanwisepoaltion(fromroot) . . . . . . 0.30 0.78center of massposition,(X@c, in. . . . . . . . -1.74 O.yl

ua Measured along elastic axis.b MeaauredperpeMicularto elastlcaxis.

.. .- -... - -.

#

-— ——. — ——. ——— — —-— ——— —— - --- -- -

Eu31=vII.— mmmEmALAlmomFm!ED Fmm!l!Rmmmmmme

cwmimE4cK wlmuImoommmwJ!m MAsa

Rmnal mde fYegpenoleei, aps, at -

!Qpe cd freqpanay

MO&l Mcda2 mcda3

ExQel’hental -1 fraqllenoy 30.9 37.9Meaeurea EllklQgfreqlency 2:2 32.6 3g.1

L

~t- -=kiStiCB

9ypaofreEluNvfJ &*pfe ‘fJmph apB

wlnMamIlel Z’eeuki, Vfe W3 o 20.1

Ad.w Imulta, nmmel-eaqpOMIlt=IhDdhlabg remllte, drstreem mthlxla

-.U?

Ane108I’eaulta, ahdream nmlllOabL? -*W &

“ 1~ -.U. 24.0

% Coefflchlt c~ = 2s .

%ft meHlcient (& = 2X cm A.

ki

MAcAm 3780

MBIJ$vlII.- ~TIC8 CiFEMRI FIGH!EERA

(EL)Plqn3ical chlm3cteriErtlca

WM@X@Kaugle,A,deg..... . . . . . . . . . . ..= . ...06@spanof~,az, in... . . . . . . . . . . . . . . . . . .238Cell. size for finite-difference structure,&, h. . . . . . . . 54Rootchord,b>,i n........ . . . . . . . . . . . . . . ..ti

Tipckn’d,b ~, &l....... . . ..s 0 . . .0..0.... 53!l?aperrati d . . . . . . . . . . . . . . . . . . . . . . ...2.%Aapectratio . . . . . . . . . . . . . . . . . . . . . . . . . . .Wingelssticexis,percentchord. . . . . . . . . . . . . . . . . 40Willgcenterofmam,perca chord . . . . ● ● . . ● . ● . . . . . 40TotalWiIlgmasseldemal of ftuelage,qf, l&sec2/in. . . . . . K).7

Fuselage mass,mf,Ib-sec2/in. . . ..; . . . . . . . . . . . ..2lFuselsgeradiusofgyration,pitch,cin. . . . . . . . . . . ..UX1Fu6elagerediu90f’gyration,rolJ.,In. . . . . . . . . . . . ...25Fuselagecenterofmass E&tofel.astic-, In. . . . . . . . . 0!lkilcenterofpresauretiof elasticaxis,in. . . . . . . . . .!hilarea, sh in...... . . . . . . . . . . . . . . . . ..3.000

Airdensi~, p,lb-sec2/in.4. . . . . . . . . . . . . . . 1.l&6(lo-7) -

(b) tiia end Etiffne6svalue8lumpedfor finite-differences&ucture

station . . . . . . . . . .

Pertit ~=......Halfchord,bb, fi. . . .Iumpedmess,3s,Ib-sec2/in.

Ilmpedpitch In&X:::

1

o.21k47.3

1.73695

2 3 4 5 6

0.357 0.500 0.643 0.786 o.g28k3.5 39.8 36.0 32.2 28.4

la Oig 03.6 o% 0.2353

J‘1(W (dy/kI) . . . . 19.8 61.3 IJ2 2@ 415 773 ---

J‘N (dy’/CW). . . . 5693 145 ag 3’65 ~ 1,260 ~ “

%kamred along elasticaxis.%asured perpendicularto elasticaxis.

cAboutelasticads.

%~ss valuesare 1- betweenmaas Statiol’la.

..— .-— — .— —— ---——- - - - --

HAC!ATIV3780

(a) ~lcal chsxacteristicO

fhnmbti a@e, A, deg...... . . . . . . . . . . . . . . ..3osemlEJpanofwblg,az, ln. . . . . . . . . . . . . . . . . . . . . @Cell she for flnlte-difference AxUcture, ~, in. . . . . . . . . slkmtchord,b ~,ln . . . . . . . . . . . . . . . . . . . . . . .. I.06Tipchord,bq, in. . . . . ● . . . . . . . . . . ● . . . . . . . 53!I!aperratio . . . . . . . . . . . . . . . . . . . . . . . . ...2.00Wingelsstlc sx18,percentcMxd . . . . . . . . . . . . . . . . . 40Wingcederofmass,percent chord . . . . . . . . . . . . . . . . 4.0Totalulngmass externaloffuselage,-f, lb-se&/in. . . . . . lh=06

Fuselsgemass,mf,lh-sec2/in. . . . . . . . . . . . . . . . . . . 21

Fuselageradiusofggration,pltch,cin. . . . . . . . . . ...100Fuselageradlus ofggration, roll, in. . . . . . . . . . . . . . . 25Fuselage center ofmass aftof elastic exts, ln. . . . . . . . . 0Tail center of pressure aft of elastic axis, in. . . . . . . . . @Tailarea, sqln . . . . . . . . . . . . . . . . . . . . . . . .,

?7Mrdensi@, p, D-sec2/in.4 . . . . . . . . . . . . . ..l.U6 10-7

(b) Inertia and stiffness values ImQPed for finite-difference structure

Statio n . . . . . . . . . 11

Perunlt spall,a. . . . . .

~choa’d,bbzti. . . .hnqpediuass,E,lb-sec2/in.

hqed pitch in&tla= 1 ; 1

2

).35743.5

1.43577

3

0.2U47.3

1. 2628 m

Jb‘l@” (dyEI) . . . . . 33.0 100 156 221 ~ 447 ----

Jado (dy/w) . . . . . n.4, 193 283 “551 400 436 224

i

)“50039.8

lg

#

4

L643%.0

l.ti4U.

516

~.mO“*52:2 28.4

O.* 0.66310 204

%msured alongelastic axis.%eamredperpendicular to elastlc _.

cAbout elastlc axis.

%ffness values are lmped between mass staticms.

.

NAcAm 3780 47

TABLEx.- CHARAc!cmmmsOP’BASICRXEERA

(a) P@sical characteristics

. SueePlmckan@e,A,deK. . . . . ● . . . . . . . . . . . . . ...0semispa?loflnlng,*2,1n. . . 0 . . . . . . . . . . ..=. .’ ..845K?ell sizefor finite-differencestructure,~, in. . . . S . . . 130

Root chord,b>, in....... . . . . . . . . ..= . . . ...200

Tipchora,%#in...... . . . . . = = = ===.==.== .80l!sperratio. . . . . . . .. m.. . . . . . . . . . . . . ...2.50Aspect ratio . . . . . . . . . . . . . ..= . . . ..=. == ..12Wingelaetic axis, percent chcrd . . . . . . . . . . . . . . . . . 40Wingcenterofmass,percentchora . . . . . . . ..=. .=.=.40Totalwingmassexternalof me-e, qf, ~-sec2/fi0 ● . ● . . 39.7Fuselagemass,mf, lb-sec2/in.. . . . . . . . . . . . . . . . . . 120Fusel.ageradiusofggration-pitch,c in. . . ..= . . . . ...240Put3elageradiusof gyration-roll, in. . . . . . . . . . . ..wpX’usel.egecenterofmass aftof elasticaxis,in. . . . . . . . . 0Tailcenter ofpressureeftof elesticaxis,in. . . . . . ...700!l?dlarea,sqti . . . . . . . . . . . . ...=...==.= 20,0(X)

Air densi~, p, Ib-sec2/in.4. . . . . . . . . . . . . . . 1.lk6(lr~)

(b) Ihertia d 6tiffnesa values lmped far flmite-difference structure

3tatio11. . . . . . ...*

PaumspaIl,* . . . . . .EaMWord,bb,in. . . .tllmpeamass,m,lb4ec2/in.

Mqped pitchtia” I 1 I

1

O-Q90.8

6.2843,300

tt

2 3b

0.3080.k610.615

81.5 72.3 63.1

5

~“7695399

o.g8

-46).ga44.6I

; ~~mt%010 (dY/kI). . . .

%.cW (ay/GJ) . . . . .=. ● ~ - -

%eaeuredalcmgelasticaxis.%easured peqpendiculerto elastic_.

cAboutelastlcaxis.%tiffhet3f3*USS ~ l-a k-~ -S stati~.

. . . . .. .— —— —— --— .—— — -—— - -

!cABIzXI.- cHARAcmRIm!ms 0FEA8~~B

(a)Physicalcharacteristics

Swe@back an@,A,deg. . . . . . . . . . . . . . . . . . . . .Saullspanofuing,az,l n.. . . . . . . ...*.... . . . .Cell stzefor finite-differencestructure,*, in. . . . . . .Root chord,b&, ti. . . . . . . . . . . . . . . . . . . . . . .

Tlpchord,bbx, in. . . . . . . . . . . . . . . . . . . . . . .

TsQerratio . . . . . . . . . . . . . . . . . . . . . . . . . . .Wingelaatic axls,p~centchord . . . . . . . . . . . . . . . .Wingcenter oflllaas,pementChora . . . . . . . . . . . . . . .Totalwlngmass externsloffhselage,-f, lbsec2/in. . . . . .

Fuselagemass,q,lh43e&/in. . . . . . . . . . . . . . . . .

Fuselagersdiusof-ation, pltch,cln. . . . . . . . . . . .Phselageradiusofgyration,roll,in. . . . . . . . . . . . .Ehselsgecenterofmassaft ofelastfcaxls, in. . . . . . . .Tailcenter ofpressureaft ofelastlcsxls, ln. . . . . . . . .!l!allarea,sqin. . . . . . . . . . . . . . . . . . . . . . . 20,000

tidensi~, p, D-sec2/in.4 . . . . . . . . . . . . . . . 1.1h6(10-7)

(b) Inertiaand stiffnessvaluesluqad for finitedtffemencestructure

station . . . . . . . . . . 1 2 3 4 ~ 6

Pertitspan,a . . . . . . 0.I.540.308 0.461.0.615 o.~ o.~EaHCho?.fl,bb,in. . . . I 77=3169.61 6L91 54=2 =5 38.8

IIIaImpaamass, iii, I I I Ilb-seG/ii. . . . . . ●13%13%%122%3d2!%12;4JImmed pitch Inertia . . .

J‘%01°(ayjm). . . . .

1%w (dy/QJ) . . . . .

I

7.12 M.8

6.31 15.5

y..o

22.6

5499

42.2

91.6

77.8

%easuredalongelastic axis.

%easured perpendicular to elastic axk.

cAtmut elastlc ~.

%tiHnass values are luqped between mass stations.

L41.o

llo.d

--

60

.

.

E=-1b.e Pi

— —

.

l’UmAm 3780

TABLEXIII.- Commmmmww cEARAcTmlmIcs Am IocATmlw

(a)Clm?acterlstlcs

Etghter Baliber

Badcmaas, lb-sec2/in. . . . . . . 2.68 15.0

Pitchradiusof gyratlon,ain. . . . 30 3515 17.56 7

Roll radl~ of gyration,in. . . . 15 ----0 ----

Basic apezxwiaeposition,bti.... ---- 390

Baaic chord.se position,cin. . . . ---- 60

a~out elasticada; alrstreamcoordlnatea.

boutboardfhmn centerlinemeasuredalongelaaticaxis.

%&ward of elaatic-s, parallelto airstreem.

(b) Iacati&

Iacations of concentratedmass,per unit span,aat station-MrpUule

1 2 3 b k.5 5 5.5 6 Tip

~ O=a 0.3% 0.500 0.643 ---- o.~ — o.g2g 1.(M

Bcmiber .W .308 .461 .615 0.692 .- 0.846 .= 1.00

%lErtancesare measuredIn per unit spanalmg elasticaxia.

.

I

.

I

I

I

1!El

8

%m~flr-B.wF

I

52 MMA!m 3780

&n MrQlmm

1.*l.go

kc1.94l.p1.762.VMO1.43

l.e

1.09

~~

2:281.1.81.812.491.491.39

1.89

:Z1.74

kg1.83E.*1.471.sL33I..ue.lg1.91.38%391.661.641.1.T?1.49Lp1.891.1.z$kq1.413.822.09

?1.17l.~2.1.3

7.45.3m.86.6

:::;.:

6:78.08.I.

;:$7.4

z.1

8:?

!;

:::IL28.7~;

is8.89.38.98.4IQ.28.0

1,o171,a*

1,*l,ol~l,m71,2Mmg

w1,227mW7

1,392

El,o@31,131847

l,om1,=*

L,*1,313801

2,0971.1#2l,olyl,3q*

l,W*

Vf,~ milt

L.gi?

1.91.881.782.o1l.gl1.972.411.36L*1.121.2

Lfl1.s1.3g

La1.%1.s2.231.381.40

1. Eb1.63L&l1.79I..’p2.lg1.461.?

::22.161.431.282.h51.X1.41l.m1.991.49

x1.1.~:.#

3:622.011.7’92.301.*2.mM

qBw

8.76.I3IL3

::;

EIQ.2

i::M.6

1%7.98.27.1

17.217.7

J..1 .5

17.0V*9

Zy4.8

z::7.4

z.2.8

;:;6.56.1

i.3●5

i.5

i;E1.411.g

lx

::;M.gIJ..213.8KL613.812.oIJ..2

2::12.4U.4

.

KAcAm 3780 53

Airplmm

1.91

e.lo

1.382.03

1.70

l.oe

wL*1.12

1:%

:%2.622.26

f f #m9.47.512.E8J?lo.e

2

H

~

17.9

1~.?

8.6m.glkyg:.

M14.218.3

l’f.~g.k7.8U.48.0m.89.69.6u.87.89.58.9

::

:!9.911..1loglo.slag8.91o.19.28.19.38.812.5Io.l?IL77.1

I.ojil.~49.80.93

:$1.36L’@

9.8

L;

GID.210.6~;

9i8.9

KJ

x?.68.8lL36.6

.

.-. .— . —.. - —-— — ---- -— —.. - —

9+ HM2Am 3780

1,s3l,=

1,*

1,*1,=L,53hl,-l,loamm

F1,L’@

:El,3@8lJml,m1,-1,3641,2611,1,z1,0231,261l,lca1,*l,=l,lm1,%

L*O741,023

E

s1,114

tg1,*L,*l,lgg

?%

ljgg

l,oho1,*1,080WUl,=1,ooo1,-983

l,t#ol,cmm

2.*l.%

E:2.002.=2.IJ.2.112.38

w1.61.44L@2.lgLgo1.982.121.762.161.1.~1.1.~

f+m

1,*71,239

l#4w1**

?%7l,Wbm

%%l,hn1#4091,333

%1,699

i%1,1881,0801,*l,=1,21M1,1JL,*l,23g1,317L.03k

kg898

1,=m

1,6631,216l#k!ll,Elk5l,rfo1,1421,*l,=1,*1,449

1,%g

1,438l,m1,*7

1-1,3861,6!391,1881,3131,2161,083. . I

~f#pm?tit

;:3Mm2.262.37%911.831.1.1%1.362.602.482.332.642.602.442.992.122.262.09

k:2.182.IQ2.62.262.232.182.671.82l.ggla?1.1.z1.60

::32.302.lh2.ui2.a2.=ala2.a2.33

x1.691A4

xi?.~2.

22.22.992.092.312.l&1.91

?--IIII

~● I

II

L-

.

,

-.2

-.1

90

.1

.2

Real

\ I .

F-

1 I

Z!ralsient response correspon&bg to mute of

/

\ .

\

o 200 400Velocity, m p h

Vmtlon of g Withvelocityfor

600

fllxtter

.- .—. .—— —.. - -.—. —— .—— ——--- - -

36 w m 3780

F42.4”

108”L

L

/0

/

I

—- 4--I

—1-- -

II

I—.—

I

II

t

--l34”

I I II

I

I I I

I

I

238”

(a) Fight; k

21.2”

* II

1

Figure 3.- Plan fonua and cell divisions of basic d.rplane

57

tJ--tio”

200”

1

.

I

t

—.

—.-

1

i

—-

65”

\\

\ ‘,’h

I\—/—‘<

\1=

165’

I

I 6I I II—-2-!1. 1

32”JI II

#

-&-*- 1

I I - + 80”I 1 I

l.- 740”

(d)Bc#iberB.

Figure3.-(%mMded.

.

. .. ---- —.- .—. — ——-— .- .— -- . .

KAcAm 3780

Iq-JIF.06 ‘--

.04 /{

/.02 “

/● A

0 +.

INo. of cells

Flmre 4.-Finite-difference

.

)

i.

errom of pixmed-plnuedbeam.

NAcAm 3780

10 ~ TheoreticalG -- Wind tunnel

>*

(a)

- 5*-

1 &4u_3=l-+

.4[ ~ -_ &

.305 1.0

Spanwise position of mass

59

Theoreticaltad Uin+-humal flutter characteristics.

m.rtter characteristics of Uniformcantileverwing with con-centratedmass●

- . ——.--—. ..— —— ----- --

60 w m 37ao

30

10, UU81

O Theoretical.

o

(b)

.5Spanwise position of

I.0mass

EulEiLog.

t)100 +f

.

10

\ 6 Ceilso“ — ~ d t

)

-lo

1“

.5Spanwise position of mass

-6.- ~utter-spe~ errors of finite~erence anaIogCantilever wing with CoulcentrateamEss .

10

of unifonu

-a . ..— —— ..— — . — ——. - - -

0

I

14XIO+0

—m.— 1

\

0W unit span

II

1.

.

70 XIO-10 9/——

60 “ E! /—— —

6’J /50 I #, I

/II / ‘

40 .

/1. .

/ 730

—I /

/

20 .

L&I10 -1

A /

L 1 I I

o .5 1.Per unit span

.

)

(a) BaBlc fighter A.

-e 7=- MIa ti StltYueaa properties of wing.

NACATN3780 63

.06

.05 9

MId .04.-N“$03~E“.02

.01

0

.16

14

N‘gin

12

.

-la 0

4

2

* I I 1

.5 1.0Per unit span

x I o-1oI I

I——~1I. — ——

GJ

0 .5 I .0Per unitspan

(b)~SiC i’i@er B.

-79- continued.

.

. . .—-. — —.. — ——.. — -—--— -

64 NAcAm 3780

120“

100 -N

NO80 - ;=m~ ~o .;a aH 40 - i

20 -

0 -;

I

.6X 10AO

.06\ . I

.05 —m——. I

.04 .

.03 \ \L: .\

.02 —- — — — — -b - — — —\

~ yl

.01—

0 .d I .UPer unit span

(c) B8slc

=e 7.-

Per unit span

bwiberA.

continued.

G NAcATlv 3780 65

60 “

50 “

40 “

%Tg

H“ 20 -

10“

o-

.06—m----

h - 1.05 “ t

Ye .04.-Cu” -,

I.6XIO-lo

ty

“ .5-

# .4 .g

.3 -

% .2 -uc

.1-

Go -

II , m ——

I 1 1 I 1 r- 1 1 1

J I I I I I I I I I I

01

.5 If)l% unit span

II

o .5 1.0Per unit span

(d)~SiC bwiberB.,

-7. - concluded.

.-. —. —-. — ..— . — — . ..— —— —- — .—— -.

1.1 1.1

/ ~~ ~

. — Fighter Aq I .0 ~ Y y + -- - h

*=otl IQ

,9 / ‘,91

ICI 1,1

-. -- . iVf I .0 ‘

Fighter B, o—//

\ A=o” “.9 “w ,~ .9

1.1 1,1

Fighter B,0

A= 45°

.9 .9

.5 1.0 2.0Wing mace denelty

(a)

I%WO 8.- meet ~

I I Ia.

— Syrnrhetri!

—- -Antlsymmetrlc

. .\ - i

\

& -

\

d iL L —

I

.5 1.0 2.0Wing pitching inertia

.

.

Wing mass deneity

I t I I ISymmetric

1.1m

\ — — — Antisymmetric

A=O” ‘“0 Ar -- A -— .-

Y h

.9i

).\

1.1

A=& 1.0

Cone, mass .9>

1.1 ‘. .,

\

A=30” ‘.OCone.mm .9 .

.0

.7 .; 11.0 1,5 2.0

Whg pitching inertia

(b) BcmQ)er A.

_ 8.- crmtinued. s’

— Symrnd?ic–—— Antisymmetrk

1.1

Vf Lo m -- - . -e -

- 1

.9

1.1 q

.9

1.1 a.

Vf Lo ~ .- b q —- ‘—— 1 h

9 “ I— .

T.5 Lo 2.0

Wing mass density

A=OO

A=300

As 45°

H

& ~I.0 i )---

.9

1.1 ‘-

- + I.9

1.1

Lo k - - + e

.9 I.5 1.0 20

Wing pltchlng tnertla

.

I

i

i

II

1.1

Vf Lo * \ ~ — —. - t“

.9\ h

1.1k

Vf Lo *..

Ib _ _. - .. _ -II

.9

LI‘

Vf I .0 -r 7L— t)

.9. ;Lo “2.0

Wing mase der@ty

(d)

A=OO

A =3OO

A =45°

— Symnetric––– AntieynmwMc

Wing pitching inertia

I .3J

I.2Vf

1.1

.9

#/-

1,1Vf1.0

.9 P/ / “

Pf

///

1.1

I,0<Vf

.9

.8)

.7 1 t.5 1.0 1.5

Per unit torsionol rigidity

Fighter A

A’ O’”

FighterB

A=o”

fighfer BA ❑ 45”

— Syrrmwtric

— — — Antisymmetric

1,1

I .0- =f= - ~ - -

.9

1,1

I .0m

.9

1.1*

I.0 * - b - A-

.9 I I

.5 Lo 1.5

Per unit bending rigidity

(8) Rl@ers A md B.

m 9.- Flti* t3pemdan a functicmof U1.llgrigidity.

1!El

1.3

1.2

1.1Vf

10

P/

1.1 ‘( /

1.0Vf

.9 {~/ ,d

7/ /1.1 /

/

Lo , 4

Vf.9

.8P

.7 .; I

R% unit tw%al rlgid#

1.1

BiEE1s

If)A. ~

Cane, mass .9

.8

.7-5I .0 1.5

‘% unit banding rlgldlty

Ixnlcent?.uted MS m *.

mlnlled.

1!la

38

I

I

1.3

1.2 ‘

Lo

,9

I df

1,1 k& N

“f 1.0

.9

1.1 ‘

‘f 1.0

.9

1.

Per unit toreional rigidity

A= 0°

A= 30°

A=450

— Symmetric

(c) Bdber B} G uln&

m 9.- Ccatlnuea.

— —— Antieymmetric

1,1

1.0— -j= - +- –. .~

.9 ‘

1.1

In - ~ +

,9

l.l

.9 .J

.8m I5 1.0 (.5Per unit bending rigidity

3

.

*

i

Ii

I

{

1.3

I .2d

4~

1.1p y

Vf I .0

.9

/$

P

1.1//

Vf I ,0

,9 ‘

{

1.1 0’ <

Vf I .0

.9

.8 + I

I .0 I .5

Per unit torsional rigidity

&00

A=300

Az~o

— Symmetric

— — Antieymmetrlc

1.1

I .0 . ~ ~ ~%.

.9

1,1

I .0

.9

1.1

lb

1.o- W

.9 ‘

,8 ,;I

I ,0 I ,5

Per unit bending rigidity ,

iiI

I

‘Wf

()()AGJ Ay——GJ 1

1.0

.5

0

\

r“ —-- --- “-

I I

r- - L- -—a ‘“— Symmetrici — — — —(Gd—— . –- -- Antisymmetric.

I —-

.i0 I

1-.1 \

‘(m

(a) FighterA.

1.0’

r - ‘-II

Ir-- “ “1

Ia

*~flf .5 - (GJ)--

(%)&)

0.—

7 *.1 r I 11

* 1 I

Sponwise position

(b)E@hter B.

X 10.- Effectof localstiffnessvariationon flutterspeed.

. .

Lo

.5 ‘

Av.f ,

()()AGJ Ay O-__T

I I I

‘ (q ‘-.5 1

1.5

1.0

AVf.5

()()

AGJ Ay10zT—

-1.0

(c) Bcaiber Aj bara ~.

I 1 I I Iz

_Concentrated massat 0.46 span

I(GJ)-

1

I

. (EI )~ 1 k <

.

I.5 I.0

Sponwise position

(d)BcaiberA; ccacentratedmass at 0.k6 span.

Fi@re 10.- Concluded.

.—. — ..———. . ——- -—.

76 NMA m 3780

1.0

.8

Vf

.6

1.6

1.4Vf

I.2

1.0

.8

1 u I s

Fighter A I Fighter B

A=OO A=OO

1. I

\1\

\ — Symmetric---- Antisymmetric

A= 45°

~

*

60 30 40 50 60Ps%nt et% PerLent chord

.

(a) Fighters A and B.

Figure111.- Flutter speed as function of center-of-mass position in per-cent chord.

-.

H- — Symmetric

—— — Antisymmetric4

.

.

I.0 \. I

\

\) \.8 \

u9- t‘r

\

.6 \ ,\J

1.0”{

I

\

.8 .

\

q

\

) \.6 \

\1 I I 1 I 1%4-1

—Ii iii.s$ I‘“”FFI$+-W’I.8 1 I 1 I I

1“\ i

“ H+F’ld+Fi.6 1 \ I

30 40 50 60Percent chord

(b)BcmtmrA.

FigureU..- ccmtlnued.

A= 0°

..

A= 00

Cone. mass,

A= 30°

Cone. mass

.---- --- .-. ..— — .- ------ .—— -— . .. . . .

78

1.1

I .0

Vf .9

.8

.7

1.[

I .0Vf

.9

.8

1.1

I.0Vf

.9

.8

HMMTN 3780

— Symmetric

— — Antisymmetric

40 50 60Percent chord

1.1

I .0

.9

.8

.7

1.1

I .0

.9

.8

1.1

I .0

.9

.8

\)d b

\\\, ~\

\’

h

A=300COnc. mass

Cone. mass

I

40 50 60Percent chord

(c) Bdber B.

- ~“- ~ncl-*

9

.

9

.&

o

. .

-.2. i

‘o Cof M40~Casel❑ Cof M45%Casell

-.4” k 1 0 C of M 50% Case 12* A c of M 60% Case IS

.8 1.0 1.2 1.4 “ 1.6 1.8 2.0

.2 ‘ /

1?

o ‘ I

/

o C of M 40% Case 72’-.2 0 C of M 50% Case 82

A C of M 60% Case 83

.8 [.0 [.2 1.4 L6 1.8 2.0v

.

IRLgura12.- Plot of daqpingfatior g a@nfi =loti@ f= s- unufmlfluttar roots.

.-. .— —— - . . ——. —-.

80 NMxim 37a0

I

Vf

I

I

Vf

I

Vf

.2 - I IFighter A

.0\\

.8

— Symmetric

——— Antkymmetric

Bomber A—A=o”

.0{\\

.8 ●

\\

\9.

) ,i\

.8 - Bombe~ A \ \

A=o” \ \-Cone. mass \

1 \

30 40 50Percent chord

I

I

I

Vf

I

I

I

.2 \ I I

Fighter B

\ ~ A=o”.0

\

.8‘) 1

.2 >~, “Fighter B—

\ A745°m

.0

.8 ‘

\ A=300.0

%

.8 k

.0

\

.8 – - Bomber BA=300

— Cone. mass”

I30 40 50

Percent chord

.

_ I-3.- Flutter weed as a function of elastic-axisposition In per-cent chord.

m

I.4

Vf 1.2

I .0

.8

I .2

“f I .0

..8

Per unit basic chord length

Symmetric

Antisymmetric

1.4’

Ay

I.2 Banber B .

‘ $ k 30°#

I .0 “

.8 4\ %

1.2‘ ~ Bomber B

A= 30° —Cane. mass

1.0” \

b.8

.5 I .0 1.5Per unit basic chord length

1!iiii?

RuurO 14.-rlutter lmeea ao functiom of Clxml length.

82

—.—

w m 37%0

SymmetricAntisymmetric

I .2

Vf I .0

.8

1.6

I .4

I .2

Vf

I .0

.8

I 1-1—Fighter A

,

\

~’

d/

\ /\ /

x/

i

Bomber A

/ Cone. moss —/

/I

\

\ ~~,//

o 25 50Sweepback ,angle, A,deg

_ 15.- Flutterspeed

I .2

I .0

.8

I.2

~f I .0

.8

I .2

I .0

.8

I I I—Fighter B

( ///

\ /\ - ~ +

\/ ‘/

— Bomber BP

— Bomber B

_ Cone. mass f

4\

/“-. ./

o 25 50Sweepback angle, A,deg

as functionof sweepbackangle.

wm 3780 83

..

.

s I I II I — Symmetric

I

Bare wing. .I I I

. .Con:. moss _‘—— Antisymmetric Stat[on 0.46

1.2 I IVf 0~ /

I .0{! -&” Basic / ,x●

..8 ‘

/,

I .2 “ /

Vfmx l/2x

.8

I.2 /a & f

Vf / /

●

I .2 ‘ # / ‘Vf

A‘0 (GJ) X 2/31.0’ L -. -+ ~’ ‘— –- >i) x ;/2” - –- :: ~

/

.8

I .2 /

Vf 01.o~~

(GJ) x 3/2~ -A

(EI) x 2/34 ‘—- -—: r-,1

.8 .

I .2 / A fVf / /

0% . /’I .C1c r

.80 c I25 50 0 25 50

Sweepback angle, A, Sweepback angle, A,degree degree

.

. ... —--— —— ——. —— —- -— -— —.— ..— —. —

1.2- 1 I I I .2 1 I 1 I (

Antleymmetric Bmnber ASymmetric

10I I I / A.ne

1.2 I ,2

10Bomber B

+ “ - A=ou 10- - -

v/v* \ _

.8Vlvfyj ~

.6 .6”

1.2 1.2

J Bomhr BI.0 A=~ ‘o

v/v~.8 vlv~ .8

!2‘

.4 6.81 2 4 68!0 !2 .46 .61 2 4 6810

fn/ff fn/f f

(a) I&am at 0.46 ctpan.

Figure 17.- Flutta ewad ma a =maof ccamntratahuus pitching

.

la

*

i

II

1.2 1 1 I IAntIsymmetric

10v/v~ / -

.8 4

r’

/

I 1,6

I I.4Vlvm

1.2

1.0

I,6

I.4~-vh~

\ ,k.2

1.0” <.2 .4 .6.81 2 4 6 810

1.2Bunk A

I I 18ymmetrlc

A=(P ID

v/va

.8~/“ /

/ “

06

1.2”Bomber BArJ& ,0

.v/va

\

.8 \

.6

1.2

8omber BA=.30° 10. e

v/vm

.8 \

\ {

6

(b) Mm at tip.

.-.2 .4 .6.81 2 4 6 810

f~/f f

1!

m 17. - Concllldad. 8?

86 NllcJlm 3780

1.1

1.(vf/vb

J

s

I -d

Uw%

Lc.[

I.;Vf/Vb i .C

.E

.6

A Case 194

Spanposition, 0.9:

— case 19CA Case.191

6.1

-.2 -J o .1 .2 .3

\ 354

(a)

qw= m- Flutter

Chordwise pos-ition

F&kt~Aj A= 00; Sylmletrlc.

‘– Case 187

tl-A ase 189

.4

characteristics against chordwise posltlonofconcentratedmss .

87

I

/I

~f Vb

1

1‘f/”b

.2

12.0

/.8 ‘ Span position,O.50

.6 10 / o Case 184/ ~ ‘lo A case 185

.2

14.0

K& @ - e.8 ~

Span position, 0.36

.6 o (@se 182A Case [83

1

-.2 -.i o .1 .2 .3 4Chordwise position

(a) Concluded.

Fi@re ti. - Contmlad..

.

.

.----- ———-- .—. .— — .—— .—. —— —- ..

88 mcll m 3780

1.2

I .0‘f\vb

.8

.6

I .4

1.2‘f\vb

I .0

.8

1.4

1.2

‘f~b 1.0

.8

.6

Span position 1.0

1? A ~

P“.

Span position 0.93

/ ‘f o Case 189– Case 190ACase [91 i

Span position 0.79

-.2 -.1 0 .1 .2 .3

Chordwise position

(h) ~ Aj A s OO; ~~_~C.

-18.- C!ontlnued.

rIAcA!my@o

. .

. .

‘f/vb

1.2 “ 1, I 1 # I #“Span position, 0.50-

11.0

.8

A-Case 185.4

.

1.2 Span position, 0.36~

1.0,

) .8 74 d

.6 “ — — — —~ Case 183

-.2 -.1 0 .1 .2 .3

Chordwise”position.

.(b)Concluded. o..

Figure18.- Continued.

————. —.. - .—. .- -

w

1 Hl+--Ll I

I

1~I.f I5 s Oan Positi!m,0.93

/ I mo Case 230+ 6.1~v - Case 231i

/

\l l\\ \

.Spanp&itioq 0U79

r T ~~~ ~1 s

o Case *27

I Id- Case 228

Q- A .Case 229

-.2 -.1 0 .1 .2 .3 4 .6Chordwise position

(C) ~ Aj A = k5°j ~iC. Ml?’,mediumfreqyencym, highfrequency.

m m.- Cbntlnued.

HAcATN~ao 91

.

vf/vb

Vf/Vb

I1

1

v2.4 -, , . ‘

2.2 “ ,-. .

2.0 .- ‘Span position,0.64

1.8s - . “, o Case 224 “ ,.1.6 .,’“ — Case 225 “

A Case 2261.4 :

1.27

1.0 - ~ — I # .

.8

1.8 0 case 221

1.6 “ —Case 222)A Case 223

t.4

1;0

/L— . — — _ _ _ _

1.4 12’

1.o-

-.3 -.2 +,,0 .1‘ Cfiokhvb position

. ..- — .._ .— .

92 NA.cAm 3780

2.0

1.8

1.6

~/~ 1.4

1.2

Lo

.8

2.4

2.2

2.0

1.8

Vf /~ 1.6

I .4

1.2

Lo

.8

Span position, LOO_

A Case 235

.Span position, 0.93-

A Case 232

-.2 -.1 0 .1 .2 .3 .4Chordwise position

NAM m 3780 93

‘f/vb

‘f/ ‘b

2.4

2.2

2.0 ‘ !

1.8 14

1.68

1.4 Span position, 0.79 _o Case 227

12

Lo

.8

2.432

2.2

2.0

1 1/ ‘

/

1.8 18Span posit ion, O.64

1.6

1.4 ‘o Case 224

1.2 — Case 22551 A case 226–

Lo

. e)0.-.3 -.2 -.1 0 .1 .2 .3

Chordwise position

(d) Continued.

FX 18. - continued.

.

. - —.. —— -- . . . .. ——. . .

NAM m 37&

.-.

‘f /“b

.

,..‘1

2.2 I I ! I ISpan position,0.50, .~

2.0 -o Case 221 \ \—Case 222 ,

!51

\

1“8“A Case 223 M

[.6 f

I .4 A I6

P r4

14 —

.8 #

3

I .6- : 1 q Icg 16 Span position, 0.36 tip

II I 0 Case218 X-A.—Case219 I

L I 1 I 1 1 1 1 n n I n n 1 n w I

,~-.? -.2 -.1 0:.1; :2 ; .31.,---- ,., ; Chordwise positioq i - ‘-~ ‘ ,

:,”

%. .’... , !-: I-, - .- -b; 1. . ,

‘., b: ,, l“

1 .[-’ (’d)&n@ded. ! -.’-:-.; -;‘! ~:”1 J .1t

,. m ti.-dont&uetL -”:...” . .

. .,

.

,

* NAM m 37a0

I.2

1.0Vf/Vb

.8

.6

1.2

.6

I .4

1.2‘f/vb

I .0

.8

I .4

1.2Vf jvb

I .0

.8

.6

I

3.7 ~q ~

7- 9 span position0.92

i I

Case262 Case255

“ /- Span.mxition 0.85

lLF# 9~1~ I IL #

I, m

, +

Case 261I ICase 254

ITF

I

/~ 14

I 0.77 I

ICase ~3

-.1 ‘--0 .1 .2 -.1 ‘o J .2Chordwise position

Figure18.- C!mMnued.

-.

HAcA!m~m 97.

vf/Vb

#

vf/%

1.6 m mMass= 0.5 basic mass “

, 1Mass= LO basic mass

1.4 ‘ ~

1.2 sSpan position, 0.69

Lo 4

Case 252.8

1.2- I I 17-Span position,(362 -@- - —

Lo I I1

Cqse 259.8

Case 25 I

.6 – y.’

//

~3m

Span position,0.461.2 IS 1I

81.0 6

1} 2 9.8 >

Case 258 Case 250F5

.6i

10 101.2’

eT

I<q

1.0 -‘8 ;0-

Span position, 0.31 8$ -g“Q.

(lye ~57 6.% / ~ < Case 249/

.6 I I Il.-.1 0 .1 .2 -.1 0 .1 .2

Chordwise position

(f) concluded.

- I-6.- continued.

. ..— — —.— —. .. —. .— ...—. — .

I

I

ass = [.0 Oasicmass.2

.0 ‘

.8 ‘

.4

.2

.0 Ii Case 262

.81

I/ ‘ }7 “

t

.4

.2— .

N Y“ I

1Span position, 0.85 /

.0 2.7~ase,261 Cpse 254

1 I

.6 .‘W-

“:1! !qy~lzl I I I I 1-

F)1 I I Ill II I I I I I I

.A .4 ~

n , m m, . m.& I I I I uSpan position, 0.77

.0 I *&

.8 Case ~60 ase ~53

-.I o .1 .2 -.r ‘

Chordwiseposition

(g) -= A; A = @j til.~C. ~,

Figure18.- Comtlnued.

.1 .2

high frequency.

.. —-

99

.

1.2

‘f/vb!.O

.8

1.4

L2‘f/ ‘b

1.0

.8

.1.2

‘f\vb 1.0

1.4

1.2‘f/vb

I●9.e

I I 1 IMass=O 5 basic mas!

, a I IMass=LObasic mass

aSpan pasitio~O.69

ciase252-

}

IG

Span POSition,9

%8I

/ Case 25S Case 251-

,

-.1 .0 .1 .2 -.1 b .1 .2

Chordwise position

(g) concluded.

Figure18.- continued.

.-— ..—. — —— .-. .— -- . .—— —— .- . - ...

100

m

1.2 J ass=O.~ basic massSpan position, 1.00

Lo

‘fivb.8

1.2

‘f/vb.8 3

.I

1.2 - — ~ .

1.0 1“f/”b : / ‘ ’12

.8 , 5“

-J o .1 .2 -.1 () .1 .2

Chordwise position

(h)Btier B; A = @; _rtC.

Figure18.. continued.

NAcA!m 101

McIss=O.5 basic mass MCJSS=LObasic mass

.

.

.

“f/”b

“f/”b

‘f/vb

I.2

10

.8

.6

1:2

1.0

.8

1.2

Lo

.8

1A I [ r I /Span position, 0.62

r6.57a

Case 272 4 Case 266

Span position, 0.46

<

/5.5 Case 271 -4.6 Case 265

.

9Span position, 0.3 I

8 6.% AI “ $

Case 270 / Case 264

-J o .1 .2 -.1 0 .1 .2Chordwise position

(h) Ccmcluded.

Figure18. - @n-k_.

..— .--— — —. - --- — — —— .. .. . .-

102 l?MYLm 3780

“f/ “1

Mass =05 basic mass Mass= 1.0 basic mass

Span position~ LOO

1.0

/ ~I I

Case 275>.8

/ s

/‘8

&mn position, 0.92

1.2

1.0.

“f/

.8 II

w . m

/ 21‘Case 26[

.6, %8 9

I

1.2

1.0% $6.2

.8

.6

&-.1 0 .1 .2 -.1 .0 .1 .2

Chordwise position

q

103

b

.

1.2

Lo

?ivb.8

.6

1.2

‘f /’b 1.0

.8

.6

1.2

1.0‘f \vb .8

.6

Mass=O.5 basic mass Mass = LO basic mass

-.1 0 .1 .2 -.1 0 .1 .2

Chordwise posithm

(i) concluded.

Figureti.- Continued.“

.

. . .-. . —. .—— .—— .— ———— —-

104 HAcA !JN3780

Vf/vb

vf/vb

1.6 I I 1 1 I 1 1 I I IMass=0.5 basicmass Mass=1.0basicmass

1.4

1.2

Lo lk7

J!y

.8

.6 P- I Id Case 289

.4

.-1.4 1 i 1

I I1 I 1 n 1 I I 1 u

//c Ll@ I I 1. I I I I I--.13L2 I m I I q.t5&t

4.2 Span position,0.92 3!2 9

1.0WIVl

I I 1 1 1 , m n “ m nm 1

Case 2881/ “

Case 28 I.8 /

.6

I

1.4 ‘ I

1.2 ‘~F~i~ .Jf (3

r I14 Span position,0.77Lo r I 1

Case 287 I Case 280.8 “ — y

I

.

.6 *-.I o .1 .2 -J o .1 .2

W, M@ frequency.

.-.

nIAcATNy@o

.

.

Moss=O.$boqic moss lyloss=l.O bosic massi

1.4 ‘ /

I$2

L2 Sponpositi( m, 0.62‘f Jvbl o

.9

Case 2[ 36 43& 6$ Cose 279.

.6 ‘

1.210

‘f Pbl.o--II Sp”on positioq O

A‘9

.8 - Cose285 Case?78

...2

WI, o / 3 -“~ ‘-1 I Spanposition,0:31 ~

-t.

/ % A.8 Cose284

●V

I

? # #

L2 I

W.Span position,() 15”

7 7 7— ~ —. ~

Case 283 Cose276-. I o .1 .2 I o .1 .2

Chordwise p;;ition

(J) concluded.

m 18. - Contimled.

...-. -.— — — —— — — -——. — ———.—- - - -

.

106 NAC.Am 3780

II I Moss= 0.5 basic mass I I I Mass = 1.0 basic mass I

S~n ‘posi;ion, ‘LO

I.0I /

/mok

.8 I

/‘f/ ‘b f19

.6 ‘Case 289

.4

I

12I,, \ A

, Span position, 0.92 III I 1 I I

m lb+/

I10 I I I A I

‘f/vb31? \ po

.8Case 268 I I ~-1 [9 I Gase-2811

/.6

. ““’tttti

1.4

1.2‘f /“b---

10 ‘ I5

1 1f3D

Cass 287 I Case 2801.8 – - , , , —

t I I I I I it)qa n

1 I I I I I1 t 1 1 (

-.1 0 ‘-”.l ‘--.2 ‘<i- o .1 .2(%ordwise position

.

NAcA m 3780 107.

.

vf/vb

vf/~b

vf/vb

1.6 1 E 1 s 1 a # 1Mass=0.5 basic moss Mass= LO basic mass

1.4

#12I

21.2 r

Span position, 0.62 ,0 /Lo I # I

I 1. 5 Case 286. . k.o 3 Case 279.8 /

A j.6 5.0 /.6

1.2“ .Span position, 0.46 ,0

Lo r— 9

6.8 A

H7 Case 285 / Case 278.6

1.2Spm position, 0.3

Lo10

Case 2844

case 277.6

1.2

Lon position, O.I 5 10 _~ w — —

10[ — ~ — -

.8.Case 283 — Y Case 276

-J o .1 .2 -J o .1 .2. Chordwiseposition

f‘,i(k) Ccm&ded.

-18.- ccmduaea.

:’- .-. —— ..— — —.—. . . -—-— . . . .

.

m m 3780

.

I VfI

1.61.41P

k.41:6

. /1.0.

I Lo

\

Forward

I I

%

LOA

I

/— - ~

I~ -

- -

‘ ;6

:a)FighterA; A = W; _ticj tiUS & ~ti~, 6 til.ovf= ~ milesper hour.

Ches;

(b) Fi@rkr A; A = @j _tiCj -US Of ~ti~, ~ tiCheSj .l.ovf= g8g ties per hour.

.

-e 19m- Contours of constantflutter speedulth concentrated mass.

..-— -.

rum TN3780

I

.

.

Vf/ \

I

%10

D L

WV&l L4

I

- - . ~

.I

I I.

(C) )?i@= Aj A = @j -ieficj -u6 Of ~tl~,l.ovf= 1,091milesper hour.

(d) FmeE Aj A = @j tii-ticj rdlu.s Of ~tia, 30 ti-ejl.Ovf= l,@l milesper hour.

.

. ——.. . —-—- —.— —-——— --

Forw

Aft

1

(e) FWer Aj A s 45°j ~cj tiUS of ~tl~, 6 tihsjl.ovf= 1,028milesper hour.

Fi@re 19. - Continued.

..— —

.

! I I I L I I ‘-l-TT-T

Forw

Aft

I.&#

0

(f) F’ldrter Aj A = 45°j -ticj -US Of ~tl~j m ti&esjl.oq = 1,028 ties per hour.

F@re 19.- ContI.med.

.. . ..— ———— — —.— — —- -- —-. — — ——- .— . . .. .

rWWLm 3780

FOI

0

‘wO

Aft

(g) == Aj A = 45°j tii~C; radlua of ~tlon, 6 inches;l.mf = 881 milesper hour.

F1.gure19. - Cent==

.. —-

5GWA !lM3780 113

.

(h) l?i@~ Aj A =.45°jtiiericj tius. Of ~ti~, n fi~Sjl.Ovf= 881 milesper hour.

---- ..— —— — .—— — —.

U4 w TN 3780

Forwai

Aft

Forward

Aft

I1.2

‘d

LoI)

I.c “— — —

.6

\.8

.

t# #I I

,I

#I I I I 1+

I II I I\ -

t

——

.

.

.

.

m 3780

. . . .

U.$

12

.-

Forwcrd ‘

I

(k) BmiberBj symetrlcj l.ovf= 1,=6 ties per hour.

1“ ..1

i

d - .9

I

. .Aft

J I m

.1 .1.. . .. ---: --– ~ ----- ---- ---

.- . .- ..- . . ---- ,- .. - -- .-J. _- -

.-, . ,.:. :“---- -,.-. ...,

(2)“B-er Bj ti~qricj l.~f ~ 1}~6 milesE -... ..-...‘ .:.

-19. - continued.

.—— — —- -— —..-.——— -

Ia6 mWL m 3700

1Vf

FbrwardI1

\

%

Aft \I

/ \ \

b .8

.— .— .

(m) BcmiberB; ~c; l.Ovf= 1,165ties per hour

r N 1Forward Vf

8r— N

Aft\

N

[

\

.

(n) == B; ~cticj l.Ovr = 1,244 ties per hour.

-19. - cmlcluaea.

—

mcA !civy@o 117

vf/vb

vf/vb

Vf/Vb

Vf/Vb

1.4 I I D s nAntisymmetric

tSymmetric

1.2 ‘I M=20

I.0 l\ \

20= y ‘

H o Cose 195.6

/

I

A ~se 197 -

1.2

I oM= 1.0 4

Lo. L *

I Y/ 0 Case 192

.8/ d ~

/

I

a

1.2/ \

\ M=O.51.0

18 w

[/ ‘

.6A use 200 IL <

.b .II

1.2

Lo 1- 10 C&e 201 \ 2_0/ HI9

.8~ %1 A CO* 203~

.6/

u4

-.1 0 .1 .2 -.1 0 .1 .2

Chordwiseposition

.

.

(a) Concentrated mm at tip.

Figure20.- EfYetiof sizeof concentratedmass on fluttercharacter-istics,Fl@ter A, A = @.

..— — .——— -— -..— —. ..-. .- .

w m 3780

1.4

1.2

‘f/vb 1.0

.8

.6

L4

1.2

“f/vb LO

.8

.6

1.2

~j”b LO

.8

1.2‘f/”b LO

.8

I I I I I 1.Antisymmetric Symmetric

1-- - f “ / 1922Q ● II

J o 4

~ 13 M=O.08 -dH

l= ‘ case 204 A II

, I

y - “ 13}/

—/ /

M=O.037

l= ‘“ Case2Q7/

/M=O.015

- Case 210. -

.- .- .-

.M=O.008

~ -~ ~ -

. . -Case 211 -- - . “- . “ ,,- ,,. I I I

-.1 “-o . .1 .2 ‘ -.1,,------ - -:=- Ctmrciwisq position --

I ,,,. L-.!---

. .

,.

0:

,..1

1.

--- -..

#

(13) Calclude!a. “

.2

1

I

vflvb

Vf /vb

1.6kJnlMlkric SylT rnetr c “. - d

I .4 -3

L A

I .21, /’

td=l.o 7

laI / I

.6Om m 166 4

A (%30 166

.6 7, 6 //

12

I10

.8 ‘/

.6

1.2

I IUIIIIIIIUWTI IIIf) .

‘ .8 4 y

.6

0.

-,3 -.2 -.1 0 .1 .2 -.2 -.I o .1 .2Ctnxdwias posltlon Chordwlse pxltlun ~

(b) Ocmcentrated nuss at O.m-epan pxitiau.

m8ureJ 20. - Ooncludaa.

1!kil

vf/vb

2.0

1.5. ‘

,

.0

I-rllz . —

Am II I .1

I I

‘f /“b

“1.5

o

-. I o .1 .2 .3 -. I o .1 .2 .3

CtKrdWise pdtkm Cha’dwlse positlcm

(a) Basic MS I?ula0.s basic mass at tip.

m =.- Caqnlrlmm of chuacterldiics for b&3iC =s and 0.5 basic~BS. RQhter A, A = 45°.

6G IuwL TN 3780

r

i.A, w m

1A II IT n

I

M=l.o.

2.0— -! .

14 0 m 3 —‘f~vb

I.5 -

6 o-Case227—Cose228

IA Case2?9

oc/

$

1 /

“ – – – – – –20

/‘fivb /

15. Ah

I.5$*1/ M=C).5 r

?o Case239

—1 —Cose24019 A Cose24 I //

wr

o \ . ~ la i ,// 40

-.I o .1 .2 -. I o .1 .2Chordwiseposition.

(b)Basicmass and 0.5 basic massat O.~-span positton..

Figurs21.- Concluded.-●

✎

NAcA-la4q FldLv&

.. .. .-. —-.. —- — .— —--- ---- .