REG NACA-TN-589-1937 Theoretical span loading and moments of tapered wings produced by aileron deflection.pdf

1、.,._.,k.e“ . _.:. -7 -: -,.i- :-,-.;*“i! . “k ,.m-4- -N.A.6.-.”TechnicalNqte No. 589 .* The.yrctilation“.,“i$:alseexpresqed b th.xetion-,.-.”: +=.-.“-.:. :.- =.- :.- ,-”-.:. ,+;-,=,-”-: -.:= .=-:-. .,!, .-. red”,sl”opeof section liftcurve, per radian,v s-ect$on.angle ,ofattack, radians. .“. - -.-,-.

2、 .,. .w downash velocity._ . , :,. .:-l r- -”For linearly tapered wings the chord at any stationis defined .b?the:exp,epsfon. . “;.”-.- “-“ . .-.-.C= r p *,(1 A) cos E/l= c= (1i=:K%COS6) (5)- -.=-.- .- -where-:x:is the roo”hord . .? -.:.-.-.,.- ,-“A; ratio of tip to root.“chQrUo is”a constant equal

3、toT wasobtained by.-.introducing-thevalues ofI,w, anclc a6 given by qUa-tions (2), (3), and (5),into equc.tion (4) and by collect-ing the t.e.rms.,.,Figurq ? gives plots of values of U. for varioustaper ratios X and aspect ratios A for m. =.6, in-Stea”pb sin b) x7sin rb ruo (1-K cos b) + sin r6 sin

4、6) .(12)L.- -“:“piq-T-pi,.-=-=a71a15a12a26a15a15a14a15.49.A-=a71. ,.=. =.B. .=.-. -.-a71-.u:.w , 1.+,3,. .=j.j - q :W - - .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N.A.C.A. Technical Note No. 589 7 It was found in applying this method that, fo

5、r taperratios from 1.0 to 0.25, the retention of four ha-mo-riicswas sufficient to determine quite accurately the Sp-ai”-lo%ddistribution. There is no objectin carrying the calcula-ti.onstoa far since the theoretical lift -“L”,.,8 “-“ lTAiC.ATechqi,calllgte,pl.8,9.“ “-“ .the even coefficients, in te

6、rms of the anglecoefficients-,are given in“table.111,.These coefficientsare.obtaineriby “solvingthe 12.set$ of four sirnultaneou hence “4bT. .%z; Uo.,.mo(ka) 42 “” “80 that%a = sin nb (17)1 *K,.cos b z uo(k6) -z-.: .Similarly, the lift coefficient at a point with aileronsneutral5.s given by ,=. ,-rn

7、oCic1 = - z% “1 sin nb1 + K.cos b U*CC (18)The total at a section is the sum of the lift coeffi-cients givetby equatimns (17) and (18). The form ofequati,”n(17) indicates that the change in %.2 due to “ailerozldeflection is directlyproportional to the equiva-lent change in angle of attack and that i

8、t is al,so for-”oddcoefficients the subscript 1,and for bothtypesno subscript. - L.:+,.-. . , .; - _- -_= ,-. +:*:-.-3-. -._._._.-.: 7.p. .vm.=,-.“:.“i;.J-.-rx, .-. .-w+ ,-*.?=.W=:,%.).ax+.gProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.u.,N.A.C.A.

9、 Technical Note No. 589 9bution is given in figure3 for a value.of (kti) equal to1.0.For any other value of (kti) the ordinates of ibisfigure need only he multiplied by the actual value of (kb)sIn an actual case the ectivechange in angle of attackat a section can be theoretically computed (rafere% -

10、or, preferably, can be o%tained from an analysis of flapdata such as that given in reference 12. At fai.rly”largevalues of 6 the experimental value of k liaf”iIiy-eTer”“- “ exceeds 0.2 and would eqtil l.O only if a whole wing sec-tion, such as one with floatig wing-tip ailerons, ive?.edeflected. .-_

11、The lift distribution due to the ailerons obtainedfrom figure 3 must.besuperposed-on the .Czl distrihu-tion due to angle of attack of the wing as a whole (equation (18) to obtain the”total distribution. The Ct1distribution and ,the Anl coefficients corresponding to,awing CL of 1.0 are given in figur

12、e 4 and table V, re-spectively. These values, taken from an unpublished re- -port, have been :A. na”= .Uo z (.-1)2 nz11 Ilaa - 1 Uoi,kThe tota,llift oh t-hewing ia given by. _ +“=,:. . 4 - .!- Lift-=/ Ctl cqdy (22) ,.-.,-b. .,+“. .-E .=%here -cl “isthat”for “:aeutral”“”osi-t”on“ofthe aileron.1 ._F=

13、-. :-., .i :7-Substituting for c; “c,and dy,.-the foregoing expreta-1sion bscomes - “.:-=., .=Lift =.24a n ZAnl .sin nd sin 6 d6 (23) .-.=. - .- -Integrng -. . .L :*”.- .,.:.s(24) -.?ssil. .,.Thus the total lift depends upon only the fir.t term Qf .t.h” ._.series. -,.-. .=Rnlli moment .- - The rolli

14、ngmoment for the semispanis determined from the expression . -.+-” ,. .- .a71IdS= J Cia Cwd.y (25)2 -.: . .,- -. ,:.5-:. . :4W-.-?-_-. .=- -=.: : ._.=,. .-4, ._. ”-” =_.;_. -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-uL. . .Ii*AeCmA.Technical No

15、te No. 589 11.Substituting for %2 c, y, and dy in (25), the valuesgiyen by equations (17), (5), and (1).Thevalue of the integral is alwayszero when the lowerlimif”is substituted,but only one value of n, namely,2; gives the integral a value when the upper limit is sub-stituted. This value is lT/4, he

16、nce11= qmo(kb) c= %2 n A22- k = k3(6) 2 (27)where.The lateral center of pressureof the cange in load dueto aileron displacement is(28)Values of Fa are tabulated in table VII and plotted infigure 7;the ratio of 32/F1 is plotted in figure.8. .INDUCED DRAG AND YAWING MOMENT DUE TO AILERONSInduced drag

17、at _Eectio- ._cient at a section is given byand substituting for w and clequations (3) and (16) with bnthThe induced drag coeffi-6(29) .the values given byodd and even coefficients411cd = i ZAn sin nb XnAn sin n6Cr(l + K cos ) sin (30)Provided by IHSNot for ResaleNo reproduction or networking permit

18、ted without license from IHS-,-,-.-=. -.:”. . -12 .“:-,$T the sec-ond group, of t-ermsin which combinationsof even termsare rc.ultiplied;and the third group, of allcombinationsof evetiand odd coefficiariis. If the aileronswere neu-traland the wing were lifting, ny terms of-,thefirstgrowould appear“W

19、ithdisplaced ailerons and the wingat zetio”lift,only terms of the second.grbup would occurqThe terms of thethird group can then be said to Wisf3from the interaction of “theneutral dis$ri+mtion and the .distribution due to the ailerons. .Induced dtiafor the“winK*-The fi.talinduced drag f-orthe wfng g

20、iven by.- , . .,.;.-hence (32) becomes after in-tegrating .“- Tli”=”nqbBXnAna (33). . .- . .This lattr expression can he divided“into”twoar-ts“.(34)in whi,ghthefirst pait represents the-ordinary induceddrag a.Q-d.$4e ecotidpart, the additional induced-drag dueto aileron displacement. The first part,

21、 rearranged, canbe written asDi = ca qs (1 + u)ITA.- ,- .=,. zIProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N.A.C.A. TechriicalNote No. 589 130 (1 + Q -the well-known formula for induced drag in whichnlAnis equal tn A17* Values of obtained from re

22、fer-ence 5 are shown in figure 9 for an aspect ratio of 6*The formula for the total induced drag becomeswhere(36) -.Table VIII and,figure 10 give values of F3 for (k) “equal to 1,0 at the various wing aspect ra”tiosand-taperratios previously used.The m.- Technical ” -”-”.:. . L- -:,.The lateral cent

23、er of prgssure of the load due toF= brolling iS given by =- ,F4 2 Equations for the drag are similar to those of theprecading secti.nse The.addi,tionalinduced drag due torolling is given by :;-*_.+.=. -.-: :“”oi”=7T Nhereas, with long ailerons, the dffere”nce “decreases as did the rolling-moment fac

24、tor.Figure 11 indicates that the ratio of yawing momenlito the rolling moment at equal values of wing CL de-creases with an increaee in wing taperl aileron span, orwing aspect ratio. Experiments (references 15 and 16)Provided by IHSNot for ResaleNo reproduction or networking permitted without licens

25、e from IHS-,-,-. .*. .7.-.: -,. -.-, . .- . . -:”4-20 “” . Tecni-calNote No. .589- - -. . -.*=: .1+.=conf,rrnthe theory that the ratio of yawing to rolling mo- .ment decreases with increase in taper bela.wthe stall.At or just above the stall, experiments(referenceM) in-dicate that the ratio increase

26、s with increasing wing”taperuntil theyawing moments become so large that they cannatbe controlled by the average rudder. This type of varia-tion is a further result of the fact that the seotionwhich.stalls first moves outward with increase In taper,The factors due to an angular velocity (figs. 12 to

27、15, ta%l.eXl)vary as do those arising from an aileron de-flection. Although, theoretically,taper and aspect ratiohave only a very minor influence on the maximum angularvelocity attainable (fig. 17), the t=aperedwing will ac-celerate to this value sooner because of bhe lower inertiaThe maximum value

28、of fb/2T likely to be obtained in con-Holled flighHn gusty air Is 0.05, according to ref8r-ence16; Thisvalue corresponds approximately to thevalue.thatwould be obtained with ailerons which are 0.15of the win? chord, cover one-third df the span, and aredeflactmd 15.SUMMARYOF-THEORETICALEQUATIONSThe c

29、haracteristicsof a wing with equally and oppo-sitely deflected ailerons under static conditions may begivenby the following equations:Chang-in lift on half tie wing, AL = 2qba(k6) Fl(fig.6)Total rolling moment, L = 23(k6) F;“(fgo).Additional Anduced drag = rrqba(k8)aF3 (fig. 10). . =:.“. -.:._g3ati0

30、” of yawningto rolling moment, =c1 G;-(fig. 11) L c=l.o:hangein lzft on half the wngj(fig. 12.)!I?otaldampin”gmoment, LD = 2qbSI. . . -.r- ,. = . .+-.: - :.;- -.,- :-*, - . .,. :.-.:.,-:-,-, :.- 1.-”-,.-_.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,

31、-,-a71a15Ratio of”yawing-.t damping moment, * ()N“LD c CL(fig:6) ., L=lcO:-. Maximum possiblevalue bf(fig. 17) p:y: J)max=”* (;. . ., ,.- - . . -.,- . -. . . . . .-LangleyMemorialAeronauticalLaboratory,-“ -NationalAdvisoryCommitteefr Aeronautics, .LangleyField,Va:, November”16;1935.“ “ -,-., . .,. .

32、 ”. .PREFERENCES ,-.,. t“1.2.?Q-*4.5.7.e.wislbarger, C;-,ahd”Asano, T.: Determination ofthe Air Forces and Moments Produced by the Aileronsof an Airplane; T*M; NO. 488, N:A,C.A, 1928- “-”. Hartshorh, A. S.: Theoretical RblC,1929. .1 ,Munk,“Max M.:Oh theDi”ribution of.Lift Along theSpah of.ah Aihfoil

33、 with Displaced Aileron”s.-. T-N*NO, 195, N.A.C*AC, 19240 “- . .-Sc”arborough. “ames B-.i.“Some Problb”rn”s”, 1925. 0: .Glauert,H.: The Elements of Aerofoiland AirscrewTheory. Cambridge University Press, 1926.Lotz, Irmgard: Ilerechnufigder Auftriebsverteilungleliebig geformter Flugel. Z.F.M., 22. Ja

34、hrgangHeft 14.Aril 1931,S. 189-195.Lippisch,A.: Method for the Determination of the7.-.-SpanwiseLiftDistribution. T-MCNO, 778, N.A,CsAo,1935.Fage, A.: On the Theory of Tapered Aerofoils. R- a.Q,.1928.-.tia;Gt; E: TheoreticalRelationshipsfur an Aero-fi-ilwith HingedFlap. R. .“-?.!.”: .-.:.=1-=Pearson

35、,Il.A.:-”-”A Methodof Esimatingthe Aerodynam-Ic Ef+ects of Ordfnary and Split Flaps on AirfoilsSimilar to the Clark Y Z!,N,No, 571, N,AoC,Az,19.36,.- - - Munk, Max M,: A New RelationBetweenthe InducedYaw-ing Moment and the Rolling Momentof an Airfoil.- in-Stia$ghtMotion.T.,RcMO, 19:7J.AC,A.,1924.? ,

36、 . ,o.rton,. H!, and Bacon,:D.”L.: PressureDlstribution”Ov8rThickAErofbils- ModelTests.T*R, Noo150j N:AscA?:!1922? .-, .-a71 ,Weick, Fred”E, 1933. -. - -.“ ,-Veick,Fred E., and Wenzinge.r,CarlJ.: Wind-TunnelResearchComparingLatprhlControl”De:ices,Par-titularlyatHigh Anglesof Att9k. - OrdinaryAileron

37、son RectangularWings, !C,R.No, 419,NsACt+.01-l?.32.-.7,. . .-a.,%.-”.,”:.-. -, ”.-:;-.-.: .”“.=?-+5R. ,:.,.=. _.- -.-.-.-. .-. .=.=. .-.-. -“.-7 -.=. =+,.d- .,.“”.-,. .t=-$-:.”=“:- “-”-=-”-y+”=”+”-“” ,TProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-

38、1 # , *TJ TABLEI.K2) uOa + (2.134- 1.333K) + ;(-2.438K +;Ka) j - 0.914 - ;(),(-7.757K+;nKa) e - 1.616u0- +1.910K= - (0.359-0.914K)(!%r-36.252K+:nf) lla+ (6.043- 3s0864K)U+f(-154749K +3TIK2)a -2.297u.- ;(16n-64.2;0K+ M Ka)a + (8.CB2-44064K) + fI1.00 0.75 - 0.50 0.25A 4 6 8CJwJtit 4 6 8 4 6 8 4 6 8Ea2

39、 1.635“1.1190.903 1.565 1.0860.878 1.477 1.037 0.847 1.370 0.977 0.807R2.4 -.539-.424-.368 -.W3 -.500-.41.5-.8E0 -.592 -.477-l?o -.712 -.556Rae -.076-.051-.038 -o -.007-.008 .108 a71055 .035 .a84 .152 ,101Rae -.035-.(E3-,017 -.0s3 -.032-.a3 -.083 -.047 -*O= -.136 -.072 -.oR44 -.3.6832.1841.5973.5452

40、.1301.5563.3942.0571.5183.2791.9981.48-446 -.KQ -.5s9-.499-1.193-.793-.618-1.660-1.034.-.7fj9-2.306-1.346-.966R4S -.135-.089-.067 .032 .Otu-.006 .288 *139 .084 .71.3.358 .227R66 6.ti343.6502.5216.3533.5442.441 6oOE33,4082.3735.8833.3192.334RG -1.058-.768-,. 204-1.4104K+0.5085Ks -0.2284K+ 0.353678.5

41、; :%76- 1.5548K+ 0.5323K= -0.2827lL+0.4673.*0.2 0.13572- 1.1842K+0.S353F3 .4i-0.0600K+0.0678 0.6372- 1.1842K+ 0.5450En -0.0576K+ 0.060866.5 .0376-0.5216K+ 0.3804K i-.!+ :2% ; :%:; % .8180-0.0614K+ 0.3089K78.5. .-;%+%37 0.2F0.7347K +.o %=3 KSL3,6 j $ %:%2-+ 0.7347K78.5 .8 -0.88CQt 0.6004K :$% $-S:EK+

42、:R% 0.4672+0.1380K - G.0848K )+0.:go.oala0.08780.0881o.053a0.0615o.osao.088S0.01530.oiloo.ti88-.3884 -.2391-.3101-.a379-.8017-.a503-.1918-.16a7-.1859-.1438-.123.3k3.5-.3802-.a6aa-.2388-.Z054-.2347-.1988-.a401-.1840-.1=1-.169a-.1302-.111079.5.a5a8,.1964.L6B4.3038.aw .193a.3541.3658.aa14.4164.?088.855

43、5Be .,%- .37 -o.17a7-.la74-.1054-.1338-.M93-.0819-.0951-0708-.0588-.0671-.0430-.0380-.1330-.0968-.0791-.oaaa-.0677-.0554-.0630-.0393-.03a5-.0164-.0127-.0108Z.5 .4531.3360.2788.4162.2088.a55a.377473.5 .a796.2!307.3W5 .a465.ao6a-.0709-.0549-.04n-.1008-.0755-.0628-.1?S5-.1OCQ-.0819-.1819-.1308-.1056 -.

44、_. -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SI.A.O.A. Teohnloal Mote Mo.369.TABLEIVValueeof AnOk)y .4.- “A. :-.064.064 .001Aa=l.a36Bn+O.314B4+0.087B6+0.0a7B.A#J.314B*+O.75aB4+0.164Be+O.043B.A6=0.067Ba+0.164B4+0.461B+0.060B.A,-O.0a7B,+O.043B4+

45、0.060B=+O.a87B.1.OCA#. 333Ba+0.403B4+0.119Be+O.036B.A.=zO.403+0.644B4+0.236B#3.060B.A6=0.119Ba+O.a36%4+0.51aBa+O.163B.A.-O.038B*+O.060B.+0.la3B.+0.313B.oaI0.1770.156.4 II0.060 -0.006;g77 .161 -,049 -.035.063 -.044: .64B -.077 .06a :Z1o.7e0.5Co.ao.7EAa=l.490B,+O.ESOB+0.169Ba+O.06aB.Aa-0.560Ba+l.016Ba

46、+O.S6 B6+0.10QB.A#O.169B=+O.366B4+0.63aBS+O.199B.A.=0.062Ba+O.1C43 0.199B+O.355B.A.8B6Ba+0.916B+O.3S7B6+0.134B.A4=0.916Ba+l.439B*+O.700Ba+.O.a23BaAI66.5?9.5 _ _I 3?0.75 “5366.5I 79.5I 37”0,50 5366.5. 79.5.1 370,25 53166,579.5 - - ,= -.4,6.8-0.2.4.6s8.066.108.14s0.02;-,064-,106.147.053.086.1190.022.051.085.1184 I0.2 0.025 0.020.4a71061 .048.6a71 104 .081s% .144 .116-0.2 !2,023 - o.olr-A.- .056. .043.6 ,098 .076s% a71 141 .1118 .-Z020.044.072 .,1000.019,043.070.099-0,017a71 040.C67.