NASA Wing reserch

7/27/2019 NASA Wing reserch

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REPORT No. 751THE MEAN AERODYNAMIC CHORD AND

THE AERODYNAMIC CENTER OF A TAPERED WINGByWALm. DmHL

SUMMARYA preliminary etudy of piikhing-naomenit data on

tupemk?mnge indicated that &ceUent agreement with tedd-da w obtained ~ locating the gy.arter-ch.ordpoint ojh acerage chord on the cwerage guart+nwhord point of theWntipan. The study was therefore art.emi%dto includemwt of the amxihrbli dab on tapered+oing rnadet?etestedby the NACA.l%e final comparisons mer: made on b bh of thedi&encee between h iixation of h aerodynamic centera8 determined by cahduti4m and by t..st. X54 agmwmentobtained when & mean quartw-chord point w locatedby geome~ aim w a8 appreciably better thun that-obtainedby introducing aerod~amic correctibmr. She th wingmoo?ak included atreme conditti of taper, eweepba.ck,and twist, it ie mio?ent that the calculations reqwir& todetermim the mean aerodynamic chmd may be greatlyeimpli$ed and at the same time giw imprmed accuracy.

INTRODUCTIONThe mean aerodynamic chord of any wing is definedby reference 1 as, The chord of an imaginary airfoiIwhich wouId have force vectors throughout the flightrange identical with those of the actmd wing or winga.The mean aerodynamic chord is required in orderthat the designer may have a ready means for evacuat-ing the wing momenta. By definition, the mean aero-dynamic ohord is to be so located that its force vectarsare identical with those of the actwd wing. Severalfactors are invohd in this substitution. In additionto the length of the mem aerodynamic chord, ite loca-tion must be determined.Conventional methmls for cdculat:hg the mean aero-

-C chord have attempted to apply corrections forall factors known to be involved. The calculationsrequired to allow for lift distribution, wing tv?ist, andtip shape tend to become wry cnmphx. Comparisonof cakndated vahws for generaUy similar wings havesometimes faikd to show ae cIosean agreement es mighthave been expeoted. C?omption of calculated +aIues

with the locations of the aerodynamic centers found inwing-model tests has indicated in some cases that eitherthe calculations or the model tests were inaco~te.The masimum deviations noted have beeu of the orderof 5 percent of the mean chord.Trial comparisons were made in several cases to findthe magnitude of the error invoIved in using the averagemean chord Iength imskead of a chord Iength adjustedon the basis of aerodynamic factars. k comps&onwith model test data, the average mean chord waqfound to give surpriaiiIy good agreement. A Sys-tematic study WEE++herefore made on aU avaiIabIe teatdata in order ta determine whether or not a revisionin the method of czdculation of mean aerodynamic .chord was indicated. As wilI be shown, a revision isindicated on the grounds of improved accuracy andmarked [email protected] of method. No mean chctrdhas been found that is better than a simple average

mean chord so located at the centroid of the semispanes to have its qusrtw+hord point coinciding with thegeometrical average quarter-chord point for the semi-span. fime~ chord ~ converted to a mean aero-dynamic chord by a simpIe fore-and-aft shift depend-kg on the factors that affect the location of theaerodynamic center.MEANCHORDOF A W ING PANEL

Site an airphme wing is symmetrically disposed withrespect to the X axis, it is neocssary to consider onlythe half-span. The determination of the mean chordstarts with the division of the semisyn into convenientpaneLs. In the &npIe case the taper is uniform fromroot to tip, and only one panel need be considmed. Ina more general case there is a constant-chord root panelwith a uniform taper on the outer pand, requiring theconsideration of the two panels separately. In comewings there may be two or more panek with ditlerentvalues of taper and dihedrd. In any ease, the determi-nation of the length and location of the mean aero-dynamic chord is based on the following assumptions:413


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414 REPORTNO. 75lNATIONALADVISORYOMI+W13EEORAERONAUTICS(a) The mean chord is equal to the area divided bythe span Z=S/b.(b) Liftand moment distributions are uniform.(c) It is unnecessary to apply corrections for minordetaik in the wing phm form such as fairings, fillete, ormoderate radii at the wing tip. (For large fakings,

fillets, or tip rounding, the actual projected area andactual local chord would be used.)(d) The mean aerodynamic chord ialocated laterally,fore and aft, and vertically by letting its quartw-chordpoint coincide with the geonietrical average quarter-chord point for the semispan.Assuming that the wing has dihedral, there are threecoordinates required to locate the mean aerodynamicchord. It will be located laterally in the zz planethat contains the center of area of the wing on one sid~of the plane of symmetry. The fore-and-aft and thevertical locations must be determined by summationof corr~ponding moments. The summaticn of themoments for a simple wing is best done by dividinginto panels, finding center of area and average chordof each panel, as shown on figure 1,and obtaining Zhwm

i~ ,1 Yb--Ic,b 4-,, .I 1- -~A~-j %.22i=. [email protected] ofa Wtngm,

-- (1)

___..--

C*

. . . ..

The same result can be ohtaincd,amdyticdly by inte-grating along a panel, since

For a.simple taper the wing chord can be expressed asc= L&-Ay (3)

and the x coordinate of the quarter-chord point byxe/t=zo+BI/ (4)

The integral of equation (2) is(5)

A graphical integration may bo used for a winghaving a plan form that is not readily divided intQpanels; for example; an elliptical plan form. The prod-ucts w and cz plotted against span give curves, the~ under wMoh is proportional to &j and SZ8respec-

tively. This method is described arid ihstxatcd by tinexample in a later section of this rcpmt.COMPARISONOF CALCULATEDAND EXPERIMENTALDATAThe comparison of a calculated mean chord with a

model @t may be made in any manner that satiafmsthe equivalence of the forcevectors, but the use of theaerodynamic center is by far the easiest method. of ac-COl@Shing this end. In this method the coordinatesof the quarter-chord point or of the aerodynamic centerobtained horn the Calculations are compared with thevaluea obtainecl from the model teat, The momentcurves from model test data define an aerodynamiccenter in aU c= where an aerodynamic center actu-ally exists. It is general practice in the cu~ent NACAreporte to include the coordinate of the aerodynamiccenter in preaenti~~ test data. The calc~at.ions OULlined in the previous section locata the quarter-chordpoint for the mean chord. The aerodynamic centernormally lies forward of the quarter-chord point by anamount depending on camber and thickness. Table Igivw the efht of thickness on the position of the aero-dynamic center for five wing sections in the NACAseries. There appears to be considerable scattming inthe data but this is probably due to the magnificationthat resulte from using the distance between the aero-dynamic center and the c/4 point. In the plotting ofthese data in figure 2, it will bo noted tht in therange-.07 - -. ~-.mtv5-.05%j +4%ti.-.~3dkoc -.02~?g:-.0/

o .04 .@ .12 .16 .20 .24 28 .32?hidmfiss, tjcFrom 2.-Aerodymd omntardntlonwiththfckmmuna camber.

Df usefd thickness ratios the maximum deviation isbout 0:4, or four-tenths of 1 percent of tho chord.The teat data on theee wing sections were obtained inthe original variabledensity tunnel with mtidcls having$inch chord, so that the uncertainty in the location ofthe aerodynamic center wsa only 0.020 inch.Comparison between calculated and experimentallydetined locations of the aerodynamic canter are


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MEANAERODYNAMICHORDANDAEEODY?NAMICENTEROFATAPESEDTKCNQgiven in tabIes II and III. Wltb the exception of the&et two wings listed, the models in table II.had sweep-back in the quarter-chord Iine. This aweepback wasdorm aIong the span for aII except the last two wings,which have a constantihord center section. C?ohnnns3 to 10 inclusive define the geometry of the wing.Column 13 is the shift in the c/4 point due to sweepbackand it is referred to the c/4 point at the root section.Column 15 is the fore-and-aft location of the c/4 pointreferred to the leading edge of the root section. Thecorresponding aerodynamic center, column 17, isobtained by app&-ing the incmnent, oohunn 16, takenfrom tabIe 1or figure 2. Note that the vfdues in cohmm17 are based on the wing geome@, without cormotionfor span distribution of lift. Column 18 contti thecorresponding values Iisted in table 1 of reference 5.Note that the values h cdunn 18axebased on the spandistribution of lift. C&mm 19 gives the Iocation ofthe aerodynamic cedar shown by the mo&I test data.Columns 20 and 21 contain the dif%rencea betweencahxdatcd values in colurom 17 and 18 and the testdata vah.s in cohmrn 19. h inspection of d-mm20 and 21 shows that the aerodynamic center is givenwith greater accuracy when the span distribution oflift is ignored than when it is considered. The maxi-mucu apparent error in cohmm 21 is about three timesg&akr, aod the average apparent error four tinmgreater than in oohImn 20.The comparison is continued in table III with agroup of winga having the quarkwohord points on astraight line. For such a wing it is ssmuned that th~quarter-chord petit of the mean chord must Jie on thissame Iine, and the aerodynamic center of the wing wiIltherefore be located ahead by the distxmce given intab~e I or figure !2. The values in the last column oitable III verify this assumption.

EFFECTOF FLAPSON AERODYNAMICCENTER.The test data on related airfoil series, as in table I,indicate that the aerodynamic center is affected verymuch more by thickness than by camber. Conse-quentiy, there should not be very muoh abift in theaerodynamic center unless there is an appreciablechange in the eflectivc wing chord as a flap is lowered,The test data in reference 6 indicate a slight forwardshift of the aerodymrnic center due ta 20 percent chordphin flaps at 20. The locatioiis of the aerodynamiccenter from the reference point were 55follows:

Flap Span~m 0 0. ao 0.60 0.70Location ofa-c. .210 .207 .201 .19:The data in reference 7 do not include aerodynamiccenters, but inspection of the moment curves onfigures tand 9 indicates a general, slight shift for split and phirflaps. The shift for Fowler fiaps, as shown on figure 11,is rearward and rather Iarge, A brief study of the

?otvIer flap data in reference 8 gives415

approxirgatexuwdynamic-center hxdions, for fir.=40, as follows:nap chord o 0.20 0.30 0.40Wing chordLerodymuuic .239 .316 .353 .378Cenier.Ax=.=. o .076 .114 .139His shift in the aerodynamic center appears ta be~bout50 percent greata than the shift in c/4 point.l?hecaktiation of mean aerodynamic-chord position -nust conaida the factors that shift the aerodynamicinter. In the case of phin or spIit flaps the shift ismall. In the case of Fowler flaps it is huge and ap-moximat.dy proportional to the shift in the quarter-hord point due to the chmge in pkm form with extendedlaps. A further analysis of these effects is highlyksirable.IFFECT OF FUSELAGEINTERFEREWE ON LOCATIONOF AERODYNAMICCENTERThere is a deihite forward shift in the aerodynamicxmter of a wing due to the addition of a fuselage or oflacelles. This effeot is due chi+ly h the unstablenomtmt on a shwandine form, so that the magnitude ofhe shift in the aerodynamic oenter w+Udepend on the

dative size and location of the wing.

LOcdfd-1 of Whq C/# Fw.& l fOkng the? f usek gs CX 7SRGUMa.-shift d aedymmio centardmatofus&ge. Nomkamumqoder,~~~.~waThe infiuenc6 of wing location along the fusekgeaxis, for one particular wing-fuselage combination, may

f


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416 REPORT NO. 75 lNATIONAL ADVISORYOMM.ITIEEORbe obtained from &we 14 and table V of reference 9.These data have bean plotted in figure 3 to show thegeneral magnitude of the &ift. Three groups of modelsare included in the plotting on figure 3. In the @tgroup, models 1, 3, 7, 11, and 13, the wing was on thefuselage axis, In the second @up, rnodele 48,42, 63,and 58, the wing was located in a plane paralkl to, and54 percent of, the chord length abouethe fuselage axis.In the third group, models 99, 83, 104, and 109, thewing was located in a plane paraIlel to, and 6.4percentof, the chord length below the fuselage axis. The for-ward shift in the aerod~amic center is given by

AX.,,.= 0.148 X/L (6)where X is the location of the quart erdord point ofthe wing along the fuselage axis and ~ @the length ofthe fuselage,It is important to recognize the limitations on thedata in figure 3 and, consequently, for equation (6).The only factors covered are the vertical and the hori-zontal location of the wing c/4 point. The test datain reference 9 include the effect ofwing incidence, whichwae small for the range normally employed, and theeffect of change in wing section, which also appeamsmall. The tests did not include systematic change inrelative size of wing and fuselage, which is likely toprove a major factor. Until such tests are made it willbe necessary to rely on figure 3 or equation (6) withsuch modification ae may appear to give reasonablecompensation for a change in relative sizeof the fuselage.The force on the fuselage should vary approximatelyas its projected horizontal area, or as W, and themoment arm of this force should vary as L . Thesefactors can be introduced as the ratios LDJS and L/c.For the wing model in reference 9 the area was S= 150square inches and the chord c= 5.00 inches. The di-mensions of the original round fusekige model were:.length, L=20.156 inches; and diameter, l?=3.44 inches.Hence, L~/S=O.462, and L/c=4,03. Introducing theseinto equation (6) gives:

A limited check on equation (7) maybe obtained fromthe tests on a rectangular fuselage model in reference 9,The rectangular model had the same length, L=-20, 156inches, as the circular-section model, but the width ofthe rectangular model was ~=2.702 inches, givingLD/S=O.363 instead of LD/S=O.462. The ratio is(L1l/S) rectangular section 0.363(L~/S) circular section -m2-0786

=.,. should be ii the same ratio.and the values of AXThe following data from table V of reference 9 are oncomparable models:

Circular-sectionmodel ----------Corresponding rec-tangular-sectionmodel=---------Circular section,measured AXe.C.-Rectangular section,calcuhk%d Ax..,.Rectangular sectio~measured AX=.c.Difference--------

AERONAUT1Ct3

7 118 165 186

204 206 208 209. 035 . 041 . 026 . 040. 027 . 032 .020 .031. 023 . 028 . 019 .034.004 .004 .001 . 003

From these comparisons it isconcluded that the ratiosin equation (7) compensate for most of lho effect due torelative size of fuselage and wing. 13encoW aerod-ynamiccenter or the mean chord, previously ddmrrninodfor aplain wing, must be moved forward by tho amountgiven by equa tiion (7) when fusehge interference ispresent.Nacelles are known to have an effect on th~ a, c. thatis generally similar to the forward shift obtained with afuselage. Analysis of wind tunnel teat data indicak,however, that the forward shift in tho a. c duc to anaceLleis inclined ta be erratic Sinci it was impracti-cable to separate aIl of the variables involved in anyfull comparison of these unrelated tests, no specific dataon nacelles wilI be presented. In the absence of testdata on a comparable, nacdle-wing arrangement., anestirntite of the effect of a nacdc can bc obtained from$quation (7), but it is necessary that any value so oh-.taiDed be regarded as an approximation subject @ ccn-~iderable modiilcation.There is a great need for a systematic invcs~igation ofthe effects of fuaelagw and nacelles on the aerodynamiccenter of a wing, since with these data it should bc pos-sible ti reduce or eliminate much of tho effort oftenrequired to obtain satisfactory longitudinal stabilityEndcontrol with unfavorable center of gravity locations.

STEPS REQUIREDTO LOCATEMEAN CHORDThe steps required to locate the mean chord of amonoDlane wing are m-follows:J.2,3.4.5.

6,

CkIculate [he mean chord Z= S/b where 8 is thawing area and b is its span.Consider @Mwing on one side of center line anddivide into a convenient number of panels.Fhid tho area AS of each panel.Find the center of area of each pane]. (See fig. 1.)Find the center of area of the wing by takingmoments about the center line

~= (As@J + (A&g,)4 i- As; (8)Locate the quarter~hord pointe for the chordthrough the center of area of each panel.


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* MEAN AERODYNAMIC CHOBD AND AERODYNMklZC CENTER OF A TAPERED WING 4177. Find the chordwise distances, ~t, ~ . . . of th-aquarter-chord points measured from some con-venient transverse *, such as the Ieading edgeof the root section.8. Fkd the fore-and-aft location of the mean quartec-chord point, Z, by taking moments about a trans-

verse axis ~=(m%) + (&z)fw+fs (9)

9. Locate the mean chord Z in a plane tbrongh thecenter of area (step 5) and, in the absence ofinterference effects, with its quarter-chord pointat the point Z calculated by equation (9).10. Calculate the totaI shift m the aerodynamic centerA&. due to flaps and to fudage interference.(See equation (7).)11. Shift the mean chord in the same direction and bythe same amount as the shift, AX..=., in the

aerodynamic center. This gives the~location ofthe mean aerodynamic chord.hi accordance with the mud conventions regadingsigns AX=.c.is negative when th~ aerodynamic centeris shifted forward.EXAMPLEOF CALCULATION

Aa an emmple of the calculation of met-m aerody-namic chord, consider the wing shown in -e 4.&

FIcmm4.TaPeredwfw. Modd No. 14h tam u.This wing k. been tested by the NACA and is listed aamodel, 14in table II. The span of the modal was b=15feet and the actuaI area was given as 5=32.14 squarefeett. If there were no rounding of the tips the areawould have been S= 32.32 square feet. The roundingat the tip may be sufficient to iduence the remits, sothat four methods of calculation must be considered, asfouows:

I. Use ,actual span and actual area, neglecting allother effects of the rounded tip.11. Use actuaI span and a corrected area corre-sponding to the extension of the taper (withoutrounding) to the extreme tip.III. Use actuaI area and a corrected span, decreasedby the amount required to compensat-efor thearea removed by the rounding.

IV. Use a graphical iUkfptiOh of the curves ob-tained by plotting the products CYand cxaIongthe span. c is the local chord, y is the distance ,from the center line, and z is the distance of thec/4 point &m thO transveree raference line.CaIcuIations required for the graphical integration,method IV, are given in table IV and the cm=respondingcurms are plotted on figure 5. The area undsr the

1 . . .2 - / t/ -.4

-flc f 2 .3 4. 5 6 7.

w curve is found ta be Ay=25.28 square inch-. Each~uare inch under this curve represents 2 square feetX1.0foot =2 cubic feet. Hence the area under the curverepresents 25.28X2= 50.56 cubic feet. .This is theproduct of the area by the lateral centroid, i!$~.Themea is 16.07 square feet, so that

In the tmne manner the mea under the m curv~ .!sfound to be kc=39.33 square inches. Each squareinch reprasanta 0.4 square footx 1 foot =0.4 cubic fopt,so that the total area under the curve wpresente39.33x0.4= 15.73 cubic feet. Hence,

The calculations by the four methods are collected intable V. Comparison of the ~ and the Xa.c. valuesshows a very close agreement. llrorn this ~eemat,ibappeara immaterial which of the four methods is used.For a simple wing, either method I or method II wouldbe fkvored, while for an eUipticaI phm form method IVwouId be easia.

DISCUSSIONANDRECOMMENDATIONSThere are two operations req&d in the determina-t,ionof the mean chord of a wing. The tit is the ques-tion of length and it has been shown that the simple~verage chord, or avaage ahord, obtained by diyidingthe arm by the span is fully satisfactory. The second


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418 REPORT NO. 751NATIONAIJ ADVISORY Committee FOR AERONAUTICS .operation is to locate the mean chord Z=S/b, so thatit becum~. a mean aerodynamic chord. This operatiomis more readily undemtood if the purpose of the meanaerodynamic chord is considered.The factor. that the designer really requires is thelocation of the aerodpamic center of a wing. Thebasic problem is, therefore, the location of the aero-dynamic centw by a summation of the free-and-aft&if& due to the major factors premmt in the design.The first step in the location of the aerodynamic chteris the calculation of the coordinates of the weightadaverage quarter


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TABLE 11,COMPARISON OF AERODYNAMIC CENTERS, OBTAINED BY CALCULATION AND BY MODEL TESTS TAPERED WINGS WITH SWEEP.BACK IN C/+ LINE

I I I chord I I Emuon IPlnnformqAOA modd T: ~r P . ..All L. R.down 2ratio

Root TIB imL TIP. . - .

1A 2 a 4 6 II 7 RIwo-o .. . ... . ~---l 21 m @ 6 001s m

2440... .. .. . . .. ..do. . . ..-. 2:1 * 2/9 6 2416 &on

H--- T 2 16 .~

------ v 21 2415 24+1-8160.. -----do-------- all 4/4 * o !?416 m

2R-M:-: v 2 2R16 aRm2R,-16-0.... ....-dn... ..... 21 4/a ,2p 6 2RI16 2R10960-16-8.46.. .-.. do.-....- !4;1 w .W 6, 2R118 2R1tQ(WH.46 ....~v 41 6 m(lhrk Y

--- a 61 p b4 QlmrkY Clnrk Y

OlnrkY ----- G a 4 4 6 wky tiy2241!4........ ~ 211 L* .WJ o mm 98012@m--- - %1 1.27 ,686 ~ ~,, ~Wing VI ---- =Ja7 2 ~n *4U 7 16 -

Avarage SPBUMLrror.... .. .. .. . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . ,

l!%%wltlr- Twlaobb~f . .9 100 0

0 016 .. . . . .

24 .. . . . .80 -a M16 -n, En

16 016 -a 4416 -~ 48

6.Sa oa. o9,87 0

%67 o*l&M o

... . . . . . . . . . . .

Lat-Lmr oralC.g,-d

11 12

o Is32s

o 1,m,!mrl Lm

.677 Lm

,677 1,W

,x 1,a?a

,m l,,a?a,m 1,6aa

,!298 L XKl

,111 1.107,04!4 J,S76,171 1,8601,

. . . . . . . ,900

..... .. . ,294

. . . . .. . . . . . . ..

(xl&.d mn rooL-rA.r ob$J

xl

13 140 0,as?

o ,6W.s67 , aaa,m ,s98,m .?aa,867 ,2s8,Ml ,8s6,867 ,m

, aza ,4JI0.124 ,417 ,06s ,wl

.lan ,a41,164 ,818

,14d ,81s

.. . . . . . , . . .. !

1,104 .-.007,6W -,lkm,wo -, me

,490 -,44m

,7Z -,010

ZO-hax tie...

. 16 16n.948 -.010

. ?aa- ,m, auo-.007

II104 -,007

LaoMOn ormmtlynamlownkwCdlnlat.ad Apparanterror

lkom MdulRfIfezenm#Olo- y :;; g; wf

mmheretry tr~;b- ~. . &y ~$&r,. q, x . * . M, ,. &e. ,,. . . . .17 18 IQ m 21 a?O.iwl a w 00W .(X?9 0 .(m mu. 12,Rep. 672. . .. . .. .. 1

.8!20 .aaa ,al? ,014, :blo m& 1%Ran.672.:..... ... 2,6$36 ,6U9 ,I?&l o -,oaa mg, 1,Rap,Ma.. . . . . . . . . .8

1,m 1,069 Lma -.011 -, w Fig.1#Rep,w... .. .... .. 4L 097 LOss 1.119 -.oaa -,061 w 8, Rep, on... . . .. . .. . 6,681 .2s8 , 6.s1 o -,018 mg, 4, Rep, 627........:. 8,w ,efa ,6s4 -. Inxl -,W Yl& &ReP,8w.. ... ..... .. 7

,ml ,444 .m ,m -,OU Big,&Rep, w ........... 8,.

~71a ,Z?rl ,784 -,m -,lm ~lg. 7, Rep, of . . . .. .. . .. . Q,M7 -, w law ...-.-. ,662 -,014 . . .. . . . ~, 4,ReII, 611. . . . .. . . .. 10

,M41,-, Om ,w ... .. .. ,861 -,011 Fjg,fi Rap,611.. ... ... .. 11. . . . . .

.C4w -,010 ;669 .Eal ,661 ,InM ,lm Fig, l% F@. ML-a . . .. .. 12,402 -, (J1O ,,472 ,46 ,466 ,037 -,CH)I Unpnblhkl NAOA tea4 la

.,,46a -,010 ,449 ,440 ,449 0 -, ml .....do...............----- 14

. . .. . . . . . . .. .- . . . . - -- - . . . . . . . . . . . . .. - . m -,0147. . . . . . .. . . . .. . . . .. . . . .. .. .. . . . . .*On4erpan97only, .

a

* . .1

., ,. -, t,, ,. #. .


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420 REPORTO. 75 1NATIONAL ADVISO~ COMN3ZTEE FOE AERONAUTICSTABLE 111.-COMPARISON OF AERODYNAMIC CENTER OBTAINED BY CALCULATION AND BY MODEL TESTSTAPERED WINGS WITHOUT SWEEP13ACK IN 2/4 LINE

~_AC!A Ma. . . _. .. - ..-c Y . . . .._- . . .._-oo.-_-..-.To.........._-:.. . . . . . . . . . . . . .. . ..- .--. .. . . . . . .-. .-. -.. . . . . . . . . . . . . . . .i-1%10: . . . . . . . . . . . . . . .Wow . . .. . . . . . . . . . . . .23112-46DID . . . . ----4412 . . . . . . . . . . . . . . . . . . .

c c!Y~92?m.2am22#.m9490104412 . .

2:12:11:3:15:11:.:::

1.0;1...--.

LstemdT

ksrodynamk canter I

From chta in tsbh 1, NACA I&p.No. 62%end table XII , NACA Rep. No. 6m.TABLE IV.DATA FOR GRAPHICAL CALC71LATIONOF CENTROID AND MEAN U$R$ER-CHORD POINT%ROR WING SHOWN ON FIG .

- :IIJ-.010.2.;. 010-.011-.010-.012-.010-. a9-. Om

CM rd I 44 ,ol ntAC

o00.67M.746L 114L6WL4S6. . . . . . .. . . . . .. . . . . . ..-- . . . . .. . . . . . . .. . . . . . . .

1--1k =2.7% o2.n :272: :%?L@M .S361.420L224 i [email protected] .------...-.- i%....._.- L u;; . . . . . . 1.96. . . . . . . . .0 ....... . - k% CY!W7.40a 74%669.19amam7. MNLm6.94011rom Dlfk-mm Rekwm~~nwingq

TABLE V.-CALCULATION OF MEAN AERODYNAMICCHORD FOR WING SHOWN ON FIGURE 4 .

Method I I

ToW U= 8 @q ft)... . . . . - -- -- -- -- -- -- -- -- 62MEm b~)..u . .. ... . ..-- ..- ~mMum cbcai c -~ .. ______________ 2,14Tlpcbard Cr (fib)..... ..... .... ..... .... .. @eE)Sm Ofowl! pauol4 (fat).... . . . . . . . . . . . . . . . 476centidofoutar,anoI Afi. .. . . . . . . . . . . . . . . . . 2.WcdrOIdofoItwpwl#$-jl +&..... . ... . . 4.72hOftiw Pared AS[. . . . . . . ..-. _... ----- 7.aAr4&ofhlier p4ndA& ... ... ... ...-... L....- &mA&Fu.. . . .. . . . . .. . . . .. . . .. .. . . . . . .. . . .. . lho7A.%A&.. . . . .. . . . . .. . . .. .-- . . . .. . ----- 4aw22A&@L+@%h... .. . . . . . . . . . . . .. . . . . . . . . .- . . . . elm6Iat.emlwntrold # . . . . . . . . . . . . . . . . . . . . . . . . . . . s , 17f18t cetrokxcdwntw Panel . . . . . . . .- .. - -- --- - .68Atsataentmld ofouter wud...- . . . . . . . . . . . . . .66M-5+Aftl Oeatrou Otonter,ael.- . . . . . . . . . L24A&ASt.. . . ..---... ---. . . . . . . . . . . . . . . . . . . Xu761U&+A~fi . . . . . . . . . . ..-... -_.. . . . . .. . . . 16.7ql

I*for>L, .. . ( rehmnm axhbkadbwtdgecd .Wqt.. . . . . . . . . -- Swtfolr). { .469

R m xv62.62 2114 s 14IAoo IL a lhalz16 ; 17 %14au am o4,ia 4066 476am LW . . . . . .&n 4.70 . . . . . .7.40 7. a 7. aa76 &67 &ml a10 16.m I&074L 6 47 4a749~ . . . . . .5L411 K1.aa . . . . . .6.18 6.16 3.16.ed .m .6s.M .M . . . . .L24 1.m . . . . . .law 10.664 :.....U.* lL W6 . . . . . ..QR6 .m .m.466 .466 .467


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421AERONAUTIC SYMBOIrS

1. FUNDAMENTAL AND DERIVED UNITS

w9mIA

8&(7baAv!ILDDoD,D,u

symbol

Le#h;----- :-------Force -------- F

Power ------- Pspeed------- 1

Metia - En@h

unit Abbrevla-1 unit IAi):i;n-in-tion

-tar ------------------ I m foot (or mile]-_---__-- ft (or mi)=mnd..------.-------- maond (or hour) ------- sec (or hr)weight aflkiloz---- L weight of 1 pound------ Ih IIIo~~wm(metia)---------------- horsepower----------- hp[Hlometerap erhour ------ kph mileaperhour --------- mphmeters persecond -------- mp feetpersecond ------- fp

lVeight=mgStandard accelerationOr 32.1740 ft/sed

I@.+

Z GENERALSYMBOLSw Kinematic viscositvof gmvi@=9.80666 111/S2 p Density (mma per finit volume)Standard density of dry air 0.12497 kg-m--e at 15 Cand 760 mm; or 0.002378 ib-ft4 sdSpeeific weight of standard air, 1.2255 kg/ma orNfomen; of inertia =mk. (Indicate axis of 0.07651 lb{cu ftradiua of ration k by prop& subscript.)To&cient o viscosity

3. AERODYNAMICSYMBOLSAreaArea of wingGapspanChord, Aspect ratio, ~True air speedI@amic pressure, ~P VLift,Aedute coefficient CL=*Drag, absolute cmflicient CD=:~fde drag, absulute Coefiient !%~=~iInduced drag, abscdute coefficient CD,=%Parasite drag, abecdute coefficient C%+Cross-wind force, abaolute coe&mient Cc=$2626

aq

Qat%r

Angle of setting of wings ,(relative to tbruet Iine)hgle of stabilizer setting (relative to thruakline)Resultant momentResultant angular veIocityReynoIds number, p% where 1ie a linear &nen-sion (e.g., for an &oiI of 1.0ft chord, 100mph,standard pressure at 15 C, the correspondingItqmolds number k f135,400; or for an airfoilof 1.0 m chord, 100 reps, the correspondingReynolds number is 6,865,000)hgle of attackAngie of downwashAngle of attack, infinite aspect ratioAngie of attack, inducedAngie of attack absolute (meaeured from zmo-lift position)Fiight-path angle


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422

zPositive dlreotiom of axes snd angles (form and momenta) are shown by arrowE

Axis Moment about axis Angle Velocities

I.ongitudinal. ---- x x La tera l--------- ; i!yfi= ti~ i$:::Normal--------- : -.. .

IJ inearPositive(Meztion ;;?& L?- n!%%%g n~]maxis)

Y+ Roll.. -.. q u; PPitah---- 8 v: qYaw----- + w r

Absolute c-cef6cientaof m~ment Angle of set of control surfaca (relative to neuhalC,=g& C.=q@ C.=.q$j position), & (@tkate surface by proper subscript.)(rolhng) (pitching) (yawing) .

4. PROPELLER SYMBOLSD I)iamet8r P Power, abscdute coefficient CF=P~P Geometric pitchpjD Pitch ratio c, dcp~v Inflow velocity Speed-power coeklicient= Etv. Slipstream velocity v EfficiencyT Thrust,absckte coefficient C==+ n Revolutions per second, rps

Q @ ()ffective helix angle= tan-l &Q 1orque, abscduti coefficient .CQ=~D5. NUMERICAL RELATIONS

1 hp=76.04 kg-m/s= 550 ft-lb/sec 1 lb= O.4ti36 kg1 metric homepower=O.9863 hp 1 kg=2.2046 lb1 mph= O.4470 mpa 1 mi=l,609.35 m=5,280 ft1 mps=2.2369 mph 1 m=3.2808 ft

u, s, 6amlaaT mm ml-ml O-tsw

NASA Wing reserch

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