1、REPORT NO. 635THEORETICAL S1ABILmY AND CONTROL CHARACTERISTICS OF WINGS WITHVARIOUS AiilOTTS OF TAPER AND TWISTBy HENRY A. F3zmaoN and ROBERT T. JoinsSUMMARYStability deriratires hare been computed fur twistedm“ngs of di.ferent plan forms that include vam”ationginboth the wing taper and the aspect r
2、atio. Taper ratiosof 1.0, OJO, and 0.I?6 are considered for each of threeaspect ratios: 6, 10, and 16. The specijic dmkaticee jorwhich results are “wn are the rolling-moment and theyawing-moment deriratiree m“th respect to (a) rollingcelm”ty, (b) yawing wlocity, and (c) angle of sidef the well-knovm
3、 wing roting-moment andyawing-moment codlkiente, Cl and C=, with respect toinstantaneous ding and yawing angular velocities (ex-pressed as hek angl-) and 19is used to designate the .-partiaI derivatives of these coefficients with respect toinstantaneous aidedip angles. In this manner the no- _tation
4、 is consideraby shortened from the usual more(J = (2T7)cumbersome expressions bCJ , W/b * , etc.Expressing the rolling and yawing momenta as the sums .451Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-452 REPORT NO. 635-NATIONAL ADVISORY COMMITTEE I
5、?OR AERONAUTICSof partial linear factors is considered valid for motionsthat are slow relative to the flight speed V and for smalldisplacements, such as occur in ordinary unstdledmaneuvers and such as are considered in the study ofstability.cc,angle between the zero-lift direction of the wingsection
6、 and the rIir velocity at infinity, radians.6, pfimmeter defining spanwiee position, y= $ cos 8( )when 0=0, y= $;- 8=7, y=: CO,Cz, (?4, cocflkients of cosine series expressingwing plan form.C, rding-moment coefficient.-0, yawing-moment coefficient.p, anguIar velocity in roll, radians per sec., anagu
7、lar velocity in yaw, radians per sec.V, flight velocity of wing along X, f. p.s.13,angle. of aideslip, radians.8, aileron deflection, radians.d, mt of change of rolhg-momenoefficieut 01with the helix angle /2V.C,P, rate of change of yawing-moment coetticlent Cmwith the helix angle pb/2 7.Cl, rate of
8、 change of rong-momet coefficit Gwith the helk angle rb/2V.C., rate of change of yawing-moment coefficient Cmwith the helix angle rb/2T.C?,rate of change of rolling-moment coefficient Clwith sideslip rmgle .C,fl, rate of chrmge of yawing-moment coefficit C,with sideelip angle B.C14,rate of change of
9、 rolling-moment coefficient CZwith aileron angle 0.450I .oolt“ 15 lat1 _LEl:w!q.901- -Elliptical wingt“432l (c)-1.0 Refai;disimce fmm wing cenfer(G)X-1.w. (b bo.lia (a) hEO.ZS.FIGGM2.Imd dfmikutkm (2) a base Iinewith a range from az_ to amuM.is laid out as in figure7 (c) with the origin of the .ordi
10、natee at a equal to zero;(3) the effect of any length of elemental rmgkwf-attackchange, da, in figure 7 (a) is found by projecting thelength of the element onto figure 7 (b) and plotting theincrements (AI) and (Az+Aa) at the angles of attackfor which these elements are drawn, as in figure 7 (c).Beca
11、use a negative angle would induce a negativo lorIdat the point in question, Al is plotted as a ncgatlvovalue. This process is continued from amt,_ to aMKand the resulting curve (fig. 7 (c) is integrtited to obtninthe total effect at 0.75, which is then plottcd in figure7 (d). The load distribution o
12、ver tho entire spnn isobtained by repeating the same proccdum fur a numberof points along the span.With the lift loading thus determined, tho inclucocl-drag distribution may be found by a simplo opera ion,namely(4)Fre 7 (d) gives the compmison of tho load-distribution curve obt inecl from tho intluc
13、ncc lineswith that computed directly by tho wing theory usingequation (3). Although the agreement is not precise,it must be remembered that the solicl curve representsa WS6”where no seriss approxirnntion wns ncccssnry;hence -it mny be concluded t.hnt tho influenco-linomethod of determining the lift
14、distribution is M nc-curate as any other for practical purposes.Mlde from other possible npplicntione, tho lend it is therefore permissible to make certain mathematicalin contrast to the system given by equation (18) ofreference 2.By means of equation (9), Fourier coefficients werecomputed for the n
15、ine tapered wings with two differentinitial anghof-at tack distributions: (1) a distributiondue to a unit angle of attack extending over the wholespan, and (2) a unit argle of attack at the wing centerco-wring half the span. In order to obtain the correctfairing of the final curves of fre 11, simiIa
16、r resultsvmre camputed for elliptical wings with six angle-of-attack distributions co-rering O M % %, %, and all ofY.J*7the wing span.As was the case viith the derimitire CXP,it is mostconwnient to give the derivative of rolling moment-. J -Provided by IHSNot for ResaleNo reproduction or networking
17、permitted without license from IHS-,-,-462 REPORT NO. 635NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSdue to yawing Cl, as a ratio in terms of a partial-spanunit angle of attack. The values of Cl, may be ob-tained from figure 11 and are to be inserted into theequationecfroi%A/4 16Fmmrt ltherefore the
18、reIative amounts contributed by thewings and td surfaces may vary considerably.Although it was not poasible to give a general chartfor det ermining the damping in yawing for symmetric-ally twisted wingg as was done with the previous deriva-tives, it can nevertheless be said that the addition ofLoad
19、toward the tips, whether by vmshin or by anincrease in taper ratio, wouId increase the wing dampingmoment due to a yavzing angular velocity.The manner in which the changes in angle of attackthat cause a rding moment are brought about duringa sidealipping motion is shown in whereas, at the opposite t
20、ip A, there ism equal decrease of the angle of attack. The portionsBB, having no dihedral, contribute no change in angleDfattack when the whg is sideshpping. At the center,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-464 REPORT NO. 636-NATIONAL AD
21、VISORY COMMITTEE FOR AERONAUTICShowever, owing to the negative angle of dihedral rc,there is an effective decrease in angle of attack overpartt C equal to Id and on C there is a similar increasein angle. Figure 15 shows the resulting effectiveangle-of-attack distribution for the particular shape ofd
22、ihedral assumed.The effect of this distribution is similar to that causedby two pairs of ailerona equally and oppositely deflectedwith the inner pnir opposing the rolhng action ofthose at the tip. Positive areas of dihedral on theadvancing wing tend ta add load onto that wing. Forthe system of axes
23、chosen, all areas with positivedihedral produce a negative rolling moment with apositive angle of sideslip, This moment, like the rollingmoment due to rcdl, is independent of the initial wingtwist as long as no-portion of the wing becomes stalled.The rolling-moment derivative due to sideslip C%may b
24、e determined from figure 16, which gives thevariation of C#r for various unit tmtisymmetricalangle-of-attack distributions (i. e., symmetrical portionswith constant dihedral) that extend out from the wingcenter and cover various relative amounts of the wingsemispan. In the usual case, where the dihe
25、dralangle r is constant along each semispan, the value ofthe rolling moment due to a sideslip angle p can beobtained from the equationat the sametime, however, the contrary moment due to the com-ponents of the Iift acte to advance the forward half ofthe wing stilI more. AR vim the case with the yawi
26、ngderivative due to rolling, the moments caused by the liftcomponents predominate and, as a result, the nettheoretical moment is an unstable one; or, in otherwords, with the system of as= chosen, a negativeyawing moment results when the dihechd and sidesIipangles are positive.The explanations advanc
27、ed in some textbooks neglectthe inward slope of the lift vectors and Iead to an incor-rect sign of the yawing moment.The yawing moment in sideslip is given by the equa-tionN,id,rtp= Cxfiflqsb (16)The deriative CXBis yen iq figure 21 as a ratio interms of cc because its value depends linearly upon th
28、emantitude of the product of these variables. The“$values of O r a have been computed for unit symmetri-cal angle-of-attack distributions that extend out oneither side of the center line and cover 0.25, 0.50, 0.75,and aII of the wing span. These curves may be used todetermine values of C,fl for any
29、initial angle-of-attackdistribution symmetrical about the wing center Ike,provided aIso that the angle of dihedral is constantacross the wing span. AIthough the rolling derivativedue to sickslip can be obtained (from& 16) for a cumi-linear variation of dihedrd along the span, it is necessaryto stipu
30、late that either a or 1? remain constant if theprinciple of superposition is to be applied in the deter-mination of C,p. The combination of variable syrumet-ricaI twist and uniform dihedral being more commonthan the converse, the computations were shortened byincluding CUTVIMfor only the case of uni
31、form dihedral.The resultant value of C,fl (to be used in equation(16) is found by either an mtegmtion or a summationof the efFects of sIements of angle of attack extendingalong the span. The process to be followed wheregraphical evahation is necessary has been illustratedin figure 10, with the ordin
32、ates of ilgure 10 (a) changedto r. The ordinates and abscissas of the remainingparts are to be changed as required. For untwistedwings with uniform dihedral, the value of cr a isobtained by muhipIying the value read at a relativedistance of 1.0 by the wing angle of attack and, inturn, by the dihedra
33、I angle.The curves of figure 21 being generally steeper be-yond the 0.5 point, the deduction of increments ofangle of attack at the tip, i. e., giving the wing wash-out, would be the simplest means of decreasing thennstable yawing moment caused by the wings in asidedipping motion.Although the predic
34、ted variation of the yawing .-. moment with dihedraI is confirmed, experiments showa residual stable yawing moment at zero dihedral thatis not predicted by the ordinary theory. This residualmoment is greater for wings with bhmt tips and isgreater at zero or negative lifts. It will be noted that .,th
35、e theoretical yawing moment is ihelf the small re-suhant of two large contrmy eiTects and is thus of thesame order as a number of possible seconda irdluenca.LANGLEY MEMOBML &RONAUTICAL LABORATORY,hTTIONAL&msoRY COMWTTEEIFOE AERONAUTICS,LANGLEYFIELD, VA., Apm”l 19, 1938.REFERENCESL Lotz, Irmgard: Ber
36、eohnung der Auftriebsmrteihmg beliebiggeformterFUigel. Z. F. M., 22.Jahrg., 7.Hef& 14.Apri.l 1931,s. 189195.2. Pearson, H. A.: Span Load Distribution for Tapered Wingswith Partial+m FIaPs. T. R. No. 535, N. A. C.& 1937.3. Betz, A., and Petarsohn E.: Contribution to the AileronTheory. T. M. No. 542,
37、N. A. (1 A., 1929.4. Peamon, H. &: Theoretical Span Loading and Momente ofTapered Wiigs Produced by Mleron Detleetion. T. N.No. 5S9, N. A. C. A., 1937.5. ShortaI, Joseph k: Effeot of Tip Shape and Dihedral onLateral-Stability Characteristic. T. R. No. 548, N. A. C.A., 1935.6. WeM, Fred E., and Jones
38、, Robert T.: R&sum6and Aalysisof N. C. A. Lateral Control Research. T. R. No. 605,K. A. c. A, 1937.TABLZL-VALUES OF COEFFICIENTS DEFINING WINGCHORD DISTRIBUTIONc. .sm e= Cmeos ndY ccYA LOO am o.= EOIpthnI.46 0.730 l.O& 1.300 L(IOO.ioo L!2i0: LOW.677 .95Z L.MO L000Cak LOOA“ +am as do. Sal o. 149 .-.- -1: . . SIs - -16 .259 -LIEI -C4+ ix LOO a60 0.25 EIUpthIA6 -y. _g -.- -_: -aM -.W .Sz3 -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
