1、REPORT No. 892DAMPING IN PITCH AND ROLL OF TRIANGULAR WINGS AT SUPERSONICBy CLINTOK E. BBOTN and MACSUMhIARYkl method is dwiced for cakula p72b )(Pitching momentpitching-moment coefficient; pvw )lift coefficient Lift force.(); J72scomplete elliptic integral(srli,11(Ithat is,This solutim, however, ma
2、y be considered the vertical orz-compommt velocity of the source-distribution potent id that is, however, it should be pointed out thwt the potcntinlof this type must be restricted to the linearizd theory and isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from
3、 IHS-,-,-9 DAMPIhG IN PTTCH AN ROL:” OF TRIANGULAR WINGS AT SUPERSONIC SPEEDS 61not of the same general nature as that of a conical fied -whichexists even in the nonlinear probems.From equation (7) the doublet distribution over the surfaceand under the assumptions of thelifting-pressure coefEcient i
4、s now=g=7(8)linearized theory theThe formation of the integral equation follows the methodof reference 1. A potential that represents a line of doubletsin the zy-plane at an angle tan-% to the z-tis is derived inthe form of equation (7). Use is made of the boundary con-ditions to set up an integral
5、equation that introduces theunknown dist.ribut ion function -f(u). The potential of thedoublet line may be obtained by following a proceduresimilar to that used in obtaining equations (3) and (4), andby substituting the expression for A given in equation (8)into equation (4). The expression obtained
6、 in the followingequation may be seen to represent a line of doublets alongwhich the doublet strength increases as (18)and, for pitching,u- y(a). du J c j(u), du_4 -tC$!h+-tt=ofim 6*O -c (a 6) 6 * (ue)Equations (18) and (19) are identical to the equations thatwould be obtained for sinilar boundary c
7、onditions on a two-dimensionsd flat plate if an amdogous process of distributingthe doublets were followed. ( .J-iowevei, weconditions of equations. (16) and (17) must be shown to .be ?)(w/x),andsatisfied. For the calculations of (w/x)p and bTthe evaluation of K, and KC, only one value of O need be.
8、conside for 6=0(22)Equations 22) and (23) may be integrated by use of(reference 6) to givep=TK, -i(f however, in application it is desirable b obtaiuthe pressure distribution and the forco and moment co-efficients for pitching about any point. A superposition ofmotions is thereforo required. The, pi
9、tching motion aboutany point w can be. made up of a pure pitching motionabout the apex of the wing combined with. a verticaltranslational mot ion .ofvelocity qrO. Tle pressure distribuionfor this translational motion corresponds to that of a wingat a constant angle of attack of (%c references 1and 7
10、.) The pressure distribution for the constant angle ofattack - isCombing equations (9); (21), (25), and (28) gives for thepressure distribution in the pitching case -.(20)/Integration of the pressures over the. wing surface and for-mation of the nondimensional derivative yields(31)where Z is the mea
11、n aerodynamic chord.Calculations of these derivatives for triangular wingshaving their leading edges outside the hhmh ccme. m mosLProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-DAMPING IN PJT.CHAND ROLL OF TRIANGULAR WINGS AT SUPERSONIC SPEEDS 63ead
12、y made by the source distribution method. In thismethod, the upper and lower sides of the wing may be con-sidered independent of each other. The source distributionfunction for the rolling wing is9,(% w) =4/1 (32)whereas thut for the pitching wing isThe calctiation of the pressure distribution is no
13、t presented,since the subject of the integration of scmrce distributionshas been weu covered in reference 3.The pressure distribution for rolling wings outside theMach cone has been crdculated to beP 4pC% (l+ce) Cos- ;pflrvW-1)*10l!?%)COS-l1+C6m (34)Integrating the pressures oer the wing and express
14、ing thederivative in nondimensional form givws(?,= (35)For the pressure distribution due to pitching about thepoint ZO,a combination of flow patterns must again be used.The preesure distribution of a wing at uniform angle ofatt ac.k is (reference 3)P= 4qqC _l1#zcervd Cosw-t?)+%if% 3)The pressure dis
15、tribution for pitching then becomesThe nondimensional derivatives CLc and C?., then become(38)(39)DISCUSSION AND CONCLUS1ONSE.spres.sions for the Ming-pressure coefficients overtriangular wings in ro are given in equations (26) and (34)and in pitch, in equations (29) and (37). Equations (26)and (29)
16、 are for wings inside the Mach cone and equations(34) and (37), for wings outside the Mach cone. Typicalpressure distributions are shown in figure 2 in which thepressure distributions for the two wings in pitch are forpitching about the apex.Expressions for the quantities C%, CL, and Cmare givenin e
17、quations (27), (30), and (31), respect iyely, for the caseof the wing inside the Mach cone and in equations (35),(38), and (39) for wings lying outside the Mach cone. ItwiH be seen that the purametera fld%, PCLC, and 19L?Cmaybe expressed as functions of fIC wheretan e/9c=tan #The stability derivativ
18、es may therefore be plotted againstthis parameter to give curves -ivhich.will hold for all triangularwings at any Mach number. These CUJWMare given infigures 3 to 5. For value: of PC approaching zero the vahes -of the derivatives closely approach those given in reference zwhich were based on the ass
19、umption of very low aspect.ratio.For values of 19CZ1 (that is, for the wing lying outsidethe Mach cone), the quantities PC% and #f?c become con-.stant and equal to $ and 1, respectively (the pitchingbeing about the c point). In comparison, the alues ofPC% and mC for infinite-span, rectangular wings
20、are and $ respectively (the pitching being about the leadiggedg=).It should be pointed out that. the pressure distributionsgiven in this paper may be used directly to caIculate thedamping in pitch and roil for wings having trailing edgescut off ahead of the Mach cone, the most interesting of thisser
21、ies being the so-called “arrow wings.”It is apparent that a suction force exists at the leadingedges of wings in pitch and roll whenever the leading edgesare s-wept behind the Mach cone. A method for obtainingthe values of these suet ion forces was derived in reference 1.-,.LANGLEY MEMORIAL AERONAUT
22、ICAL LABORATORY,NTATIONAL ADVISORY COMWMIEE FOR AERONAUTICS,LANGLEY FIELD, VA., December 1$2,19.47.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-64.REPORT NO. 89=NATIONM” ADVISORY “XMMDTEE I?OR AERONAUTICS .Rolling - . . Rolling.?-,., - -(4 Pitchin
23、g m Pitching.-(a) Leading edge behind Mach WM. . .(01 kad!% edgQahead Of pach.cone,.L.FIQURE 2.Pressure distributions for rolling and pitching about apex.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-DAMPING IN PITCH AIWDROLL OF TRIANGULAR WINGS AT
24、 SUPERSOMC SPEEDSI t“ I I I I t I a- 1,-s t-e SUI / ./wnce 2) k /.24 4/ /+C,p- ,6 /.08I-7v Io I.2 .4 .6 .8 LO L2.FIGURE 3.-Stability derivative CL=for triangular wings.2 I L I I I bi.4 /I.2 / mo .2 .4 .8 .8 Lo L2BcFIGURE 4Stabilit derivative CLaabout the c point for triangularwings.65.- :./.2Ribnws
25、resuttlo kefer-?me 2)” l w!/I/ ,.8 I / 1 /1 /+Cq .8If/tI.4 . t/1!.20 2 .4 .6 .8 L.o f.2PCFIGGHE5.-StaMity derivative Cmaabout tile c point for triangularwings. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPENDIXMETHOD FOR DIFFERENTIATION OF EQUA
26、TION (12)The expression for w (equation (12) cannot be used directly when z is set equal to zero bccrmse of n troublesome singularityin the term and the occumence d an indeterminate form under the integral sign.J To obtain the vtiluo of w on th. _.surface, however, it is possible to integrate and th
27、en set 2 equal to zero. The troubkaome parta of equation (12) come fromt These terms, written out, may be integrated as follows:the terms involving .Introducing the imits and theu settingz= O gs_-!.m -_pj(C) (119ec)4m_ll?(-Q (l+Wm+ /* N (;q) (l /9%)(!y;)- d(pu) /32(3/3u+XM+fw%)j(u) fwlf() Pm)d(l #a)
28、 (p”) + +=FwB-/m2 w- Ill Bwl-B2d (ITheodorsen, Theodore: General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Rep. No. 496, 1935.De Haan, D. Bierens: houvelles Tables DInt6grales D&inies. G. E. Stechert & b. (New York), 1939, pp. 474-476.Stem-art, H. J.: The Lift of a Delta J
29、Ving at Supersonic Speeds. Quarterly APPI. Math., VOJ.IV, no. 3, Oct. 1946, pp. 246-254. - “Glauert, H.: The E1ementa of Aerofoil and Airscrew Theory. Cambridge Univ. Prw, reprint of 1937, pp. 83-93.* a71Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
