1、REPORT 1268THEORETICAL CALCULATIONS OF THE PRESSURES, FORCES, AND MOMENTS ATSUPERSONIC SPEEDS DUE TO VARIOUS LATERAL MOTIONS ACTINGON THIN ISOLATED VERTICAL TAILS By EENNBTH MARQOLIS and PEECYJ. BorwmrrSUMMARYvehI poteniiab, pre8 cotangent ofsweepback angle of leading edge (see fig. 1)=1dH%7Bm*(Bm)=
2、mm=aE (k)E(k) complete elliptic integral of second liiud withmodulus JTJ2h-(1-H) sid n dnoK (k) complete elliptic integral of fist kind withmodulus ,JT/ dn0 J1(1H) sinnr, u, CO arbitrary constants (G=uBm)W+k)K(k)+( l4c+k)E(k)“= (z.-k) (12k)E(k)+k( l+k)K(k)E(k) kK(k)2(l+)(l +k)E(k) 2kK(k)“=- 2(2-k)(1
3、 2 simple charts are presented which permitrapid estimation of the 12 stability derivative for givenvdues of aspect ratio and Mach number. Tabulationof the derivatives for subsonic-edge trianghr tails withtrailing-edge sweep are also presented.Three systems of body axes are employed in the presentre
4、port. For plan-form integrations and in the derivationand presentation of velocity potentials and pressures, theconventional analysis system shown in figure 2 (a) is utilized.In order to maintain the usual stability system of positiveforces and moments, the axes systems shown in figures 2(b) and 2 (
5、c) are used in formulating the stabili derivatives.A table of transformation formulas is provided which enablesthe stability derivatives, presented herein with reference toa center of gravity (origin) located at the leading edge of theroot chord (fig. 2(b), to be obtained with reference to an arbitr
6、ary center-of-gravity lomtion (fig. 2 (c).BASICCONSIDERATIONSThe calculation of forces acting on the vertical tail essen-tially requires a knowledge of the distribution of the pressuredifference between the two sides of the tail surface. Thispressure-difference distribution is expressible in terms o
7、f theperturbation-velocity-potential d.iiference or “potential jumpacross the surface” Aq by means of the linearized relationship(1)Inasmuch as for the present investigation thin isolated tailsurfaces are considered and thus no induced effects arepresent from any neighboring surface, the perturbatio
8、nvelocity potentials on the two sides of the tail are equal inmagnitude but are of opposite sign. Equation (1) may thenberemittenin terms of the perturbation velocity potential p asfollows:(qwhere q is evaluated on the positive y-side of the tail surface.The basic problem, then, is to find for each
9、motion underconsideration the perturbation-velocity-potential functionp for the various tail regional areas formed either by plan-form or plan-form and Mach line boundaries. (See, forexample, the sketch given in table I.)For time-independent motions, such as steady rolling,steady yawing, and constan
10、t sideslip, the potential functionsare” of course independent of time (i. e., the last term ineqs. (1) and (2) vanishes) and may be determined for thesubsonic-leading-edge cases by the doubletdistributionmethod of references 14 and 15. The details of the methodand its application are given in the ap
11、pendixes. The super-sonic-leading edge ccdgumtions are analyzed by the well-known source-distribution method utilizing the area-can-cellation-Mach line reflection technique of reference 16.The mathematical details are not presented herein, becauseit is felt that previous papers dealing with wing pro
12、blems(e. g., refs. 17 to 20) have applied the basic method in sufli-cient detail. The main difhwence to be noted is that theroot chord of the isolakd vertical tail is, in effect, anotherfree subsonic edge similar to the tip chord and must betreated accordingly. Actually, tail regions I and III (refe
13、rto the sketch in table I) are not affected by the additional.Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-d388 REPORT 1268NATIONAL ADVISORY COMMITTEE FOR AERONAUTICStip, and wing results in these regions for constant angleof attack (ref. 20), st
14、eady rolling (ref. 20), and steady pitch-ing (ref. 18) are applicable to constant sideslip, steadyrolling, and steady yawing motions, respectively, for theverticid tail, provided appropriate changes in coordinatesare introduced and the proper sign convention is maintained.The timedependent motion co
15、nsidered in the presentreport, that is, constant lateral acceleration, can be analyzedin a manner analogous to that used for a wing surface under-going constant vertical acceleration (e. g., refs. 21 to 23).By following this procedure, the basic expressions for theperturbation veloci potential and p
16、ressure coefficient(evaluated at time t=O) may be derived as follows:(3)=-E-(+%dAp( (9 +% P $=+ and thus the first integrationwith respect to z in equations (5) and (7) yields p; hence,equations (5) and (7), when applied to steady motions, reduceto essentially a single integrationfunction.involving
17、the potentialThe nondimensional force and moment coefficients *andcorresponding stability derivatives are directly obtainablefrom the definitions given in the list of symbols. I?orexample,l%(YLv J r-loInasmuch as the various pressure coefficients are linear withreference to their respective angular
18、velociticw, attitude, oracceleration (i. e., linear in p, r, P, or ), the partial clerivn-tive in the preceding example may be replaced by the stability derivatives due to rBu(BmBu) tanh-270 (1ll%qfl d(Bu) (C5)It might be mentioned at this point that the integrands of expressions (C-4) and (C5) are
19、finite and continuous over the intervalO to Bm and therefore must yield a iinite quantity -when integrated.The integration of expression (C4) by parts givmIntegration of expression (C5) by parts givesProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.
20、.396 REPORT 1268NATIONALCombining expressions (C6) and (C7) results inADVISORY COMMIITE.E FOR AERONAUTICS(C8)The fit term of equation (C8), w-hen evaluated at the limits, is either zero or iniinity. The integrand of J, aa ma noted,is finite over the whole interval; tharefore, infinities introduced a
21、s a result of parts integrations must, in the end, canccdthemselves.The second term of equation (C8) is an elementary integration which when evaluated (with infinities neglected) yields(T 2r+4Z3TBm5GBm 4Z5uBm+2r+3rBm - .HGrn ) (C9)It is now convenient in integrating the third term in equation (C8) t
22、o introduce the variable substitutionso that Bm and k are related by2km=l+h?The third term in equation (C8) when transformed by equation (C1O) maybe written in the formwhere(Clo)(011)(C12)I,= J kT(I+k)4(+)dx_, lI,= J k(7-+dF)F(#)d#-tI,= J :, (Ui?lj) (*)d*The integrals I, 15, and 1, are elementary an
23、d may be determined by an integration by parts. If the multiplicative factorProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL AERODYNAMICS OF THIN ISOLATED VERTICAL TAILS AT SUPERSONIC SPEEDS 397before the summation sign in equation (C12) i
24、s neglected until all the components are totaled, these three components become1,+15+17= 7k(l+lT_w(l+l i?)T1P lh?Consider the integration required for 1*, that is,Let #=k sin O; then expression (C14) becomesJ12 d-%m=wl-X12 ah P sin%It can be shown that(C13)(C14)(C15)(C16)This fact allows expression
25、(C15) to be written asThe last, two integrals of expression (C17) cancel each other and leave(C18)After the inverse hyperbolic tangent is replaced by its logarithmic equivalent and the additional variable transformation1Pi =lF sine (C19)is introduced, expression (C18) becomesLLi4+%LJb9It is now conv
26、enient to make the substitution=-1Expression (C20) then becomesJ(+log. )4l+sin v dv6 lsin v sir? v sin% (C21)which is exactly in the form of the fourth integration formula of table 335 in reference 28. This formula gives the value ofexpression (C21) as u-K(k). The integration of IJ may now be expres
27、sed asI,= lc(T+W) TK (k)Using the same integrating procedure for I, 14, tmd 10 as justoutlined for 1, and the integration formulas in tables 335 and336 of reference 28 leads to,=+(l+m +3 Wjl+K(k) E(k)71FProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,
28、-398 REPORT 1268NAIZONAL ADVISORY COMMITTEE FOR AERONAUTICS 1I,+l,=mfi(l-kycm(1P),-,; z(l/P)k%K(k) W% +K (k) E (k)1Pwhere II 1(1), isacomplete elliptic integral of the third kind With modulus = and parameter (1W).s umming all the various parts contributing to the third term in equation (C8), includi
29、ng the common factor, gives thefollowing expression:17k3+mz (C23)The addition of expression (C23) to expression (C9) completely evaluates Expression (C3) gives the evaluation of , ,and ). Before writing the total integration, that is the sum of expressions (C23), (C9), and (C3), it is desirable to c
30、ombineexpressions (C3) and (C9), which are functions of . 1(C25)By use of the process commonly known as interchanging the amplitude and parameter (see pp. 133 to 141 of ref. 2fI) the elliptic“k) This operation permits he exprcw-integral of the third kind appearing in equation (C25) is found to be eq
31、uivalent to .Idsion for u/x (eq. (C25) to assume the form given in equation (A20).REFERENCESI. Ribner, Herbert S.: The Stability Derivatives of Lon=Aspec 6 is positive; V m shown is in plane)$(zb) (Bm+l)5(l+Bm) m(z+Bz) 01- xv I z (11+111I) I1 TTBH(Bm)4_”v BmProvided by IHSNot for ResaleNo reproducti
32、on or networking permitted without license from IHS-,-,-. . . . . - REPORT 1268NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTABLE II .FORMULAS FOR PRESSUREDIFFERENCE COEFFICIENT DUE TO CONSTANT/-y(Tail is in zz-plane; is positive; V as shown is in -plane)/won (sees etch) y (X,z)I I I1 II I 4flm orlm z
33、(2Bm 1)x B%F-1 mxzI III I 4fim ml mz-z+2(l+Bm)(z-b) mzIV z (11+111-l_)2flzH(Bm)vBSProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL AERODYNAMICS OF THIN D30LAD VERTICAL TAILS AT SUPERSONIC SPEEDSTABLE IIIFORMULAS FOR POTENTIAL DISTRIBUTION
34、DuE TO STEADY ROLLING(Tail is in positive rolling)Region(see sketoh) p(z,o+,z)I Pz(il)tfl (mZ-z)-m+z(2B%l- 1)_ m(4B?n-1) +3%#-2BT?a-3) /7?az(z-Bz)_II x 3(B%P-1) Bm+ 1(mZ-;)mz-z(2B%#-1) 08_l mz+s(l-2Bm)2(B% l)ai mz 1P(?nX-z)-mZ+z(2B%X-1) W_, mz+2(l+Bm)(z)X(*2 l)m 2 111 m z(wb)(ml)+(z b)(32B% positive
35、 rolling)/T(sees “etch) yp (Z,z)I I I 4pmm*z- mz)v(f- 1):211 rv (E1) l=4pm2(zB%m) rl mz+z(l-2Bm)(*,1)3B m z 14pm(fiiz-m) cog-l z+2(l+Bm)(z)_III .V(mn?-lp mz-z2Bm(z-b)(Bm+l) (l+13m) nz(z+Bz) IIV z(II+ III I)2p m positive yawing)(se!%h) p(z,o+, z)I 2(lPm l)m (2m+) +tiZ-#m z(5B%i+4Bm-6) B%z(2Bm+l)B- 3(
36、17m2-1) J%H+11(me z)mzlF+cc(mEP-2) CO*., m zBm- 1)2(mfP-1)3 mz 1 wc(5B%i-4Bm- 6)+ B%:(z-b)(2Bm-1) +Bmb(4+Bm)r 3m(B%i l)zfl l(zb) (Bm+l)b(l+Bm) m(z+Bz)+III(mc-z)fi(mz+z) 2mz or, mzz+2(l+Bm) (zb)2m(B%g l)Jfl m z 1IV z(II+III-1)v Br(.,z+ UJW) /z(mxz)Provided by IHSNot for ResaleNo reproduction or netwo
37、rking permitted without license from IHS-,-,-. . . _REPORT 1268NATIONAL ADVISORY COMMTITEE FOR AERONAUTICSTABLE VIFORMULAS FOR PRESSUREDIFFERENCE COEFFICIENT DUE TO STEADY YAWINGvzr/-Y/Re “onf(sees etch)IIIIIIIVv I(Tail is in zz-plane; positive yawing)$ (X,2)4r mdm-2) + 21Cos+ n=-fry: 1) +TV(E%P 1):
38、2(B%?+B? 1) Jl) (ZBZ) 4r z.+mZ(B%n-2) CoS-ti-z+2(l+Bm)(z-b)+TV(B%P l):E mz z2(B%F-Bm-1) 4(zb)(Bm+l)P(l +Bm) m(z+llz) z(II+III-1)2Br .,m + 3m*wzz2nw$v /z=)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL AERODYNAMICS OF THIN ISOLATED VERTI-
39、CAL TAILS AT SUPERSONIC SPEEDSFORMULASvTABLE VIIFOR POTENTIAL DISTRIBUTION DUE TO CONSTANT LATERAL ACCELERATION/-Y/(Tail is in zz-plane; positive in positive irection)r I IRe onT(see s etch) I $+,()+, 2!)I I _j(mz-z) v_mW(mx-z)fi 2(B%-1) 1+ 2tV+Wmx(4-Bm) WmBz(2Bm+ 1) +II T %11 3B(B%# 1) 1orlmzz(2Bm1
40、)mz 1 Y%zY+vl (=-) hmz)+vt-* *Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT 1268NATIoNm ADVISORY COMMIT1EEl FOR A13RONAUITCSTARLE VIIIFORMULAS FOR PRESSUREDIFFERENCE COEFFICIENT DUE TO LATERAL ACCELERATIONRe onr(sees etch)IIIIIIIVv(Tail is i
41、n zz-plane; positive j in positive irection)$(.,.)4(mz-z)(l+mv(B%#-1)34b) 4B(2AB + 5) _B(8AB3)r AB(All+2)p9 f3A3Clr (TAB 6 T,+% ) 23AW+ 9AB+ 5) 3AB 14 3AB(AB+2)Pfi 3A,Al%4 .+2#-$;$m)_Cyp 8 DABI _2(-W ./-W.Clb 2 fl+AsBz+3AB+3 P+23ABABr =2U,J+; rlW ii%”)7(3AB + 2Bm) H(Bm)6Bm%ZEP H(Bm); 2Bm(zB4ti T, +-
42、J- )(32Bm 16AB * )J-+ -!- - “J w%gi%ati:%ur%tan- q (positive forward) and zo(pceitive downward) from the tailapex (see fig. 2 (o)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-410.REIPORT 1268NATIONg ADYfSORY COMMTIT13E FOR AERONACS(o) “E(b)/44/ ,/
43、/A (a) Plan forms of ve.rtkal tails analyzed. (Trailing edge may beeither mveptbaok or mveptformwl provided it remains supemom-o.)(b) Speoial cases (rectangular and half-delta verlioal tails) for vddchstabl%ty-denvative curves are presented.IFIGUREI.Tail plan forms and associated dataIzr, Nv(o)(b)r,
44、 NTzvy, YT-p,Lx z0 “o Tr,N(c)z(a) Body-ax system used for analysis. Free-stream velooity V.(b) Principal body-axes system used for prmntation of stabilityI derivatives. Entire system moving with tight velooitv V,(o) Same type of axea system as (b) with origin tranelatod.FIGURE2.-Syetema of body axea
45、. Positive direotione of axes, foroes,and moments indicated by arrows.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-THEORETICAL AI?RODYNCS OF TEIN ISOLA!IIDD VERTICAL TAILS AT SUPERSONIC SPD3!lDS 4111.141.101.061.02.98.94.90.78.74.70.66.62.58.540-0
46、4-.08-J 2-J6, , , , I , , , 1 , 1IllI I I I I I I I I I-20-.24-.223-.32-.36-.40-94-.48w-.52HII-56-.600 J 2 .3 9 .5 .6 .7 .8 .9 1.0BmFIQUIi 3.Variation of the parameters , UP, Bnti, .= and 7, with Bm.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-412
47、 REPORT 1268NATIONAL ADVISORY COMMITTEE FOR AERONAUTW3Fmu, , ./ L.I/I / Y -cn,., /3/ -/ -/- -H10 I 2 3 4 5 7 8 9 10 20 Qprincipal body-area system tith origin at leading edge of root section.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TE3iORETICAL AERODYNAMICS OF THIN ISOLATED VERTICAL TA
