1、nL .NATIONAL ADVISORY COMMITTEEFOR AERONAUTICS/TECHNICAL NOTE 3030METHOD FOR CALCULATING THE SUBSONIC STEADY-STATELOADING ON AN AIRPLANE WITH A WING OFPLAN FORM AND STIFFNESSW. L. Gray and K M. SchenkBoehg Airplane CompanySeattle,WashWashingtonDecembr 1953Provided by IHSNot for ResaleNo reproduction
2、 or networking permitted without license from IHS-,-,-iCONTEN!FIPage11266771516181919222239444456:636464676980Em M!.RY.INTRODUCTIONSYMIX)LS. . . . . .a71a15.a71a15a15a15a15a15a15a15a15a15a15a15a15a15a15.a71a15a15a15a15a71a15a15a15a15a15a15a15.a71a15a71a15a15a15a15a15a15a15a15.a71a15a15a15a15a15a15.a
3、71a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15.a71a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15a15.a71a71a15a15a15a15a71.a71a15a15a15.a71PRESENTATION OFAssumptions .METEODa71 . . .Basic Equations . . .Syme;rical flight conditionsUnsymmetrical flight conditionsSte
4、ady roll . . . . oRoll initiation . . . . . . .Roll.termination . . . . . .DISCUSSION . . . . . . . . . . . .JWPEN-DIXA - AERODYNAMIC mAmwusTheS-JMatri x . .CompressibilityCorrections . . . .AEPENDIX B - TEE ELASTICITY MM!KICESDevelopment of the 1S2 Matrix .JS212 and. .a71. . . . .rS2Development of
5、the Auxiliary ElasticityAssumed pressure distribution . . . .Rolling-moment correction at stationPitching-moment correctionat stationShear correctionat station n, LSnModification of 1S2 nmtrix . . . .-.Mtrix.a71a71. .a71a15a15a15a15a15a71ny n.AEPENmx c - COMPUTATION OF iag MATRICES . . . .APPENDIX D
6、 - DERIVATION OF EwlmNAL-smm M4TRIcEsAPPENDIXE -WING-FUSIILMECE . . . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-iiAPPENDIX F - EQUATIONS FOR TAILLESS AND TAIJJ-EOOMAIRPLANECONFIGURATIO.,+.,.“,;. ,.;Provided by IHSNot for ResaleNo reproduct
7、ion or networking permitted without license from IHS-,-,-NACA TN 3030as spoiler deflection11 dimensionless spanwise station, PYe airplane pitching angular acceleration,positive for nose up,radians/sec2A local sweep angle of elastic axis, radians44 equivalent local sweep angle including compressibili
8、tyeffects, radiansP mass densi of anibientatmosphere, slugs/cu in.(P = 0.4679 x 10-6 lb-sec2/in.4 at standard sea-level conditions)Matrix notation:I_l sqpare matrix, elements of which we designatedby.use of1-1 mibscripts; for example, element j is in ith row andjth COIUIIIlrow matrixcoluma matrixo1d
9、iagonal matrix, which is a square rmtrix in which allelements me zeros except those on the principaldiagonal an, , a33, . . . 1s aerodynamic-inductionor downwash matrix in which ele-ments aij relate downwash angle at station i tounit running lift at station j on wingClf%? elastici matrix in which el
10、ements aij relate changesin streamwi.seangle of attack at stationrunning lift at station j on wingi tolmitc1si fuselage image-vortexmatrix relating imageeffects at station control points to unit(see appendix E)downwashrunning lifts1s: fuselage “overvelocity”matrix (see appendix E)Provided by IHSNot
11、for ResaleNo reproduction or networking permitted without license from IHS-,-,-61I,NACA TM 3030identity matrix; that is, diagonal matrix in whichdiagonal elements are equal to unityPRESENTATION OF METHODJh this section of the report the basic equations necessary to themethd are outltied and discusse
12、d in a general way. Details of thederivations are contiined in the various aendixes.AssumptionsIn the development of the method certain assumptions that arecommon to atifoil.theory apply, namely:(1)The flow is potential; that is, boundary-layer effects, separa-tion, and compressibilityshocks are abs
13、ent or(2) The wing thickness is smlJ.(3)A stagnationpoint exists at the wing(4) The angles of attack a are smll sonegligible.trailing edge.that tana=stia=a(where a is measured in radians) and cos a s 1.(5)m tiag-load effects except those due to nacelles and storesare neglected entirely in deterndnin
14、n the deformations of the wing usedin obtaining the equilibrium spanwise airload distribution.With regard to the structure the following assumptions are made:(1) Camber changes arising from twisting and bending of the are neglected entirely.(2) The elastic twist of the control surface is the same as
15、 that ofthe adjoiningwing structure.(3)The =gles of structural deflection e are small so thattane=sine=e (where e is.measured in radians) and cos El= 1.(4)Although the angle-of-attack changes, including those due tobending and torsional deformations of the wing, are accounted for inthe determination
16、 of the equilibrium spanwise airload distribution onthe wing, this f= atiload distribution is applied to the geometry of .the unreflected wing in computing the bending and torsionalmoments. . . . _ . . . . . ,_. .,-,$ ,. , .,:,.- . .,-. ,-, - . J,-. ., -,. :.; .v- .-. . . . .,. - . , : .,-,. . . . .
17、 . . . .: . ,: , ,:a,.: “Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3030Basic Eqllfltions.thesymmetrical flight conditions.- The fundainentalproblem involved isdevelopment of a series of equationswhich relate the spanwise liftdistributio
18、n for an arbitrary wi plan form in a given fli-t conditionto the properties and attitudes of the individual sections that form the-*If the two-dimensionalwing is considered first, the followingrelationships for lift and downwash behind an airfoil are availablefrom most stanikrd textbooks on aerodyna
19、mics:z = Pvrrr=27rrThe circulation i is taken to be such that, at atance r behind the lifting line, the resultant of thei Wr and the flight velocity V isline; that is, no flow exists normal toThen,Wr=v %and from equations -(1)and (2),%bstitutingequation (5) into equation c/2wr= 27c rparallel to the(
20、1)(2)(3)specified dis-downwash veloc-section zero-LLftthe zero-lift line at this petit.c(3)results in%eV(4)(5)(6).V . . -.z.! . . . . . . . . . . . . ., ., , . . . . . - , ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8orNACA TN 3030(7)% c/2 equa-
21、In order to satis eqyation (4),the expression 2YC rtion (7)must be eqyal to 1.0. Since the theoretical section two-dimensional lift-curve slope is eqwl to 2Yr, r must equal c/2, whichis the distance between the lifting line and the three-quarter-chordpoint. the development of the method presented in
22、 this report, equa-tion (7)is always used in the formThis shplification reqyires that the sectionthe two-dimensionalvalue (i.e., the value ofan unswept two-dimensionalwing) and that thecontrol point D (see fig. 2) be one-half ofto the rear of the quarter-chordpoint, or at(8)lift-curve slope bethe li
23、ft-curve slope forlocation of the downwashthe local streamwise chord3c/4.The essential differencebetween a two-dimensionalwing and a wingof finite aspect ratio srises from the nonuniform spanwise loading wchproduces the trailing vortices of the finite-aspect-ratiowing. Theequationspresented thus fsr
24、 are considered to apply to the finite-aspect-ratiowing when the effects of all the vortices, both bound andtrailing, have been taken into account.lkpation (8) timatrix”form is(9)TMs matrix relation represents a series of equations, each applicableto a particular station on the semispan of the wing.
25、 The values of()wv 3c/4 every one of which is affectedby the bound and trailingProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 3030vortices at all of the wing stations, can be evaluated from.-Lsl rv 3c/4 4flvwhich, in conibtiationwith equation
26、 (l),results in(lo)The 1s matrix in these eqyations is the aerodynamic-inductionordownwash matrix which is derived in appendix A.Combining equations (9)oroand (10) VeSoL mythen be written with smll error asloads and nose-up nmnents aremay be appropriately included(19)(20)consideredpositive.in equtio
27、ns (16),inwhich fuselage upThis lift and moment(17),and (18)to getthefollowing more complete set of equations(seefig. 2):For the wing load distribution,for the s,ummtion of vertical forces,(a)(22)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA T
28、M 3030 13and for the sumation of pitching moments,H+(%)a-4%)J%+%=(23)The twist term PT appearing in equation (21)has been includedat this point to provide for the possibility that tail loads may enterthe wing at some point along the span, as for a tail-boom type of con-figuration, for instance. !Ehi
29、s PT contribution is otherwise con-sidered to be zero. A method for handling the tail-boom type of airplaneas well as the case of the tailless airplane is described in aen F.In consideringthe effects of the external stores (alteration (b),as in the case of the fuselage, the lift and moment character
30、isticsofthethedixstores in the presence OF , ., -= :,-, . . . . . . .-, . . . . . . . . . . . . . . ., ., .,-., . ”., .“. , ., .-. ?. . ,- . . ” .“. .-. . .,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.NACA IN3030 151Si 1matrix (seeappendix E) to
31、 the S1 downwash matrix. The secondpart of the effect is the increment in vertical velocities over theexposed wing due to the presence of the fuselage at an angle of attack.This effect is calculated as an interference twist of the typeand is expressed as a function of fuselage angle of attack times
32、the (“overvelQcity”matrix o . See appendix E.) The lift distribution(eq. (21)when altered to include these fuselage-interferenceeffectsbecomes10 -(27)o.-where the elements of the ISo I matrti give the increments in verticalL -Jvelocities along the span.The calculation of these fuselage effects would
33、 not be required ifappropriate data were available tiom tid-tunnel tests of a scaled modelof the mibject airplane. A method of deterdning these and other aero-C Msts as welJ-as the applicable values of section lift-curveslope from appropriate wind-tunnel data is given in appendix G. Themethod utiliz
34、es equation (12) to obtain aerodynamic coefficientswhichsre free of model wing flexibility effects and which are therefore appli-cable to the full-scale airplane having a w$ng flexibili different fromthat of the model.Unsymmetrical flight conditions.- In addition to the symmetricalflight conditions
35、already outlined, a nmiber of unsymmetrical flightconditions are usually investigated in structuraldesign. Among theconditionswhich may readily be investigatedby the methods of thisreport sre those which arise through the use of roll-productig devicessuch as ailerons or spoilers. The load distributi
36、ons on an elastic wingassociated with roll-control deflectiotimay be thought of as the suma-tion of distributions from the following specific loadhgs:- .- _ _ .,., .- y-,r :-. . . . .: ) “ “ “ areSince a linear relationship existsparticular horseshoe vortex j and thebetween the strength Ij of aaownw
37、ash velocity wit at aputicq point i on the wing plan form due to that hmsebe vortex,the following general equation canbe written:where K is a constant.then causes the followingto 8:ij = “Kijj (fA particular horseshoe vortex, such as rlyvalues of d.ownwashveloci at controlpoints 1EL = %.A 51 = %lrl 1
38、a = arl 61 = K61rl1(A3)W31 = K31h Wm s rl41 = zlrl a = 81r1 1Similar relationships exist between r2, !?3,. r8 and the controlpoints I to 8,thatis,12 K12r2 22 =K=r2 . . . . . 82 = 82r21. . . . . . . . . J . . . . . . . . . . I18= %8r8 28= K28r8 “ “ “ “ “ 88= K88r8J(A4) . - ., _ T . . - -v -. - ., , ,
39、 .4. . . - . . . . . . . ,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-26 NACA TN 3030If the horseshoesof the wing plan form,typical wing sketched:K31 =%8K41 = 58%1= 48% = 3871 = %8 ,81 = 18axethesymmetricalwithfollowing valuesKM .%2=K32 =K42 =32
40、=%2 =Kp =K82 =Further, for symmetricalpossible to obtain a spanwisewith respect to the plan-form87K%7%7K4737%27%7K13 =%23 =33 =K43 =K53 =%3 =Kv =K83 =respect to the center lineof K are equal for theK%76%56%6KM36%616%4 =K-%24 = K75K34 =%5Ku = %5 = K45 = K35K74 =%5K for example,u .4i.e . - .,. T.-:,.
41、,- ,., ./ , . . . . . . . . . . . . . . . . ,. . . . . . . . . .$.-.;. ”,:,- , =”- , ,”.,.-. . ,.:.:.- ,. . ,-, -. . .,.! d -.,.” “,-.-”-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TN 3030 27At each spanwise station the S coonent is that due
42、 to the symmetricaldistribution of load, the A component is that due to the antisyunetricaldistribution, and the U component represents the algebraic sum of theS and A components,that is, the unsymmetrical distribution of loadacross the span. This division offers a considerablereduction in theamount
43、 of work required in that, for either symmetrical or unsymmetricalflight conditions,airload distributions needbe determined on only one-half of the wing, provided, of course,sign of the circulations existing overFor aand for ansymmetrical distribution of1 = r8r2 = r7antisymetrical distribution1 = -r
44、82 = -r7The total downwash velocity at abydownwash-velocity contributionsat thatof the horseshoe vortices in the systemis,that praper account is taken of thethe other half of the wing.airload aver the span3 = r6r4 s r5of ah-load3 = -r6r4 s -r5(A6)(A7)control point is the sum of thepoint that are inducedby eachthat represents the wing; that1 +ww+w=W 13+ 14+ 15+ 16+ 17+ 18 1=W2 21+w22 + 23 + 24 + 25 + 26 + 27 + 28I(A8). . .“ . . J- . n- . -m - .-. . -., . . - . ” . ,.4 -”- ,.-. , . . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
