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UI I, ACR No. L4L13 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WATRTIMIE REPORT ORIGINALLY ISSUED December 1944 as Advance Confidential Report L4LI3 CHARTS FOR THE DETERMINATION OF WING TORSIONAL STIFFNESS REQUIRED FOR SPECIFIED ROLLING CHARACTERISTICS OR AILERON REVERSAL SPEED By Henry A. Pearson and William S. Aiken, Jr. Langley Memorial Aeronautical Laboratory Langley Field, Va. t: tItt~~ A ..1 WASHINGTON NACA WARTIME REPORTS are reprints of papersoriginally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre viously held under a security status but are now unclassified. Some of these reports were not tech nically edited. All have been reproduced without change in order to expedite general distribution. L 187 Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/chartsfordetermilang NACA ACR No. LTL13 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADV..!i;CI_ CONFIDENTIAL REPORT CHARTS :F, THE DETEREI.i..TIlON OF vVING TORSIONAL STIFFNESS REQUIRED FOR SPECIFIED ROLLING CHARACTERISTICS OR AILERON REVERSAL SPEED By Henry A. Pearson and William S. Aiken, Jr. A series of charts are presented by which the wing torsional stiffness required to meet a given standard of rolling effectiveness may be quickly determined. The charts may also be used to obtain quickly the aileron reversal seed and the variation of the loss in rolling effectiveness with airspeed. The charts apply to linearly taoered wings and elliptical wings of tubular shell construction having various aspect ratios with aileron span and location of ailerons as variables. In the derivation of the charts, induced lift effects have been taken into account and the form of the wing torsionalstiffness curve has been assumed. INTRODUCTION In order to insure adequate rolling control at high seeds, present structural requirements for Army airplanes (reference 1) specify that the computed aileron reversal and divergence seeds be at least 1.15 times the terminal velocity of the airplane. The accuracy of such compu tations depends on the availability of aerodynamic data, on a knowledge of the wing torsional stiffness, and to a smaller extent on the method of computation used. Since the terminal Mach number for fighter airplanes is approximately 0.85, aerodynamic data should be available at a Mach number of about 1.0 if accurate results are to be obtained in rollingperformance calcu lations. A yach number of 1.0 is.considerably higher than that at which highspeed windtunnel data are available or to which they can be extrapolated. 2 CONFIDFNTIAL NACA ACR No. 14L13 Data available on the torsinoal stiffness of wings ave Indicated that the calculated values of wing Yorsio nal stif" necs arc likely tu vary considerably from be toet values. Unoublished test data have indicated that, for wings of the ane model, significant differences .n tri >ual stiffness can be expected as a result of differ.rces in "abrication. heny methods are available for commuting speeds of aileron reversal and divergence. (See references 2 to 10.) These rmethods differ mainly in the combination of assumnntions used rad in the degree of exactness attempted. Tn Lmot cases, the foir of the wingt;.vt curve has been assaued. to be linear or carabolic and the induced lift effects have been eil'hr enLir.ely negected or antroxi mr.ated. In one fc t0 e I;: roc's (refeur..ces j, 6, 8, andl0) the actual ::ingtrslIonal3ti stress distribution is required and eii 1 i le i est ilisl cd between elastic end arod..ic :'re s all *lon th. .:ing scan; with the excetion of ti r.t. of re:c .r .nce ., however, these tl.etithod ei Ler it lIuced lift u ffcts or apn2oximate them. Ihe iToplctatin of ti c:act r:.thod of reference 4 requires, cor....g the a,~tin r, g100 man hours. Tnasi.luch a the accuracy to ee ~' :ined by che use of the rest exco ime tno, w vi ich includes induced lft effects, is only a'bot 6 percent as cowparedi with methods employing ctrin ti ry, the use of cuhe ':1 exact rme;hod is cons.cidred I'icacti al when th e nossiEle inaccuracies in the ard "na ni.c and structural quantities are considered. Soe3 'cro'ver.ent in the roui'.rercnts can be obtained Ly s ecif n t hat the loss in rolling, effectiveness due to a>~i'n tw A. at either terminal velocity or highsoend level fli'lt shall not exceed soe i fixed percen'_.. of that at I F:. eo,ed. ch a c'sc afication .iould then confo)rm better to maneuverabiliTy reire!iemets and would not require as r:uch :xtraolation of the aerod':.. ic data, P rther irrr.ow:,..ent could be o.tainmd by requlring that, fter fabri cat'ion, tbe torl onal stiffness of each lwng nanel be greater than a specified value. ChMarts re ,ivn in the CruP.ent laoer ironic which the rc.q:iirtca inn tar:iona:.L st!ifness may be readily obtained if a given loss in rolling ability is not to be excci ed The charts .,:, bt used to predict the aileron reuve rsal so:ed ani tni los: in rolling effectiveness at any other speed. The charts arc tased on the application C ONFI I. 'AL NACA ACR No. LlL15 of the usual liftingline theory to wings of tubular shell construction and allow variations in wing tanpr, aileron span, and aileron position to be taken into account. In the derivation of the charts, the form of the wingtorsionalstiffness distribution has been assumed and as a result the twist curves for a given wing vary with the aileron span and position in a manner to be expected. The advantages of the present method over others is mainly in the speed with which results can te obtained. More nearly accurate results are obtained than with methods that assume the shape of the wingtwist curve to be linear or parabolic. SYM BOLS Symbols used in the Dresent paper are defined in the following list (see also fig. 1): S wing area, square feet c mean geometric wing chord, feet (S/b) c wing chord at any soanwise station, feet ca aileron chord b wing span, feet A wing aspect ratio (b2/S) y ,risoan station, feet k nondimensional semis'an station \b/2/ Swir. t.aer ratio; that is, ratio of fictitious tip chord, obtained by extending wing leading and trailing edges to tip, to root chord PG wing torsional stiffness as obtained by application of concentrated torque near win tip; footpounds per radian COIPT DEI7TTAL CONFIDENTIAL NACA ACR ::o. LLl5 t torque per unit span, footpounds per foot Z accurflated or total torque, footpounds o angle of '::ing tw:"st, radians 5 aileron deflection, radians c/cr/d ra2te of c ' e o section pitching.conient coefficient with aileron engle as obtained for incot..pressible flow, per radian da/d6 rate of change of wing angle of attack v;ith aileron ari~l3 as obtained for constant lift at section pb/27 heli: angle of roll, radians Le rolling mo::rnt due to antis..:strical wing t.' st, I'o tpound CL rcllfingi:. :..nt coee 'fficient C5 rate of change of rolling_nir.ent coefficient with aileron deflection, per radian CI rzate of c: f e )f rolling~ Orent coefficient : ith heiix an.le pb/2;, per radian C7: rate of cha nge of rolli ::m.orent coefficient with wing twIL.st at reference section, oer rad an 7 derived constant for airplane (see appendix) She1lixanle arar:. ter t fr ction of ri gd ..: rolling effectiveness to be retainkdc7 by flexible wing C rna;'.c )r.cassr, r inc Dor square foot pV2) p ocnslit, sl s nor cubic foot V true ~lrs;d, fxct oer second Syndicated airsoeed, miles _r hour C )H :i.. AL CO:; IDE ,TIAL NACA ACR ":. LLL13 cpeed of sound, feet per secor.i .a ch nunuber (V/a) Glauert c r.essibility coirection factor Subscripts y and k denote a oartfcular saanwise station utboirdd n of aileron inboard end of aileron reference station taken as r idaileron c..n refers to nartlcular sran le of aile ~n san, region 1 refers to oartici.ar span.vise of aileron, region 2 refers to particular saanui:se of aileron snan, region 5 station inboard station in way station outboard L limit diving speed R reversal seed CF..7 The desin of ilerons that will provide adequate rolling control at high speed requires the determination of (1) the w.in': torsional stiffness r, uired to meet a :ven standard of ro'liii effectiveness, (2) the aileron reversal :.d, cnd (5) the variation wi ch airspeed of helix angle of roll ocr unit aileron deflection. The charts oresen. herein hae been prepared in order to facilitate the comutation of these factors. L.. results aoply to wings of tubularshell construction, of various asoect ratios and taper ratios (including elliptical wings), and .'.it: various lr. ths and positions of the aileron alo.: the soan. .. various olan 'orms considered are shown in f.ie 2. CONFIDENTIAL COi FIDE"I7IAL /VTF2 NACA ACR No. L4L13 The derivation, which is given in detail in the anrendix, is based on an aoolication of the liftingline theory to the wing plan forms chosen. (See figs. 1 and 2.) In the derivation, the wing section aerodynamic coef ficients were assured to be unaffected by torsional deformation and the shape of the torsionalstiffness curve was assumeed to be inversely proportional to the cute of the distance from the wing center line. It was also assumed that the ratio of the wing chord to the aleron chord was constant and that no twist occurred about the aileron hinge axis. e results, which apoly to wings having aspect ratios ranging from 5 to 16, are presented in the charts of figures 3 and i. Tn1 figure 3 the nondimensional coefficient T is given as a function of ki and ko. This coefficient T is directly prooortional to the rollinigmolent loss due to the torsional deformation of the wing and is inversly proportional to the dynamic pressure. Tf it is desJ ed to determine the wing stiffness required at the reference section (mida leron) to retain a specified rolling effectiveness, the following equation may be aoolied: dem/di b3 q Tr = T (1) r da/d6 ?Ad( ) (14 The d Tnaic pressure at reversal speed may be determined from the equation 5. (2) da/d /A If the actual variation of dcm/d5 with Mach number is know, n, this variation may be substituted in equations (1) and (2) in place of the Glauurt coropressibiiity correction factor 1/ I 2 CO 'I DEINTIAL NACA ACR No. LL15 The coefficient '7 given in figure 4 is a nondimen sional quantity representing a helixangle parameter for a rigid wing. This parameter may be used in the equation pb/2V d () 6 db6 to obtain the helix angle pb/2V of a flexible wing for a unit aileron deflection. The results given in equations (2) and (5) may be used to determine the loss in aileron effectiveness with airspeed due to wing flexibility. Because the relation between b/2V and q/V 2 is linear, a straight line drawn between the point representing the value for b/2V at q/lM2 = 0 and the point for zero helix angle will yield the value of nb/2V at all intermediate values of q. APPLICATION OF CHARTS Determination of torsional stiffness required at the reir..:n': e t Li illn requirement. It is desired to determine the torsional stiffness required of the wing at the reference section (midaileron station) in order that at limit diving speed at sea level the airplane will retain 0.25 of the rolling effectiveness of the rigid wing. ''e following values are given: Fraction of rolling effectiveness to be retained, / (by stipulation) . .25 Limit diving sneed, VL, miles per hour 553 Mach number, M . . 0.728 Wing span, b, feet .. .. 41 Aspect ratio, A . . 5.6 Plan form . . Elliptical CONFTDiEIT'IAL C0iI DEUITIAL G CONFIDEI.TIAL TACA ACR No. LL13L Distance to inner end of aileron, k$, fraction of seispan . .58 Distance to outer end of aileron, ko, fraction of semisnan . 0.945 Dynalic pressure at limit diving speed, qL. pounds per square foot . 782.8 dc/dI . . 0.?6 dc,/dfi . . 0. 2 From figure 5, for an elliptical wing and with the given values of ki and iko, T is determined as 0.249. ! Ln these values arc subs!ttuted in equation (1), S = 0.2 x x 732.8 r .7 2 x (5.6)2 x 0.75 0.685 = L6,'0i footpounds per radian = 102,000 inchnounds nor degree Thus, if a concentrated to.: 'e of 10,2CO inchpounds applied outboard of the midaileron section produced less than 0.10 wing twist of the reference section relative to the wing Ioot. the wing would exceed the required stiff Iess. Dtterniration of aileron reversal speed. The same quantities uned ic thet previous examilte, together with an experimentally determined value of the wing stiffness mn (equal to 527,jOO ftlb per radian, r = 0) may be used to obtain from equation (2) _R 2 x 527,000 V\ C.2 9x x 1 .65 pouns p urf.6) 1652 pounds per 'iuare foot CONPIDEITTIAL NACA ACR No. LLLl5 From figure 5, which gives the relations between q/l'12 1 and VC the reversal speed at sea level is determined to be 619 miles per hour. Determination of variation of helix angle of roll with s Cee.d . 1 iL i l ..* ::. _. p 1 .i ,. j r airgI': r the rigid wing ( = 1.0) is found from figure 4 and equation(3). Figure 4 applies to all normal taper ratios and aspect ratios. For the airplane of the example y = 0.91, and since da/d6 was assumed to be 0.56, pb/V for the rigid wing is 0.328 radians 6 per radian deflection or 0.00575 radians per degree of ob/2V q aileron deflection. at this value of 0. qR If  = 1652 pounds per square foot, the corre Vl 2 spending helix angle is zero. The helix angle for any intermediate value of q/vi?,: or airspeed may be determined by drawing a strai._ht line through these two points. Discl.: r1n :,f e::''.i:s. In the application of the charts to :.et.crr.':ne n ...r stiffness, reversal speed, or helixangle variation, certain quantities will be obtained from the geometry of the wings; other quantities will be determined either ... Performance sDecification or by the aerodynamic characteristics of the reference airfoil section, which is located at the midspan of the aileron. The values of b, A, ki, and ko can easily be determined from the geometry of the wing. In equation (1), the value of must be specified and q//l'2 must be known. An alternative method for determining the aileron reversal '..: nmay be used instead of equation (2) in cases in which the variation of dcm/d6 is not a function of 1/1172. This procedure involves the calculation of at speeds greater than the limit divin r speed by the use of equation (1). The values of dcm/d6 and da/d. corresponding to the Mach numbers of the airseeds chosen are used and a plot of J against Mach number is made. As the Mach number is increased, s a oiroaches zero and when reaches zero, the aileron reversal speed is determined. These computations may be made for various CONFIDENTIAL COITFIDENTIAL NACA ACR Ho. L4L15 altitudes to determine the variation of aileron reversal soeed with altitude. In equation (2) the altitude must be specified and the value of the torsional stiffness mr must be known either from tests or calculation. At present, exoerimental values of mY are greatly referred or e since the calc'.ulted values may be considerably in error. The experimental value of the wing stiffness may be easily obtained by applying a unit torque of about 20,000 inchpounds to a section near the win tip and determining the angle of twist at the reference section relative to the wing root. It is recommended in reference 9 that in order to obtain best results an antis:trmetrical torque be applied to the other wing tip. The quantities du/d3 and dcm/d6 are aero dynai c quantities that anuly at the reference section. In general, they ;ill vary primarily with flaochord ratio and Mach number and secondarily with other variables such as nose shape, gap, aileron section, and cngle of attack. Figure 6 has been preonred to show: average variations of these quantities viLh flap chord and gap ratio. More accurate values can, hc'aever, be obtained from specific tests of the aileron section used or from reference 11, which n'al"es Lhe data obtained from a large number of windtunnel tests. DISCUSSION The agreemrert that may be obtained between the calculated valves of v.ing t:2ist and actual values of wing t.iist :!hen the form of the1 wingtorsional stiffness distriLtion is assuc~d is illustrated in figure 7. Co. nationss ee node for the P!.7B ?:"i.yS by usne of stifns c .ba fwurnished b, the Army Air esi r oces r Technical Service Command, ;ri.,'.t Field, Ohio. The twist curve resulting from C(.".'1fr:'TTAL CONFIDENTIAL NACA ACR Yo. LJLl5 CO:. IIDZ.:7:AL 1 these conutations is compared in figire 7 vith a twist curve cormuted by assuming that wing torsional stiff. ness varied inversely with the cube of the distance from the airiane center line. In calculations involving the use of a *'ingtvist curve, more consistent results vill be obtained b5 assuring u cubic stiffness distribution than b assui,:ing tne wing twist curve to be either linear or parabolic. Tr a practical case the trailingedge portion of the wing usually contributes a negligible amount to the wing torsioral stiffness, but the twist curves for a given wing will differ considerably with aileron span and position since the typist is dependent upon the magnitude and position of the applied torques. Such a variation is included in the results given by the charts. Equation (2) shows that, other things being equal, an", increase in T lowers the aileron reversal sr ed. From figure 5, an increase in T is seen to occur when the aileron sPan is decreased about a given reference position. For an ellipticall wing with ailerons exter., i.g from ki = 0.1q to ko = 0.8, T is therefore 0.L67; whereas, for the same wing with ailerons exten.1ing from ki = 0.2 to ko = 1.0, T is 0.588. In both cases the reference section is located at 0.6 semisnan. In order to determine the wing stiffness required to insure a specified rolli.. effectiveness, the largest value of q/Vl2 obtainable should be used. If the present Ar. Air )rce specification is used as a guide namely, that the reversal speed should be 1.15 times the terminal velocity of the airplane calculations show that the value of / to be used in equation (1) would in .rneal yield overly conservative results for wicr torsional stiffness. Another procedure for deter. inilg the wing stiffness would be to specify a value of ( at highspeed level flight. Either specifi cation is believed to be more useful than the current COI.' UTID EITIAL 12 CONFI .. iTAL NACA ACR Lo. LI.T one (VR > 1.15 V,) that yields results at Mach numbers beyond which data would be comnletely lack ng and that ouldntrodue omlicatio inLrcce c licti in the equaSions. As ;n illutration, if an airplanee wore capable of reaching a Yach nu:nber of 0.87, the design aach namnber vould be 1.0. Vith Glauert's approximation, which Is usd v:ith the present method, the required .ing stiffness would then be infinite. In recognition f this di fficuly, Victory has introduced in reference 9 the concept of an equivalent Yach number while still retaining the require ment that the reversal soeed be 1.15 times the terminal velocity of the airplane. Reference 9, in introducing this concept, interprets the present requirement that VR > 1.15V L as referring to an indicated speed in an inconipre ssible flow . Grinsted, in refi;reno. 12, has Si..ested an alternative procedures nam ily, that the wing stiffness be determined so that aileron reversal would occur at limit diving coped and that this value of aing& stiffness thn be increased b the factor (1.15)2. From a consideration of references 9 and 12, together with current iirmy tir Force requirements, the foll~ ng values of V/ and q/i h2 are recomnended for use with the charts presented herein: Method I. th lir:it diving speed as a basis, 1 i i use 2 = and q/'1L at limit diving seed at sea level. Method IT. ith highspeed level flight as a basis, use and the largest levelfliZht value of q/"'i, 5 regardless of the altitude at which it occurs. e enrpl snocifications that have been advanced, the wing stiffness at the reference section has been coirOuted for the air plane used in the example. The following table shows the sbiffnosn as obtained the use of the various require ments: CO 1 LYT L NACA ACR ro. LJL.L13 Requirement Method I L at VTi .ethod TI =f sats ma: fliLt seed ximuin le Army iJr Forces, VR = 1.15 VL Victory (reference 9), i'IR = i.0.9,o6 r Grinsted (reference 11), m 0 = 1.32 m9L where S is the stiffness tas:i = assuming VR L \L Sealev bas"c so 553 el .25 1.15 x 5 1.0 b x 1553 _..._] .~ el [1 (tlb r rdia ) ) :Jed i 353 Also, for comparison, the follo : in nwuaerical values of mn are listed for the airplane used in the exa.iole: Source Conditi on (1) EzXerimental Amm'unitlon doers closed E.."r. rilental Am'uni tion doors open Calculated ammunition doors omen (f cl /ralc an) 527,000 351 ,ooo 209,0oo 'E::rerirmental data furnished by Army A Fr Force., Air Tecinical Service Colmnand, :iI ; t Fiold, Ohio. Calculated data r.., iepoublic Aviation Corporation. COI; 'iDENTIAL CO~~ICF~TT~L St .iffn:,.. 01, 000I 721,ooo i1 CO'PIDITTAL NA.CA ACR T'. L . CO'CL7DTNG ThAR S Ch,,rts I ave beea r~reorred for use in ildterminin.l; the wing torsio nai stiffn3sc foar v.i:Lig of ti'l .r shll corns ruct ioi wi th 1as3ct ratios rarnirn fro! 5 ico Vol an! taper rat os ranging fromr 0 to I ircluiin rYe el3 1nt: C 1. The loss in rolling effectiveness Ean the al .lcr)A reversal srced *iaaT also'be clc.ullte 1 chioai advantf.ge of the present method over areviuns net.cds is tle seced witi w:i ch ic; r eSrlts ya y o obt ained. M or accurate resv.lts ray :e ouai neu y tri use of this imiethod than by the use of nmet] ods thait Esruie the shape of the wangtrl st curve to bro linear r: ya.ic Langley etemorial Aer tacticall Ia'oraory iutiona] Advisor7 Colia'ttecs for .ronautics Laxgl] 3y THl Va ,.. C :' :) : :A NACA ACR No. L)L 13 APPFINDIX DERIVATION OF CARTS Although there are a number of types of air loadni and inertia loading that contribute to the wing twist about the elastic axis of wings in flight, so far as the problem of rolling effectiveness is crncerned, Jnly the twist due to aileron deflection need be considered. In fact, since the distritutoan due to darpi n in roll is likely to be almost the saie3 a3 the s&onw.ise air load distribution resu!ltt. from aileron deflection, only the increase in section pitching mo:6cnt in may of the aileron need be taken into account in deterIi ning the wiLngr twist. A strio of the wnrg dy in vay of the aileron (see fig. 1) will have acting on it an Inccrement in torque as follows: cm (lc2 At c.y = 5d7 (AI1) The factor 1/ i_2 is introduced in equation (Al) in order to increase lowsoeed values of dm/dc for ilach number effects. Tf the correct variation of dcm/d i s available, the quanti: (dcm/d6)(l/!d2) may cb replaced by the actual variation with I.ach number The accumulated increr.ent in torque at a particular station yo in way of the aileron is :7o AT = At diy (div 1t2 and, similarly, the accumulated increment in torque at any station Yl inboard of the aileron is Yo ATy = at dy (a4) 'i CC IDENTIAL COI7 iL.'AL 16 CONFIDENTIAL r GA ACi No. TI T 1 Tn thei der1E tion of the ch~rtl fIr t.e z dte r..l ITi: n cf vig torsi)nal scifnf,e required for specified. ro li*ng characteristics or aii ron .reer.,o':al soeeo, i c desirable to use the win centr line ).s the reCference n3. to define ;he to 'ioial stir or a Wn of tu ; ,:ll rsFell c ist cti on as the co ( centC.ated tr which, wxhen aplied outb>o.rc o' a ,vcn .nction, v)  pri duce a unit reflection vjith resnc ct to the rcfeir nc.. section. A]tihough this ,diflntion of *h. torsii:ul stifess r':es the eariLcal devoiopier;. somc, at longer, it is better cuitec to the test Droce4ures elat are nowv in use ihen tle bor ionaltiff ness varia tiJn along toi :sran is to 'c duetr4 lCd. The aiglo ol tu.lst 72 L T' o c t a t j n" i 1 due to aincron cef.tctaion i a It n nation 13 way of t sa:"or, (rcgin :. 1) 's .tus *iv n bi =  I t + cy ( +) i The t';ist at any station Yi in rein 1 inboard of tLe a i r 'n is 9 n n 1. O ,:i Tn region c itboard of the allron :c t1he ': it is r5 t / k d ; ,..c) t> o Gince r' antitsv.rctri cal trquj; ti. cng outbor'Cd of t'i. ail < n tiol . is constant to thi in; ti:). .y C subst fiuttin,( e taiion (A!) in .equation (. .) and r tru ci , a < "[ u'l'J.i *l)n ::,.:y b: obtai :.id. This equation a; tl :i e nut 'i.. r' c rcnient fjr', by miltiolyir.I eac. t r.i.: ,' th, ri,tio of th s 1il'n.j:.' mn,> to te square of tro r.ean geriomtric C) D IAL !:AC. ACR TT. LjJ.L13 CO"TF'IDENTTIL 17 chord c. The resulting" e 1..tion iay be rearranged to give the following equal tion arplyingr to region 2 (for convenienc the factor 1/ 'I/1:7' will be 'rouped with instead of with dcm/di): Sk2 2 kc\2 1 . ; dk + / M () /.Cj eT eiquations for thL wins twis at stations irboard and outboard of the a1leron (qu.ations (A5) and (Ao)) siilarly become __ __ /c\ (we S/C i dC  q/ b L'k^ *c/ G.O 6k3 do Equations (AT), (AS), and (, ) define the anrle of twist in the three regions in tecrns of tho chord and stiffness distribution, subject to the assur.ptions that dcm/d6 is a constant &alon the aileron oan and that the aileron does not tvist about its hinge axis. Inspection of figure 6(b) indicates that the factor dc;/, d is essentially constant for flaDchord ratios from 0.2 to 0.3 and, since the variation of allronchord ratio along the span will normally fall within this range, the assumatio.n is justified. In order to evaluate equations (A7) to (A9), the tvist carves will be obtained in terns of the twist er at a reference section, which will be taken at the midspan of the aileron. From equation (A7) the twist at the reference section becomes COP I''Tl !7 AL !GIkC' '73. T'l 'i U / + i .i th .e sti ff.'i s 3 s ef 'i d I the ..res; it r ' the torsii.l s. iffne'.s is in nite tc t. Li c  1.i.e a Jecrea:'cs : th Jist&nc to 'or .e iriit3 v tL.le at t}.e cbip.. i.' ely' i o data or c.ic fi i'rrr ... c.l:.iec 'ndlcates thct :;f li, 1le ,rr os :ill rest in t.ist. c~n:p.tatflons if tle o'1sl.hal stzlf no.ts i': trl :t tion aa1 on the svran i' as,.;l:..ed to bo 1' ... (All) ;,hen equation (All) i" substltit:d nd (AQ. L:U follo'w:n; raios fr 4 n the various r. 1n:: 1: c' C' :i t\ / SL r II o bLrS?.cd ~~~ #: \~' i ~".~ '; \ S ,iA ,2. I i! I I I 7 * +  1  1  r \ H1 CI I N 1j *~ WI A '1 ' " *l t :ii +' S'. I r 1'7 I' ,. *7 i I ( I 1 i 1I\:j i I ir? F C = 'TIAL 3 , ir:llr r :L"i C3 (,,10) ~ ,I I i i 1:ACA ACR I'o. LL13 CO:' TDETTr!AL 1 It will : noted in equations (A12) that only :;:etrical terms such as spanwise extent of aileron scan and chr ratios c/c occur and that, in order to determine tle resultant twist distribution, only these values need be specified. In reference 15 influence lines are prcsented f r a series of tapered wings (see f2. 2) of several asecrt ratios, which nmae postble the co:putation of a cef ficient of rollingmoment loss Cla ..e to any sort of twist distribution. As a first step in the evaluation of a loss coefficient Cl., the ratios of t6/6r were evaluated for the series of wi.s shown in fi.u c 2 with ailerons cf various span. The loss in roll'..r mo.,ent due to a twist , at the reference section, was then defined by the equation Rolling mon:ent loss = L where dC 9 d9r (Al4) dCL/ dBr e/9, 9r The results shown in figure 16 of reference 15 were used to determine C69 for the twist variations computed from. equations (A12). T.,e coefficient C w was also determined for elliptical win's of aspect ratios 6, 10, and 16, with values of ki of 0.2, 0.5, 0., 0.5, 0.6, and 0.7 and for values of ko of 0.8, 0.9, .:. 1.0 as well as for the win; ulan 'orms shown in figure 2. 1. numerical results of these steos are not iven herein because they are only intermediate steps in the procedure. CC .r :T.T IAL 20 CONFID IAL NACA ACR No. Li L.1 In the steady rolling condOition, the damping moment equals the moi:et ii pressed ' t the ailerons minus the loss in omnen; due to twist. In coefficient forln, tilis relation rnay be expressed as C b Co 2 qjb = C Sb CLerqb (15) from which the helix anile per unit aileron deflection 6 is obtained as C 7 C (A16) C 0 The coefficients C0l, C7, T and Cl will v.:' with Mach number but of these coefficients onl the variation of C viith 1,ach number can readily be determiined from windtunnel tests. At onresnt C' must be obtained r) either from results of lowsoced te.ts or front results of computations and C T rust al: ays be obtained by coumutation. For this reason it would anpear reasonable to use consistent values and to assu:re that each varies with ;:ach nu:1ber according to 1/j Fro;"i equation (al0), for a particular wing aileron coo>bination, to r1 ( Al17) where the constant e1 equals tle ri :htiand sde of equation (`10). Also, f! eq :ton' (Al) and (Al), C9 is s cn to Ie conc' .nt for a .art cular :rngaileron conoination. '.1.hn these value:; of CA@ and Or are substituted in equation (Arl), the f)llowinf equation re sults: C ON17TL i:'"T TAL NACA ACR No. L4L13 rnb/27V 6 where the constant CONFIDE1NTTIAL ~C, ____ _ C Bt = C 3, At the aileron reVersal seed, d cm_ C6 2mr C5 nr, 01 Q "z~r qi T 2 and the value of the dynamic pressure is q lM 2m r dcm2 dCb d6 C2 ':'2 Mr 0 dcm b3 d6 A2 Cl5 da/d5 da/di B2 By setting B2 !da C T Sd the dynamic pressure at aileron reversal sPeed is IR 1 ' 2m, dcm/dO b3 da/d5 A2 (A20) (LA21) CONFIDENTIAL r l8 (A19) 22 COFFIDE 'TAL T .C. ACR No. L In o:e dterni'ntion of the values of T the ~ ces: ry nrur cci ci v all.e of L 'ere obtal b7 the } roc \re outlined a.d the nezcs. lcs  .. ere ten rctl fr: ur 6 of ref ence 1. le; te cf r .:re plotted the results ;v. ee ouId t3o I es'nt'Illlc th3 sauis for aspect rato. o"' u, an. 16 aver.,e deviation a lessa t 1ha: 1 rcen. Te nval to of di how ver, vary vit lli.ln p ltion sn1 '.in. taper as fo2o1 in f l.2re 5. Dr an inrini ly ri d ..r.c, x .,; per de:ree aileroI n eflectoin can 'e (obtaned from equation (Aa8) as vthlrc C;7, wa cbtair..ed fron fi' ... 3 of refr ncc I. Fi ur e } ires t:e vloues oif r i.. .i 0 >X:3 & . Ey seccif ng t? Liat the file :tn1 r etain o.1ic fraction of the ri id vin ro i i1 effcctl venes; at;I spe ifid ca ynar,.ic orcsrs u (scay, t ....nal vel.cit'), tht. fol ':.in tO.qu tion r;s.ult3 y substitutin results frc e:j tt: n, (20) san (A. ) into cqualion (f.iE), the ;inj :;tit. ~ required at tihe reference s ecti n :n retn n i "1n i I 'fied valie C : 7 L.I NACA ACR 2o. L4TL13 CO:'.; 'TIAL of ro!li ; ability at e 'ven value of qi is given by dc,/d5 b3 q mer d c./d 2A2 F_) _2 NACA ACR No. L4L13 RT7 .. :;'ES 1. Anon.: Eandbook of Instructions for Airplane Designers. Vol. I, Materiel Div., Army Air Comns, 3th ed., Revision 7, Nov. 1, 1945, sec. II, pt. V, par. 601, p. 654. 2. Pugsley, A. G.: The aerodynamicc Characteristics of a SemiRigid .~ing Helevant to the Problem of Loss of Lateral Control Due to Wring Twisting. R. ,j 1 No. 1I90, British A.R.C., 1952. 3. Ccx, H. Roxbee, and Ougsley, A. G.: Theory of Loss of Lateral Control Due to .ing Twisting. R. & 1 No. 1506, British A.R.C., 1955. 4. Pugsley, A. G., and Brooke, G. R..: :e Calculation by Successive Approximation of the Critical Reversal Soeed for an Elastic ling. R. & M. No. 1508, Sritish h.R.C., 1955. 5. Hirst, D. !.: On the Calculation of the Critical Reversal S1eeds of Lings. R. & M. No. 1568, British A.R.C., 19!T. 6. Shornick, Louis H. T'he Computation of the Critical Soeeds cf Aileron Reversal, Wing Torsional Divergence and WingAileron ivergence. Meno. rep., 3er. o. ENCi51/7'F'l, add. 1, Materiel Center, Army Air :9rces, Dcc. 19, 1942. 7. orton, '. H.: Critical Reversal Speed. Aircraft Engineering, vol. XV, no. 177, Nov. 1943, pp. 319524. 8. Rosenberg, Reinhardt: Loss in Aileron Effectiveness Because of Wing .ist and Considerations F.e.. riding the InternalFreosure Balanced Aileron. Jour. Aero. Sci., vol. 11, no. 1, Jan. 1' 14, op. 4147. 9. Victory, I:ary: The Calculation of Aileron Reversal Seed. Reo. ko. S.7.E. 57, Eritish R.A.E., 1944. 10. Harmen, Sidney V.: Determination of the 'fect of .'ing Flexibility en Lateral maneuverability and a Conmarison of Calculated Ro'lln Effectiveness with Flight results. NACA nRR "'*. 4A28, 19i4. C'NFI 7.jTTAL CO.UITDF',TIAL NACA ACR No. L4L135 11. Purser, Paul E,, and Toll, Thomas A.: Analysis of Available Data on Control Surfaces Having Plain Overhang and Frise Balances. ..'A ACR No. LilI, 19.44 12. Grinsted, F.:; Ti Effect of Compressibility on the Estimation of Aileron Reversal .need. Rep. No. S.M.E. 3192, British i.A.E., 192. 15. Pearson, Henry A., and Jones, Iobert T.: Theoretical Stability and Control Characteristics of VWings with Various Amounts of Taper and Twist. NACA Rep. No. 635, 1958. COU'IDENTTAL COC 0"'T ?i DEiNT AL NACA ACR No. L4L13 Fig. 1 ^J a C F 1 i z q C I ! 80 1 2 iCS ? __ c 4Od Z .2 NACA ACR No. L4L13 CONFIDENT AL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 5//'gure 2. /Winy p/o/7 forms used /'n oao/ys3A. (/rwn reefere,/ce /3.) CONFIDENTIAL Fig. 2   wwlww NACA ACR No. L4L13 Figs. 3a,b = l~ 7 L   I t. N i, i. f i..i X I! f I I I I 'l /I     c 1 1 n < s ij iiiiin 111111 j ii' ti iii iiiin iiiiiiiii i n i 1 1111111 iiiiiiiiiii iiiiiiin  't! ^^ ^ /^Z  E^9P NACA ACR No. L4L13 < * ~tj  t z o 0 [ Figs. 3c,d It :,I I   L4) I NACA ACR No. L4L13 CONFIDENTIAL Fig. 4 /6  5    /2 .2 .13 3 .,5 ," /Upre 4~ Fo/ai or dkoq/e \ rael for \ a N \i_ _ . 6z  ex ooea a f   .v~ I _  ______' ^^^^1ii)    _ 8 02 3 __ /' tper /os A _lct /lio Ill ro/ny .a 5 b .16 0 :^ ^   CO IDEN TI\^ AL /'i .id^V^^V^ ..ki xr. .o iern l t/ _ __ EACA ACR No. L4L13 I 4 4 "   \ !  i i i idz IX, z 0 a % I ^1$ oft/w'^>/ Fig. 5 I NACA ACR No. L4L13 f I I _ /iI ____ / I I  _ ^  __ K ^ $  * u \ iy~4 '1 ") ^ h u N Na \^^ Y N Fi g. , I k 1 NACA ACR No. L4L13 Fig. 7 S. ck. 1 I_ __o_  S II* ^   C I ^ ^r ^ .S' ^ *^ V ^ ~ ^ \  c y '\ ^^  l f ^ ^ > ^  1 ^ ^ + * ; am yo who w UNIVERSITY OF FLORIDA 3 1262 08106 556 6 ! ,X ,1 1  T I , I ~ I f 
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