yf\chL'i'^i / ACR No . IALI3 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT OfilGlNALLy ISSUED December I'^kk as Advance Confidential Report IALI3 CHARTS FOR THE DETERMINATION OF WING TORSIONAL STIFFNESS REQUIRED FOR SPECZFIED ROLLING CHARACTERISTICS OR AILERON REVERSAL SPEED By Henry A. Pearson and William S. Aiken, Jr. Langley Memorial Aeronautical Laboratory Langley Field, Va, I 1 \\\\ft:nc:-r\' r.'^ H O-^JDA N ACA ri^ WASHINGTON NACA WARTIME REPORTS are reprints of papers originally 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 tlie Internet Arcliive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/chartsfordetermilang :^t^ NACA ACR No. lI|L15 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE CONFIDENTIAL REPORT CHARTS FOR THE DETERU! I NATION OF ffING TORSIONAL STIFFNESS REQUIRED FOR SPECIFIED ROLLING CHARACTERISTICS OR AILERON REVERSAL SPEED By Henry A. Pearson and '«illlain S. Aiken, Jr. SUIVIMARY A series of charts are presented by which the vvlng torsional stiffness required to meet a given standard of rolling effectiveness may he quickly deterinined. The charts may also he used to obtain quickly the aileron reversal speed 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- torsional-stif fness curve has been assumed. INTRODUCTION In order to insure adequate rolling control at high soeeds, present strv:ctural requirements for Army airplanes (reference 1) specify that the computed aileron reversal and divergence soeeds be at Itast I.I5 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 com.putation used. Since the terminal Kach number for fighter airplanes is apTDroximattly O.85, aerodynamic data should be available at a Mach number of about 1.0 if accurate results are to be obtained in I'olling-perf ormance calcu- lations. A Mach number of 1.0 i 3. considerably higher than that at which high-speed wind-tunnel data are available or to which they can be extrapolated. CONFIDENTIAL NACA ACR No. I.^LIJ Da.ta aveilible on the torsional stiffness of v^lngs have indicated that the calculated values of wing torsional stiffness are likely to vary considerably from the test values. Unpublished test data have indicated that, for wings of the same model, significant differences in torsional stiffness can be exr^ected as a result of differences in "fabrication. L'any methods are available for comr)uting speeds of aileron reversal and divergence. (See references 2 to 10.) These methods differ mainly in the combination of assumntions used and in the degree of exactness attempted. In most cases, the form, of the wing-t'A'lst curve has been ass-^xmed to be linear or parabolic and the induced lift effects have been either entirelv ner^lected or apiDro.<:i- miated. In roms of the methods (references L, 6, 8, and 10) the actual 'vving-torsional-sti f fness distribution is required and equiliorlum is established betv^een elastic and aerodynamic forces all along the wing span; with the exception of the metliod of reference b^, however, these methods either omit induced lift effects oi' approximate them. The application of the a::act miethod of reference I4. requires, according to the autnor, 100 man hours. Inasmuch as the accuracy to be gained by the use of the r.ost exact' metnod, which includes induced lift effects, is only about 6 percent as comipared with methods employing atrin theory'", the use of che most exact method is considered impractical when the possible inaccuracies in the aerodynam.ic and structural quantities are considered. Some improvement in the requirements can be obtained by specifying that the loss in rolling effectiveness due to wing twist at eithei' terminal velocity or high-speed level flight shall not exceed somic fixed percentage of that at lov/ speed. Such a specification would then conform better to maneuverability requirements and would not require as much extrapolation of the aerodynamic data. Further improvement could be obtained by requiring that, after fabrication, the torsional stiffness of each wing panel be greater than a specified value. Charts are given in the present paper from vi/hich the required wing torsional stiffness may be reaaily obtained if a given loss in rolling ability is not to be exceeded. The charts may be used to predict the aileron reversal speed and the loss in rolling effectiveness at • any other speed. The charts arc based on the application CONFIDENTIAL NACA ACR No. lIjXIJ CONFIDENTIAL of the usual lifting-line theory to wings of tubular- shell construction and allow variations in v>?ing taper, aileron span, and aileron position to be taken into account. In the derivation of the charts, the form of the v.'ing-torsional-stif fness distribution has been assumed and as a result the tv/ist curves for a given \;in^ 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 be obtained. More nearly accurate results are obtained than with methods that assume the shape of the v;ing-twlst curve to be linear or Darabolic. SW.BOLS Symbols used in the present 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 v;ing chord at any soanwise station, feet c^ aileron choi'-c b wing span, feet A wing aspect ratio (b^/s) y semisoan station, feet o k nondimensional semispan station ' ■ " j ^ wing t&?er ratio; that is, ra-cio of fictitious tip chord, obtained by extending wing leading and trailing edges to tip, to root chord 1^-0 wing torsional stiffness as' obtained by application of concentrated torque near wing tip; foot-pounds per radian CONPIDEflTTAL k CONPIDEJITIAL NACA ACR No. ll^Ll^ t torque per unit span, foot-pounds per foot T accurfiulated. cr total torque, foot-pounds 6 angle of wing tv/ist, radians 5 aileron deflection, radians ^Cm/do rate of change of section pitchlng-xT.oinent coefficient with aileron angle as obtained for inconpressible flow, per radian V da/d5 rate of change of wing angle of attack with aileron angle as obtained for constant liJ at section pb/2V helix angle of roll, radians Lg rolling moment due to ant i sATKnie trie al wing twist, foot-pounds Cj rclling-rnoraent coefficient Gj- rate of change of rolling -moment coefficient with aileron deflection, per radian Cj rate of change of rolling-moirent coefficient with helix angle pb/2V, per radian Cj rate of change of rolling-iaoment coefficient ° with v/ing twist at reference section, per radian T derived constant for airplane (see appendix) Y helix-angle parameter fraction of rigid-wing rolling effectiveness to be retained by flexible wing q dynanic pressure, pounds oor square foot (— PV 1 p density, slugs -oer cubic foot V true airsTDeed, feet per second Va^ indicated airspeed, miles per hour CONFIDENTIAL NACA ACR No. li;L13 COIIFIDEITTIAL 5 a speed of sour.d, feet rier second K V.acli number (V/a) l/Vl-M^ G-lauert conpressibility correction factor Subscripts y and k denote a oarticular spanwise station outboard end oi aileron i inboard end of aileron r reference station taken as midaileron span 1 refers to particular span'vise station inboard of aileron soan, region 1 2. refers to oarticr.lar spanv/ise station in way of aileron, region 2 5 refers to particular spanwise station outboard of aileron span, region 3 L limit c"ivin£; speed R reversal soeed CHARTS The design of ailerons that will provide adequate rolling control at high speed requires the determination of (1) the wing torsional stiffness required to meet a given standard of rolling effectiveness, (Z) the aileron reversal speed, and (3) the variation wibh airspeed of helix angle of roll oer unit aileron deflection. The charts oresented herein have been prepared in order to facilitate the computation of these factors. The results aoply to v/ing3 of tubular-shell construction, of various asoect ratios and taper ratios (including elliptical wings), and with various lengths and positions of the aileron along the span. The various plan ferns considered are shown in figure 2. COI'TIDENTIAL COKPIDEImTIAL NACA ACR No. lI+LIJ The derivation, which is given in detail in the a-Dpendix, is based on an aoplication of the lifting-line theory to the wing plan forrus chosen. (See figs. 1 and 2. In the derivation, the wing section aerodynamic coef- ficients v;ere assunied to be unaffected by torsional deformation and the shape of the torsional-stif fness curve was assumed to be inversely proportional to the cute of the distance from the vi/ing center line. It was also assumed that the ratio of the wing chord to the aileron chord was constant and that no twist occurred about the aileron hinge axis. The results, which apoly to vjings having aspect ratios ranging from 5 to 16, are presented in the charts of figures 5 and 1+. In figure 3 the nondimensional coefficient t is given as a function of kj_ and kg. This coefficient t is directly proportional to the rolling-moment lo3s due to the torsional deformation of the wing and is inversly proportional to the dynamic pressure . If it is desired to determine the wing stiffness required at the reference section (midaileron) to retain a specified rolling effectiveness, the following equation may be apolied: dCjn/dS b5 q da/d5 ^A^(l->2f) Vl-M The dynamic pressure at reversal speed may be determined from the equation qo '^^^. \/l -M 2 i ^dc^/do N^ /t5 (2) If the actual variation of dcm/d5 with Mach number is known, this variation may be substituted in equations (1) and (2) in place of the Glauert compressibility correction factor l/\/l-Ji2. CONFIDENTIAL NACA ACR No. li|L15 CONFIDENTIAL 7 The coefficient y given in figure ij. 1 s a nondlmen- sional quantity representing a helix-angle parameter for a rigid wing. This parameter may be used in the equation P±Z«=,^^a (5) 5 a5 to obtain the helix angle pb/2V of a flexible wing for a unit aileron deflection. The results given in equations (2) and (3) may be used to determine the loss in aileron effectiveness with airsoeed due to wing flexioility. Because the relation between ^ — ^ and q/Vl-M^ Is linear, a straight line drawn between the point renresenting the value for •2--/ — I - - .. e g at q/'j'l-M^ =^ and the point for zero helix angle will yield the value of 2-*^/ — at all intermediate values 6 )f q/>^&^. APPLICATION OP CHARTS Determ-ination of torsional stiffness required at the reference 3ectTon~to fulfilT a sp "e" ci7Ted rolling requirement . - It is desired to determine the torsional stiffness required of the vising at the reference section (midaileron station) in order that at limit diving speed at sea level the airnlane v/ill retain 0.25 ^^ the rolling effectiveness of the rigid wing. ^The following values are c-iven: 0-' Fraction of rolling effectiveness to be retained, )2f (by stipulation) 0.25 Limit diving sneed, V^, miles per hour ..... 555 Kach number, M . O.728 Wing span, b, feet .... = ,.....,...., IlI Aspect ratio, A .....,...,....,,. 5.6 Plan form ......,,.,. . . Elliptical CONFIDENTIAL 8 COKFIDSI'JTIAL FACA ACR No. LI+LI5 Distance to inner end of aileron, k^ , fraction of sen.ispan 0.^33 Distance to outer end of aileron, ko , fraction of semi span 0.9^+5 Dynaiaic pressure at limit diving speed, q-j^. pounds per square foot . ^6c..Q Qc/d5 0.-,6 dc,„/d5 0.1^2 rrom figure 3> f'""!' an elliptical wing and with the given values of k^ and ko , t is determined as 0.21+9 V;hen these values are substituted in equation Cl)^ = 0.2^. xi^iSx 4il^ kI%1 Pr ■ o.:;6 2 X (3,6)^ X 0.7s 0.6t)5 = 1^86,000 foot-pounds per radian = 102,000 inch-DOunds per degree Thus, if a concentrated torque of 10,200 inch-pounds applied outboard of the miuaileron section produced less than 0.1° wing twist of the reference section relative to the wing root, the wing vi/ould exceed the required stiffness . Determiration of ailero n reversal speed .- The same quantities ul-ed in the previous example, together with an experimentally determined value of the wing stiffness m^ (equal to S27,J00 ft-lb per radian, ^ = 0) may be used to obtain from equation (2) __5_R_ _ 2 X '327,0 00 VUP " c.2l,Q X ±^ x-i-iil - 1652 pounds per square foot CONPIDiillJTIAL NACA ACR No. lLl13 CONPIDEITTlAL /■{i Pro^ figure 5> v/hi ch gives the relations betv/een q/yl-M^ and Vgs, the reversal speed at sea level is determined to be 619 r.iles per hour. Deten .- ;! nation of variation of helix angle cf roll with soeed". - The helix angle ob/'2V per unit aileron angle for the rigid wing (j2f = 1.0) is found from figure i\. and equation (5). Figure Ij. applies to all normal taper ratios and aspect ratios. For the airplane of the examole y =^ O.9I, and since da/d5 was assumed to ■ be 0.56, -^ — - — - for the rigid wing is O.328 radians 5 per radian deflection or 0.00573 radians per degree of DO ^2'^^ q aileron deflection. At this value of ^^^ — '- — ~, , - 0. If , = 1652 Dounds oer square foot, the corre^ \/l-M^ sponding helix angle is zero. The helix angle for any intermediate value of q/l/l-M^ or airspeed may be determined by drawing a straight line through these two points. Discussion of examples .- In the application of the charts to determine the wing stiffness, reversal speed, or helix-angle variation, certain quantities will be obtained from ohe geometry of the wings; other quantities will be determined either by performance specification or by the aerodynamic characteristics of the reference airfoil section, which is located at the midspan of the aileron. The values of b. A, kj_ , and ICq can easily be determined from the geometry of the v/ing. In equation (1), the value of ^ must be specified and q/'/l-M^ must be known. An alternative method for determining the aileron reversal speed may be used instead of equation (2) in cases i n wh ich the variation of dc^/dS is not a function of l/Vl-M^. This procedure involves the calculation of ^ at speeds greater than the limit diving speed by the use of equation (1). The values of dcjr;/d5 and da/d5 corresponding to the Mach numbers of the airspeeds chosen are used t^nd a plot of 0" against Mach number is made. As the Ivlach n^uraber is increased, ^ approaches zero and vvhen yf reaches zero, the aileron reversal speed is determined. These computations may be made for various CONFIDENTIAL 10 COi^FIDElJTIAL NACA /'.CR No . Li;L15 altitudes to determine the variation of aileron reversal speed v»lth altitude. In equation (2) the altitude Tiust be specified and the value of the torsional stiffness itiq must Up be known either froi.a tests or calculation. At present, experimental values of m are greatly Dref erred ^r since the calculated 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 inch-pounds to a section near the wing tip and determining the angle of twist at the reference section relative to the wing root. It is recomniended in i-efertnce 9 that in order to obtain best results an antis^'num^etrical torque be applied to the other win;; tio. '■b The quantities da/dQ and dC;vi/d5 are aei'O- dynaii.ic quantities that a^oply at the reference section. In general, they will vary primarily with flao-chord ratio and Mach number and secondarily with other variables such as nose shape, gap, aileroxi section, and angle of attack. Figure 6 has been prepared to show average variations of these quantities wirh flap chord and gap ratio. More accurate valueg can, however, be obtained from specific tests of the aileron section used or from reference 11, which analyzes the data obtained from a large nuinber of wind-tunnel tests. DISCUSSION The agreement that may be obtained between the calculated values of v/ing twist and actual values of wing twist when the form of the wing-torsional- stiffness distribution is assumed is illustrated in figure 7. Computations were made for the P-I-I.7B wing by use of stiffness data furnished by the Army Air Forces, Air Technical Service COiiimand, V»right Field, Ohio. The twdst curve resulting from CONFIDENTIAL NACA ACR ^To. L!;Li3 COIIFIDENTIAL these co'-riDutatlons is compared in figr.re 7 v.itli a twist curve computed by assuming that wing torsional stiffness varied inversely with the cuhe of the distance from the airnlane center line. In calculations involving the use of a wing-twist curve, more consistent results will be obtained by assuming s cubic stiffness distribution than by assuming the wing- twist curve to be either linear or pcirabolic. In a practical case the trailing-edge portion of the wing usually contributes a negligible amount to the v/ing torsional stiffness, but the twist curves for a given wing will differ considerably with aileron span and position since the twist is dependent upon the magnitude and position of the applied torques. Such a variation is included in the results given by the cxiarts. Equation (2) shov/s that, other things being equal, any increase in t lowers the aileron reversal speed. Prom figure pj ^^ increase in t is seen to occur when the aileron span is decreased about a given reference position. For an elliptical v/ing with ailerons extending from. kj_ = O.L to Icq = 0.3, T is therefore O.L67; whereas, for the same wing v;ith ailerons extending from k:;?_ = 0.2 to ko = 1.0, T is 0.388. In both cases the reference section is located at 0.6 semisoan. In order to determine the wing stiffness required to insure a specified rolling effectiveness, the largest value of q/Vl-M'^ obtainable should be used. If the present Arm.y Air Force specification is used as a guide - naUiely, that the reversal speed should be I.I5 oimes the terminal velocity of the airplane - calculations show that the value of )2(' to be used in equation (1) would in general yield overly conservative results for wing torsional stiffness. Another procedure for determining the wing stiffness would be to specify a value of ^ at high-sneed level flight. Either specifi- cation is believed to be more useful than the current CONPIDSFTIAL 12 CONFIDENTIAL NaCA ACR I^o . I.L L' ^ one (Vf{ > 1.15 Vj,) that yields resiJlts at Mach nurnbei^s beyond which data would be completely lacking and that would introduce complications in the equations. As on illustration, if an airplane were capable of reachinj-3 a Mach number of O.87, the design Mach noinber Y/ould be 1.0. With Glauert's approximation, which is used v;ith the present method, the required v.ing stiffness would then be infinite. In recognition of thia difficulty, Victoi'y has introduced in reference 9 the concept of an equivalent Mach number while still retaining the require- ment that the reversal speed be 1.1'3 tiniss the terminal velocity of the airplane. Reference 9j I'T^ introducing this concept, interprets the present requirement that Vjl > I.15VL as referring to an indicated speed in an incomoressible flovi'. Grinsted, izi reference 12, has suggested an alternative procedure - namely, that the wing - stiffness be determined so that aileron reversal would occur at limit diving speed and that this value of wing stiffness then be increased by the factor (1,15)2. Prom a consideration of references 9 ^^^"^ 12, together with current Army ^Ir F orc e requirements, the follov/ing values of GC and q/Vl-M are recomirnended for use with the charts presented herein: Method I.- 'with limit diving speed as a basis, use i2f = Y and (i/\jl-V.^ at limit diving speed at sea level . Method II.- With high-speed level flight as a basis, I, use - — and the largest level-flight value of q/Vl-M*^, regardless of the altitude at v/hlch it occurs. By employing the detailed results and the equations gi^^en in the present report together with the various specifications that have been advanced, the wing stiffness at the reference section has been comiputed for the air- plane used in the example. The follovving table shov;s the stiffness as obtained by the use of the varlo\;s require- ments J CONFIDENTIAL NACA ACR No. lIlLIJ co:t7ID"^:.^itial Require.nent Method I j2f = ^ at V> Method TI ) ^ - —■ at maximum le\el- flight speed Arn\y hi r ?o r c e s , Vr = 1-15 Vl Victory (reference 9)» ?3a- level basic sooed (; ■x-h' ^53 i.'i R o86 M L Grinsted (reference 11), rriQ = 1 .^2 rriQ whei-e ^r ' ^L iTiQ is the stiffness 525 1.1b X 555 1.0G6 X 555 555 assiirriing V •R 1/. Stiffness ITi: ■ft- lb per rjidian) LSd,000 1|75,000 601,000 1^78,000 i|8l,G00 Also, for coniparison, the follo'.ving numerical values of mg are listed for the airplane used in the exarr.ple ; Louroe (1) ;;cnditicn 1 ^^?r i (fc- lb/radian) Experimental | Ammunition doers closed j Experiinental j A;nmiinition doors o^en \ Calculated > Aiiniuniticn doors ooen I 527,000 555,000 209,000 ■'-Experimental data furnished by Amy Air Forces, Air Tec-inical Service Coi-nmand, V'/ri.jht x^isld, Ohio. Calculated data from Republic Aviation Corporation. CONEIDEKTIAL li; CO!TPIBENTIAL KA.CA ACR Fo . T.':T, c r c UjDI ng ri^v: a r k s ' T -! 7, Charts have been prepared for use in deter.ainln^ tlio Vising torsional stiffness for wings of tubular-shell construction with aspect ratios rangln;^ from 5 '''-o l6 and taper ratios ranging from to 1 including the elliptical. The loss in rolling effectiveness and the aileron reversal soeed i'aay also' be calculated. The chief advantage of the present method over previous methods is the speed with which tix results i::ay oe obtained, Kore accurate results nay 'oe obtained by the use of this laethod than by the use of methods tliat assume the ohape of the wing-tv/ist curve to bo linear or parabolic. Langley Memorial Aeronau^tical Laooratory National Advisory Comijiiittee for Aeronautics LangD.ey Hold, Va . . CONFIDENTIAL NACA ACR No. Li^LlJ CONFIDENTIAL 15 APPENDIX DERIVATION OF CtlARTS Although there are a number of tvpes of air loading and inertia loading that contrloute to the wing twist about the elastic axis of v.'ings in flight, so far as the problem of rolling effectiveness is concemea, only the twist due to ailfron deflection need be considered. In fact, since the distribution due to damping in roll is likely to be almoet the same as the spanwise air load distribution resulting from aileron deflection, only the increase in section pitching mioment in way of the aileron need be taken into accounb in determining the wing tv/ist. A strin of the wing dy in way of the aileron (see fig. 1) will have acting on it an increment in torque as follows: ^t c.y = -__5-=.:=:jdy (-••1) The factor l/^'l-I,:^ is introduced in equation (Al) in order to increase low-speod values of dCnj/dS for liach number effects. If the correct variation of dci-,i/d5 is available 5 the quantity (dcni/d5 ) ( l/'/l-l.I'^) may be replaced by the actual variation with Mach number. The accumulated increment in torque at a parti cu].ar station j-^ in way of the aileron is AT,. = / At dv (A2) and, similurly, the accumulated increiaent in torque at any station y2_ inboard of the aileron is ATv^ = / at dy (Ap) CONFIDENTIAL l6 CONFIDENTIAL NACA ACli No. iIlI^ In the derivetlon of the charts for the deterniinstion of '.ving torsional stii'fness required for specified rolliiig cht.racteristics oi' ailoron reversal speed, it is desirable to use the \7in,£^ center line as the reference end to define the torsional stiffness for a vising of tuDuls.r- shell constriiction as the concentrated torque which, when applied oufooai-d of a given statioii, would produce a unit deflection Vi/ith respect to the reference section. Although this definition of the torsional stiffness makes the analytical developrrient somewt.at longer, it is better suited to the test procedures that are now in use v/hen the torsional-stif f ness variation along the span is to be determined. The angle of twist due to aileron deflection Gy-, at any station y^. ^^ way of the ailt.ron (region 2, fig. 1) is thus ^j;,l\'^n by ("■>'o ey ^ — -— /^ '■i'o '^^'2 2 i;yp //v: A J- 'yp - -._-„ / /\t dv + — d:v (a1+) •'IQ The twist t^t any station y-[_ in region 1 inboard of the aileron is ; -"o ., - ~~ I it dy (a5) -.1(3 9,. = 1 In ref-ion ^ outboard of the aileron end the twist is J 7 ^ ^^^ = ' ' ^ dy (a6) Since no anti synrrietrical torque is acting outboard of the aileron tip, 9-r is constant to the wing tin. Ey ■' J> substituting equation (Al) in equation (aI^) and introducing k ~ — ^ a nev/ eq.ia'cion ^riay be obtained. b/2 This equation :nay then be nut into rnoi'o convenient form by multiplj'-ing each term by the ratio of the stiffness mp, to the square of tne niean geometric CONFIDENTIAL NACA ACR No. Lh.Ll3 CONFIDENTIAL 1? chorl c. The resultiiir- eiuation ri.ay be rearranged to give Ghe follovylng equ&tion applying to region 2. (for convenience, the factor 1/ !/l-M'^ Viflll "be -I'ouped with instead of with dc^n/^o}: gl/i/l-M2 S^2y^ k. ^^> J^i d5 ^ ■" ""^ I •- ' .n -^ \c; The equations for the wing twlsb at stations inboard and outboard of the aileron (equations (a5) and (Ao)) sliTilarly become q ^^^9v. \"^' ^ '^l - ' '-£-) dk (Ati) Qq/i/'l-M^ £2:1^2 b i>k| IJ) 5^/1/1^^2 dc^2^ J,^^ vcy "^^e ' dk (A9) Equations (A?), (aS), and (.^9) define the angle of tv;ist in the thi^ce regions in ter-rus of the chord and Etifi"ness distribution, subiect to the assur.otions that dc;n/d6 is a consta.it along the aileron opan and that the aileron does not tv;ist about its hinge axis. Inspection of figure 6(b) indicates that the factor dcrn/dS is essentially constant for flap- chord ratios from 0.2 to 0,3 andj since the variation of aileron-chord ratio along the span will normally fall within this range, the assujnption is justified. In order to evaluate equations (A7) to (A9), the twist curves v;ill be obtained in terns of the twist 8r ^^t a reference section, v/hich will be talcen at the midspan of the aileron. From equation (A7) the twist at the reference section becomes CONFIDENTIAL 18 V CONFIDENTIAL NACa ACR No. L1I15 G-. 2m, nk / o ^^^' dk P -^r ■6. ^-) (AlO) V/ith the stiffness as defined in the pi'-esent report, the torsional .stiff ness is infinite at the vring center line and decreases with distance to sor.e flnitG value at the tip. Analysis of data for "^jjTjical fighter airplanes indicates that negligible errors Viill result in tv/ist computations if the torsional stiffness distribution along the span is a33U!:.ed to bo m.p _ Constant k3 (All) V/hen equation (i\ll) is substituted into equations (A/) to" (AlO), the following ratios of B/Op will be obtained in the various regions; ^ x^-r/ / i— ) Ck -! -^ " -r [""^(^^ dl. . -4 / .I'kp W kr> (i;ci ~-\ k3 dk j / f:^^i dk + — ■■ nkp 02 _ /V2\3 UkZ \ri TrJ -?'' Jki (^^ k5 dk" / h i"'^ /c\2 ^'■■' k.^ 4. ^^^ \o/ k^ dk /K ? _ { o dk ko" i'k.j_ ^^^^O /,.^2 • ...J, [~l dk + ^/ (tr^'- t/;c, > (A12) CONI'^IDLNTIAL NACA ACR No. lJL:-L13 CONFIDENTIAL 19 It will be noted in equations (A12) that only geometrical terms such_as spanwise extent of aileron spf.n and. chord ratios c/c occur and that:, in order to determine tiie resultant tv.-ist distribution., onljr these values need be specified. In reference IJ influence lines are presented for a series of tapered wings (see fig. 2) of several aspect ratios, v/hich make possible the computation of a coef- ficient of rolling-moment loss Cj q "^U-® to any sort of twist distribution. As a first step in the evaluation of a loss coefficient Cjn, the ratios of B/Gp were evaluated for the series of wings shown in figure 2 with ailerons of various span. The loss in rolling moment due to a twist 9p , at the reference section, was then defined by the equation Rolling moment loss - Lf (A15) CjgSrqSb where Cz 9 ~ ds7 dCj/de^ g (All,) The results shown in figure 16 of reference 13 were used to determine Cjq for the twist variations computed from equations (A12) . The coefficient Cj was also determined for elliotical v;ings of aspect ratios 6, 10, and l6, with values of V.± of 0.2, O.J, O.U, 0.5, 0.6, and 0,? and for values of \o o- 0.8, O.9, and 1.0 as well as for the wing plan forms shov/n in firure 2. The numerical results of these steos are not given herein because they are only intermediate steps in the procedure. CONFIDENTIAL 20 CONFIDENTIAL NACA ACR No. Li Ll^ In tiie steady rolling condition, the damping moment equals the moment im.pressed by the ailerons minus the loss in moment due to twist. In coefficient form, this relation may be expressed as Db/2V from which the helix angle oer unit aileron deflection - — 5 is obtained as O L 7 The coefficients Cj , Cjp, e.nd Cj Vi'ill vary v/ith Mach number but of these coefficients only the variation of C^,^ with r.:ach number can readily be determined from wind-tunnel tests. At ore sent C? must be obtained "'P either from results of low-speed tests or from results O- computations and C/ must alv;ays be obtained by computation. For this reason it would appear reasonable to use consistent values and to assume that each varies with Mach nijmber according to l/vl-M . From equation (^10), for a particular wing aileron combination. 9r = -S^-— :7--=-3i (Al?) im^ r fl- Vi-'"^ where the constant B]_ equals the right-hand side of equation (AlO). Also, from equations (Al2) and (Alii), Cy^ •'9 is seen to be a constant for a particular wing-aileron combination. When these values of Cjq and O^ are substituted in equation (Al6), the following equation results: CONFIDENTIAL NACA ACR No. L[^L15 CONPIDSIITIAL 21 5 ' "■ C7 where the constant ^2 ~ ^je'^l At the aileron reversal speed. -c b and the value of the dynamic pressure is q 2'^Gr '^io '^^'Qr ^^5 da/do */iIm2 ^^-^^3 ^'2 £^ ^_ Qa/d5 ^2 d6 ^ q6 ^-1 By setting Bo /da ^^s/dS (AI8) (AI9) (A20) the dynamic pressure at aileron reversal speed is 2r, ^ - "^ (A21) \/^2 ^ ^^^m/d5 t3 da/dS a2 CONFIDENTIAL 22 COFFIDENTIAL NACA ACR No. lI'.IJ In the deterailnation of the values oT t the necessrry nuitisricai values of b? v/ere outalaed bv the -orocodure outlined and the neoetsarv values of -.-— v/ere t-J:en GO./ do clrectly from figure l6 of reference 15 . Vihen the -valus-j of T were plobted the results were found to be essentially the sariie for aspect ratios of 6, lU, and l6; the average deviation was less than 1 percent. The value of T did_, however, vary vlth tileron position an: taper as shown in fi^Tjre 5- ,-j For an Infini-cely ri^id wing, uhe helix an^le per degree aileron deflection can be obtained from equation (Al8) as do. {pb/2V ( _ d a/db ad \ /rijid '^1, (A22) where Cj was obtained from figure S of reference IJ . Figure l^. gives the values of y in terms of k^ and k^ By soecifying that the flexible v.'ing retain soue fraction of the rigid wing rolling effectiveness at a specified dynar'vic pressure (say, terminal velocity'-), the following equation results (A23) By substituting results from equations (a20) and (A25) into equation (.^l8), the wing stiffness n^ required at the reference section to retain a rr'scified value COKPIDENTIAL NACA ACR KOo ri+Ll3 CO^PIDS'ITIAL of rolliag atllity at e given value of q/vl-M- 1: given by "9r dc./c'^e 2a2(1-^) fT-wJ CONFIDENTIAL 2k CONFIDENTIAL FACA ACR No. l1;L13 REFERENCES 1. Anon.; Handbook of Instructions Tor Airplane Designers. Vol. I, Materiel Div., Armv Air Corns, 3th ed., Revision 7> Nov. 1, 194-5, sec. II, pt. V, par. 60-l, p. 65I+. 2. Pugsley, A. G.: The aerodynamic Characteristics of a Serci-Rigid Wing Relevant to the Problem of Loss of Lateral Control Due to l/Ving Twisting. R. & M. No. li^^O, British A. R.C., 1952. 3. Cox, H. Ro.xbee, and °ugsley, A. G.: Theory of Loss of Lateral Contiol Due to *ing Twisting. R. &M. No. 1506, British A. R.C., 1953. U. Pugsley, A. G., and Brooke, G. R.: The Calculation by Successive Approximation of the Critical Reversal Sueed for an Elastic Vving. R. & M. No. I5G8, British h.R.C. , 1955. 5. Hirst, B. i: . : On the Calculation of the Critical Reversal Speeds of Vaings. R. & M. No. I568, British A.R.C. , 195ij-. 6. Shornick, Louis H.; The Computation of the Critical Speeds of Aileron Reversal, Wing Torsional Divergence and Wing-Aileron Divergence. Memo, rep., Ser. No. ENG-M-5I/VPI8, Add. 1, Materiel Center, Army Air Forces, Dec. I9, 19[j.2 . 7. Horton, 'A'. H.: Critical Reversal Speed. Aircraft Engineering, vol. XV, no. l??? Nov. 19^4-5, PP« 519-5214-. 8. Rosenberg, Reinhardt; Loss in Aileron Effectiveness Because of Wing Twist and Considerations Regarding the Internal-Pressure Balanced Aileron. Jour. Aero. Sci., vol. 11, no. 1, Jan. I'il+k, np. lj-l-l|7. 9. Victory, L'ary: The Calculation of Aileron Reversal Speed. Rep. No. S.M.E. 5279, Eri tish R . A. E. , 19i|l4.. 10. Harmon, Sidney M.: Determ.ination of the Effect of Wing Flexibility on Lateral Maneuverability and a Com.Darison of Calculated Rolling Effectiveness with Plight Results. NACA ARR No. i|.A28, 19kk' CONFIDENTIAL NACA ACR No. l1i,L13 CONFIDENTIAL ?S 11. F. , Purser, Paul Available Da Overhang and ajid Toll, Thomas A.: Analysis of ;a on Control Surfaces Having Plain- Prise Balances. N^CA ACR No. Lif.Iil3, 12. 13 Grinsted, P.: The Effect of Compressibility on the Estimation of Aileron Reversal Sneed. Rep. No. S.M.E. 3192, Bri tish R . A. E. , IQl Pearson, Henry A., and Jones, Robert T.; Theoretical Stability and Control Characteristics of Wings with Various Amounts of Taper and Twist. NACA Rep. No. 655, 193 8. I GONPIDENTTAL NACA ACR No. L4L13 Fi§. 1 .^u D -43 Xi c> y- -Q 1? ^ «s a: — ) <=> II .^ ^ NACA ACR No. L4L13 CONFIDENTIAL Fig. 2 A -6 L//?e of quarter -c^ord po/ntsS - so ^S A'/6 CONFIDENTIAL NATIONAL ADVISORY COMMIHEE FOR AERONAUTICS /^/pure 2 — y^/np /)/(7/7 /vrms used /n o/jo/y| "^ ^ M '^J \ 1^ II §^ el- T I NACA ACR No. L4L13 Figs. 3c, d < O o II ^ ^ «Q ^ P 7/ II ^ 1 II u 1 'n 1 1 Ihi y \i ' t i .%'> // ft 1 J / ' u j J 1 1 ' 1 1 1 t 7- : 1 1 / 7 i '' 'i -- / \/ ■ 1 / // V \i - \ - /' v/ /J ' / / 1 i / V f/ // ''/, n / , 1 / 7 / '/ 1 , f /, / // '/ /J / /; A / y / ' / ' / s. /' V ■'/■ / • / 1 .<\ / J /, 1-/ /. / ' } r 1 /. ' / / / / / / / / A ' / / / ' / / V / y A \/ / A / / / / ' / y / V / / / / / / / / ^ /" ■'/\ / ^V / 7 _j <0 K vo ><> ^ ^ ") ^i \ o I I NACA ACR No. L4L13 CONFIDENTIAL Fig. 4 y y^CX ^ \ '^ ^ V. \ 1 /^ ^ N \ X ^ ^ "^ •^ ■^ V. \, --^ ^ N \ \ \ \ \ k. ^ \ V \ /^ \ "x ^ V N ^ N :i^ s V \ \ N \ \ \ \ s \ x^ N X s ' v N N N \ \ \ k\ \^ \^ \ /^ X s \ \ \ \ N k , \\ \ \ \ N^ \ N^-\ \ \ \ ^ > \ \ ^ N k^ \\ K \ ' V ^ k'^ \ > A \N s. .<9 ^ \ '' \ . \ \N \' v\ \ vN ^ \ > .^ 1 N \ \ > A ^ 1 \^ A \ \ \^ \ > s \^ A \ .e K, ^' \ - N s> S^ \' \ \'^ .N nN \ N^ ', N ■0 N^ \ \ s \ s"" \ .■^ \ \^ \ L\ \, \^ \' .£ \ NAT! }N AL A m%{ m CO Mf 4mi F FO R AERn NAU1 IC s /? , \ % .3^ % .e .z .3 :^ .^ .e .7 /ndoorc/ ex ten/ ofo/Zeron jpo/?^ z^- CONFIDENTIAL f/^6/te ^.- l^r/af/o/? c^f /?e//x-a/?^/e /X7r(7/?pe/er for a y/^/c/ m/?^ /^//^ ex/e/?/ or a//eron ^/)a/? fc^r a// c7S/?ec/'r(2f/06 arc/Zs/^er raZ/os. /^6pect ra//os ror^/r^ from 5 to J6. NACA ACR No. L4L13 Fig^ 1 «o 1 ^ ^ c^ >- < A 1 ^ 1 o Z o A 1 > o « ?? \ 1 o UJ 1 1 t— \ \ 8 \ \ ) \ \\ \ \ 1 \ \\ V \ \ n\. ,\ \ V \ 1 \ < 1 — I E- Z tii Q b. O O \ ^ \ \ \ V \ \ ^ \ ^ N X, -~~- 1 ^ ^ ^ ^ § \ 1 ^ 4 i ^ \ ^ ^ o ^ i/c/^'^Ji^ I I ll I I I I 'i: NACA ACR No. L4L13 Fig. 790 I NACA ACR No. L4L13 Fig. 7 J r 1 U3 < 1 1 > < c 2 O QC a. z O \ c •< ae ^- ULI CJ \ c \ \ N ^^ V O X ^\ k.^ S^ •j^ ^ \ 1 1 \ ^ *l II \ 1 :^ \ ? 1 ^^ ^ w \ \ H \ 0) '^) ^ vs A: S t «) ^ >i«^ t-s ^ ^ ? I 8^ ^ o o