'Hk L'ii NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED August 19^5 as Advance Eestricted Eeport L5F23 LTFTUKJ-SaRFACE -THEORY VALUES OF THE BAMPIBCr Df EOLL AHD OF THE PARAMETER USED Hi ESTIMATIUG AHJEROU STICK FORCES By Roljert S. Swaneon and E. LaTeme Prlddy Langley Memorial Aeronautical Laboratory Langley Field, Va. ^ MAC A 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. I L - 53 DOCUMENTS DEPARTMENT Digitized by tlie Internet Arcliive 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/liftingsurfacethOOIang 7/2. (f(7(^i- ^kCA ARR No. L5F23 RESTRICTED NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT LIFTING-SURFACE-THEORY VALUES OP THE DAMPING IN ROLL AND OF THE PARAMETER USED IN ESTIMATING AILERON STICK FORCES By Robert S. Swanson and E. LaVerne Prlddy SUMMARY An Investigation was made by lifting- surf ace theory of a thin elliptic v/ing of aspect ratio 6 In a steady roll by means of the electroraagnetic- analogy method. Prom the results, aspect-ratio corrections for the damping in roll and aileron hinge moments for a wing in steady roll ?;ere obtained that are considerably more accurate than those given by lifting-line theory. First-order effects of com- pressibility Vifere included in the computations. The results obtained by lifting- surface theory indicate that the dam.ping in roll for a v/ing of aspect ratio 6 is 13 percent less than that given by lifting- line theory and 5 percent less than that given by lifting-line theory v;ith the edge-velocity correction derived by Robert T. Jones applied. The results are extended to wings of other aspect ratios. In order to estimate aileron stick forces from static v/ind-tunnel data, it is necessary to knov/ the relation between the rate of change of hinge moments with rate of roll and rate of change of hinge moments with angle of attack. The values of this ratio were found to be very nearly equal, v.'ithin the usual accuracy of wind-tunnel m.easurem.ents, to the values estim.ated by using the Jones edge-velocity correction, vifhich for a wing of aspect ratio 6 gives values 4.4 percent less than those obuained by lifting-line theory. An additional lifting-surface-theory correction was RESTRICTED NAG A ARR No. L5P23 calculated but need not be applied except for fairly large high-speed airplanes. Simple practical methods of applying the results of the investigation to wings of other plan forms are given. No knowledge of lifting- surf ace theory is required to apply the results. In order to facilitate an imderstanding of the procedure, an illustrative example is given. INTRODUCTION One of the many aerodynamiic problems for which a theoretical solution by m.eans of lifting-line theory might be expected to be inadequate is the case of a wing in steady roll. Robert T. Jones has obtained in an unpublished analysis similar to that of reference 1 a correction to the lifting-line-theory values of the damping in roll that amounts to an 8-percent reduction in the values for a wing of aspect ratio 6. Still more accurate values m.ay be obtained by use of lifting-surface theory. A method of estimating aileron stick forces in a steady roll from static wind-tunnel data on three- dimensional models is presented in reference 2. This method is based upon the use of charts giving the relation between the rate of change of hinge momient v/ith rate of roll Gj-, and the rate of change of hinge of Ch moment v/ith angle of attack G]^ in the form of the parameter lay.) = -^ , which is determined by m.eans of lifting-line theory. It was pointed out in reference '. that the charts might contain fairly large errors which result from neglecting the chordwise variation in vorticity and from, satisfying the airfoil boundary condi- tions at only one point on the chord as Is done in lifting-line theory. A more exact determination of the parameter (a^^. p is desired. In reference 3 an addi- tional aspect-ratio correction to Ci, as determined from lifting-surface theory is presented. In order to evaluate the possible errors in the values of (ctpV IIACA ARR No. L5P23 3 as dsterir.ined. by lifting-line theory, it is necessary to determine similar additional aspect-ratio corrections to Gi-,^. A description of the methods and equipment required to solve lifting-surface-theory problems by means of an electromagnetic analogy is presented in reference 4. An electromagnetic-analogy model simulating a thin elliptic wing of aspect ratio 6 in a steady roll v;as constructed (fig. 1) and the magnetic-field strength simulating the induced downwash velocities was measured by the methods of reference 4. Data were thus obtained from which additional aspect-ratio corrections to Chi^ ^ov a wing ■ of aspect ratio 6 were determined. Because of the small magnitude of the correction to (^r,\n introduced by the lifting-surface calculations, it was not considered v;orth while to conduct further experiments on wings of other plan forms. An attempt was therefore made to effect a reasonable generalization of the results from, the available data. Inasmxuch as the theory used in obtaining these results is rather complex and an understanding of the theory is not necessary in order to make use of the results, the m.aterial presented herein is conveniently given in two parts. Part I gives the results in a form suitable for use without reference to the theory and part II gives the development of the theory. SYMBOLS angle of attack (radians, unless otherwise stated) section lift coefficient f Mli qc c I C^ wing lift coefficient (-i^i^i L ^ \ qS / C^ hinge-mom.ent coefficient i^^inge momentA V qca ha ^ r, TT. 4. wr.j^' • +- /Rolling m.oment\ C, rollmg-mom.ent coefficient f — ::^ -^r ) AC A ARR No. L5P23 i\ a slope of the section lift curve for incom- pressible flow, per radian iinless otherwise stated ph/2V viflng-tip helix angle, radians r circulation strength C|, darnping coefficient: that is, rate of charge P of roll ing-iTioment coefficient with rate / ^.C^ \ of roll '- ) \6(pb/2V)_/ C}^ rate of change of hinge moment with rate of 'a (">0c. ro 11 I \d(pb,-^27) ; Cyi rate of change of hinge moment with angle of attack 1 -^^^ j \oa J Cj^ rate of change of wing lift coefficient \6a / _ with angle of attack absolute value of the ratio ( -— i- 1 c wing chord c_ wing chord at. plane of " syrrmei:ry c-]-, balance chord of aileron Cg. chord of aileron Ca_ aileron root-mean- square chord X chordwls-e distance from .v;ing leading, edge y spanwrsc distance from plane of -symmetry ha aileron span b/2 wing semi span NACA ARR No. L5F23 5 S area of \ving W ;veight of airplane Pg stick force, pounds 9g stick deflection, degrees 5^ aileron deflection, degrees, positive downward A aspect ratio A_ equivalent aspect ratio in compressible / flow (a/i - ir ) A. taper ratio, ratio of fictitious tip chord to root chord M fz'-ee- stream Mach nur.ber v/ vertical component of induced velocity V free-stream velocity q free- stream dynamic pressure ; -^pY^ ' E edge-velocity correction factor for lift E' edge-velocity correction factor for rolling morae nt F hinge-moment factor for theoretical load caused by streamline-curvature correction (reference 5) T| experimentally determined reduction factor for P to include effects of viscosity trailing-edge angle, degrees 1 n r> n +". i rtT. / o r. ?? ~ ■'- b/2^ 8 paramieter defining spanwise location (cos"-'- — ^ K-]_, Kg constants Subscripts: LL lifting-line theory 6 ivIACA ARR Ko. L5F23 LS lifting- s'tirf ace theory EV edge-velocity correction SC streamline curvature max maximum o oufooard i inboard e effective c compressibility equivalent I-APPLICATION OP METHOD TO STICK-PCRC3 ESTIMATIONS GENERAL METHOD The values of the damping in roll C^ presented in reference 2 were obtained by applying the Jones edge-velocity correction to the lifting-line-theory values. For a wing of aspect 6, the Jones edge-velocity correction reduces the -values of Gj ^ by about 8 percent "P '' ^. _-_p( J. could be calculated. The damping In roll was found to be 13 percent less than that given by lifting-line theory. The results were extended to obtain values of C^-Q for wings of various aspect ratios and taper ratios. These values are presented in figure 2. The / ~ parameter \/l - H^ is included in the ordlnates and abscissas to account for first-order compressibility effects. The value of a^ to be used in figure 2 is the value at M = 0. The method of estimating aileron stick forces requires the use of the naram.eter (a-n\ - \ , P \ ^/Ch ICh a 7 NAG A ARR ITo. L5F25 3ecau.-e Ch can be found from the static wind-tunnel data, it is^'possible to deteritdne Ch^ and thus the effect of rolling upon the aileron , sticl: forces^ -:f In \ 1^ known. Tn order to avoid measuring 0-J3 at all points to be computed, the effect of rolling Is uBuSlly'^accounted for by estimating an effective angle of attack of the rolling wing such that txie static ^ hinge moment at this angle is equivalent to ^}^l f-_^Jf moments during a roll at the initial angle 01 ^ttac^. The effective angle of attack is equal to the ^^^tiai angle of attack corrected by an incremental angle (Aa)^^^ that accovmts for rolling, where £^ (1) (^«^Ci, = («p)ct, 2V The value of (Aa)c^^ is added to the initial a for the downgoing wing and subtracted ^^f "-^..^^^^^'^L^^^^J^^aLg for the upgoing wing. The values of o^ corresponding to t^ese corrected values of a are then determined and are coS?;?ted to stick force from the known dynamic p?essSre, the aileron dimensions, and the mechanical advantage . The value of Fb/2y to be used in equation (1) for determining (Aa)c. -^ i-^ explained in reference .) the estimated value for a rigid unyawed wing; that is, pb ^ _SL 2V Ct,^^ The value of C7 to be used in calculating pb/2V ■should also be corrected for the effect of rolling. ?he calculation of pb/2V is therefore determined by .ScceJ5?v; apnroximatlons. For the first approxi- matLnpthe static values of C, are usea with tne _ value of C^ from figure 2. From the f irst-approzi- mat-: on valuel of pb/2V, an incremental ^""^l^t^f r^uvv^ses Ittack (Aa)c, i^ estimated. For all practical purposes, (''p)ci = ("p)ch 8 NACA ARR No. L5P23 and from equation (1), Second-approximation va?.ues of O-j can be determined at the effective angles of attack a + Aa and a - Aa . The second-approximation value of pb/EV obtained from this value of C7 is usually sufficiently accurate to make further approximations urjiecessary. In order to estimate the actual rate of roll, values of pb/2V for the rigid unyavv-ed rnang must be corrected for the effects of wing flexibility and airplane yawing motion. An empirical reduction factor of 0.8 has been suggested for use when data on wing stiffness and stability derivatives are not available to make more accurate corrections. Every attempt should be made to obtain such data because this empirical reduction factor is not very accurate - actual values varying from 0.6 to 0.9. The improvement in the theoretical values of C^ obtained by use of lifting- surface theory herein is lost if such an empirical factor is used. In fact, if more accurate corrections for wing twist and yawing motion are not made, the empirical reduction factor should be reduced to 0.75 vi^hen the more correct values of C7 given in figure 2 are used. (. p <•- o The values of (a-r\ presented in reference 2 were obtained by graphically integrating som.e published span-load curves determined from lifting-line theory. Determination of this parameter hj means of the lifting- surface theory presented herein, however, gives somewhat more accurate values and indicates a variation of the param.eter v/ith aspect ratio, taper ratio, aileron span, F Ti Mach number, Ch„, and the parameter — ;— 7^ '^ ('■ca/c\2 In practice, a value of (a„\ equal to the ' ^h lif ting-line-theorv value of (a^jp' (see appendix) tiro.es the -Jones edge-velocity correction A + 4 A S„ + 2 parameter — — 2 is probably sufficiently accurate. The incremental angle of attack (Aa),'^, is then ^h NACA ARIt Ifo. L5F25 (Aa)c pk/: (^p)cb,,. h " 2V A^ + 4 A^E^ + 2 LI. - / \ ^'^c + 4 A ^Eg + 2 pb - \"P;Ch A^+ 2 IcE'c + 4 2V ^^ If further refinement in estimating the stick force is desired^ a small additional lifting- surf ace- theory correction AC^ = A (Cii_\ •^y ma^T- be added to the hinge moments determined. For wings of aspect ratios of from about 4 to 8, values of this additional lifting- surf ace- theory correction are within the usual accuracy of the measurem.ents of hinge m.oments in vvind tunnels; that is, ACh = A(Ch \ || - 0.002 for a pb/27 of 0.1 and therefore need not be applied e:^cept for very accurate work at high speeds on large / \ ^c + 4 airplanes. A/alues of (cCpl p 7 — ^^'^ given in figure 3. The effective aspect ratio Ac = AV 1 - M^ is used to correct for first-order compressibility effects and values of are given as a function of Ap in figure 4. Values of the correction ^K).,^'^-%^/-''^ LS are given in figure 5 as a function of A^ and values w of — — are given in figure 6. The value of n is (ca/c.)2 approximately 1 - 0.0005JZ) . The values of Cb/ca given in figure 6 are for control surfaces v/ith an external overhang such as a blunt-nose or Frise overh.ang. For shrouded overhangs such as the internal balance, the value of cb/ca should be multiplied by about 0.8 before using figure 6. If the wind-tunnel data are obtained in low-speed wind tunnels, the estlm-ated values of Gj and (ttp')p should be determined for the w/ind-tunnel Mach nuiriber 10 2^ AC A ARR (a&sujne M = 0) . Othervfise the tunnel data must be corrected for corfipressibllity effects and present methods of correcting tunnel data for compressibility are believed unsatisfactory. ILLUSTRATIVE EXAJ,iPLE Stick forces are computed from the results of the v/ind-tunnel tests of the 0.40-scale semdspan model of the viflng of the same typical fighter airplane used as an illustrative example in- reference 2. Because the wind-tunnel data were obtained at low speed, no corrections Vifere applied for compressibility effects. Because this exam.ple is for illustrative purposes only, no com.putations were made to determine the effects of yawing motion or wing twist on the rate of roll but an empirical reduction factor was used to take accoiont of these effects. A dra'jying of the plan form of the wing cf the miodel Is presented In figure 7. The computations are m_ade at an indicated, airspeed of 250 miles per hour, which corresponds to a lift coefficient of 0.170 and to an angle of attack of l.S*^. The data required for the computations are as follows; Scale of model 0.40 Aileron span, ba, feet 3.07 Aileron root-mean-square chord ^a> feet . . . 0.371 Tralling-edge angle, 0, degrees ... 13.5 Slope of section lift curve, aQ, per degree . 0.094 Balance-aileron-chord ratio, c-b/cg, 0.4 Aileron-chord ratio, Ca/c, (constant) , . . . . 0.155 Location of inboard aileron tip, -^-4- 0.58 b72 Location of outboard aileron tip, -2^ ...... 0.98 b/2 Wing aspect ratio, A 5.55 V/ing taper ratio, a. 0.60 Maximum aileron deflection, Sa^vjax' degrees .... ±16 Maximum stick deflectlo.n, 9 Smax' degrees ±21 Stick length, feet '. 2.00 Aileron-linkage- system ratio It 1 VJing loading of airplane, Vs/s, pounds per square foot 27.2 NACA ARR No. L5P23 11 The required wind-tunnel test results Include rol.ling-momej.it coefficients and hinge-moment coefficients corrected for the effects of the Jet boundaries. Typical data plotted against aileron deflection are presented In figure 8. The'^e same coefficients cross-plotted against angle of attack for one-fourth, one-half, three- fourths, and full aileron deflections are given in figure 9. The value of C^ /aQ as determined from figure 2 is 4.02 and the value of Cj is 0.573. The A + 4 A„E,, + 2 value of /a-^\ -r-- ^ ■ „ ■ .^r ~ used in equation (2) ^ ^)Cy, ^c +^ A^^'c + 4 to determine (Aa)^ Is found from fip-ures 3 and 4 to ^h be 0.565 and is used to compute both the rate of roll and the -stick force . In order to facilitate the computations, simultaneous plots of Cy and {^a)r against pt)/2V were made ""h (fig. 10). . The steps in the computation will be e.xplained In detail for. the single case of equal up and dov/n aileron deflections of 4°: (1) From figure 9, the valuesof C^, corresponding to 6a = 4*^ and 5a = -4° at a = 1.5° are 0.0058 and -0.0052, respectively, or a total static C7 of 0.0110. (2) A first appro xlm.at ion to ('^^Iq taken at the value of pb/2V corresponding to Cj = 0.0110 In figure 10 is found to be 0.95°. (3) Second-approximation values of C-j (fig. 9) are determined at a = 0.35^^ for 5a = 4°' and at a = 2.25° for 5a ---4°, which give a total Cj of 0.0112. (4) The second approxim.ation to (La) n is now ^h tovxi-d from figure 10 to be 0.96°, which i.s sufficiently close to the- value found in step (2) to make any additional approxim.ations ijnnecessary. 12 NACA ARE Ko . L5P23 (5) By use of the value of C^, from step (3^ , the value of £^ = 0.0300 is obtained from figure 10. (6) From figure 9 the hinge-moment coefficient corresrionding to 5g^ = 4*^ and the corrected angle of attack a = 0.34^ is -0.0038 and for 5^ = -4° and a = 2.26° is 0.0052. The total G^ is there- fore 0.0090. (7) The stick force in pounds is calculated from the ailerpn-linkage-system data, the aileron dimensions, the increment of hinge-moment coefficient, and the lift coefficient as follovi/s; Stick force x Travel = Hinge moment x Deflection where the hinge mom.ent is equal to ^v^qbg^c - and the motion is linear. Substitution of the appropriate values in the equation gives ^s -57T3- - 5T3 C^qb^Ca^ and the wing loading is 1 = qC, S ^ L = 27.2 Therefore, p - S'^'S Q 3.07 /0.371^^ 16 x 57.3 , _3^07 ( 0.57lY 'h 0.4 y 0.4 / or s - Ct i^ 0.4 \ 0.4 / 2 X 21 X 57.3 F„ = 68.4 _S s Cl Thus, when Cy^ - 0.0090 and C;^ = 0.170, ^s - 68.4 X Q^^^Q = 3.62 pounds NACA ARR No. LbFPZ 13 This .•^tick force is that due to aileron deflection and has been corrected by ('ap\ as determined with the Jones edge-velocity correction applied to the lifting- line-theory value. (o) The small additional lifting- surface correction to the hinge moment (fig. 5) is obtained from = 0.0207 ind since gf = 13.50, •n = 1 - 0.0005(13.5)2 = 0.91 Prom figure 6, ^ =0.55 (Ca/c) Therefore, A/Ch ) = 0.0207 X 0.91 x 0.55 = 0.0103 and "(=h)Ls = 0.0103 X 0.03 = 0.0003 (9) The AP^ due to the additional lifting- surface correction of step (8) may be e::pressed as = 0.124 pound 14 NACA ARR No. L5P23 Then, Total stick force = F^ + AP3 = 3.62 + 0.124 . ^ = 3.74 The stick-force computations for a range of aileron deflection are presented in table I. The final stick- force curves are presented in figure 11 as a function of the value of pb/'^V calculated, for the rigid unyav/ed wing. For comparison, the stick forces (f irst-approxlraation values of table I) caloulatedi by neglecting the effect of rolling are also presented. Stick-force characteristics estimated for the flexible airplane with fixed rudder are presented in figure 11. The values of pb/2V obtained for the rigid unyav/ed wing were simply reduced by applying an empirical factor of 0.75 as indicated by the approxi- mate rule suggested in the preceding section. No calcu- lations of actual v/ing twist or yaw and yawing motion- were made for this example. II - D E V S L P M E N T OF The method for determining values of C7 and Ch„ p -t^p is based on the theoretical flow around a wing in steady roll with the introduction of certain empirical factors to talce account of viscosity, wing twist, and minor effects. The theoretical solution is obtained by means of an electrom.agnetic-analogy m.odel of the lifting surface, which simulates the wing and its wake by current- carrying conductors in such a manner that the surrounding magnetic field corresponds to the velocity field about the v/ing. The electromagnetic-analogy m.ethod of obtaining solutions of lifting- surf ace- theory problems is discussed in detail in reference 4. The present calculations were limited to the case of a thin elliptic wing of aspect ratio 6 rolling at sero angle of attack. 1\^ACA ARR Wo. L5F23 ELECTROMAGNETIC -ANALOGY MODEL Vortex Pattern In order to construct an electromagnetic-analogy model of the rolling wing and wake, it is necessary to determine first the vortex pattern that is to represent the rolling wing. The desired vortex pattern is the pattern calculated "by means of the two-dimensional theories - thin-airfoil theory and lifting- line theory. The additional aspect-ratio corrections are estimated hy determining the difference between the actual shape of the wing and the shape that would "be required to sustain the lift distribution or vortex pattern determined from the two-dimensional theories . For the special cases of a thin elliptic wing at a uniform angle of attack or in a steady roll, the lifting-line-theory values of the span load distribution may be obtained by means of simple calculations (refer- ence 6) . The span load distributions for both cases are equal to the span load distributions determined from strip theory Vi/lth a uniform redaction in all ordinates of the span-load curves by an aerodynamic- induction factor. This factor is -,; p^ for the wing A -r d at a uniform^ angle of attack and -, — '■ — 7 for the wing in A + 4 steady roll. The equation for the load at any spanwise station — 'V- '^^ ^ thin elliptic v/ing at zero angle of b/2 attack rolling steadily with unit wing-tip helix angle pb/2V is therefore (see fig. 12) cc b/2V) - A + 4 (^/2)^ \W2j ^ ' where a^ = 2Tr. 16 NACA aRR IIo. L5F23 The chordwise circulation function _£l_ from thin- CCjV airfoil theory for an inclined flat plate is 2V cc^V 1 TT /f - (f ) ' + cos"Vl -f^ (4) where x/c is measured from the leading edge fig. 15 for values of 2V \ ccjVy / I: ee The vortex pattern is determined from lifting- line theory as the product of the spanwise- loading CCi function : — '^ and the chordwise circulation Cg(pb/2y) function — ^ CC^v wake ; thus , r^ for all points on the wing and in the q p cc- CgV(ph/2V) Cg(pb/2V) cc^V Contour lines of this product determine the equivalent vortex pattern of the rolling wing. Ten of these lines are shov/n in figure 14. The contour lines are given in term.s of the Toarameter 2r c<,V(pb/SV) 2V CoV(pb/2V) -i max which reduces to r "■ max Construction of the Model Details of the construction of the model may be seen from the photographs of figure 1. The tests were made under very nearly the same conditions as were the tests of the preliminary electromagnetic-analogy model, reported in reference 4. The span of the model v/as !TACA ARR No. L5P23 17 tv/ice that of the model of reference 4 (6.56 ft Ir. stead of 2.28 ft), but Flnce the aspect ratio is tv/ice as large (6 Instead of 3), the maxiraurn chord is the rame. In order to simplify the construction of the model, only one seriispan of the vortex sheet v/as simulated. Also, in order to avoid the large concentrations of wires at the leading edf;:e and tips of the wing, this sem.ispan of the vortex sheet v/as constructed of two sets of Vi^ires; each of the wires in the set representing the region of high load grading simulated a larger increment of A ( — - — ] than the wires in the set representing the region of low load grading. Downwa sh M e a sur erne nt s The m.agnetic-f ield strength was neasured at 4 or 5 vertical heights, 15 spanwise locations, and 25 to 50 chordv/ise stations. A number of repeat tests vv'ere made to check the accuracy of the measurements and satisfactory chec'rs were obtained. The electric current was run through each set of v/ires separately. With the current flowing through one set of v;ires, readings were taken at points' on the model and at the reflection points and the sum of these readings was multiolied by a constant determ.ined from the increment of vorticity h (•— — \ represented by that set of wires. Vrmax / Then, with the current flowang through- the other set of wires, readings v'ere taken at both real and reflection points and the sum of these readings was multiplied by the appropriate constant. The induced downwash was thus estimated from the total of the four readings. The fact that four separate readings had to be added together did not result in any particular loss in accuracy, because readings at the missing semi span m^ere f airier small and less influenced by local effects of the Incremental vortices. A more accurate vortex' distribution was made possible by using two separate sets of wires. The measured data were faired, extrapolated to zero vertical .height, and converted to the downwash function' -^-^ as dis- ^max cussed in reference 4. The final curves of -^ are "^■^max 18 HACA ARR No. L5F23 presented for the quarter chords half chord, and three- quarter chord in figure 15. Also presented in figure 15 are values of -^ calculated by lifting-line theory ^ '- max and values calculated oy lifting-line theory as corrected hy the Jones edge-velocity correction. DEVELOPMENT OF FORMULAS General Discussion Lifting-surface corrections .- The measurements of the magnetic-field strength (induced downwash) of the electromagnetic-analogy model of the rolling wing give the shape of the surface required to support the distri- bution of lift obtained by lifting-line theory. Correc- tions to the spanwise and chordwise load distributions may be determined from the difference between the assumed shape of the surface and the shape indicated b:/ the downwash measurements. Formiulas for determ.ining these corrections to the span load distributions and the rolling- and hinge-moment characteristics have been developed in connection vflth jet-boundary-correction problems (refer- ence 5) . These formulas are based on the assumption that the difference between the two surfaces is equivalent at each section to an increment of angle of attack plus an increment of circular camber. From figure 15 it may be seen that such assumptions are justified since the chordwise distribution of downwash is approximately linear. It should be noted that these formulas are based on thin-airfoil theor7/" and thus do not take into accoujit the effects of viscosity, wing thickness, or compressi- bility. Viscosity . - The complete additional aspect-ratio correction consists of two parts. The main part results from the streamline curvature and the other part results from an additional increment of induced angle of attack (the angle at the 0.5c point) not determined by lifting- line theory. The second part of the correction is norm.ally small, 5 to 10 percent of the first part of the correction. Some experimental data indicate that the effect of viscosity and wing thickness is to reduce the theoretical streamline-curvature correction by about 10 percent for airfoils with small trailing-edge angles. ■?;ACA ARR No. LtP23 19 Essentially the same final answer is therefore obtained v/hether the corrections are applied in two parts (as should be done, strictly speaking) or v^hether they are applied in one part by use of the full theoretical value of the streamline-curvature correction. The added simplicity of using a single correction rather than applying it in t".vo parts led to the use of the method of application of reference 3. ■ The use of the single correction worked very well for the ailerons of reference 3, which were ailerons with small trailing-edge angles. A study is in progress at the Langley La.boratorles of the NACA to determine the proper aspect-ratio corrections for ailerons and tail surfaces with beveled trailing, edges . For. beveled trailing edges, in which viscous effects may be much m.ore pronounced than in ailerons with si.iall trailing- edge angles, the reduction in the theoretical streamline- curvature correction may be considerably more than 10 percent; also, when Cho; is positive, the effects of the reduction in the streamline-curvature correction and the additional downwash at the 0.50c point are additive rather than compensating. Although at present insufficient data are available to determine accurately the magnitude of the reduction in the streamline- curvature correction for beveled ailerons, it appears that the simplification of applying aspect-ratio correc- tions in a single step is not allowable for beveled ailerons. The corrections v/ill therefore be determined in two separate parts in order to keep them general: one part, a streamline-curvature correction and the other, an angle-of-attack correction. An examination of the experimental data available Indicates that more accurate values of the hinge moment • resulting from streamline curvature are obtained by multiplying the theoretical values by an empirical reduction factor n, wh^ch is approximately equal to 1 - 0.000502 where ^is the trailing-edge angle in degrees. This factor will doubtless be modified when further experimental data are available. Compressibility .- The effects of compressibility upon the additional aspect-ratio corrections were not considered in reference 5. First-order compressi- bility effects can be acco\aited for by application of the Prandtl-G-lauert rule to lifting- surf ace- theory results. (See reference 7.) This method consists In 20 NAG A ARR No. L5F23 determining the compressible -flow characteristics of an equivalent wing, the chord of v;hich Is Increased by the ratio where M Is the ratio of the free- / 1 - M^ stream velocity to the velocity of sound. Because approximate methods of extrapolating the estimated hinge-moment and damping-mom.ent parameters to wings of any aspect ratio will be determined, it Is necessary to estimate only the hinge-moment and damping parameters corresponding to an equivalent wing with its aspect ratio decreased by the ratio /l - M^ . The estimated param^eters for the equivalent v;ing are then Increased by the ratio \/l - M'^ The formulas presented subsequently in the section "Approxim.ate Method of Extending Results to Wings of Other Aspect Ratios" are developed for M = 0, but the figures are prepared by substitutin-g Aq = A \/l - M for A and multiplying the param.eters as plotted by yl - M^ . The edge-velocity correction factors E^, Eecj E'c, and E'ec ^^® '^^^'^^ factors corresponding to A^ The figures thus include corrections for first-order compressibility effects. Thin Elliptic Wing of Aspect Ratio 6 Damping in roll C? .-In order to calculate the correction to the lifting-line-theory values of the damping derivative C7„ it is necessary to calculate the rolling m.oment that would result from an angle- of -at tack distribution along the wing span equal to the difference betv/een the measured downwash (determined by the electromagnetic-analogy method) at the three- quarter-chord line and the downwash values given by lifting-line theory. (See fig. 15.) Jones has obtained a slm.ple correction to the lifting-line-theory values of the lift (reference 1) and the damping in roll (unpublished data) for flat NAG A ARR Fo. L5F23 21 elliptic wings. This correcticn, termed the "Jonjs edge- velocity correction/' is applied by multiplying the lifting-line-theory values of the lift hj the ratio An + 2 AcEc + 2 of the dam.ping and the lifting-line-theory values A, in roll by + ^c^'c + 4 with values of E, and E'^ as given in figure 16. As may be seen from. figure 15, the dov/mvash given by the Jones edge-velocity correction is almost exactly that measured at the 0.50c points for flat elliptic v/ings. This fact is useful in estim.ating the lifting- surface corrections because the edge-velocity correction, which is given by a simple formula, can be used to correct for the additional angle of attack indicated by the linear difference in downwash at the 0.50c line. The variation in doi^nv/ash between the 0.25c line and 0.75c line, apparently linear along the chord, indicates an approxim.ately circular stream.line curvature or camber of the surface. The increment of lift resulting at each section from circular camber is equal to that caused by an additional angle of attack given by the slope of the section at or the tangent at 0.50c Because this difference 0.75c relative - that is. to the chord line (v)o.75c " ^"^/0.50c' in dov;nwash does not vary linearly along the span, a spanwise integration is necessary to determine the stream.line-curvature increment in rolling moment; that is, r-i) sc ST' max-^c bV(AcE'c + 4) Jo ri / wo l2r max -/0.75c An evaluation of T m.ax 0.50c m berms ox Cs b/2 pb/2V (5) IS necessary to determine the correction to the damping-moment coefficient C: 'P The lifting-line-theory relation 22 NACA ARR IIo. L5F23 between F ^^^ and p'!d/2V is, from equation (3), _ 2\T3(pb/2V) "max ~ A + 4 V/lth the edge-velocity correction applied ^ _ 2Vt(pb/2V) ^ rnax - "A^E'c + 4 (6) The value of the streamline-curvature correction to C;^ is therefore P (f^^p)sc 16 Ae .'1 ^(_wb_^'^ \ QP niS-X, 0.75c ^ wb ^■ ,' WD \ \2^ max/ 0.50c _£. _Z_ d/-l b/2 \b/2 (7) A graphical integration of equation (7) gives a value of C.022 for /aCt By the integration of equation (3) , the value of (Ct I for incompressible flovi' is found to ^e -? -r-~—7 - 0.471 'for A = 6. 4 A + 4 Application of the edge-velocity correction, for A = 6, p;ives N ''p lA EV 4(AE' + 4) = 0.433 and, finally, subtracting the streamline-curvature correction gives a value of ^l-r.) ^^^ A = 6, as follows : Cl. = Cj ^ \ p/ev = 0.411 AC- 'P NACA Aim Ho. L5P23 'j'he value of '^l for a Y/ing of aspect ratio 6 5,3 therefore 13 percent less than the value given by lifting-line theory and 5 percent less than that given by lifting- line theory Tivith the Jones edge- velocity correction applied. ^ling.e-moriient parameter Ch .- The rtreamllne- curvature correction to Cv, for- con3tant-"Dercentage chord -\llerons i; ■P Oi max ■^" .ven ;, from reference 5 and with the value .n equation (6) , ^h P/SG l'Fr\ r i 6(x/c) . C \" \C; . ( J \ \^ + 4 (8) '7^ where the integrations are made across the aileron span. Because the downvirash at the 0,50c point is given satisfactorily by applying the edge-velocity correction to the lifting-line- theory values of the downvvash, the part of the correction to Cv,p -which depends upon the downwash at the 0.50c point raay be determined by means of the edge-velocity correction. The effect of aerodynamic Induction was neglected in developing equation (8) because aerodynam.ic induction has a very sraall effect upon the hinge -m.o3r.ent corrections caused by streamline curva.ture. Values of the factor various (oa/c)2 chord ratios and balan^^e rati.os as determined from thln- a.irfoll theory are given in figure 6. As mentioned previously, rj is a factor that approximately accounts for the combined effects of wing thickness and viscosity m in altering the calculated values of experim.ental ^ta available at pre 5^ en t indicate that T] r 1 - O.OOOSjZf'^. Results of the integration of equation (8) for the elliptic wing of aspect ratio 6 are given in figure 17 as the parameter .^2 f^Chp) /SO (V + ij\/i - m' Values 24 NACA AKR No. L5F25 of (AC^ ^ 'hrr j __ p^?r — (Ac + l) /l - 1^ determined as in refsreiice 3 are given in figure 18. is The value of fChS] f^p) /LS 'Mll ^'P)ci Ac + 4 ^^LL '^e^V — ^(^^^P). Since A,3 + 2 AcEc + 5(=ha),,= (=^-.),S-e=^-a)sc then (°bp)^3 = (-P)ch A + 4- A. f^ + 2 j^^ A^E'^ + 4 A^, - 2 \ -a/rs + (^°'^p)sc- ('°'^a)3, Nc^^I Af, + 4 A.^Ep + 2 or The formula for the parameter + (=-"p) ("p)< LS • (9) is derived VChLT. for elliptic wings in the apjpendix, and numerical values A_ + 4 ^ in figure 3, ChLL ^c + 2 are given in the form (a \ together with values for tapered wings derived from the data of reference 2. It -CTi&j he noted that use of the parameter (o.j}\ determine the total correction for rolling would he impractical because (Cj-^ \ is not proportional to '^LS To. L5F23 25 ;>, \ . Although the numerical values of /j-ni ^ /IS . ,. V ^Z < varv considerably with y^\y ) , the actual effect on \ Vls the stick forces is small because /ctp\ changes most with Ch ) when the values of [0,^^ \ are small. < VLS \ «/LS This effect is illustrated in figure 19, in which numerical values of ('cip\ for a thin elliptic wing of aspect ratio 6 are given, together virith the values obtained by lifting-line theory, the values obtained by applying the Jones edge-velocity correction, and the values obtained by using the aileron midpoint rule (reference 8) . The values obtained by the use of the Jones edge-velocity correction are shown to be 4.4 percent less than those obtained by the use of lifting-line theory. The right-hand side of equation (9) is divided into the following two parts: Part 1 = (ap) (ch„) ChEv V 7lS Part II = Afch„) \ VLS Part I of the correction for rolling can be applied to the static hinge -moment data as a change in the effective angle of attack as in reference 2. (Also see equation (2) ,) Part II of equation (9) , however, is applied directly as a change in the hinge-moment coefficients. AC h Inasmuch as part II of equation (9) is numerically fairly small { Aq, =0.002 for £| = o.l for a wing of aspect ratio 6y , it need not be applied at all except for fairly large airplanes at high speed. 26 I-TACA ARR No. L5F23 Approximate Method of Extending Results to Vvings of Other Aspect Ratios Damping in roll Ct,^.- In order to make the results of practical value^ it Is necessary to formulate at least approximate rules for extending the results for a thin elliptic wing of aspect ratio 6 to wings of other aspect ratios. There are lifting- surface-theory solutions (references 4 and 9) for thin elliptic wings of A = 3 and A = 6 at a uniform angle of attack. The additional aspect-ratio correction to Cj; was computed for these cases and was found to be approximately one-third greater for each aspect ratio than the additional aspect-ratio correction estimated from the Jones edge-velocity correction. The additional aspect-ratio correction to C7 P for the electromagnetic-analogy model of A = 6 was also found to be about one-third greater than the corresponding edge-velocity correction to C^,--. A P reasonable method of extrapolating the values of Cj to other aspect ratios, therefore, is to use the variation of the edge-velocity correction with aspect ratio as a basis from v/hlch to work and to increase the magnitude by the amount required to give the proper value of C^,^ for A = 6. Effective values of E and E' (^Eg and E'e) were thus obtained that would give the correct values of Ct, for A = 3 and A = 6 and of Ct, for A = 6. The formulas used for esti- mating Ee^ and E'q_ for other aspect ratios were See = 1.55(Ec - l) + 1 = 1.65 (s'c - l) + 1 Values of Ee^. and E'e^. ^-^^ given in figure 16. are presented in figure 2 as a function of Ac/ao Values of — ^ v'l - M'^ determined by using E'e^ aQ c NACA AP.R No. L5F25 27 where A^ = A /l - M"^ and a.^ is the incompre ssfhle slope of the section lift cjirve per degree. Hinge-moment parameter Ch .- In order to deter- 2_P mine Ch for other aspect ratios, it is necessary to estimate the formulas for e.xtrapolating the streamllne- 7sc \ '''^ysc' of (ACh^'l for A = 3 and A = 6 are available in SC curvature corrections (ACh \ and (ACh ' • Value; reference 3. Values of (AC]-, \ might he expected to be ^ / 3C approximately Inversely proportional to aspect ratio and / \ K-| an extrapolation formula in the form ! ACv, i = -; =rr— V N SC ^ + ^2 is therefore considered satisfactory. The values of K]_ and K.2 are determined so that the values of (ACh i \ V SC for A = 3 and A = 6 are correct. Values of K]_ and K2 vary with aileron span. The values of li2, however, for all aileron spans less than 0,6 of the semi span are fairly close to 1.0; thus, by assuming a constant value of Kg = 1.0 for all aileron spans and calculating values of llj^, a satisfactory extrapolation formula may be obtained. It is impossible to determ.lne such a formula for (ACh I because results are available \ P/SC only for A = 6; however, it seems reasonable to assume the same form, for the extrapolation form.ula and to use the same value of Kq as for (ACv, \ . The value ^ \ S'sc of K-| can, of course, be determined from the results for A = 5. Although no proof is offered that these extrapolation formulas are accurate, they are applied only to part II of equation (9) (values of A(Cit \ ), v/hich is numeri- \ V "P/ls/ cally quite sm.all, and are therefore considered justified. 28 IJACA ARR No. L5F2; CONCLUDING REMARKS Prom the results of tests made on an electromagnetic- analogy model simulating a thin elliptic v^ing of aspect ratio 6 in a steady roll, lifting-surface-theory values of the aspect-ratio corrections for the damping in roll and aileron hinge mom.ents for a wing in steady roll were obtained that are considerably m.ore accurate than those given by lifting-line theory. First-order effects of compressibility were included in the computations. It v;as found that the damping in roll obtained by lifting-surface theory for a wing of aspect ratio 6 is 13 percent less than that given by lifting-line theory a.nd 5 percent less than that given by the lifting-line theory with the Jones edge-velocity correc- tion applied. The resul.ts are extended to wings of any aspect ratio. In order to estimate aileron stick forces from static wind-tunnel data, it is necessary/- to Irnow the relation between the rate of change of hinge moments with rate of roll and the rate of change of hinge moments vi'lth angle of attack. It was found that this ratio is very nearly equal, within the usual accuracy of wind-tumiel m.easurements , to the values estimated by using the Jones edge-velocity correction, which for an aspect ratio of 6 gives values 4.4 percent less than those obtained by means of lifting-line theory. The additiorial lifting-surface-theory correction that was calculated need only be applied in stick-force esti- mations for fairly large, high-speed airplanes. Although the method of applying the results in the general case is based on a fairly cornxDlicated theory, it may be applied rather simply and without any reference to the theoretical section of the report. Langley Memorial Aeronautical Laboratory National Advisory Committee for Aeronautics Langley Pie id, Va. lACA ARR llo. L5P23 29 APPENDIX EVALUATION OP /ar,\ FOR ELLIPTIC V/INGS It was shown in reference 2 that for constant- percentage-chord ailerons the hinge moment at any aileron section is pi'oportional to the section lift coefficient multiplied by the square of the wing chord; for constant- chord ailerons, the hinge moment at any aileron section is proportional to the section lift coefficient divided by the v/ing chord. The factor /a-n\ is obtained V"°-^LL , by averaging the tv;o factors CjC^ and Cy/c across the aileron span for a rolling wing and a wing at constant angle of attack. For elliptic wings, with a slope of the section lift curve of 2Tr, it was shown in reference 6 that strip-theory values multiplied A A by aerodynamic-induction factors -, ^^^—^ or A^ + 2 A^ + 4 could be used. (Note that A„ is substituted for A to account for first-order effects of compressibility.) Thus, for constant-percentage-chord ailerons on a rolling elliptic wing, ^£_, sin^e c 22TT M o ^l'^" ~ Ac + 4 ""'' ^ "s '^" ~V = — — ^ — 2 ££ sln"^9 cos 9 Ac -i- 4 2V and for the same wing at a constant angle of attack a A cz,c2 = - — -2__ sinSe 03^2 Ac + 2 Tra r In order to find /ap\ , the integral /CtcS dy across the aileron span must be equal for both the 30 ".;TACA ARR No. L5P23 rolling v/ing and the wing at constant a. Thus, jcic^ dy "-^ i V^m^e .0= 9 *, STrcs^Ac /■. 2q ^ — — £: — — a /sxn'^Q dy dy = 2 d(cos 0) § sin e de Let ^ = Ho^.„, i ^^LL («p) Ac + 2 (7 /sin'^e cos 6 d9 Ch LL Ac + 4 Ac + 2 3r d9 T^i^^eje 1 ■ijsln^B cos 9+2 cos 9j I'o where 9^ and 9^ are parameters that correspond to the outboard and Inboard ends of the aileron, respectively. / \ A^ + 4 Values of (a-n\ — ?t were calculated for the outboard end of the aileron at —7— = 0.95 and plotted b/2 in figure 3. JACA ARR No. L5F23 A similar development gives, for the const ant -chord aileron, ^^- SrrAr pb /cos G c /A + 4 \ 2V ^ s {^^c J sin ^y JttA, r C3(^^C + 2) ph / \ / ; \ 2V V^'P/ChLi^ "^'^^ dy Ln G fa. Vci ^^c + + _4 _ < 2 ycos 9 ae ^-c /de r i'° sin G ^ [^r These values are also presented in figure 3. 32 I'lACA ARR No. L5P23 REFERENCES 1. Jones, Robert T.: Theoretical Correction for the Lift of Elliptic Wings. Jour. Aero. Sci., vol 9, no. 1, Nov. 1941, pp. 8-10. 2. Swan:^on, Robert S., and. Toll, Thomas A.: Estimation of Stick Forces from Wind-Tijxinel Aileron Data. NACA ARR No. 2 J2 9, 1943. 3. Swanson, Robert S., and Gillis, Clarence L. : Limitations of Lifting-Line Theory for Estimation of Aileron Hinge-Moment Characteristics. NACA CB No. 3L02, 1943~". 4. Swanson, Robert S., and Crandall, Stevifart M.: An Electromagnetic-Analogy Method of Solving Lifting- . Surface-Theory Problems. NACA ARR No. L5D23, 1945. 5. Swanson, Robert S., and Toll, Thomas A.; Jet-Boundary Corrections for Reflection-Plane Models in Rectangular Wind Tunnels. NACA ARR No. 3E22, 1943. 6. Munlc_, Max M. : Fundamentals of Fluid Dynamics for Aircraft Designers. The Ronald Press Co., 1929. 7. Goldstein, 3., and Young, A. D.; The Linear Perturbation Theory of Com.pressible Flow with Aiopllcations to Wind-Turmel Interference. 6865, Ae. 2262, P.M. 601, British A.R.G., July 6, 1943. 8. Harris, Thomas A.t Reduction of Hinge Moments of Airplane Control Surfaces by Tabs. NACA Rep. No. 528, 1935. 9. Cohen, Doris: A Method for Determdning the Camber and Twist of a Surface to Support a Given Distribution of Lift. NACA TN No. 855, 1942. NACA ARR No. L5F23 33 00 B o h o tn o fs o 6h a < m g > n o K *-• o o O 3 > « U) ■H E 2 CO Pc O I n o o O 0] « a i.>J o lO lO CM o CM o> lO CM o o * • o> o o • o o CM 1 o 1 o o rH n lO o in to 00 rH o ■* o> to to r-1 to ^^ to CM 0) IO o CM iH fi to ^ to o CM r-K CM CO CM u H o o o • * o • o O • • o ■H • • to H • •* • • CO IM d 1 o o I-I 1 o 1 o O o o C4 CM o c- t- 43 CO c-l Oi n I-I to I-I (0 0» CM t- n 00 8 .H I-t I- to o CM i-i 1-1 0* i-l •H o o • • o • o o • • ^ • • • t* o • to • « t- 00 o o 1 O o • o 1 o o C o ^^^^ c 00 00 to CO CM o iH to to •I" 8 o to to 0> CM ■* g •H 8 8 g to • CM • 8 8 to • • 4> • • • 10 o • CM • • to to 4J o c o o o o s e 3 1 1 d Si Si I- _ > ol o o U) 0) o (i rt CM lO t- o o T5 o o H 0> to ■a la iH to CM 5S1 to to t- (0 09 CM <»« O to a CM CM CO ej rH c- 00 rH o O o O •H • O • o • o o r-\ • o M (, • » • • to o CM * ^ • • • to • O o. o o o o e 1 o o o o o *> a • CO 1 « 3 43 (O c- to CO Oi o o> o IO •O 10 to ra A Vi +1 O o O o • • o • O O o • o o ^ • • • I-I O • to • • « rH • o C> o o o o o o o o 4j c 1 1 00 CM o m CO rl" CM o lO 0) ^ lO Id l-« O) in iO to to lO i-\ o to CM If •H 8 8 I-I o CM o • to • § CM • o o iH o 8 0> • CM O a • • • • o o • CM • • • o • « o *? o o o o 1 o o o 43 • a a m > W a o t > Si • <0 o 1 > A ^ rO CM o lO s. • ^ — N^ 1-3 .?» o ■e ■o .^ i-a o \ ^^mt ro ^-^ r^ o \ ^-* Ol o o p "b tJ o o o p tJ IO ei a < •— ' ^— -^ *.-• ^^-^ Ol ■o i> o I-H I-t CM H iH " p-l CM • o m o OK CO o % o •H 43 o « rH Vi • •H «H IS 4) " % « > d *< «H 43 t£ «H »H «1 F< O o. ol 43 CM ■O Si « Ol o •D 3 43 rH Ol j CO t:) cd cd > QJ <—{ tm O, a o to U ■-1 (d 0) cd c C <^-i 0) (d o o 1 o tio — ■ •H C n3 +J -H ■ — (U > c bD K a 2 O i <2 o •:^£ Z UJ ii o s VD ^ W +3 «} u o p. to ClJ -H •s nJ •^'^ > •H >v ^ d >^ UJ H^ ^ ^ 5 5- M « + s^ ^ < «>>^ ti-i ^ \ C o V ■^ ^ a! .^ •H ^^j > • ^ 3 bo Fig. 5 NACA ARR No. L5F23 M S iH ,-» 5i CM CM is. O •\ P o OS iK o J • ^ -' o ^ \ //. / / / / / ■^ / / / / / / / .s> i '/ f / / / / > 1 / / / ^ / 1 .8 1 / / / / 1 / / / / 1 / 7 1 7 / / / /I h / / / / J / .6 1 / / / / 1 / / / r ,S ll t / / / 1 V / / / / .^ / t / / 1 / / / .3 I / i / / / I / / .Z h 1 / / 1 / ./ / NATIONAL ADVISORY COMMITTEE F0« AERONAUTICS 1 1 r> ./ .^ .3 ff- .5 Ai/eron -chord rai-/o , Ca/c w/th a//eron- chord rat/o ard ex.terr}a/-o/er/7a/o^ aerodj/n^ I Hy?/ K \ \ . ex. r^ >/¥^ n\ \ (deq) / 7 p€\ V -4 3 // V / Ail V 1 I o ./_ A 4.5. □ 8.9- X /J.5_ 17-7 '/ / 7 \ // V ;/ m \ // V \^ \: // / N^ ^^ \ / H M p5q K W' \ i V\ f J . c \a I r A Y ^ r Lii • «#■ jV [^ . T — ^ rm- h — > r -( .02$ -.0/^ I ,03 -26 -20 -15 -10 -5 O 5 10 15 20 25 Aileron deflection, 6q, deg Figure d.- Aileron hinge- and rolling -moment characteristics of the 0.40 -scale nnodef of the airpfane used for the ittustrative example. Characterfst/cs plotted against aileron deflection. Fig. 9 NACA ARR No. L5F23 -/6 -IP i — . NATIONAL ADVISORY COMMITTEE FOB AERONtUTICS ./2 (c ^) r' .08 / / y ^ -8 -4 .04 // / r' ~ — - -^ 6 ^ 1 iH ^ 1 — h-7- =i =^ i^= -^ ~~~ ^ f ~r- -^ ■ pT^ ~ ^- ;:rr rr -.04 ^ ^ / . >> _ - - -- ^ -^ ""^^ ~— > ' "'^-^ --, _^ ~^ ■ ^ ^ -- ^ -.08 4 /, // / "~~ — - — , "■ 8 /. / --- . ■J2 12 16 _/ .03 icg 16 /^ ^ ' .oz ^ ^ ^ ^ ^-' .0/ ^ ^ ^ ^ 5 "^ ^ 3 ^ ^ -^ 1 ^ ^ -^ -^ ^ ^ NATIONAL ADVISORY COMMITTEE FM AERONAUTICS '^ .0/ ./o .oz .03 .04 .05 .06 .07 .08 .09 He//x ang/e ,pb/ZV^ rad/an f/^ure /O .-Prehm/naru cutve5 useof /r? f/?e conipufaf/o/i of the ay/n^'t/p /)e/jx ang/es anc/ 7/ie sf/cA forces for 7/?e j7fusfraf/ve examp/e. II Fig. 11 NACA ARR No. L5F23 Aileron det/ection (deo ' ■ l(o r / cO / / /(> (a) Effect of rollin^^ yaw^ yaw/ny., and win^ fwisf neq/ecfed /n bof/i F^ and ph/ZV. y y y /z '}z ^ ^ 6 ^ ^ ^ 4- ^ ^1^ _^ ^ ^ (b) Effect of rol//n^ accounted for m hot /i fl and pbJZV; effect of yaiAZ-iYawmi^.^ and wm^ twist accounted for jn f^ but not m ph/Zlf. [ 17 /& ^ ^ d 1. s:: ^ -" ^ -"""'^^ 1- ^ -^ /z (c) Effect of roll my accounted for /n both E and oh/ZV \ effect ofyaWyyawmj, and i 1/1 np twist accounted ^or r-\ 6 in F^ and assumec / to __ . — ■ — -t- IZ- ' — ~ r' 1^ decrease phj Z5/:>ercent. '/iV by -d j± ^ y NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS "r- ^ ^4 n ^ -^ ^ to J I .01 .OZ .03 .04- .OS .06 .07 .06 .09 hte//x an^/e^ ph/^V^ relief /an F/^ure // .'5t/ct< -force ctiaractenst/c5 esf/matecf for ftie a/rp/ane used for //?e /J/astraf/^e e){a/nple. NACA ARR No. L5F25 Figs. 12,15 1 >- •- _ a 3 §1 l§- -■s Z uj ou;- s- \ \ \ V \ \ \ \ \, \ \, s X ^ ^ . Q o V 3 o U-, Ui ^ c \ c a -C o E o ft O (w VJ (-. ■0 \ ^ . c • * u VI -.J 0) 3 ^* Vv ra ^.^ —1 T3 ♦J 11 CO P. S _ ~D u wlox: Q X «0 vo ^V o -^ 'C/0/-P syu/?y U0/^0/^3J/Q 10 ii / ( \ \ \ ■V \ X \^ - va ■a ^ ^ 00 'uo»f9U^j pbo/ uodg^ I Cb o Fig. 14 NACA ARR No. L5F23 V^Q 5i 0" I i u > K Si I Z uj I ^ I I 5^ 5 50 N. I NACA ARR No. L5F23 Fig. 15 wb Z.8 1 1 ' / Liftina- //na f/ieo/^y lu/ t/) -Jonas / // Zi' (so//d ///7e) '" -\ ' -^ V / meaiurecf at j -7 ^c 1 ' -> ^^ / ^ / 7/ ./ / ZO V *~- ^^ ,/ / ^ // 7 /6 '(' / //a ' / bf^JM -V. /.-^ r I.Z /// / 7 /*lf Ofy /// .3 ' / .^ / "/ > f NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 1 i ri / /^ Figure 15.- Induced downwesh f\anctlon wb/z/^gx ^°^ an elliptical lifting surface of aspect ratio 6 in steady roll. Fig. 16 NACA ARR No. L5F23 s i^ 1 ^^ i ?K1 i| 1 £; , 1 < c: II II ' •^; 1.. -I- 1 ^ / 1 / 1 / ^^/ / / / 7 / / / / / ^V / y Y / / / L .. J / ^ / / / / 7 •3 ^ ^ ^ ^ ^ y / / / ^ ^ ^ 1 "^ ^^ ^ ^ y / / -^ ^ ^ y ^ ^ S5 O ■< < k 1 f 3^^ '■'3 ;^ ■N, ■^ ( NACA ARR No. L5F23 Figs. 17,18 uci//pe>J >