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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARI'TI ME RE PORT ORIGINALLY ISSUED August 1945 as Advance Restricted Report L5F23 IJTINGSURFACETHEORY VALUES OF THE DAMPING IN ROLL AND OF THE PARAMETER USED IN ESTIMATING AILERON STICK FORCES By Robert S. Swanson and E. LaVerne Priddy Langley Memorial Aeronautical Laboratory Langley Field, Va. 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. DOCUMENTS DEPARTMENT frAA 1 3 L 53 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/liftingsurfaceth001ang 7/. 2If <, NACA ARR IHo. LEF23 RESTRICTED TIATIO!IAL ADVISORY COMMITTEE FOR aERO.IA.UTICS ADVANCE RESTRICTED REPORT LIFTINGSURFACETHEORY VALUES OF THE DAMPING III ROLL AND OF THE PARAIT.ETER USED III ESTIMATING AILEROni STICK FORCES By Robert S. Swanson and E. LaVerne Priddy SUMMU1ARY An investigation was made by liftingsurface theory of a thin elliptic wing of aspect ratio 6 in a steady roll by means of the electromagnetic analogy method. From the results, aspectratio corrections for the damping in roll and aileron hinge moments for a wing in steady roll were obtained that are considerably more accurate than those given by liftingline theory. Firstorder effects of com pressibility were included in the computations. The results obtained by liftingsurface theory indicate that the damping in roll for a wing of aspect ratio 6 is 13 percent less than that given by lifting line theory and 5 percent less than that giren by liftingline theory with the edgevelocity 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 windtunnel data, it is necessary to know the relation between the rate of change of hinge moments with rate of roll and rate of change of hinge moments with angle of attach. The values of this ratio were found to be very nearly equal, within the usual accuracy of windtunnel measurements, to the values estimated by using the Jones edgevelocity correction which for a wing of aspect ratio 6 gives values 4.4 percent less than those obtained by liftingline theory. An additional liftingsurfacetheory correction was RESTRICTED NACA ARR No. L5F23 calculated but need not be applied except for fairly large highspeed airplanes. Simple practical methods of applying the results of the investigation to wings of other plan forms are given. No knowledge of liftingsurface theory is required to apply the results. In order to facilitate an understanding of the procedure, an illustrative example is given. INTRODUCTION One of the many aerodynamic problems for which a theoretical solution by means of liftingline 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 liftinglinetheory values of the damping in roll that amounts to an 8percent reduction in the values for a wing of aspect ratio 6. Still more accurate values may be obtained by use of liftingsurface theory. A method of estimating aileron stick forces in a steady roll from static windtunnel 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 moment with rate of roll Chp and the rate of change of hinge moment with angle of attack Cha in the form of the parameter ) = which is determined by means parameter pCh rCh of liftingline theory. It was pointed out in reference 2 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 liftingline theory. A more exact determination of the parameter p )Ch is desired. In reference 3 an addi tional aspectratio correction to Cha as determined from liftingsurface theory is presented. In order to evaluate the possible errors in the values of (ap)Ch NACA ARR IT. L5?23 as determined by liftingline theory, it is necessary to determine similar additional aspectratio corrections to C p. A description of the methods and equipment required to solve liftingsurfa:cetheory problems bj means of an electromagnetic analogy is presented in reference 4. An electromagneticanalogy model simulating a thin elliptic wing of aspect ratio 6 in a .teady roll was constructed (fig. 1) and the magneticlield strength simulating the induced downwash velocities was measured by the rnethods of reference 4. Data were thus obtained from which additional aspectratio corrections to Chp for a wing' of aspect ratio 6 were determined. Because of the small magnitude of the correction to (aP)h introduced by the liftingFurface calculations, it was not considered worth whilee to conduct further experiment on wings of other plan forms. An attempt was therefore made to effect a reasonable generalization of the results from the available data. Inasmuch as the th6ery 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 material 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 IT gives the development of the theory. SYrIEOLS a angle of attack radiants, unless otherwise stated) cb section lift coefficient L q i CL wing lift coefficient ( L\ q3 / H/inge moment Ch hingemoment coefficienTL n o ent CL rollingmoment coefficient Rolling moment NACA ARR No. L5F23 ao slope of the section lift curve for incom pressible flow, per radian unless otherwise stated pb/2V wingtip helix angle, radians p circulation strength CL damping coefficient: that is, rate of change P of rollingmoment coefficient with rate of roll 6'  6(pb/2V)/ Chp rate of change of hinge moment with rate of roll ( \ (pb/2V) Cha rate of change of hinge moment with angle of attack 1( aC rate of change of wing lift coefficient _with angle of attack (T) Vap)C absolute value of the ratio Ch Ch Cha/ c wing chord cs wing chord at. plane ofsymmetry cb balance chord of aileron ca chord of aileron Ca aileron rootmeansquare chord x chordwise distance from wing leading edge y spanwise distance from plane of symmetry ba aileron span b/2 wing semispan "'ACA nRI; ! S W' 8, Oa A Ae w 'F' V q E E' I "n KI, K2 Subscript T LL 0o. LEF2S 5 area of wing weight of airplane stick force, pounds stick deflection, degrees aileron deflection, degrees, positive downward aspect ratio equivalent aspect ratio in conpressible flow (A/ 2 ) taper ratio, ratio of fictitious tip chord to root shord freestream Mach number vertical component cf induced velocity freestream velocity freestream dynamic pressure (a1pi edgevelocity correction factor for lift edgevelocity correctiDn factor for rolling moment hingemoment factor for theoretical load caused by streailinecurvature correction (reference 5) experimentally determined reduction factor for F to include effects of viscosity trailingedge angle, degrees parameter defining spanwise location (cos1 \ \ b/2/ constants liftingline theory 'AC!.', ARR No. L5F23 J~ii 2 LS liftingGcrface theory EV edgevelocity correction SC streamline curvature max maximum o outboard i inboard e effective c compressibility equivalent I AP P LI C ACTION OF METH OD T O S T I C K F 0 R C E E S T I M AT I 0 N S GENERAL METHOD The values of the damping in roll C2p presented in reference 2 were obtained by applying the Jones edgevelocity correction to the liftinglinetheory values. For a wing of aspect 6, the Jones edgevelocity correction reduces the values of Cjp by about 8 percent. From the data obtained on the electroma~nreticanalogy model of the elliptic wing of aspect ratio 6, a more accurate correction to CLp for this aspect ratio could be calculated. The damping in roll was found to be 13 percent less than that given by liftingline theory. The results were extended to obtain values of CLp for wings of various aspect ratios and taper ratios. These values are presented in figure 2. The parameter \/ M2 is included in the ordinates and abscissasto account for firstorder cormpressibility effects. The value of ao to be used in figure 2 is the value at M = 0. The method of estimating aileron stick forces requires the Use of the parameter a Ch FTAc. AFR Do. LJoZ3 Because Ch can te fo'.nd from the static windtunnel data, it is possible to determine Chp and thus the effect of rolling upon the aileron stick forces if (P h is known. Ir. order to av:,id measuring C at all points to be computed, the effect of rolling is usually accounted for by estisni.tin, an effective angle of attack of the rolling wing such that the static hinge moment at this angle is equivalent to the hinge moments during a roll at the initial a;,gle of attacik. The effective angle of attack is equal to the initial angle of attack correctted by an incre.nental angle (Aa)Ch that accounts for rolling, where a)Ch (aP) C (1) The valne of (Aa)Ch is added to the initial a for the downgoing wing and zubtracted from .the initial a for the upgoing wing. The value., of h corresponding to these corected values of a. aie then determined and are converted to ctick force from the knowr:n dynamic pressure, the aileron dimensions, and the mechanical advc.ntage. The value of pb/2V to be 'ised in equation (1) for determining (&a)cU is (s explained in reference 2) the estimated value fnr a ric'.i unyawed wing; that is, pbt 7 2V C L,, The value of CL to be used in calculating IApb/2SV should al.o be corrected for the effect of rolling. The calculation of pb,/V is therefore ceterimined by successive approx.imatiz ns. P For the first approxi mation, the static values of C, are visd witlh the value of CL from figure 2. From the fI r tappro:li matlon values of pb/2V, an incremental anle of attack (Aa)C is estimated. For. all practical ourrosec, (ap)CI = (aP)Ch NACA ARR No. L5F23 and from equation (1), ) = (P) pb ePO Ch 2V Secondapproximation values of C0 can be determined at the effective angles of attack a + Aa and a Aa. The secondapproximation value of pb/2V obtained from this value of CL is usually sufficiently accurate to make further approximations unnecessary. In order to estimate the actual rate of roll, values of pb/2V for the rigid unyawed wing 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 veryaccurate actual values varying from 0.6 to 0.9. The improvement in the theoretical values of C p 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 when the more correct values of C0 given in figure 2 are used. The values of (a)Ch presented in reference 2 were obtained by graphically integrating some published spanload curves determined from liftingline theory. Determination of this parameter by means of the lifting surface theory presented herein, however, gives somewhat more accurate values and indicates a variation of the parameter with aspect ratio, taper ratio, aileron span, Fr Mach number, Ch., and the parameter IcF (ca/c\2 In practice, a value of ( Ch equal to the liftinglinetheory value of h(p) ChL (see appendix) times the Jones edgevelocity correction A + 4 AE + 2 parameter Ac + c AEc + 2 is probably sufficiently Ac + 2 AcE'c + 4 accurate. The incremental angle of attack (Aa)Ch is then rTcAM. AR. ; Ic. L55 192 (AQ)Ch = h A + 4 cEc + 2 pb \ P)/ChLL Ac c 2 cc + 4 2V If further refinement in estir.atinr the stick force is desired, a s.nall additional liftingsurfacetheory correction Ch = A pChp L may be added to the hinge moments determined. For wings of aspect ratios of from about 4 to 8, values of this additional liftinz surfacetheory correction are within the usual accuracy of the measurements of hinge moments in wind tunnels; that is, Ch = pb 0.0 02 for a pb/27 cf 0.1 and therefore need not be applied e:,cept for very accurate work at hih rped: on lar e A^ + 4 airplanes. Values of la) Ac + are riven in S/ ChLL Ac + 2' figure 3. The effective aserect ratio Ac 1 Mv is used to correct for firstorder compressibility effects ArEc + 2 and valves of c are given as a function of Ac AcE' c + 4 in figure 4. Values of the correction 2 'hd p) (a/c) 12 LS are given in f ig're 5 as a function of Ac, and values of are given in figure 6. The value ,o ni is (ca, c) approximately 1 0.000~5". The values of ctb.Ca given in figure 6 are for control surfaces with an external overhang such as a bluntnoce or Frise ov.ert.,ng. For shrouded overhangs such as the internal balance, the value of cb,/ca should te multiplied : about 0.8 before usin figure 6. If the windtunnel data are obtained in lowspeed wind tunnels, the estimated values of CZp and aPn)C should be determined for the windtunnel Mach number 27ACA ARFr 14o. L5F23 (assume M = 0). Otherwise the tunnel data must be corrected for compressibility effects and present methods of correcting tunnel data for compressibility are believed unsatisfactory. ILLUSTRATIVE EXAMPLE Stick forces are computed from the results of the windtunnel tests of the 0.40scale semispan model of the wing of the same typical fighter airplane used as an illustrative example inreference 2. Because the windtunnel data were obtained at low speed, no corrections were applied for compressibility effects. Because this example is for illustrative purposes only, no computations 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 account of these effects. A drawing of the plan form of the wing of the model is presented in figure 7. The computations are made 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 1.30. The data required for the computations are as follows: Scale of model . . Aileron span, ba, feet . . Aileron rootmeansquare chord ca, feet . Trailingedge angle, 0, degrees . Slope of section lift curve, a, per degree Balanceaileronchord ratio, cb/ca .. Aileronchord ratio, ca/c, (constant) . Location of inboard aileron tip, .. .. b/2 Location of outboard aileron tip, Yo b/2 Wing aspect ratio, A ...... .. Wing taper ratio, A . Maximum aileron deflection, 6amax, degrees Maximum stick deflection, esmax, degrees . Stick length, feet .. .. Aileronlinkagesystem ratio . . Wing loading of airplane, W/S, pounds per square foot . . . 0.40 . 3.07 S0.371 . 13.5 S0.094 . 0.4 . 0.155 . 0.58 . 0.98 * 5.55 . 0.60 . 16 S. 21 . 2.00 . :1 . 27.2 NACA ARR Ho. L5F23 The required windtunnel tect results includJ rol inr~moimenit coefficients o.ndj hingemoment coef'icientts corrected f r the effects of the jet oourdariecs. Typical data plotted against aileron deflection are presented in figure 8. Thee same coefficints crossplotted against anile of attacU: for ornefi~rth, onehalf, three fourths, and full aileron deflectfons are given in figure 9. The value of C'Z/ao ~c determined from figure 2 is 4.02 and th'e valIu of Clp is 0.373. The A + 4 A + 2 value of a used in equation (2) CL Ac + 2 A c + 4 LL c c tol determine (a) is fund from fiPures Z andh 4 to be 0.565 and is used to compute both the rate of roll and the stick force. In order ra facilitate the ,onputati:ns, simultaneous plots of C? and (ia)Ch against pb/EV were made (fig. 10) . The stepor in the computation will be explained in detail for the single case of equal up and down aileron deflecuions of 40: (1) From figure 9, the valuesof CL corresponding to ba = 40 and 6a = 40 at a = 1.'' ae 0.005 and 0.0052, respectively, or a total :tatic C, of 0.0110. (2) A first approximation to (j,)Ch taken at the value of pb/2V corresponding to Cj = 0.0110 in figure 10 is found to be 0.9g. (3) Sec:ndapproximation values of CL (fig. 9) are determined at a = 0.350 for 6, = 40 and at a = 2.250 for 6a=40, which ;ive a total C0 of 0.0112. (4) The second approximation to (Aa)C is now found front figure 10 to be 0.96, which is sufficiently close to the value found in step (2) to make any additional approx::iatILons unnecessary. 12 NACA ARR No. L5F23 (5) By use of the value of Cj from step (3), the value of p = 0.0300 is obtained from figure 10. 2V (6) From figure 9 the hingemoment coefficient corresponding to 6a = 40 and the corrected angle of attack a = 0.34 is 0.0038 and for 6a = 40 and a = 2.260 is 0.0052. The total Ch is there fore 0.0090. (7) The stick force in pounds is calculated from the aileronlinkagesystem data, the aileron dimensions, the increment of hingemoment coefficient, and the lift coefficient as follows: Stick force x Travel = Hinge moment x Deflection where the hinge moment is equal to Chqbac 2 and the motion is linear. Substitution of the appropriate values in the equation gives 2 x 21 16 a FsF3 57.73 Chqb a and the wing loading is = qCL = 27.2 Therefore, 27.2 3.07 (0.371\2 16 x 57.3 s CL h 0.4 0.4/ 2 x 21 x 57.3 or Ch F = 68.4 h s CL Thus, when Ch = 0.0090 and CL = 0.170, 0.0090 F = 68.4 x 0.0090 s 0. 170 = 3.62 pounds [TACA AARE ,. LF...F This tick fcrce is that due t c al.er,.n deflection and harE beer: corrected by (ap)0 as determirEd with the EV Jone edge'elocity correction apflisd to the lifting linetheoir value. (,) The small add.tionl lift ngurtfacc correction to the hine moment (fig. 5) is obtained fror: ,LS (ac) a = 0.0207 r i' and nce = 13.50, : = 1 0.00.'5(13.5) = 0.91 From figure 6, F0 (ca/c) Therefore, S(Chp) LS = 0.0207 x 0.91 x 0.55 = 0.0103 and C Ch )LS = 0.010.. x 0.(3 = u. .U 5 (9) The AF, due to the addiition.l lifting surface correction of step (8) may : be e:presred as 60 l'' /LS = 0.124 pound NACA ARR Io. L5F23 Then, Total stick force = Fs + AFs = 3.62 + 0.124 = 3.74 The stickforce 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/2V calculated. for the rigid unyawed wing. For comparison, the stick forces (firstapproximation values of table I) calculated by neglecting the effect of rolling are also presented. Stickforce characteristics estimated for the flexible airplane with fixed rudder are presented in figure 11. The values of pb/2V obtained for the rigid unyawed 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 wing twist or yaw and yawing motion were made for this example. II DEV EL O PM 1ENT OF METHOD The method for determining values of CLp and Chp is based on the theoretical flow around a wing in steady roll with the introduction of certain empirical factors to take account of viscosity, wing twist, and minor effects. The theoretical solution is obtained by means of an electromagneticanalogy model 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 wing. The electromagneticanalogy method of obtaining solutions of liftingsurfacetheory 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 zero angle of attack. UACA ARR 11o. LEP'2 15 ELECT ROM AGI ETICA!NALOGY MODEL Vortex Pattern In order to construct an electr.nmagneticanalogy model of the rolling :jing and wake, it is necessary to determine fir.t the vrtex pattern that is to represent the rolling .ing. The desired vortex pattern is the pattern calculated by means of the twodimensional t1 eries thnairfoil theory and liftingline theory. The :ad'ri tional a:pectratio corrections are estimrnted by.r deermrrining the difference between the act'i~l .sh~re of the wing an. the shape that would be req.:.irecia t.:. sustain the lift distribution or vortex pattern determined from tie twodimensional theories. For the special cases of a thin elliptic wing at a uniform angle of atcick or in r. steady roll, the liftinglinetnor '.lues of the sp1:n load distribution may be obtained bt, mriears. of C imle oal..:.lcatiCons i.refer ence 6). The span loa diw;,ri.o:.iaLjon, for bUth cases are equal .. the span lead distributions deter.,;i.ned from strip theory with a uniform reduction in all ordinates of the spanload curves by an aerodynamic induction factor. This factor is  for the wing A + at a uniform anic of attack and for the winr in ,q + 4 steady roll. The equation for the load at any spanwise station J of a thin elliptic win at zero angle of b/2 a k ,J C attack rolling steadily with unt wingtip helix angle pb/2V is therefore (see fig. 12) cc0 *rrA \ / 2 c = 1 (3) c (b,'2V) A + 4 b2, where a = 2rr. NACA ARR No. L5F23 The chordwise circulation function 21 from thin ccIV airfoil theory for an inclined flat plate is 2r 1 [2 /X 2 + Cos 2x ccV + D (4) / where x/c is measured from the leading edge. tSee fig. 13 for values of 2 V The vortex pattern is determined from lifting line theory as the product of the spanwiseloading ccti function and the chordwise circulation cs(pb/2V) function for all points on the wing and in the ccLV wake; thus, 2r ccI 21 cgV(pb/2V) c,(pb/2V) ccV Contour lines of this product determine the equivalent vortex pattern of the rolling wing. Ten of these lines are shown in figure 14. The contour lines are given in terms of the parameter 2P c V(pb/2V) S2r F .V(pb/2V) ax which reduces to  Smax 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 electromagneticanalogy model reported in reference 4. The span of the model was :ACA ARR No. LEF23 twice that of the model of reference 4 (6.56 ft ir stec. of 3.28 ft), but since the aspect ratio is twice as large (6 instead of 3), the maximum chord is the same. In order to simplify the consLtruction of the modJl, only one semrispan of the vortex sheet was simulated. Also, in order to avoid the large concentrations of wires at the leading erfe and tips of the wing, this semispan of the vortex sheet was constructed of two sets of wires; 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. Downwash I.easurer.:ents The magneticfield strength was measured at 4 or 5 vertical heights, 1 spanwise locations, and 25 to 50 chordw.ise stations. A number of repeat tests were made to check the accuracy of the measurements and satisfactory checks were obtained. The electric current was run through each set of wires separately. With the current flowing through one set of wires, readings were taken at pointL on the model and at the reflection points and the sum of these readings was multiplied by a constant determined from the increment of vorticiuy A ( \ represented by that set of wires. \ max / Then, with the current flowing throu.g the other set of wires, readings rere 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 semispan were fairly 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 b as dis max cussed in reference 4. The final curves of wb are rmax IACA ARR No. L5F23 presented for the quarter chord, half chord, and three quarter chord in figure 15. Also presented in figure 15 wb are values of 2 calculated by liftingline theory max and values calculated by liftingline theory as corrected by the Jones edgevelocity correction. DEVELOPMENT OF FORMULAS General Discussion Liftingsurface corrections. The measurements of the magneticfield strength (induced downwash) of the electromagneticanalogy model of the rolling wing give the shape of the surface required to support the distri bution of lift obtained by liftingline 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 by the downwash measurements. Formulas for determining these corrections to the span load distributions and the rolling and hingemoment characteristics have been developed in connection with jetboundarycorrection 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 thinairfoil theory and thus do not take into account the effects of viscosity, wing thickness, or compressi bility. Viscosity. The complete additional aspectratio 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 normally 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 streamlinecurvature correction by about 10 percent for airfoils with small trailingedge angles. :.ACA ARE: i'. LEF23 Essentially the same final answer is therefore obtained whether the corrections are applied in two parts (as should be done, strictly speal:ing) or whether they are applied in one part by use of the full theoretical value of the stream?.inecurvature 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 trailingedge angles. A study is in progress at the Lanley Laboratories of the ITACA to determine the proper aspectratio corrections for ailerons and tail surfaces with beveaed trailing edges. For beveled trailing edccs, in v',hich viscous effects may be much more pronounced than in ailerons with small trailing edge angles, the reduction in the theoretical streamline curvature correction may be corsierably more than 10 percent; also, when Cha is positive, the effects of the reduction in the rtraaralinecurvature correction and the additional down:ash at the 0.50c point are additive rather tnan 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 aspectratio correc tions in a single tep is not allowable for beveled ailerons. The corrections will therefore be determined in two separate parts in crder to keep them general: one part, a streamnlnecturvacure correction and the other, an angleofattack correction. An examination of the experimental data available indicates that more accurate values of the hinge moment resulting from streamline curvature are obtained 1by multiplying the theoretical values by an empirical reduction factor n whch is approximately equal to 1 C'.CO00502 where is the trailingedge angle in degrees. This factor will doubtless be modified when further experimental data are available. Compressibility. The effects of compressibility upon the additional aspectratio corrections ..;ere iot considered in reference 3. Firstorder compressi bility effects can be acco noted for by application of the PrandtlGlauert rule to liftingsurfacetheory results. (See reference 7.) This method consists in iTACA ARR No. L5F23 determining the compressibleflow characteristics of an equivalent wing, the chord of which is increased by the ratio where M is the ratio of the free 1l M2 stream velocity to the velocity of sound. Because approximate methods of extrapolating the estimated hingemoment and dampingmoment parameters to wings of any aspect ratio will be determined, it is necessary to estimate only the hingemoment and damping parameters corresponding to an equivalent wing with its aspect ratio decreased by the ratio /I M2. The estimated parameters for the equivalent wing are then increased by the ratio .... /1 M2 The formulas presented subsequently in the section "Approximate Method of Extending Results to Wings of Other Aspect Ratios" are developed for M = 0, but the figures are prepared by substituting Ac = A/ M2 for A and multiplying the parameters as plotted by Y M2. The edgevelocity correction factors Ec, Eec, E'c, and E'ec are the factors corresponding to Ac. The figures thus include corrections for firstorder compressibility effects. Thin Elliptic Wing of Aspect Ratio 6 Damping in roll Cp. In order to calculate the correction to the liftinglinetheory values of the damping derivative CLp it is necessary to calculate the rolling moment that would result from an angle ofattack distribution along the wing span equal to the difference between the measured downwash (determined by the electromagneticanalogy method) at the three quarterchord line and the downwash values given by liftingline theory. (See fig. 15.) Jones has obtained a simple correction to the liftinglinetheory values of the lift (reference 1) and the damping in roll (unpublished data) for flat "ACA ARR ito. LEF23 21 elliptic wings. This correction, termed the "JonAs edgevelocity correction," is applied by: multiplying the liftinglinetheory values of the lift byr the Ap + ratio A+ 22 and the liftinglineLheory values AcEc + 2 A + , of the damping in roll by Acc + with values of Ec "cE' c+ 4 and E'c as given in figure 16.. As may be seen from figure 15, the dov;nwash given by the Jones edgevelocity correction is almost exactly that measured at the 0.50c points for flat elliptic wins. This fact is useful in estimating the liftingsurface corrections because the edgevelocity correction, w'jhich is given by a simple formula, can be ued t. correct for the additional angle of attack indicated by the linear difference in downwash at the 0.50c line. The variation n dovrnwash between the O.25c line and 0.75c line, apparently linear along the chord, indicates an approximately circular streamline curvatLure or camber of the surface. The increment of lift resulting at each section from circular camber is equal to that caused by an additional anrle of attack given by the slope of the section at 0.75c relative to the chord line or the tangent at 0.JOc that is, ( (. 'O. 7c 'O.50c Because this difference in downwash does not vary linearly along the span, a spanwise integration is necessary to determine the rtreamlinecurvature increment in rolling monent; that is, / c B iQc '1 wb (C)SC bV(AcE'c + 4) max/0. :~nS (^\ L C d/^ (5) naxOc.5c e b/2 \b/2) An evaluation of rmax in terms of pb/2V is necessary to determine the correction to the dampingmoment coefficient CLp. The liftinglinetheory relation TfACA ARR No. L5F23 between rmax and pb/2V is, from equation (3), n 2Vb(pb/2V) "max A + 4 ,7ith the edgevelocity correction applied 2Vb(pb/2V) .max AcE'c + 4 The value of the streamlinecurvature correction to Cp is therefore A 4T r( 1. wb \ SC cc + 4) 2 [ Kmax/O.75c wb d (7) rn0 /2 /2 2) KPmaxO.50c b/2 x/ A graphical integration of equation (7) gives a value of 0.022 for (AC . \ p SC By the integration of equation (5), the value of (CL) for incompressible flow is found to \ P/LL be H A 0.471 for A = 6. Application of the edgevelocity correction, for A = 6, gives P EV 4(AE' + 4) = 0.433 and, finally, subtracting the streamlinecurvature correction gives a value of Cjp, for A = 6, as follows: C7p = (9 E) P C EV = 0.411 TATCA ARR No. L5F23 '  ri . f 7 fl o' wing of 1 sc.c.t r: tlo 6 " th.icrefore 1z percent les than the vavlu jiveI by lifti line thec.. and nercent e' :. tha. that given by, lift ingline theory e .it'h the .Toners ede velocit:,r ; c rect.or. applied. invr enmoment per'eter Ch . The trear'lin currvatiire corretion to 0h for c n: c;r~t: fr i.'ntao chord .;lerrrsn is. ifrm reference 5 an'r. wvith the value of Pn c;lvern eia ntion (C), .. vi . .,...wrI/ i H f T ( ) c( (x'c) (\cs f ACE'c + 4 d . where the inctecrA tonss Ear e mn.de acrcs t:.e .Lleron spCn. Pec.au'. e the .down .ash at the C.' 0 c jcint' is ,'.en satis ac'tori ly b appl.y.ing the edgev:locit.;, .cctior to the lift inglnetheory values :f thF d.: 'wnwa'.'h, the part oD the correction: to C' which dependics upm: the downwash at the 0.53c pcint .ay be i.eter.minc bby rEisanc of the eFde'.eloc ty correctior. The effect f aFre...d.rinamic induct ion .'tas ner'.ec ted in ie" el3oiin' ecqu ;.. .r ) tec i; aerodynamic inductio,, has a verr small effect .'c.on the binrermo.ent, crec ti.;n au ca))sd oy; .tr,'&arline cu.rvatuLre. V.]ueS ,of the factor  for, ,arious tiler:.n (I2/c )' chord ratios ard ban st ,s as detcrr.ine'd fr;., thin airfoll theory are gIven in r'l.re 6. ..s ;uent i ned p~evi.1sl r, is a factor th.st r.ap3 roiris ely aer:cou. nt. fvr the cornmir ed effects of wvir:. thick ne s an'd visc:3si t in "lte. in ch.e calciladted values of .. The experiimcrntil &Seta vaivlatrle at .'rE'ent inidica3te that rn 1 C0.'C0'5. Re lts th: nt ationr of equation (8) for the ellipt.i winl of .ect ratio 6 arc iven in f' qur e 17 w. t r p.a ancter ,] +  V, 1u s A C ( ) + \/ l V.iL.L s 24 "A. ARR No. L5F23 (ca//c2 ___ of (ACh)  (A + 1) l M2 determined as in reference 3 are given in figure 18. The value of (Chp) is (ChphLS h ( ( LL A+ 4 + AChp)5 a~I ~ ChLL Since Ac + 2 AcEc + 2 ,Ch LL c LL : h~) LS  (Ch) SC A + 4 AE + 2 = (P)ChLL Ac' + 4 Ac + 2 (Ch!LS + (LCh) SC (h) LS  / Ch Ac + 4 ArcE + 2 SC /SC pchLL AcE'0 + 4 Ac + 2 SC LL (P (EV a) L Chp) Oy 'Lb*L The formula for the parameter ChLL is derived for elliptic wings in the appendix, and numerical values are given in the form (a \ c in figure 3, \ /ChLL Ac + 2 together with values for tapered wings derived from the data of reference 2. It may be noted that use of the parameter (p)ChLS determine the total correction for rolling would be impractical because ChpL is not proportional \Op is no rpotoa 'Jib then (Ch )LS :"ACA i7r. :o. LEF23 25 to (bh \. Although the numerical values of (.)ChL vary considerably with h the actual effect on / js the stick forces is small because r C changes most with h when the values 1of Ch l are small. This effect is illustrated in figure 19, in 4hich numerical values of (apCh for a thin elliptic wing of aspect ratio 6 are given, together with the values obtained by liftin.iline theory., the values obtained by applying the Jones edcevelocity correction, and the values obtained. b using the aileron midpoint rule (reference 8). The values obtained by the use of the Jones edgevelocity correction .re shi.w.n to be 4.4 percent less than those obtained by the use of liftingline theory. The righthand side of equation (9) is divided into the following two parts: Part I (ap)ch (Ch Part I = hEV LS Part II = A SChp)L Part I of the correction for rolling can be applied to the static hin7erorritet data as a change in the effective angle of attack as in reference 2. (Also see equation (2) .) Part II of equat in (9), however, is applied directly as a change in the hingemoment coefficients, 'Ch A(11) LS ib Inasmuch as part II of equation (9) is numnerically fairly small Arh = 0.002 for = 0.1 for a wing of aspect ratio 6), it need not be applied at all except for fairly large airplanes at high speed. TACA ARR No. L5F23 Approximate Method of Extending Results to Wings of Other Aspect Ratios Damping in roll Cp. 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 liftingsurfacetheory solutions (references 4 and 9) for thin elliptic wings of A = 3 and A = 6 at a uniform angle of attack. The additional aspectratio correction to CL was computed for these cases and was found to be approximately onethird greater for each aspect ratio than the additional aspectratio correction estimated from the Jones edgevelocity correction. The additional aspectratio correction to C1p for the electromagneticanalogy model of A = 6 was also found to be about onethird greater than the corresponding edgevelocity correction to C p. A reasonable method of extrapolating the values of Cp to other aspect ratios, therefore, is to use the variation of the edgevelocity correction with aspect ratio as a basis from which to work and to increase the magnitude by the amount required to give the proper value of CL for A = 6. Effective values of E and E' (Ee and E'e) were thus obtained that would give the correct values of CL_ for A = 3 and A = 6 and of C;p for A = 6. The formulas used for esti mating Eec and E'ec for other aspect ratios were Eec = 1.65Ec 1) + 1 E'ec = 1.65 (E'c 1 + 1 Values of Eec and E'ec are given in figure 16. Values of i /l M2 determined by using E'ec are presented n ure 2 as a function of A/ao are presented in figure 2 as a function of Ap/ao NACA ARR No. L5F23 where Ac = A /,1 M2 and a3 is the incompress'ble slope of the section lift c'uve per degree. Hingemoment parameter Ch . In order to deter mine Chl for other aspect ratios, it is necessary to estimate the formulas for extranplating the streamline curvature corrections (ACCh and ACh Values s' c \ P/sc of (ACh S) for A = 3 and A =6 ere available in reference 3. Values of Cha) might be expected to be approximately inversei:, proportional to aspect ratio and an extrapolation fcrm.,ula in the for( ICh, L j C ha A + K is therefore considered satisfactory. The values of K1 and K2 are determined so that the values of LCh for A = 3 and A = 6 are correct. Values of K and K2 vary "with aileron scan. The values of K2, however, for all aileron spans less than 0.6 of the .emispan are fairly close to 1.0; thus, by assLuniin. a constant value of K2 = 1.0 for all aileron spans and calculating values of I1, a satisfactory extrapolation formula may be obtained. It is impossible to determine such a formula for Ch~ because results are available only for A = 6; however, it sels reasonable to assume the same form for the extrapolati :in formula and to use the same value of K. as for (iCh The value of K1 can, of course, be determined from the results for A = 6. Although no proof is offered that these extrapolation formulas are accurate, they are applied only to part II of equation (9) values of Ci which is numeri cally quite small, and are therefore considered justified. JACA ARR No. L5F23 CONCLUDING EEMARkS From the results of tests made on an electromagnetic analogy model simulating a thin elliptic wing of aspect ratio 6 in a steady roll, liftingsurfacetheory values of the aspectratio corrections for the damping in roll and aileron hinge moments for a wing in steady roll were obtained that are considerably more accurate than those given by liftingline theory. Firstorder effects of compressibility were included in the computations. It was found that the damping in roll obtained by liftingsurface theory for a wing of aspect ratio 6 is 13 percent less than that given by liftingline theory and 5 percent less than that given by the liftingline theory with the Jones edgevelocity correc tion applied. The results are extended to wings of any aspect ratio. In order to estimate aileron stick forces from static windtunnel data, it is necessary to know the relation between the rate of change of hinge moments with rate of roll and the rate of change of hinge moments with angle of attack. It was found that this ratio is very nearly equal, within the usual accuracy of windtunnel measurements, to the values estimated by using the Jones edgevelocity correction, which for an aspect ratio of 6 gives values 4.4 percent less than those obtained by means of liftingline theory. The additional liftingsurfacetheory correction that was calculated need only be applied in stickforce esti mations for fairly large, highspeed airplanes. Although the method of applying the results in the general case is based on a fairly complicated 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 Field, Va. 7TACA ARR No. L5F23 APPENDTX EVaLUATIOI7 OF o p) FOR ELLIPTIC WINGS ChLL It was shown in reference 2 that for constant percentagechord ailerons the hinge moment at any aileron section is proportional 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 wing chord. The factor (ap is obtained LL by averaging the two factors cjc2 and cl/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 2n, it was shown in reference 6 that striptheory values multiplied Ac A by aerodynamijcinduction factors or c 'c +2 A, +4 could be used. (iHote that A is substituted for A to account for firstorder effects of compressibility.) Thus, for constantpercentagechord ailerons on a rolling elliptic wing, S2 A sin28C 22n c Ac + 4 s V 2nrc 2Ae b sin20 cos 9 Ac + 4 2V and for the same wing at a constant angle of attack a cLc2 Ac sin2r Cs22na A 0C + 2 Uc In order to find C ,the integral c d across the aileron span must be equal for both the :TACA ARR No. L5F23 rolling wing and the wing at constant a. Thus, c Lc2 dy u. 21 cs2Ac pb sin2e cos dy Ac + 4 2V /d 2=Acs2Ac a sin28 dy C Ac + 2 dy= 2 d(cos 9) b sin e d9 2 Let a = (ap cLL Then (p) ChLL Ac + 2 sin3e cos e do Ac + 4 in3e do Ac + 2 Ac + 4 S[sin4eol S1 o 7isin2o cos 0 + 2 9o cos 90 4 i where e0 and 9i are parameters that correspond to the outboard and inboard ends of the aileron, respectively. Values of A + 4 a L Ac + were calculated for the PChLL Ac + 2 outboard end of the aileron at Y = 0.95 and plotted b/2 in figure 3. :IACA ARR 1:0. LEF2 A similar develo.i.mrnent iv,.'e:, fc::r the c.nstantich:,rd ailer.ni,r cl 2nrAc pb f + 4 =  Sc c ch + 4^ 2V a c cos 6 J II; c s (A + 2 V 2 hLL s In 0 / \ + + 4 S os q d9 ' ChLL c I'L in 6 These value are aL so presented in fic.re 3. ITACA ARR No. L5F23 REFERENCES 1. Jones, Robert T.: Theoretical Correction for the Lift of Elliptic Wings. Jour. Aero. Sci., vol 9, no. 1, Nov. 1941, pp. 810. 2. Swanson, Robert S., and Toll, Thomas A.: Estimation of Stick Forces from WindTunnel Aileron Data. NACA ARR No. 3J29, 1943. 3. Swanson, Robert S., and Gillis, Clarence L.: Limitations of LiftingLine Theory for Estimation of Aileron HingeMoment Characteristics. NACA CB No. 3L02, 1943. 4. Swanson, Robert S., and Crandall, Stewart M.: An ElectromagneticAnalogy Method of Solving Lifting SurfaceTheory Problems. NACA ARR No. L5D23, 1945. 5. Swanson, Robert S., and Toll, Thomas A.: JetBoundary Corrections for ReflectionPlane modelss in Rectangular Wind Tunnels. NACA ARR No. 3E22, 1943. 6. Munk, Max M.: Fundamentals of Fluid Dynamics for Aircraft Designers. The Ronald Press Co., 1929. 7. Goldstein, S., and Young, A. D.: The Linear Perturbation Theory of Compressible Flow with Applications to WindTunnel Interference. 6865, Ae. 2252, F.M. 601, British A.R.C., July 6, 1943. 8. Harris, Thomas A.: Reduction of Hinge Moments of Airplane Control Surfaces by Tabs. NACA Rep. No. 528, 1935. 9. Cohen, Doris: A Method for Determining the Camber and Twist of a Surface to Support a Given Distribution of Lift. NACA TN No. 855, 1942. NACA ARR No. L5F23 0 o 0 0 0 14 14 N ID 10 '1 K' 0 01 * 0 0 0 01 8 * 0 0 O0 0 14 40 'II t O 03 .4 I 0o .4 t0 DIO to 0 0 N 4 0 0 0 SO ' 0o o  0 11 n0 4m ca I 0 0 N 1i r 0m I4 .0 0 0 * o to t* 0 ! 1 0 0 0 O N O0 SCQo 0 .0 . o N 1o 'o C, U a0 0 0 N 0 . 0 I + o o o to t 0a t3 CO to  1 0 0 0 0 0 0 CM toN 0 r 0 w 0 0 o H I 'r O r r4 01o 002 D * O 0 00 0 !4 ON * 1 0 0 00 0 H 1 N 0N UT rI O r4 N 0) m 4 o r4 rm 14 CI LO m . 0 a 0 0 0 0! .o O OO LO 0 t0 t 03 1 0 0 00 0 . O NO O O O N 0 0 0 0 a O 0 O roa cI 03 0 co 9 0 0 0 9. 4 0 o 0 8 0 0 0 0 .4 * e 0@ a a I SbO 0 3t E4) 0 O : *U0 0 D 0D :4 OS 00  m  NACA ARR No. L5F23 Fig. la 4 040 &4 e to ,i ..,Co 4iZ* 4 o bD v o . .: .. 4 O4 ., a, 0 ,i b ~I rD NACA ARR No. L5F23 Fig. lb N.. o C * t 0 4 %I (i .i ' ~!. e: . NACA ARR No. L5F23 a, I I I / NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 60 ao 60 /00 , l/9d f 0 /2C /'fo /&0 Figure 2. VariaLion of damping coefficient wilth aspect ratio and taper ratio in Lerms of slope of section lift curve for incompressible flow (per degree). tLift nelinetieory values of reference 2 wi t an effective edgevelocitj correction applied.) Fig. 2 NACA ARR No. L5F23 Fig. 3 dv'Jc 3Ii 4 4 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS O .2 .4 .8 /o Re/af/ve /oc4ton of /ndoorad ,//ron f/ , Ac + 4 Figure 3. Values of (ap)hL Ae + for various aileron tip locations. The corrections of figure 4 must be used with these values. NACA ARR No. L5F23 Fig. 4 0, Z \ o O(d  l  I c   i 0.    '4 %   cg. sL 4 _______ ______ ______ ______ ______ ______ _______, ______ ______ _______ C . ,_ V t3 V Fig. 5 NACA ARR No. L5F23 01 0 p r 1 4 o S 0 >g \ o I O / 0 bo r14 9 7 a O a   ^ s ^' s< /^o~ 9 h ___ ^____ ___ ___ ___ ___ ___;SP <      o ,'" I u ,; ^ ^ J . NACA ARR No. L5F23 /.f A/ /. /0 .7 I ' .6 .', c 3.  . AY/ ,/// ____ _ / / NATIONAL ADVISORY CO{mITTEE roR MAIE AUTICS //! / / z __ i_ /// i / / /^ I_ LL l _,1"_ _/_ ___'1 /z ,z /_ _ _^ z _/Z _ lil / i Ill/ i 1^  lr i"  ; / ONHITEE 0 MlO~tTC /i I /zzzz A,/eron ch/ord ra foo, ca/C Faw., 61. /o frtation o the IAre malewl correc/on factor le ,^s wi th a leron chord rat/o ,and e ternd/a'oyerhany aerodynawnl/c o/ance. chord raf/o. For ilrternal/ erod namIc bo/aoce, use an effectf/e c e 0.8 c, (reference 5). Fig. 6 Fig. 7 NACA ARR No. L5F23 // 0  I< > Z I C o IV 1 o .U *1 I o ri / s I 1 I 0 I / ^ l i t < I / ri I f PL NACA ARR No. L5F23 .O4  0 QJ .08 Qa S./6 25 20 15 10 5 0 5 /0 /5 20 25 Aileron deflection, SC, deg Figure 6. Aileron hinge rind rolling moment characteristics of the 0.40scale mode/ of the airplane used for the ///us/roaive example. Characteri/sthc p/ol/ed ogo/nsf o//eron def/ection. Fig. 8 NACA ARR No. L5F23 Fig. 9 .1C C) :.08 S.04 0 08 S.03 .0 o :08 02 .03 4 2 0 2 4 6 8 /0 /2 /4 /6 1/ Angle of attack, cr, deg Figure 7.Aileron hinge and roll//ngmomnenl characferisic of the 0.40sca/e mode/ of the airp/one used for the i//ustra//ye example. Character/stics p/oiled against ang/e of attack. NATIONAL ADVISORY COMMITTEE FO AEROIIAUTICS NACA ARR No. L5F23 .08 .07 .06 o .0 .03 F 0.02 S EE N, 82 k n Fig. 10 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 0 .,' .03 .04 .05 .0o .07 .0 .09 ./0 .1 h'e/l/x ange .ob/2V rad/an Fqure /0 .Pre//l, .., ary cur ves5 ,sed n .he corr.oua/'a ..c of the a ..! t,o Ae,,x ang'/es andA /ne //c c forces :or //e d//a/sra/lve examp/e. NACA ARR No. L5F23 A/eron def/lecon (de9) 20 /  /6 (a) Effect of ro//in,yaw yawiny, and wng twist neq/ected /n both and p6lV. , /26  / 8______ 0 * O   Ile 0 Z 1(b) Effect of ro/llly accounted for in both / and pb/2Ky, effect of yaw,yaw/ny, and wing twist accounted for in Fs bu t not in b/ / (c) Effect of ro/ly account ted for in both {and p, 2V; effect of yaw,yawny, and wgnj twist accounted for n s and assumed to  25percent  NATIONAL ADVISORY COMMITTEE FOlR AERONAUTICS 0 __I 0 .0/ .O2 .03 .04 .05 .06 .07 .08 .0? ./0 .// Hel/x anyle, fL 2V, radian Fare P/. Stc/A force char~acterlsics esatmated for the airplane based for /1e I//astra11;e ex amo/e. Fig. 11 NACA ARR No. L5F23 Figs. 12,13 o Ai I o= A 0 7 *! t0^3 ^ UlSC/ft^/ F9^j? to~ ^ pI uodL N a *1? a U . 1, ,L t L. L .  C i C j. C e  L ,s 1 " 1  . F?. '4 'I j ; 3 2 I Fig. 14 NACA ARR No. L5F23 IC tot 4, w//ox   ^ \\^^^  ^ \V ^^_ __^ \>. \ \.  \ \ ^ ^____ ^ NACA ARR No. L5F23 Liftiny/iu theorA y wnrP'A Jones/ edgeve/ocftu correction  S(so/d //ne)  .,5v c / Downwash \ / / measured at .75c // '' lilfH /,Ape NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS I I I I/ I I oa*v.v/se 1/s3c / Figure 15. Induced dowr.wash function wb/2.8ax for an elliptical lifting surface of aspect ratio 6 in steady roll. wb 2/3,Id Fig. 15 Fig. 16 NACA ARR No. L5F23 i \  i i~rrn O g k ts   p 4 + o '3t ,, ', Q K VC o l ix i c, / l i  y ~. /^ 144  q G _' a __ ^ ^^^^^   ^ ^  r y    ^^ ^ ^ ^ ^ ^ 37 ,'3c '7 NACA ARR No. L5F23 Figs. 17,18 a .3 .3 C  .3 ..I .3 .3  3. SUJ  .3 .* .3  0.3 C. C. ) 3 U I. , a C .3 .3 CL. 0C ) , .3 .E 3. C 1 C. 3l .3  a .3   . ii, s J 3S ^w^^s^'P9 rq^{^ 7P)(a>?) 1J ~r (o3v) NACA ARR No. L5F23 r',# / Indoad aleronh/2.~ Figure 19. Values of parameter (ap)Ch from aileron midpoint rule, liftingline theory, liftingline theory with edgevelocity correction applied, and lifting surface theory for an elliptic wing of aspect .ratio 6. M = 0. Fig. 19 UNIVERTY OF LORIDA S262 0814 954 5 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY RO. BOX 117011 GAINESVILLE, FL 326117011 USA 