Minimum Drag Due to Lift for a Delta Wing with Sonic Leading Edges Citation Strand, Torstein (1954) Minimum Drag Due to Lift for a Delta Wing with Sonic Leading Edges. Engineer's thesis, California Institute of Technology. doi:10.7907/Y3KY-N097. https://resolver.caltech.edu/CaltechETD:etd-12122003-094215 Abstract Using linearized theory a close approximation to the optimum camber and twist distribution of a Delta wing of zero thickness with sonic leading edges has been obtained. The optimum wing has a camber line that is negative over the front quarter of the total planform area and is positive over the remaining area. A certain amount of washout is exhibited. Compared to the flat plate a saving in induced drag of slightly over 7 percent is shown. The leading edges are relieved of a considerable portion of the total lift. It is shown that this lift has been re-distributed over the central part of the wing. Item Type: Thesis (Engineer's thesis) Subject Keywords: Aeronautical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Lagerstrom, Paco A. Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1954 Record Number: CaltechETD:etd-12122003-094215 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-12122003-094215 DOI: 10.7907/Y3KY-N097 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4956 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 15 Dec 2003 Last Modified: 13 Jun 2023 23:14 Thesis Files Preview PDF (Strand_t_1954.pdf) - Final Version See Usage Policy. 696kB Repository Staff Only: item control page

MINIMUM DRAG DUE TO LIFT FOR A DELTA WING WITH SONIC LEADING EDGES Thesis by Torstein Strand In Partial Fulfillment of the Requirements For the Degree of Aeronautical Engineer California Institute of Technology Pasadena, California 1954 -i- ACKNOWLEDGEMENT The author wishes to thank Professor Lagerstrom for suggesting the subject of this thesis and A. M. Rodriguez for several helpful discussions. -ii- ABSTRACT Using linearized theory a close approximation to the opti- mum camber and twist distribution of a Delta wing of zero thickness with sonic leading edges has been obtained. The optimum wing has a camber line that is negative over the front quarter of the total planform area and is positive over the remaining area. A certain amount of washout is exhibited. Compared to the flat plate a saving in induced drag of slightly over 7 percent is shown. The leading edges are relieved of a considerable portion of the total lift. It is shown that this lift has been re-distributed over the central part of the wing. PART I. Til. Iv. Vi. -iii- TABLE OF CONTENTS TITLE ‘ Acknowledgement Abstract Table of Contents List of Figures Explanation of Symbols INTRODUCTION STATEMENT OF THE PROBLEM BASIC DISTRIBUTION ORTHOGONALIZA TION RESULTS Percent Drag Reduction Optimum Camber and Twist Distribution Optimum Pressure Distribution CONCLUDING REMARKS References Appendix I Appendix II Table I Table II PAGE ii iii iv 14 14 14 15 17 18 19 20 21 24 FIGURE | -iv- LIST OF FIGURES TITLE Wing Planform and Axis System Angle of Attack Distribution Camber and Twist Distribution ; Lift Distribution Along the Trailing Edge Lift Distribution over the Root Chord PAGE 25 26 27 28 29 Ap'(x, y) Ap OB P(u) (x, y) X,Y,U,V £7, bv '0,9,6,6 I m,p m,n,p,q b,,b3,b Ca2Cyrdy n(x; y) 4? -V- EXPLANATION OF SYMBOLS wing lift coefficient wing drag coefficient L Sp angle of attack distribution (slope in x-direction of mean camber line) lift drag pO naneannny normalized drag = ZS local static pressure static pressure at infinity local lift distribution = 2(p-p, ) Ap’ 4qS normalized local lift distribution = basic angle of attack distribution nth Legendre polynomial perturbation velocity potential rectilinear coordinates dummy rectilinear coordinates variables 1-6 J wT 1 sin (2m4+1) 3 Po (cos 6} d® 0 _ . (a) _ — i) sin (2n41) FP. (cos O) 1, (0) dB wT constants camber and twist distribution -vi- Subscripts identify particular wings. Subscript opt stands for optimum (min. drag) configuration of the four ortho- gonal wings considered. I. INTRODUCTION It is well known that in supersonic flow using linearized theory it is permissible to consider drag due to lift and drag due to thickness separately and obtain the total pressure drag by super- position. . The drag due to lift is computed from the zero thick- ness mean camber wing at angle of attack and the wave drag is found by considering the symmetric thickness wing at zero angle of attack, This thesis is concerned with the problem of minimiz- ing the drag due to lift of a wing of a specific planform. In 1952 E. W. Graham (Ref. 1) developed a method of drag reduction using superposition of "orthogonal" loadings. Since this is the method used in this thesis a very brief description of the theory is given below. Consider a free stream Mach numberM> 1 and a wing planform of zero thickness with a certain angle of attack distri- bution. For this wing it is then possible to calculate pressure, lift, drag, moment, etc. If another wing of the same planform, but with a different kind of angle of attack distribution in a stream ‘of the same Mach number is superimposed on the first wing, a new wing is obtained which has angle of attack distribution, pres- sure and lift equal to the sum of the two angle of attack distribu- tions, pressures and lifts. The drags, however, do not add up in this manner. Since the resulting drag is the integral over the planform area of the product of the sum of the two pressures and the sum of the two angle of attack distributions, cross product -2- terms (interference. drag) appear. For some special loadings these cross product terms will disappear. By definition such loadings are called orthogonal. If the convention is made only to consider superposition of mutually orthogonal loadings it now becomes possible to maximize the non-dimensional parameter which is a characteristic of the type of angle of attack distribu- tion of the wing. If the first wing is defined by the subscript 1, the second wing by-the subscript 2, and so on it can be shown that by superimposing several of these orthogonal wings in a manner so as to obtain minimum drag for a given lift, the resulting op- timum wing has a parameter L as follows: 4 oe = 71 th thst ccecrees Furthermore Rodriguez, Lagerstrom and Graham in 1954 (Ref. 2) showed that for the wing thus obtained Cc o iL L k a = DL — opt £ opt k=l D,. “ik Cc eo L ; Lf, k and Ap! = HH a Apt opt Loot k=] D, k where a and Ap' are defined as respectively the angle of attack and pressure distribution over the planform. -3-. Il. STATEMENT OF THE PROBLEM The problem discussed in this thesis may be stated as follows: Given a Delta wing of zero thickness and sonic leading edges at a cer- tain lift coefficient, find the camber and twist distribution for mini- mum drag. For convenience the Mach number has been chosen equal to 2. In order to have sonic leading edges the leading edge sweep- back is 45°. By the Prandtl-Glauert rule the results can be applied to any sweepback angle or Mach number as long as the leading edges are sonic. To obtain the optimum angle of attack distribution it would be necessary to superimpose an infinite number of orthogonal loadings. Since it is not possible to find the sum of the series of orthogonal loadings used due to the somewhat complex expression of each loading, a true optimum cannot be obtained. It was decided _ to employ only four orthogonal loadings, this being a compromise between the amount of labor involved and the desirability of obtain - ing a large drag reduction. The camber and twist distribution found is therefore not a true optimum. It would be correct, how - ever, to call it a restricted optimum since it represents the best possible combination of only four orthogonal loadings selected from an infinite number of orthogonal loadings. Knowing the angle of attack distribution of a particular plan- form the pressures, lift, drag, moment, etc. can in theory be cal- culated using linearized supersonic wing theory. However, in prac- tical application it is often very difficult to perform the required integrations analytically. -4- For the planform discussed in the present report the analyti- cal work is comparatively simple. A complete set of basic angle- of-attack distributions may be found, using Legendre polynomials, such that lift, drag and interference drag may be calculated in closed form. This method is due to A. M. Rodriguez. These basic distributions will be discussed in the following section. Ill BASIC DISTRIBUTION The coordinates x,y and u,v (and the corresponding dummy variables of integration §,m and ,,v) are shown in Fig. 1. Let the basic angle of attack distribution be defined as foal B an = 5 [ PP tv) + Pu) Pt) From supersonic wing theory the perturbation velocity po- tential (6) corresponding to the flow of free stream velocity U over this wing is (from Ref. 4) mnt 4 son # | - for M =)Z ¥ (x-€)"-(y-m)* ' where R = region of the forward Mach cone from the point (x, y). Also from linearized theory (Ref. 4) -p = -o 2% P-P pU >= Let the local lift, that is the differential pressure between top and bottom surface of the wing (positive in the positive lift direction), be written as dfdy AP inn ~ 2(P-Pyy) = T a x- £)¢ ~(y- nm)“ Since from Fig. I: x= (ut+v) y {u-v) a — y2 -6- it can be easily verified that put (2 + 2) i J BronihY dpa? ou av s Yu-n v-v AP ant! v) =- A new set of variables (0,0 and $,¢) for ease of integration is now introduced: u = cos 9 f= cos > v =cos 0 V=coso The following expression is then obtained | 2 van8indsingdgag . mn ang | yess0- -cosoyeos0- cos The Legendre polynomials may be expressed as ° 9 cos(2n+1)>5d0 p_(coso) = ye \ é 7 0 ¥ cosp-cos6 which when substituted into AP oan yields 5 _ g [cortenstgeintameny cos(2m+ )esin(2ntl)> = .1)1 AP inl 9) ~ 2m |sind 2n +1 + 2mt+1 \0 - 9 9 1 sin(2nt1}scos2(m41}5 sin(2m+1)$cos(2n41}> | + + Sint tmti Zntl The above expression has been normalized by dividing by 4pU* (or 4qS since S= 2). This Ap, is now the normalized pressure -7- distribution of a wing with angle of attack distribution Ban - In Appendix I the four chosen basic loadings and their pressure dis- tributions are given. It may be noted that Boy and Pio are not used since they give zero lift and are hence useless for drag re-- duction. : Since the cross product terms previously mentioned must be set equal to zero to satisfy the condition of orthogonality, it is advantageous to obtain a general expression for the pressure dis - tribution of one loading acting on the angle of attack distribution of another loading. Let the interference drag between the angle of attack distribution Ban and Pog be denoted by D'(m,n;p,q).. This quantity is then evaluated as: D' (mp;pa)= fart, Pq 3S = 4a8 D(m, nip, a) S where S = total planform area. Note that when m = p and n= q, D(m,n;m,n) is the normalized drag of the B loading. Then mn | j 0 w-9 [costantiig sin(em*tyy cos(2m+1)}$ sin(2n +1)5 D(m, n;p, q) = 5— J ont * amt wT wT x [P,(coso)P (cos) +P (cos 0)P, (c059)] sinddodo Performing the integration the following expression is obtained: , _ 1 1 D(m, n;p,q) = 4 \aA [Kn atlsp, q + Kin, ntl, q.P + Kn, -n,p,q + K,, -n,p,q + K., -n,q,p + 2mi+l IK. mtl,p, qi? K.,m4l, q;p +K +K ) { n,-Mm,p,q n,-m, q;P -8- where by definition 0 =1 | sin(anti)2 Pp (cosd)1 _(B)ad m,n,pP,q 2 °q cos m, p\®) T and > = , Q Ln, pl?) = sin(2m+1) z P,(cos0)d6 te The recurrence formulas for the K's and I's are: 2q-l q-1 K = Si - m,n,p;,q q [x nt+l,p,q-l + *m,n-1,p, 4-1] q Kian, p, q-2 =~, _ 2p-l p-l , It is seen that if the first few K's are found by regular integration the rest can be obtained from the recurrence formula. A "dic- tionary" of the values of the K's required to solve for the cross product drags of the four orthogonal loadings considered here is ‘found in Table I. -9- IV. ORTHOGONALIZATION So far no orthogonal loadings have been considered. It may be mentioned that the concept of orthogonality is not at all necessary for drag reduction calculations. For the special case of conical a-distributions S. H, Tsien (Ref. 3) has minimized the drag of a Delta wing with sonic L.E. without using orthogonal loadings. It is, however, believed that by using orthogonality the work involved in obtaining drag reduction is considerably lowered. The general theory of orthogonality has been amply described by E. W. Graham (Ref. 1) and will not be repeated here. Now from the basic angle of attack distribution it is desired to construct orthogonal loadings. Let by definition a, = Poo a2 = Bog +271 O4 =Pog + bg Pyyteg Patty Pay a, = Pog + bg By, + ¢3 Faq | where the a's are the orthogonal loadings and the constants bys bg, by, Ca» Cy and d, are to be determined from the condition that each a distribution must be orthogonal to any other a distribution. Having defined the angle of attack distributions as shown above it immediately follows that: AP, = APog AP2 = APog t P24?) AP; = APog t b34P1) + C34P 29 it AP, = APgg + AAP], + C44P29 + 444P2y ~10- where AP, is the pressure distribution of the wing with angle of attack distribution a,3 AP, is due to a5, etc. As an illustration of how the constants above are determined consider the superposition of the first two loadings (a, and ay). The resulting angle of attack and pressure distributions are respectively (a, + a,) and (Ap, + Ap,). Then the drag of the combination is 1 2 1 2 J tanytar,v ta,)ds = J aoyoyas Jango sforyapaeefangnt where it is understood that the products as before are integrated over the planform area. The last two terms on the right-hand side repre- sent the interference drag which must be set equal.to zero. This means that Jaro Bootes By, )ds + Sos299%24P13) oo 0 or collecting terms: [D(o, 0;1,1)+D(1, 150, oy by = -2D(0, 0;0, 0) Hence b, is determined. Similarly b3 and C2 are determined from the following two linear equations: 1) [D(o, 0;1,1)4+D(2,1;0, 0)] b, + [Do, 0;2, 0)+D(2, 0;0, 0) c,=-2D(0, 0;00) 2) [D(0, 031, 1) + D(1,1;0,0) + 2b, D(1, 150, 0)] b3 {+ D(0, 032, 0) + D(2, 0;0, 0) + b, LD(1, 1:2, 0) +D(2, 031, 1)]{ c = -2D(0, 030, 0)-b, [D(0, 01,1) +D(1, 150, 0)] -ll- For the determination of by; cy and d, the equations below must be solved. 1) [D(0, 031, 1) + D(1, 150, 0)] b, +[D(0, 032, 0) + D(2, 030, 0)] cy +[D(0, 032, 1) + D(2,1;0, 0)] d, = -2D(0, 050, 0) 2) [D(0, 051, 1) + D(1, 150, 0)+2b, D(1,131,1)] by +] (0, 0:2, 0) 4D(2,0;0,0) +b, [D(1, 152, 0) +D(2, 051, mi Cy + }D(0, 032, 1) + D(2, 130, 0) +b, [D(1, 132, 1) i211, hay = -2D(0, 0;0, 0) -b, [D(0, 031, 1) 4D(1,1;0, 0)] 3)| D10, 031, 1)+D(1, 1;0,0)+2b, D(1, 151, 1) +c [D(1, 152, 0)4D(2, 051, il}, +}D00. 0;2, 0) #D(2, 050, 0)+b, [D(1, 152, 0) 4D(2, 051, 1)] +2c, D(2, 0:2, 0)} cy +)D(0,0;2,1) +D(2, 1;0, 0) +b, [D(1, 152, 1)+D(2, 131, 1)], +e, [p(2, 032, 1) +D(2, 1;2, 0)| d, = -2D(0,0;0,0) -b, [D(0, 031, 1) #D(1,1;0,0)] -c, [D(0, 032, 0)+D(2, 030, 0)] With the aid of the general expression for D(m,n;p,q) and Table I, Table Z has been prepared. Using Table 2 it is now easy to show that the above system of equations become -10b, = -240 -10b, + i5c, = -240 470b, + 87c. =0 -12- a , 10b, + I5e, + 19d, = -240 4 470b, + 87c, - 149d, = 0 4l7c, - 74d, = 0 ~N which upon solution give these results: b, = 24 b, = 29/11 b, = -3. 62385 C4 = -470/33 c, = -2.26300 dy = -12.75229 The first four mutually orthogonal loadings are now completely de- termined, i.e.: a) = Poo a = Bog + 24B 1 a, = Bot 29/1184, - 470/338 55 | a, = Bog - 3.623858), - 2.2638, - 12. 752298.) | From these the corresponding pressure differences across the wings are determined: AP; = 4Po9 AP, = AP go + 24Ap,, AP = APgg + 29/11 AP), 7 470/33 AP 54 AP, = APgg - 3-62385 Ap,, - 2-263 Apy, - 12.75229 Ap,, Using Table 2 it is then just a simple calculation to obtain lift and drag of the orthogonal wings. Note that since Poo =], D(m,nj0,0) = lift of the m,n loading. Thus -13- L L = L, = 49S 1 2 3 i L, = 4q8 { Ap,ds = D(0, 0;0, 0)-12. 75229 D(2, 030, 0) = (. 14985)4qS s Similarly Dj = D(0, 050, 0) = 4qS DS =D, 4q5 = D(0, 050, 0)+(24)*D(1, 131, 1)+24 D(1, 1;0, 0) +D(0,0;1,1) = (47) 4qS es N 24 (49S) o il 1.78448 (4qS) -14- V. RESULTS Percent Drag Reduction The four orthogonal loadings are now combined so as to give the minimum drag using the non-dimensional parameter L as defined on page 2. Since p. - 49s)” (o(0, 030, 0)1* _ , 1 gS 4q5 D(0, 0;00) 4, = 4/47 = .08511 4, = 4/24 = .16667 L = 4 = + 05035 the parameter for the wing of minimum drag is Soot =A, +h, +4, +L, = 4.30213 This means that the percentage reduction in drag from the flat plate (a,) drag is 4 lL _ 4 _ fe) a Oe ELIE 0313 = 7-023 /o Since t, is larger than 4, and f 4 the a3 loading gives the most drag reduction. Optimum camber and twist distribution The optimum angle of attack distribution is given by opt opt k=l Di. & opt k=1 Ly, vk Cc _ L = 37-3033 (a, + 1/47 a, +1/24a, + .084 a,) which in terms of x, y coordinates become wopt _ 35765 + .088x - .09007x2-. 1831 ly2-. 132x2+. 132xy" <r e . * ° ° -15- and is found plotted for various spanwise stations in Fig. 2. The expressions for Q,2 Az and a, are given in Appendix 2. In order to obtain the camber and twist distribution of the optimum wing the angle of attack distribution was integrated with respect to the x variable. Let (x,y) be the camber and twist distribution. Then x (x,y) = - \ Goot Ox tf Cy) 0 where f(y) is a constant of integration that has been determined (=0) by requiring the wing trailing edge to lie in the plane z = 0 (2 being the axis perpendicular to x,y). The above integration results in X(e¥) ) = -x(.35765 + .044x - .03002 x7 - .18311 y* 7 7 - .033x> +.066xy*) which is found plotted in Fig. 3 for several constant values of y. It is noted that the wing has washout which is in accordance with the usual aerodynamic practice. The wing has S camber over the . center section. Optimum pressure distribution Similarly the optimum pressure distribution is given by C,qS 4 £ L k APG =o COT OP opt Loot Kel Ly, k Cc +—3097 (Ap, + 1/47 Ap, + 1/24 Ap, + . 084 Ap,) -16- which can be written as AP, t 4 opt 8C, qo = = aoagapg (AP, t1/47 Apst+1/24 Apst. 084 Ap,) or u Cc t L - “Fo acre (Ap, + 1/47 Ap,t1/24 Ap, + .084 Ap;) In the u,v coordinates this becomes C, u 1: opt = 0077481 { TAY (21, 60304 - 6.34892 u ru +14, 07988v + 2,04256uv - 12.7969u~ - 11.12586 v7 It+u o 2 2, 21.4162u°v = 12, 8497uv-) +\/ (21. 60304 -6. 34892v + 14.07988u + 2.04256uv - 12.7969v~ -11.12586u" - 21.4162uv" - 12. 8497u“v) where 1 1 , us - (x+y) and =— (x-y) y2 2 Fig. Pehows the distribution of local lift along the wing _ trailing edge compared with the flat plate distribution and the dis- tribution obtained by S. H. Tsien (Ref. 3) by the conical flow theory. | | In Fig. 5 the distribution of local lift along the root chord has been plotted. The flat plate lift distribution over the root chord is also shown. -17- VIII. CONCLUDING REMARKS It has been shown that a reduction of induced drag of 7.023 percent can be obtained by suitably cambering and twisting the flat Delta wing. The procedure used can be continued to obtain even higher drag reduction by taking additional basic loading into ac- count. For conical distribution S. H. Tsien (Ref. 3) obtained a drag reduction of 8°/o. By combining this loading with the non- conical loading of this report it should be possible to find an ap- preciable drag reduction. Due to lack of time, this has not been done here. However, the present report contains all the basic material needed and only some numerical work is required. In Figs. 2 to 5 the angle of attack, local lift and the camber and twist distribution of the optimum wing are shown. The wing has washout and has negative camber over approximately the front quarter wing area, while the rest carries positive camber. The leading edge droops down and reaches a maximum droop ata point about 65 percent along the half span from the root chord. Compared to the flat plate the optimum wing carries a higher lift distribution over the center section and less along the leading edges. 3. -18- REFERENCES E. Ww. Graham, "A Drag Reduction Method for Wings of Fixed ‘Planform", Douglas Report No. SM-14441, July 15, 1952. A. M. Rodriguez, P. A, Lagerstrom and E. W. Graham,. "Theorems Concerning the Drag Reduction of Wings of Fixed Planform", Jour. Aero. Sc., Jan. 1954. S. H. Tsien, "The Supersonic Conical Wing of Minimum Drag", Thesis, Cornell University, June 1953, A. Ferri, "Elements of Aerodynamics of Supersonic Flows". Macmillan, 1949. -19- APPENDIX I mUN) So am] Ih I = cos0cos0 ae) — l " + [3cos"0+3c0s“6-2) _l 2 F132. ox ay Pay = z [3cos Ocos0+3cosO0cos 9 -cos0-cosd | cos 8 in® sin? cos A 1 2 2 + 2 2 Poo "F sin 0 ind _ sind _ cos3 = sin3 2 sin3 cos3 9 A _l 2 2 4 2 2 Pil 30 sind _ > sin 9 @ ..@ e. 9 A 4 5cos 5 sin5 z + cos5 > sin > P20 ~ T0m Sino a) a) an) a) : sin > cos5 5 + 5sin5 5 cos z _ sind 8. is) Q.. 0 s 4 Scos35 sin5 z + 3cos5 3 sin3 > P21 ~ 300 ; sing 3sin3 9 cos5 $ + 5sin5 5 cos3 5 sin@ + -20- APPENDIX II» =1 1 + 24uv = 1 + 12(x*-y%) 8.1212 - 9.3636x~ - 12y* 9. 56422 uv (utv) + 3.18807(utv) = 2.1315 +4. 50857x - 3.00918x” + 6.76286xy~ - 6.76286x? 8.1212 + 2: 6364 uv - 10. 6818(u“+v) 2.1315 - 3.62385uv - 1.69725 (u-4v*) - 0. 38532y 2 -21- TABLE I .. = 0 except as stated below m,n;,;p,q K | = (-1) when m-n m,n,0,0 2émt =." mn ami ___(-1)" _ Kin,n,1,0 > ~ {Zme1 when m-n =- (-1)™ m-n 2m+3 __(-1)™ = Wam-T mtn ~_(-))™ ~ 2am-+3 mtn _ 3(-1)™ Kin,n, 2,0 7 8(2m-3 when m-n = (-1)™ m-n 4(2m+ = 3(-1)™" m-n 8(2m+5 =. 3(-1)™ mn * 8(2m-3) — (+1) ~ ~ 4(2m-41 mtn mY =- oat mtn __ (-1)™ Kin,n, 0,1 > Bama when m-n __-(-1)™ ~ 2(2m4I mt+n mtn -2 m,n,2,l m,n,0,2 m,n,1,1~ U ~ Té(am+ Gu" | 1, 1 4 2mt+3 “ 2m-1 ae) ee ae 4 (2m-1) 3(-1)"_ ] m-3 + \m 3(-1) | 3(-1) 16(2m+5) _ _-3(-1)" ~ 16(2m-3 (-1)™ 8(2mt]1 (-1)" 2mt 3(-1)" ~ B(2m41) ~~ Té6(Zm-3) -3(-1)" | ~ Té(2mt5) 3(-1)_ ~ 16(2mt5) 3(-1)" m+1 (-1) 4(2m+41) (-1)" 2mt+l when when m-n = -2 m+n = -l mtn = 1 mtn = -3 m-n = 3 m-n=l m-n = -l m-n = -3 mtn = -2 mtn = 0: mtn = -2 min = -4 m-n=+t2 m-m = 0 m+n = | ‘, mt] = -] m,n,1,2 3(-1)" ~ 16(2m-1) : 3(-1) _(-)) m+3 2m- -23- _ 2 (I | 3-1)" ~~ 8(2m+3) ~ Té(2m-T) ___3(-1)™ ~ ~ Té(2m43) __3(-1)™ iT em-1 - 0", 3p" ~ B(2m-1) | 16(2mF3) __(-1)" 3(-1)" ~ 8(2m43) 16(2m-T) 3¢a)™ = Té(am+¥3) when m-n m+n m+n m+n mtn -2 -24- TABLE I D(m, n;p, q) 1 -1/12 1/8 11/120 0 l/l2 1/24 ~23/840 0 -1/60 1/8 -1/84 1/15 -13/420 -5/168 7/240 A FIG.| — WING PLANFORM AND AXIS SYSTEM NOILASIYLSIG MOVLLIV SO JTIONV WNWildO GQ3SL0IN4ISSY - 2 Old a Xx —~> 0 Z20- v0- 9g0- 8O- = Ol--— sg I- bl- ! —~<— Xx o) ~~ (O'ZA-)> ( 2'0) 7 YU NOILNGINLSIO 1SIML ONY HYSEWV)D -€'914 20- vO- = 9'0- 8°0- Ol- Z1- t'l- | | -28- 3903 ONITIVYL ONOTV NOILNEIYISIAO 1417 1v9071-¥v Old <—_ My, 0} 80 90 £0 20 0 of O 3 | oS? | ao (pjeu) 4equod joo1u0D -___ | ysoday yuasaig —---— | | 840igd $OlI3. ——— -29-— QYOHD 1008 ONONV NOILNEINLSIG L317 1V901-S ‘9ls 8'0- 90- vO- 80- : a . : Pn oe meme mers ts en at os fp et ‘ - i eee rr rea — + _ ' 4 ; oot — a: | <A : : t- 4 t i | + yuoday juasasg —~ — 8}1Did $Ol4 —— 20 oe) n'9 nu 01 | G|