NASA Contractor Report 3140 

Minimization Theory of Induced Drag 

Subject to Constraint Conditions 

LOAN C-:.-‘-Y: RETURN TO 

AFWL TECHNICAL LIBRARY 

\&IRTLAND AFB, N. .w;;/ 

John DeYoung 

CONTRACT NASl-13500 

JUNE 1979 

i 

wEA . 
-- 

TECH LIBRARY KAFB, NM 

NASA Contractor Report 3140 

Minimization Theory of Induced Drag 

Subject to Constraint Conditions 

John DeYoung 

Vought Corporation Hampton Technical Center 

Hampton, Virginia 

Prepared for 

Langley Research Center 

under Contract NASl-13500 

National Aeronautics 

and Space Administration 

Scientific and Technical 

Information Office 

1979 
L 
TABLE OF CONTENTS 

Page 

No. 

SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

INTRODUCT 

SYMBOLS 

ION .............................. 2 

................................ 3 

MINIMIZED INDUCED DRAG OF PLANAR WINGS WITH THE CONSTRAINTS OF 

LIFT PLUS WING BENDING MOMENT . . . . . . . . . . . . . . . . . . . 7 

Force and Moment Coefficients in Terms of Fourier Coefficients . . 8 

Solution for an/a, Fourier Spanwise Loading Coefficients . . . . . 10 

Wing Bending Moment Minimized Solution Loading Characteristics . . 11 

Solution in infinite series 11 

A limit on the magnitude of t' : : : : : : : : : : : : : : : : 13 

Solution in closed functions . . 13 

Solution in closed functions for Ab'='O' : : : : : : : : : : . 15 

Numerical values of loading characteristics for various 

nbandt . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

Comparison with Elliptic Loading . . . . . . . . . . . . . . . . . 17 

MINIMIZED INDUCED DRAG OF NONPLANAR WINGS WITH EITHER THE CONSTRAINT 

OF LIFT OR OF LIFT PLUS BENDING MOMENT . . . . . . . . . . . . . _. 21 

Biplane and Wing in Ground-Effect Solutions . . . . . . . . . . . . 21 

an'al coefficients for minimized induced drag . . . . . . . . 24 

an/al coefficients for minimized induced drag and bending 

moment 

Antisymmetr;c'loading'an/a2'coeff;c~ents ior minimiied ' . * 

. 25 

induced drag . . . . . . . . . . . . . . . . . . . . 

Example numerical'solution with 5 = l/2 of biplane and 

25 

wing ground effect models . . . . . . . . . . . . . . . . . . 26 

Flow-Field Solution of a Flat Vorticity Sheet . . . . . . . . . . . 34 

Cruciform Wing Solution . . . . . . . . . . . . . . . . . . . . . . 35 

iii 
Page 

No. 

an/al coefficients for minimized induced drag ........ 38 

an/al coefficients for minimized induced drag and wing 

bending moment, and antisymmetric loading .......... 38 

Cruciform wing with y = 90 degrees .............. 39 

Flow Field of a Y Banked Plane Induced by a Flat Vorticity 

Sheet . . . . . . . . . . . . . . . . . . . . . . . . . ...39 

V-Wing . . . . . . . . . . . . . . . . . . . . . . . . . . . ...40 

Recurrence formula for Q, and Qnn* .............. 41 

an/al coefficients for minimized induced drag ........ 43 

V-wing in ground effect ................... 44 

Planar-Wing Winglet Configuration . . . . . . . . . . . . . . . . . 45 

Planar-wing winglet aerodynamic characteristics from 

reference9.........................45 

an/a 1 coefficients for minimized induced drag for a given 

lift . . . . . . . . . . . . . . . . . . . . . . . . . ...47 

an/al coefficients for minimized induced drag and wing 

root bending moment . . . . . . . . . . . . . . . . . . . . . 49 

Example numerical solution with +. = 5~/32 . . . . . . . . . . 49 

RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . 53 

Planar Wings . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 

Constraint that either lift or rolling moment is specified . . 53 

Constraints of lift and of wing bending moment about nb . . . 53 

Derivation of t for minimal induced drag and wing span . . . . 55 

Nonplanar Wings .......................... 57 

Other Application of the Analysis ................. 58 

Flow Field Solution of a Thin Wing Chordwise Vortex Sheet ..... 60 

Formation Flying of Wings with Wingtips Linked .......... 62 

CONCLUSIONS .............................. 64 

APPENDIX A. - MATHEMATICS OF TRIGNOMETRIC SUMMATIONS .......... 66 

iv 
Algebraic Series ........................ 

Trigonometric Series ...................... 

Example summations of trigonometric series ......... 

Spanwise Loading Distribution .................. 

The k-Factor Summation ..................... 

Spanwise Center of Pressure, Qcp ................ 

Induced Downwash Angle in the Wake ............... 

APPENDIX B. - EVALUATION OF In AND In,,* INTEGRALS .......... 

Integral of Equation (67) .................... 

Integral of Equation (70) .................... 

Numerical example of Inn* for 5 = l/2 ........... 

APPENDIX C. - EVALUATION OF Pn AND Pnn* INTEGRALS .......... 

Integral of Equation (125) .................... 

Integral of Equation (128) ................... 

REFERENCES .............................. 

Page 

No. 

66 

71 

74 

77 

81 

84 

86 

90 

90 

93 

98 

100 

100 

102 

104 

TABLES................................ 106 
SUMMARY 

The method treats wing configurations with trailing vorticity sheets of 

arbitrary shape. Induced angles normal to the wake are analytically formulated. 

'These, factored by the vorticity in the sheet, form the induced drag integral. 

The vorticity is represented as a Fourier series in loading, then the unknown 

Fourier coefficients can be determined after specifying the constraint conditions 

then minimizing with respect to the loading Fourier coefficients. Exact analytical solutions in terms of induced drag influence coefficients can be attained 

which define the spanwise loading with minimized induced drag, subject to specified constraint conditions, for any nonplanar wing shape or number of wings. 

Closed exact solutions have been obtained for the planar wing with the constraints 

of lift plus wing bending moment about a given wing span station. Compared with 

a wing with elliptic loading having the same lift and same wing bending moment 

about a span station, the induced drag can be of the order 15 percent less for 

the wing with the inboard minimized-solution loading. Example applications of 

the theory are made to a biplane, a wing in ground effect, a cruciform wing, 

a V-wing, and a planar-wing winglet. For minimal induced drag the spanwise 

loading, relative to elliptic, is outboard for the biplane and is inboard for 

l-the wing in ground effect and for the planar-wing winglet. The theory can be 

applied to determine the loading for minimal induced drag of wings in formation 

flying, banked wings, flying in or near wake, and linked wingtips. No-roll 

can be an additional constraint. A spinoff of the biplane solution provides 

mathematically exact equations for downwash and sidewash about a planar vorticity 

sheet having an arbitrary loading distribution. 
INTRODUCTION 

Induced drag minimization theory is in a rapid development and use phase. 

This is because of widespread interest in performance and structural advances 

that possibly may be realized with unconventional aircraft configurations, or 

by imposing structural and/or performance constraint conditions. Numerical 

solutions of minimization theory have been programmed for computers, which 

extends the application range of the theory. Unconventional aircraft include 

such configurations as a wing with winglets; a biplane; a tandem wing; a cruciform; a V-wing, swept box wing; a strutted wing; a skewed wing; and a nonplanar 

wing designed to lessen wing and fuselage bending moments. A structural constraint condition can be that the wing bending moment about a given span station 

be minimum. A performance constraint condition can be an aircraft flying near 

ground, a surface flying vehicle, or an aircraft in a flying formation. 

Minimum induced drag solutions for planar wings with different structural 

constraint conditions have been developed by several investigators. Dr. L. 

Prandtl (ref. 1) in 1933 obtained a solution using the constraint that the 

integral of the local bending moment be specified. Compared with an 18.1 

percent smaller span wing with elliptic loading, and the same lift and constraint 

magnitude, the induced drag is 11.1 percent less. In reference 2, the constraint 

condition is the wing-root bending moment. The solution for spanwise loading is 

obtained after the induced angle function is determined. Compared with a 25 

percent smaller span wing with elliptic loading, and same lift and wing-root 

bending moment, the induced drag can be 15.6 percent less. In reference 3 the 

constraint condition also is the wing-root bending moment and the solution is 

somewhat similar to that of reference 2. For approximating wing structural 

weight, the integrals of wing shear force and of wing local bending moment are 

used as constraint conditions in reference 4. With these two constraints, the 

induced drag can be 7.1 percent less when compared to the 13.8 percent smaller 

span wing with elliptic loading and same lift and constraint magnitude. 

Numerical solution vortex-lattice methods for computing the optimum spanwise loading which results in minimum induced drag, have been developed in 

references 5 and 6. These are quite adaptable to complicated nonplanar 

2 
configurations. In reference 5, the optimum loading calculation is based on 

Munk's theorem III in which conditions are defined on the induced velocity. 

In reference 6, the numerical procedure is generalized in terms of constraint 

conditions. An application of a nonplanar wing numerical method is made in 

reference 7 to the investigation of wing-winglet characteristics. Examples of 

nonplanar wing analytical or classical theory solutions for loading are given 

in references 8 and 9. 

The principal objective of the present study is to develop a generalized 

analytical induced drag minimization theory to which aerodynamic and configuration constraint conditions can be imposed. The term generalized implies direct 

solutions are possible for arbitrary configurations without having to resort 

to complex conformal transformations. Here generalized also means that minimization is done without resort to other induced drag theorems such as Munk's. 

Aerodynamic constraint condition examples are, for a given lift, for a given 

rolling moment, and/or for a given wing bending moment about an arbitrary span 

station. Configuration constraints include, multi-wing, wing-winglet, or 

flying near wake of other aircraft. It was found that a generalized theory could 

be formulated by expressing the spanwise loading distribution as a Fourier 

series in which the Fourier loading coefficients are unknowns. With constraints 

imposed and minimization, equations are evolved in terms of the unknown loading coefficients and induced drag influence coefficients. Solutions of these 

equations leads to an analytically exact evaluation of the loading coefficients. 

In an age of numerical solutions, available exact solutions provide an accuracy 

check. The present theory also serves as such a check for the evolving numerical minimization methods. 

SYMBOLS 

A aspect ratio, b2/S 

a a n' n* Fourier coefficient of spanwise loading, equation (1) 

b wing span 

'b ratio of wing bending moment coefficient to lift coefficient, 

Cmbb/CL. equation (11) 
'bc 

'Di 

cL 

CL 

C mbb 

C 

C 

av 

5 

Di 

D nn* 

e 

er 

,f, , fn 

G 

In 

I nn* 

Jn 

k 

k. 

kl 

L 

L n*w 

Mbb 

m,n,n*,N 

Cb of wing with elliptic loading, fl/4n 

induced drag coefficient, Di/qS, equation (13) 

lift coefficient, L/qS, equation (3) 

rolling moment coefficient, L/qSb, equation (78) 

wing bending moment coefficient about lateral point nb, Mbb/qSb, 

equation (7) 

wing chord 

average or mean wing chord, S/b 

section lift coefficient, equation (5) 

induced drag 

induced drag influence coefficient for planar-wing winglet, 

equation (164) 

induced drag efficiency factor, equation (15) 

induced drag efficiency factor of rolling wing, equation (79) 

wing bending moment influence coefficients for planar wing, 

equations (9) and (10) 

dimensionless circulation, r/bV, equation (1) 

downwash influence coefficient for biplane, equations (67) and (118) 

induced drag influence coefficient for biplane, equation (70) 

sidewash influence coefficient for biplane, equations (119) and 

(B12) 

planar wing bending moment constraint parameter of Cb and e, 

equations (26) and (37) 

planar wing bending moment constraint spanwise loading function, 

equation (31) 

planar wing.bending moment constraint parameter of ncp, equation 

(33) 

wing lift, also, rolling moment 

induced normal velocity influence coefficient for wing winglet, 

equations (165) and (166) 

wing bending moment about span station nb 

integers in Fourier series for loading and in influence coefficients, 

odd only for symmetric loading 

4 
'n 

P nn* 

Qn 

Q nn* 

9 

S 

S 

'e 

Tn 

t 

5 

V 

V 

S 

W 

Y 

2 

/ 

I 

1 “w 

I 

r 

Y 

5 

rl 

nb 

nO 

ncp 

P 

induced velocity influence coefficient 

equation (125) 

induced drag influence coefficient for 

(128) 

induced velocity influence coefficient 

induced drag influence coefficient for 

dynamic pressure,; pV2 

for cruciform wing, 

cruciform wing, equation 

for V-wing, equation (141) 

V-wing, equation (143) 

wing area 

spanwise coordinate along nonplanar wing surface 

spanwise semispan along nonplanar wing surface 

wing bending moment constants, equation (171) 

arbitrary constant, proportional to how much wing bending moment 

constraint is imposed 

limit condition for t, equation (29) 

free stream velocity 

induced sidewash, tangent to surface 

induced downwash, normal to surface 

lateral coordinate, from midspan station 

vertical coordinate 

induced angle normal to the wake 

circulation, also dihedral angle 

angle between wings in cruciform, also, dihedral cant angle of 

winglet 

influence coefficient constants used for evaluating In0 and In,, 

equations (B39) and (B40) 

dimensionless vertical coordinate, 2Z/b or Z/s, 

dimensionless lateral coordinate, 2y/b or y/s, 

span station about which bending moment is taken, equation (6) 

span station at which winglet starts, no = COS+~ 

semispanwise center of pressure, equals 4Cb when +b = IT/~ 

air density 

5 
0 spanwise trigonometric coordinate, cos-ln 

Subscripts 

av average 

b bending moment 

C wing with elliptic loading 

m,n,n* series integers 

S symbolizes, from vorticity sheet 

6 
MINIMIZED INDUCED DRAG OF PLANAR WINGS WITH THE 

CONSTRAINTS OF LIFT PLUS WING BENDING MOMENT 

Relations between spanwise loading quantities can be established in the 

wake or Trefftz plane of the aircraft. Then solutions reduce to two-dimensional 

problems and are independent of the chordwise shape of the wing and of how the 

loading is developed. Quantities dependent on spanwise loading include, induced 

velocity normal to the wake, induced drag, spanwise center of pressure, wing 

bending moment, and rolling moment when the loading is antisymmetric. When 

constraint conditions are imposed and induced drag is minimized, then from the 

relation between loading and induced drag, the optimum spanwise loading can be 

evaluated. A simple method for establishing the relationships between the 

spanwise loading quantities is by expressing the spanwise loading as a Fourier 

series in which the Fourier loading coefficients are unknowns (see eq. 1). This 

method permits a manageable, viable, analytical solution, regardless of the 

complexity of the configuration or of the constraint conditions. The method 

can be demonstrated by showing the solution for a planar wing with simple 

constraints. With the constraint that lift is specified, then (by eq. 15) 

aC, 

TrA 

'Di = 4 (al2 + n = 2 n 

y na2),Uj= 

aa m 

e (2m am)m,l = 0, G = a, sin 4 

that is, the loading for minimum induced drag for a given lift is elliptic 

loading. Also, for a planar wing with the constraint that rolling moment is 

specified, then 

aCD 

CDi = p (2 a22 + ? n an2), $ = p (2 m am)mf2 = 0, G = a2 sin 24 

n=l#2 m 

which is the loading for minimum induced drag for a given rolling moment. 

In this section solutions are developed for the constraint conditions of 

lift and wing bending moment about a given span station such as at a wing strut 

connection. It can be noted as pointed out in reference 2 that the induced drag 

of a wing having a given lift and a given spanwise load distribution is not 

affected by the compressibility of the air at subsonic speeds. At supersonic 

speeds an additional drag associated with the formation of waves arises. 

7 

A ‘r 

i 
Force and Moment Coefficients in Terms of 

Fourier Coefficients 

The wing spanwise loading distribution expressed as a Fourier series has 

proven to be a versatile analytical tool (see e.g. refs. 10, 11, 12, and 13). 

In Fourier series 

c c 

G(4) = k = 7%~ = n F = , an sin n+ 

where for symmetric loading n is an odd integer, and $I is related to span 

station by 

n = 2y/b = cos 4 

Lift coefficient is 

L 1 

f 

b/2 

f 

1 

CL=q=$ pVrdy = A Gdn = A 

f 

z sin $ d+ 

-b/2 -1 0 

then with equation (I), equation (3) becomes 

CL = pa, 

When CL is known the loading distribution can be written 

CRC 2A -=- 

'L'av cL 

G = p (sin $I + 

odd 

Wing bending moment coefficient about the span station 

nb = 2yb/b = cos +, 

is 

C Mbb _ 1 

mbb = - 

-- 

qSb qSb 

b/2 

I 

1 

(Y - Y&JVrdY = $ (n - nb)Gdn = 5 

J 

nb 

(2) 

(3) 

(4) 

(5) 

(6) 

I 'b 

(cos 4 - cos $1 I 

0 

G sin + d$ (7) 
. . . . ..__._. ._, ,,.._ . . . .

. .._... . _ 

then inserting equation (1) and performing the integration 

C mbb 

odd 

where 

f 1 = 2(sin $b - + sin3$b - $b cos $b) (9) 

(8) 

fn =i[ sin (n-2)$h _ sin (n+2)9h _ 2 cos 9b sin (n-l)$b + 

n-2 n+2 n-7 

2 cos +b sin 

n+l 

At $b = F: f, = 4 ; fn = 

An important parameter is the ratio of wing bending moment to wing lift. From 

the ratio of equations (8) and (4) 

odd 

where fl and fn are given in equations (9) and (10). When the loading extends 

over the semispan, then +b = IT/~, and Cb is l/4 of the spanwise center of 

pressure9 'cp' Then, combining equations (lOa) and (11) 

I 

1 n-l 

SC 

ncp = 

- ndn = 

o 'L'av 

k [1 - 3 Y ($',' $1 

n=3 

odd 

Induced drag coefficient is 

Di 1 

'Di = tj%= 5 

I 

b/2 

-b,2iW pVI'-dy = 'jl"wdn=~j,""wsin~db 

(12) 

(13) 

9 
_,_._.__. --- --...._ - .._.. ._..___ ,. ._._. - -_.-. - _... _ - .._ ._. .-. - . ..- --._-- --- 

where aw is the downwash angle in the wake given by 

TldG 

1 =- 

id$* d+* 1 

aW IT cos~*-cosl#l = - sin+ ? n*a 

n* = 1 

n* sin n*$ 

0 

Inserting equations (1) and (14) into equation (13) leads to 

'Di = p y n an2 = 

n=l 

where for symmetrical span loading 

e= 

1 

1+ Y n ($1 

2 

n=3 

odd 

(14) 

(15) 

(16) 

Solution for an/al Fourier Spanwise Loading Coefficients 

The problem is to evaluate the infinite number of an/al Fourier coefficients 

such that the spanwise loading is defined which has minimum induced drag per wing 

bending moment about nb' Minimal combined induced drag and wing bending moment 

per unit lift can be realized by minimizing the parameter Cb/e. From maxima 

and minima theory for a function of many variables, the necessary conditions for 

an extreme value are that all the partial derivatives with respect to each 

variable be zero. Taking partial derivatives of Cb/e with respect to as/al, 

as/al, . . . . am/al, results in 

a(cb/e) _ l aCb 'b ae 

a(a,/al)-ea(a,/al) 

-- 

e2 a(a,/al) =0 (17) 

The partial derivatives of equations (11) and (16) are 

10 
(18) 

*=- 

2m(am/al 1 

= -2 e2 m 2 

Cl + B n (z)212 

n = 3 

odd J 

With equation (18), equation (17) becomes 

&fm+2mCb$=0 Ii 1 

which can be solved for am/al. Then the solution for the Fourier loading coefficient, with m referred to the general n, becomes 

a 

n - fn -- _~ 

a1 8aeCb (19) 

Equation (19) is in the form of a constant times (-f,). Let t be an arbitrary 

number, then the minimization condition is 

a n -= 

al 

- t fn 

A better parameter for the minimization process is Cb2/e since it indicates 

induced-drag bending-moment efficiency. However, the minimization of the term 

Cbr/e where r is any constant leads to (an/al) = -(r/8meCb)fn, that is, a 

constant times f, and can be expressed as in equation (20). 

Wing Bending Moment Minimized Solution Loading Characteristics 

Solution in infinite series. - Equation (20) inserted into equations (5), 

m, (12), (14), and (16) leads to the following spanwise loading characteristics: 

11 
where 

cRc -= 2A G = $ (sin 4 - t 

'L'av 'L 

F 

n=3 

fn sin n$) 

odd 

n-l 

2 

ncp 

= g- [l + 3t Y $1 4 fn] 

n=3 

odd 

A -ci 

cL w 

= p (1 - _t Y n fn sin n$) slw n = 3 

odd 

cb = F = k (fl - kt) 

e = cL2 -= 1 

rAC 

Di 

1 + kt2 

k = ? n fn2 

n=3 

odd 

(21) 

(22) 

(23) 

(24) 

(25) 

(26) 

where fn is given in equation (10) and f, in equation (9). 

A limit on the magnitude of t. - A limit can be defined by the condition 

that the spanwise loading remains positive at all span stations. This condition 

is satisfied by requiring that the slope of the loading distribution be zero 

at the wingtip. Taking the derivative of equation (21) with respect to 4, settin 

to zero, and solving for t at $ = 0 

(27) 

12 
where t is the value of t at which the loading is positive at all span stations. 

If t exieeds tl the spanwise loading becomes negative in the region near the 

wing tip. The summation in equation (27) is readily found by expanding n f,. 

From equation (1iD) 

? n f, = sin $b + $ sin 3 $b 

n=3 

+ & sin 5 +b + +- sin 7 f$b + . . . 

odd - k sin 5 $b - f sin 7 +b - . . . 

- 2 cos $I~(: sin 2 $b + i sin 4 $b + k sin 6 $b + . . - 

- t sin 4 I#I~ - i sin 6 $b - . . . ) 

= sin I$~ + i sin 3 $b - sin 2 $b cos +b = $ sin3+b 

then equation (27) becomes 

3 

(28) 

(29) 

Solution in closed functions. - The infinite series summations of equations 

(22) and (26) converge quite rapidly, however, those of equations (21) and (23) 

do not, especially that of equation (23). Summation methods are developed and 

presented in appendix A which includes the functional evaluation of the summations given in equations (21), (22), (23), and (26). Using equations (A60), 

(A74), (A87), and (A95) the closed function form of equations (21) through (26) 

become 

cRc - = fl G = : (1 - n2)4 (1 - kot) 

'L'av 'L 

(30) 

13 
where 

k, = - 5 (1 + 2 rlb2) (1 - nb2)k+ (%+d2 

ncp 

= $ (1 - kit) 

where 

(32) 

kl = ; (1 - ; nb2) (1 - nb2)' + $ nb4 cash-l t (33) 

TA 

2CL aw 

Cb = Fy = k (fl - kt) 

where f, = $ (2 + nb2) (1 - nb2 > a 

- 2nb cos-lqb 

k = 2nb6 - hb4 h nb2) ; (1 - 6nb2 + 3nb4 + 

e - 

cL2 1 -- =- 

aACD l+kt2 

i 

(31) 

(34) 

(35) 

(36) 

(37) 

(38) 

The functional behavior as nb approaches unity of the coefficients k, kl, 

and fl can be made as done in the development of equation (A75). 

14 
For nb -f 1: 

k = F (1 - $)4 [l - $j (3 + 11,) t1 - $)I 

kl = R (1 - $)5'2 (1 + nb)% cl - + I1 - $11 

fl = E fl (1 - ,-,b)'j2 [l - & (1 - nb)] 15 I 

(3% 

Solutions in closed functions for nb = 0. - In this case the solution 

simplifies.The condition that bending moment is taken about the wing root is 

that nb = 0. Then the k, k,, and fl factors simplify considerably. For nb = 0; 

k = 2/9, kl = l/6, fl = 4/3. With these values the aerodynamic characteristics 

are, for nb = 0 

Cc -= 

'L'av 

p C(1 t s, (1 - $)% 

ncp 

=+-(1 -k, 

TrA 

q aW = 1 + t (4 - ITIn/) 

'b =k(l -;, 

e= 1 

1+ 

- t n2 cash-l (40) 

(41) 

(42) 

(43) 

(44) 

Numerical values of loading characteristics for various nb and t. - In this 

wing bending moment minimized solution several generalized constants appear. 

These include k, k,, and f,. The term k, is a generalized function of the spanwise coordinate. Values of k, k,, f,, and the ratio k/f, have been computed 

from equations (37), (33), and (36) for a range of nb values. These are presented 

in table 1. Values of k, determined from equation (31) appear in table 2. 

15 
The spanwise loading distribution is given in equation (21) in the form of 

an infinite series, and in equation (30) in the form of a closed function. It 

is of interest to know the magnitude of the loading harmonics, that is, the 

factors of sin 3+, sin 54, . . . Values of fn computed from equations (10) and 

(lOa) are presented in table 3. These data show that the optimized loading is 

made up primarily of elliptic loading and the sin 34 harmonic, and much less 

due to higher harmonics. 

The value for t such that the spanwise loading gradient is zero at the wing 

tip is tl given by equation (29). For these t = tl values, the aerodynamic 

characteristics are listed in table 4. These are computed by use of equations 

(32), (34), (35), (38), and table 1. The parameter (Cb2/e)/(Cb2/e)c' c denotes 

t = 0, is a measure of the induced drag efficiency compared to a wing with 

elliptic loading having the bending moment ratio Cbc/Cb. The optimized loading 

is elliptic when t = 0. The parameter of the induced angle in the wake has a 

constant value from n = 0 to nb, then it is a straight line joining the points 

at nb and at 1. 

The bend ing moment ratio is obtained from equation (35). Then 

"bc 1 -= 

'b l-kt 

f 1 

Solving for t 

t = & 0 - CbjCb) 

(45) 

(46) 

For comparison of data it is convenient to keep the parameter Cbc/Cb constant. 

Then t is evaluated from equation (46). The aerodynamic characteristics for 

Cbc/Cb = 1.1, 1.2, and 4/3 are presented in tables 5, 6, and 7. These values 

are determined with the t of equation (46) and using equations (32), (34), (35), 

(38), and table 1. 

16 
The spanwise loading distribution due to the specified conditions on 

loading in tables 4 through 7 is obtained by using equation (30) with the listed 

k, values in table 2, and t values of tables 4 through 7. Values of spanwise 

loading are presented in table 8. For positive values of t the loading is 

inboard of elliptic loading. 

For nb + 1, mathematical limits need to be taken. W ith equations (39) and 

(29) the terms kit, fit, kt, and kt2 can be evaluated and are all zero at nb = 

Then n,, = 4/31~, and e = 1, that is the values for ellipt ic loading. However, 

(kt/fljr+ 5/8, and ~(1 - nb)t -f 31~fl/8(1 - nb)', as nb -+ 1. Thus for the t of 

equation (29) as nb + 1 

1. 

?TA TrA 3&? 'bc 8 

zy aw b-l) -+ 19 T aw (1) + 1 - 8(1-~ (47) 

b 

)4 3 q-- + YJ 

Since from equation (39), (k/fl) -t (5a/6)(1 - nb)3'2, then t of equation 

(46), as qb -f 1, becomes 

t+ 3&? (, 'b ) 

-- 

5(h-,b)3/2 'bc 

(48) 

where cb/cbc is a specified value. With the t of equation (48) then, as before, 

ncp 

-f 4/31~, e + 1, and a,(l) at nb = 1 approaches minus infinity. The values 

for nb = 1 are incorporated into tables 1 through 7. 

Comparison with Elliptic Loading 

A measure of the efficiency of a wing with inboard spanwise loading can be 

attained by comparing this wing with a wing having elliptic loading and the 

same lift and wing bending moment. Since elliptic loading is the most efficient 

in terms of induced drag it serves as a good standard for comparison with the 

inboard loading characteristics. The induced drag ratio and ratio of aspect 

ratios are 

Di A 

-= 

D. 1C 

(b)2 ($)2 $, 

C 

6 = g ($2 (49) 

17 
where subscript c refers to the wing with ellipt ic load 

moment coefficient ratio is 

ing. The wing bending 

'b 'mdCL Mbb Lc bc 

~=-~=MbbcmC 

(50) 

for the same lift and bending moment, and since for elliptic loading e, = 1, 

then equations (49) and (50) reduce to 

-2E b 

bc 'b 

(51) 

(52) 

The ratio of aspect ratios depends on S/SC. It can be noted that when these 

wing surfaces are the same then friction drag will remain about equal. In 

equations (51) and (52), Cb and e are given in equations (35) and (38), 

respectively. The wing bending moment parameter for elliptic loading is 

equation (35) with t = 0. Then 

(53) 

where nbc = ybc/(bc/2) is the span station of the wing with elliptic loading at 

which wing bending moments are taken. Two conditions on nbc can be made. These 

two moment conditions are pictured as follows: 

nbc = nb: 0 ‘;bc 

2 

I I Y 

bC b 

/ 2 2 lwingtip, for hingtip, for 

elliptic load- inboard loading 

ing \ \ 

ybc=yb: 0 

I I 

./rb b _ bc 

Y 

g b 

-nbS - nbc2 2 

18 
Then for the same yb 

-- 

b 

'bc = 5;- nb 

C 

For these two cases of moment definition, equation (53) becomes 

fh$ 

same nb: cbc = 4T 

Same yb: Cbc = 

with equations (55), (56), and (35), equation (51) becomes 

b _ ‘1(+- ‘b) 

Same yb: F - fl -Ckt 

C 

Solving for t leads to 

Same nb: t=&y -I ) - -- 

b/ bc 

(54) 

(55) 

(56) 

(57) 

(58) 

(59) 

Same yb: t = &- - 

fl$j- $1 

k b/b, (60) 

where k and fl are given in equations (36) and (37), and in table 1, and 

flbnb/bc) is 

f& 'Ib) = ; [2 + (k qb)2] [1 -(k nb)21'- 2 $ qb cos-'(g nb) (61) 

C C C 

19 
When b/be is specified then t can be determined from either of equations 

(59) or (60) for a given value of nb. With this value of t, induced drag 

efficiency, e, is computed from equation (38), induced drag ratio from equation 

(52), ncp from equation (32), Cb from equation (35), and induced velocity is the 

wake from 

(62) 

When t is specified, such as for the condition of wingtip zero-slope 

loading of equation (29), then b/b, can be determined from either of equations 

(57) or (58). An iteration for b/be is required in equation (58) when t is 

specified. 

The data presented in tables 4 through 8 applies also to the case of same 

nb when comparing this data with that due to elliptic loading. This can be 

seen by noting that equations (46) and (59) are identical when Cbc/Cb = b/b,. 

From the relations of equations (51), (52), and (62), the data of tables 4 

through 8 can be written as 

'bc b RA -=-.-a 

Cb bc ' 2CL w 

Cb2/e (Cb'/e) 

C 

(63) 

which relations apply when comparing with elliptic loading which has the same 

value of qb, nbC = nb. 

Data for the case of same nb is tabulated as a function of b/b, in table 

9. A similar table for the case of same yb is presented in table 10 with 

corresponding spanwise loading values given in table 11. 

20 
MINIMIZED INDUCED DRAG OF NONPLANAR WINGS 

WITH EITHER THE CONSTRAINT OF LIFT OR OF LIFT 

PLUS BENDING MOMENT 

In the yz-plane or cross section plane the trailing vorticity system is 

assumed to analytically follow the downstream projection of the aft view geometry of the aircraft. Thus a staggered wing biplane will appear as two 

planar vorticity sheets separated by a vertical distance. The term nonplanar 

means that the vertical position of the vorticity sheet or wake is a function 

of the lateral coordinate. Nonplanar can also mean multiple independent 

vorticity systems which, however, can influence or interfere with each other. 

A flat straight vorticity sheet is the simple planar wing. 

The -general approach to a minimization solution is first to analytically 

determine the induced angles normal to the wake. These angles factored by 

the vorticity in the sheet form the induced drag integral. The vorticity in 

the wake is represented as a Fourier series in loading. Then the drag integral 

develops into an infinite series in terms of unknown Fourier loading coefficients 

and induced drag influence coefficients. Constraint conditions are specified 

on the Fourier loading coefficients. Then the induced drag series is minimized 

with respect to the Fourier loading coefficients, that is, partial derivatives 

are taken with respect to a given am. Solutions for the Fourier loading coefficients, an, are obtained from these minimization equations. Constraint 

conditions can be as simple as that for a given lift, then al is constant in 

the derivative process, or for a given rolling moment, then a2 is constant, or 

for wing bending moment with a given lift, then the induced drag series times 

the bending moment series in a, is minimized. Thus, exact analytical solutions 

in terms of induced drag influence coefficients can be attained which provide 

the spanwise loading for minimized induced drag, subject to specified constraint 

conditions, for any nonplanar wing shape or number of wings. 

Biplane and Wing in Ground-Effect Solutions 

The biplane configuration provides an example for a nonplanar minimization 

solution. The wing in ground-effect is the same but with opposite sign of the 

vorticity from the lower wing. Shown on the right are the trailing vorticity 

21 
sheets from wings 

downwash at sheet 

ticity sheets is 

W(YJ> = k 

a and b. The normal induced 

a, due to both trailing vor- 

'b 

a'2 2 dyl 

i 

bb/2 drb 

1 (Y-~1) dyl dyl 

+- 

-b Y-Y1 2lT (y-y$ + 22 

a/2 -bb/2 

(64) 

For the same wing span and vorticity strength, in trignometric coordinates, 

equation (64) becomes 

*(costs-cos$) G'(+l)d+L 

0 0 

(cos$q-cos$J2 + 52 (65) 

where G and cos$ = 0 are defined in equations (1) and (2), and 5 = 2Z/b. The 

Fourier series for G from equation (1) is 

~(4~) = ? an sin n$l, G'($,) = y n an cos 31 

n=l n=l 

The first integral is 

1 - 

I 

Tags n+l d@l = sin n+ 

IT cos~l-cos~ sin I$ 

0 

and the second integral is defined by 

(66) 

(67) 

where n = cos$ and In is evaluated in equations (B5) through (Bll) in appendix 

B. With equations (66) and (67) equation (65) can be reduced to 

QW 

= F 

n=l 

(w + In) n an (68) 

22 
The induced drag coefficient (eq. 13) becomes 

cDi = (1" awsin+d+ = $]In f ,an sin n+ ,*t l($~n~m+ I,,*) n*aneo 

Tr 

A sin$d@ = 2 Y t B ($ n = 1 n an2 + n = 1 an n*- 1 n* a,.,* In* sin$sin n@d$) 

0 

Define 

I 

l-r 

I l I nn* = T n* sin+sin n+ d+ 

0 

then with equation (4) for CL, equation (69) becomes 

TA CDi 1 a a an* 

cL 

2 = -= 

e 

Y n(J)2 + 2Y n 

n=l al 

Y n* -1 

n = 1 7 n*= 1 al nn* 

(70) 

(71) 

These lift and drag coefficients and aspect ratio are based on wing area and 

wing span of one wing only in this equal winged biplane. The Inn* 's represent 

induced drag influence coefficients due to a second wake vorticity sheet. The 

1 * integral of equation (70) is evaluated in equations (B25), (B26), (B27), 

(::9), and (B40) of.appendix B, also in table 12 for 5 = l/2. 

For symmetric spanwise loading only odd numbered n and n* apply, then 

equation (71) can be written 

1 

-=1+2111+n=3 e al H n(k)'+ ,'r 3(n Iln 

odd odd odd odd 

n* I an* 

nn* a 1 

(72) 

Equation (72) applies for a biplane with equal wings. Since Inn* denotes the 

influence of one wing on the other, then for a monoplane wing in ground effect 

the I's in equation (72) are all preceded by a negative sign. For elliptic 

23 
loading, an = 0 for n>l, then e-l = 1 + 2111 for the wing in a biplane, and 

e-l = 1 - 21 11 for the wing at h/b = s/4 above the ground. In ground effect 

the ground vortex images are in opposite sense to the vortices of the model. 

an/al coefficients for minimized induced -_ drag. - -- F - Objective is to minimize 

equation (72) for a given lift but without any other constraint. Following 

the procedure leading to equation (17), taking partial derivatives of l/e 

of equation (72) with respect to a3/al, a5/al, . . ., am/al, results in 

a(a - 2 m !L + 2(m Ilm + Iml aWe> - 

ml al 

)+2 Y n* Inn* 

n*= 3 

odd odd 

an* -= 

al 

2[m Ilm + Iml + m am + 

al 

? (m Inm + n I,,) $I (73) 

n=3 

odd 

where m is an odd integer greater than one, and the I's are from equation (70) 

or appendix B with corresponding subscript integers. For minimal induced drag, 

equation (73) is zero, then 

a N 

m Ilm + 'ml + m < + n z 3 (m Inm + n I,,) : = 0; m=3,5,7, . . .N (74) 

odd 

where N is an odd integer made large enough to insure accuracy. There are 

(N-1)/2 m-equations with (N-1)/2 an/al unknowns. A simultaneous solution of 

these equations leads to evaluation of the an/al ratios. As an example, for 

N=5, equation (74) appears as 

3113 + I31 

a3 a5 

+ 3(1 + 2133) a + (3153 + 5135) a = 0 

1 1 1 

5115 + I51 + (3153 + 5135) 2 + 5(1 + 2155) $ = 0 

1 

J 

k (75) 

For the wing in ground effect a negative sign must be inserted before the 

I values of equations (74) and (75). Examination of equation (74) shows that 

minimum induced drag is not attained with elliptic loading. For elliptic 

24 
loading, an = 0 for n>l, then equation (74) is not satisfied because m Ilm + 

I ml is not zero, except when c= m where I = 0. With the an/al ratios known, 

then T/e can be determined from equation (72), induced angle in the wake from 

equation (68), wing bending moment ratio from equation (11), ncp from equation 

(12), and spanwise loading by equation (5). 

an/al coefficients for minimized induced drag and wing bending moment. - -- 

The ratio Cb/e is minimized as in the development leading to equation (17). 

Then 

a tcb/e) a(l/e) 1 aCb 

a(a,/al)=Cba(a,/al)+e-a(a,/al)=o (76) 

where a(l/e)/a(a,/a,) is given in equation (73) and aCb/a(am/al) is given in 

equation (18). With these values, equation (76) becomes 

a 

m Ilm + 'ml + m < n i 3 N Cm I,, + n I,,) $ + m t fm = 0; 

odd 

m=1,3,5,. . .N (77) 

where t is a constant similar to that in equation (20), and fm is given in 

equation (10). Equation (77) is equal to equation (74) plus the added term 

m t fm. The solution procedure for determining an/al is the same as that 

of equation (74). 

Antisymmetric loading a,/ap coefficients for minimized induced drag. - ------ 

The wing rolling moment coefficient is given by 

1 

LA 

'E = qSb 2 -1 

A 7T I I Grldn = 4 TA G sin 2+d$ = 8 a2 

0 

(78) 

where L is the wing rolling moment, and G is defined in equation (1). With 

this C, and a2 relation, equation (69) can be written as 

CDi ITA CD 

. 1 1 

= = 

=- 

T&a 2 32C,l 2 nn* er 

2 2 

n 

(79 > 

25 
For antisymmetric loading only even integer values of n and n* apply. Then 

1 1 y n(%)2 + 

- e = ’ + 2122 + 7 n = 4 a2 y (n12n 

r n=4 

+ 21n2) > + y an 

2 n=4a2 

even even even 

a n* Y n*Inn* a 

n*= 4 2 

(80) 

even 

where er is a drag efficiency factor for loading due to rolling. The partial 

derivative of equation (80) leads to the equations for minimized induced drag. 

Thus, 

aWe,) 

a(a,/a2) = m12m + 21m2 + m $ + ! (mInm + nImn) 2 = 0, 

n=4 2 

even 

m = 4,6,8 . . . N (81) 

which applies to the biplane. For wing in-ground effect the signs of the I’s, 

are changed. 

Example numerical solution with.<= l/2 of biplane and wing ground effect 

models. - For N = 9, equation (74) is written as 

a3 a5 a7 a9 3I13+I31+3(1+2I33)~(3I53+5I35)~(3I73+7I37)~(3I93+9I39)~ = 0 

1 1 1 

a3 a5 a7 a9 5~15+~51+(3I53+5I35)~5(1+2155)~(5175+7I57)~(5Ig5+9I5g)~ = 0 

1 1 1 1 

a3 a5 a7 a9 

(82) 

7I17+I71+(3I73+7I37)~(5I75+7I57)~7(1+2177)~(71g7+9I79)~ = 0 

1 1 1 1 

a3 a5 a7 a9 911g+Ig1+(3Ig3f9I3g)~(5Ig5+9I5g)~(71g7+9I7g)~9(1+21gg)~ = 0 

1 1 1 1 ! 

where In,,* is given in equation (70) and in appendix B. Using the I,,,,* for 

5 = l/2 give n in table 12, equation (82) becomes 

26 
-.139103 + 3.348175 - a3 - .155810 --+ a5 .009726 - a7 - .004216 - a9 = 0 

al al al al 1 

-.004910 - .155810 - a3 + 5.179748 - a5 - .078668 - a7 + 

al al 

.010699 - a9 = 

al al 

0 

- a3 .078668 --+ a5 - a7 - a9 

t 

(83) 

.001489 + .009726 i 7.077076 - .044194 = 

al al al al 

0 

-.003196 - .004216 - a3 •t- .010699 - a5 - .044194 --I- a7 .- a9 = 0 

al al al 

9.008733 

al J 

The simultaneous solution of these four linear equations provides the Fourier 

loading coefficients. Convergence accuracy can be assessed by also doing the 

solution of the top three, top two, and top equations. These are 

a3 a5 a7 a9 -- 

al al al al 

4 eq. solution .041649 .002196 -.000241 .000370 

3 eq. solution .041649 .002197 -.000243 

2 eq. solution .041648 .002201 

1 eq. solution .041546 

From equation (5), the spanwise loading distribution is 

Cc ~ = 

'L'av 

: (sin+ + n F 3 Z$ sin n+) 

(84) 

(85) 

odd 

where an/al values are given in equation (84), and $ = COS-~~. This loading 

is that for minimum induced drag of a biplane wing with the wings vertically 

separated by 5 = l/2. This optimized loading is outboard relative to elliptic 

loading. 

With the 5 = l/2, In,,* values of table 12, the drag efficiency factor of 

equation (72) becomes 

27 
1 

- = 1 f .422074 + ; n (an)2 f C-.278206 - 

a3 

- .009820 3 + 

e n=3 al al al * 

002979 

odd 

a7 a3 a3 

a + a (.348175 a - 

a5 a7 a9 a5 .274717 --+ .019452 - - .008431 -) + - 

11 1 al al al al 

(.179748 ; - 

a7 a9 a7 a7 .157336 - + .021397 -) + - (.077076 - - .088388 

al al al al 

a9 2 .008733 (-) a9 

al 

- .006391 -1 

al 

(86) 

where in the f signs, plus applies for the biplane an/al's and minus for the 

ground effect an/al's. 

The spanwise center of pressure of equation (12) simplifies to 

(87) 

With the an/al values of equation (84) used in equations (86) and (87), 

the 4 through 1 equation solutions compare as follows: 

4 eq. 3 eq. 2 eq. 1 eq. elliptic 

sol. sol. sol. sol. loading 

1 

e 1.416271 1.416272 1.416273 1.416294 1.422074 (88) 

ncp 

.434873 .434879 .434885 .434993 .424413 

which shows 0.4080 percent less induced drag than that with elliptic loading. 

The minimum induced drag and root bending moment solution for the biplane 

is obtained from the simultaneous solution of the equations of equation (77). 

That is, to add mtf, to the left side of equations (82) and (83). For the 

wing root case, nb = 0, and using equation (lOa) 

m=3 m=5 m=7 m=9 

mtfm = Lit 5 = - St = & = - + (8% 

28 
~~__ .

. 

I 

Choosing t = 1, these values are added to equation (83), and its solution 

for an/al results in 

a n=7 n=9 n 

n=3 n=5 

a = -.195876 = .031633 = -.012113 = -005932 

1 n-l 

and for odd n > 9 an 

a 

nLg 4(-l) 2 

1 n(n2-4) 

(90) 

(91) 

where for n > 9, an/al is obtained from equation (20). With equation (91) 

the following summation is: 

2 n (an)2 = 2.04061 x 1O-4 (92) 

n = 11 al 

odd 

Also 

;= (;) + t2 

eq. (82) 

Y n (aR)‘, ncp = (ncp) + 

n=l al eq. (83) 

odd 

;y r; t2 n f lln (>-I21 

1 

odd 

Then with the an/al values of equation (90) and t = 1 

1 

- = 1.613214, 

ncp 

= .372175 

e 

1 

- = 1.422074, .424413 

eC %pc= 

e 

5 = 1.134410, 'b 

n,,=O 

e c--- = b = .876764 

bc ncpc 

(93) 

(94) 

(95) 

With equations (49) and (50), for same lift and same wing root bending moment 

k = 1.140558, 

D. 

C 

+-- = -872038 

ic 

(96) 

29 
The spanwise loading distribution for these minimized conditions is equation 

(5) in which the an/al's are those given in equations (90) and (91). These 

results show the minimum induced drag for a given root bending moment due to 

t = 1 inboard loading of a wing in a biplane with 5 = l/2. Compared with 

the biplane wing with elliptic loading and same lift and wing root bending 

moment, it has 12.8 percent less induced drag, 14 percent greater wing span, 

and spanwise center of pressure is 12.3 percent less, that is loading is inboard. 

For ground effect solutions the Inn* 's in equation (82) change sign. This 

sign change can be taken into account in equation (83) by replacing the 

numbers 3.348175, 5.179748, 7.077076, and 9.008733 by -2.651825, -4.820252, 

-6.922924, and -8.991268, respectively. With these numbers substituted into 

equation (83), the 4 through 1 equation, simultaneous solutions result in 

a3 a5 a7 a9 - 

al a 1 al al 

4 eq. solution -.052494 .000675 .000136 -.000331 

3 eq. solution -.052495 .000676 .000134 

(97) 

2 eq. solution -.052495 .000678 

1 eq. solution -.052455 

where as before in equation (84), excellent convergence is demonstrated. The 

an'al 's of equation (97) inserted into equation (85) gives the optimized spanwise loading distribution for minimum induced drag of a wing in ground effect 

at h/b = c/4 = l/8 above ground. This optimized loading is inboard relative 

to elliptic loading. For elliptic loading, an/al = 0 for n > 1. 

With the an/al values of equation (97) used in equations (86) and (87), 

the 4 through 1 equation solutions compare as follows: 

4 eq. 

sol. 

3 eq. 

sol. 

2 eq. 

sol. 

1 eq. 

sol. 

elliptic 

loading 

1 e .570628 .570629 .570629 .570630 .577927 

(98) 

QcP -411014 .411008 .411004 .411056 .424413 

30 
which shows 1.2627 percent less induced drag than that with elliptic loading. 

The minimum induced drag and root bending moment solution for the wing 

in ground effect is obtained by the simultaneous solution of the equations of 

equation (83) which is changed by substituting the four numbers listed above 

equation (97), and by adding equation (89), t = 1, values with changed signs. 

The solution results in 

a n 

n=3 n=5 n=7 n=9 

q = 

-.357148 = .050279 = -.013734 = .005717 (99 

n-l 

and for odd n > 9 an 

al 

"Ip 4(-l) 2 

n(n2-4) 

These an/al values are inserted into equation (93) for t = 1, then 

ncp 

= .330000 

n = .424413 

cpc 

(100) 

(101) 

e 

-L = 1.429613, 

e 'b $=O 11 

'bc 

_ = = .777544 

ncpc 

(102) 

With equations (49) and (50), for same lift and same wing root bending moment 

k = 1.286101, 

D. 

+- = .864307 

ic 

(103) 

The spanwise loading distribution for these minimized conditions is equation 

(5) in which the an/al's are those given in equations'(99) and (100). These 

results show the minimum induced drag for a given root bending moment due to 

t = 1 inboard loading of a wing in ground effect at h/b = c/4 = l/8 height. 

Compared with the wing in ground effect with elliptic loading and same lift 

and wing root bending moment, it has 13.6% less induced drag, 28.6% greater 

wing span, and spanwise center of pressure is 22.2% less. 

31 
The antisymmetric loading solution for minimum induced drag for the 

biplane with 5 = l/2 is obtained with equation (81) in which the I's are the 

even n and n* integer values of Inn* listed in table 12. Then for an N = 6 

solution 

-.158416 + 4.256811 - a4 

a2 

- .107350 - a6 = 

a2 

0 

.004993 - .107350 - a4 + 6.121520 - a6 = 

a2 a2 

0 

Then 2-equation and l-equation solutions are 

2 eq. sol. -- a4 - 

a2 

-037211, - a6 = 

a2 

-.000163 

1 eq. sol. 3 = 

a2 

.037215 

(104) 

(105) 

The induced drag efficiency parameter of equation (80) with even integer 

values of n and n* for Inn* of table 12, becomes 

1 a4 2(-) 2 a6 2 a4 a6 

- = 1 t .215435 + 

e 

r a2 

+ 3(-) 2 c-.158416 -t .004993 -t 

a2 a2 a2 

a6 2 a4 a6 + .060760 ($ - .107350 ar] 

2 2 

(106) 

where in the + term, the + sign is for biplane, and the - sign is for ground 

effect values. Then with equation (105) for 

2 eq. sol., 1 = 1.212487; 1 eq. sol., L = 1.212487 (107) 

r er 

that is, the same to the sixth decimal place. This is 0.253% less than the 

elliptic loading equivalent sin 2$ loading value which is l/e, = 1.215435. 

The spanwise loading is given in equation (1) which for antisymmetric loading 

is 

SC a 

- = 

2ba2 Y n sin n$ 

n=2a2 

(108) 

even 

32 
where from equation (78), a2 = 8C,/aA. The spanwise loading resulting from 

inserting an/a2 values of equation (105) into equation (108), is outboard of 

the sin 2$ loading. This minimized induced drag loading is for a biplane with 

only antisymmetric loading. 

The antisymmetric loading for minimum induced drag for the wing in ground 

effect at h/b = c/4 = l/8 height from ground is obtained with equation (81) 

but with sign changes on the I's from table 12. 

.158416 + 3.743189 -t a4 *lo7350 - a6 = 

a2 a2 

0 

-.004993 + .107350 ?t 5.878480 - a6 = 

a2 a2 

0 

2 eq. sol. -- a4 - k = 

a2 

-.042368, 

a2 

.001623 

1 eq. sol. a4 -= 

a2 

-.042321 

(109) 

mm 

Using equation (106) 

1 2 eq. sol., e-- = .781205; 1 eq. sol., & = .781213 (111) 

r r 

which is 0.428% less than sin 24 loading value which is l/e, = .784565. The 

spanwise loading is equation (108) with the an/a2 values of equation (110). 

This loading is inboard of the sin 24 loading. 

Unsymmetric spanwise loading is a combination of symmetric and antisymmetric 

loading, that is, it includes both odd and even integer values of n and n* in 

the loading series. A solution for minimized induced drag starts with equation 

(69) 5 written as 

-frA 

'Di = 7j- ( Y na2+2 Y a 

n=l n n=l 

F n* Inn* an*) 

n n*= 1 

(112) 

33 
Then 

aCD 

---L 

aam 

9 [m am + Y 

n=l 

(mInm + "I,,) an], m = 3,4,5 . . . (113) 

For a minimized induced drag solution, equation (113) is equal to zero, then 

the equations are solved simultaneously to obtain a3, a4, . . . an. The coefficients a, and a2 are specified for given values of CL and CE of equation 

(4) and (78). For a single wing, the I's of equation (113) are zero, then 

a 

n 

= 0 for n > 2, which means that minimal induced drag is with a, sin @ and 

a2 sin 2+ loading combinations. 

Flow-Field Solution of a Flat Vorticity Sheet 

A spinoff of the biplane drag m inimization solution provides mathematics 1lY 

exact equations for downwash and sidewash about a vorticity sheet having arbitrary loading distribution. The downwash at point (y,z) due to the vorticity 

sheet is the second integral of equation (64), then 

i 

b'2 

Ws(YJ) = & 

(y-y1 > $ dY1 

l-s+ 

j-b/2 

where subscript s symbolizes, from sheet. Similarly, the sidewash is the 

horizontal component of the induced velocity, that is, z/(y-y,) times the 

integrant of equation (114). Thus 

-b/2 

(114) 

where the vs is positive in the y-direction. In terms of the dimensionless 

coordinates of equations (1) and (2), equations (114) and (115) become 

I 

Tr (cod, -q) cos n+l d+, 

(cos$q-r++ <2 

0 

016) 

34 
2 (T-M> = -5 n T , n a, i 

Tr 

cos n+l d9, 

(cos$q-T-l)2 + 52 

0 

(117) 

where an are the Fourier coefficients of loading, given in equation (1). The 

integrals of equations (116) and (117) are defined in equations (67) and (B12) 

of appendix B, then 

w 

s- y -- 

V n=l 

n In a 

n 

V 

-= 

Vs Y -' n = , n Jn an 

(118) 

(119) 

where for arbitrary values of n and 5, In is determined from equations (85) 

through (Bll), Jn from equations (Bll), (Bl3), (Bl4), and (B18) through (B20). 

It should be noted that a sign change in n influences the sign on In and Jn 

as shown in equations (B6) and (B16). 

For n = 1 the loading distribution is elliptic and equations (118) and 

(14) can be expressed as a ratio so that 

W 

S 

-= 

W 

W 

I1 (120) 

where ww is the downwash at the wake and is constant spanwise for elliptic 

loading. The elliptic loading solution for ws/ww is given in the literature, 

for example,solutions by conformal transformation in references 10 and 11, 

however, it appears as a function in elliptical coordinates. The general 

solutions for ws/V and v,/V such as those of equations (118) and (119) were 

not found in the literature. Computed values of equation (120) for various 

n and 5 correlate with values in the table on page 150 of reference 11. 

Cruciform Wing Solution 

The cruciform wing or x-wing is a nonplanar type configuration. An aft 

view of this wing and coordinate system is shown on the next page. The induced 

35 
velocity normal to the upper right wing 

panel due to the crossing wing with dr 

vorticity is -'he 

dwwv = 

dr 

- G sin A (121) -se S’ 

V 

‘s 

e 

where r2 = sv2 + sw2 - 2 svsw cosy, sin A = - $ (sw - sv cosr) which relations 

come from the cosine and sine formulas, respectively. The total normal velocity 

along the upper right wing panel is the sum of that induced by the wing on 

itself and equation (121). Thus for the x-wing the induced angle in the wake 

is 

I 

Tr 

aw(+) = + _G’(h) d$l + 1 

I 

“( cosycosd+-n) G'(&) dQL 

cosl$l-cos~ 7r (cos~l~~cos~)2 + 772sirG-f 

0 0 

(122) 

where the dimensionless terms are 

cos+ = q = S/Se, G = r/2seV (123) 

With the Fourier series for G in equation (1) and with equation (66), equation 

(122) simplifies to 

aW 

= y ( sin n@ 

n=l sin4 + P& n an 

where 

Pn = + 

I 

7 COSY COS+Q) cos nh dh 

(COS~~-~COSY)~ + ~2 sin% 

0 

This integral is evaluated in equations (C2) through (C8) in 

Pn integral is similar to the In integral of equation (69). 

can be derived directly from equation (68) 

by a transformation of the variables. From 

(124) 

(125) 

appendix C. The 

Equation (124) 

the components of n,, shown at the right, the 

relations for n,, and 5, are 

36 
OV = ‘I1 cosy = cosy COS+~, 5, = n1 siny = siny COS+~ (126) 

These values of n,, and 5, are substituted into equation (67) for COST, and 5, 

respectively, in the term which factors cos n$l. Then In of equation (67) 

becomes Pn as given in equation (125). This transformation method can be 

applied to any two or more spatial arrangements of separate planar vorticity 

sheets. 

The induced drag expression, derived in a similar manner to that of equation (69), is 

ITA 

'Di = 4 ( Y na2+2 Y a Y 

n=l n n = 1 n n*= 1 

n* P nn* an*) 

where 

P ,.,* sin@ sin n+ d$ 

(127) 

(128) 

where P,.,* is given in equation (125), but with n = n*, and n = cos$. The drag 

influence coefficients, P ,,n*, are developed into a recurrence formula in 

appendix C, e 

The lift 

wing from the 

lift upwards 

uations (C15) through (Cl9). 

coefficient given in equation (4) is with the lift normal to the 

aft view. For a cruciform wing or a tilted or rolled wing, the 

s a component of the value given in equation (4), then 

CL = +J a, cos z (129) 

where CL, A, and CD are based on one wing of the crossed wings of the cruciform, 

i 

i.e., the lift, induced drag, wing area, and width (-se to se), of one of the 

two equal wings. Thus 

(2se)2 D. 

S= 2secav,A= s , CL = &, CD 

i 

= -& (130) 

For symmetric spanwise loading only odd integers n and n* apply, then 

equation (127) with (129) becomes 

37 
a 

T (nPln + Pn,) $ 

n = 3 1 

odd 

a 

+ 2 Y J Y 

n = 3 al n*= 3 

n*P y 

nn* a 

odd odd 

(131) 

an/al coefficients for minimized induced drag. - Derivatives of equation 

(131) are obtained in the same procedure as that leading to equations (73) 

and (74). Then 

mPlm + Pm1 + m 

a 

J+ 

al 

! (mPnm + nP,,) $ = 0; m = 3,5,7, . . . N (132) 

n=3 

odd 

where N is an odd integer. A simultaneous solution of these (N - 1)/2 equations 

results in the evaluation of the an/a, ratios. These an values, through 

equation (1), define the spanwise loading distribution for minimum induced 

drag for a cruciform wing where the wings are angled Y to each other in the 

yz plane. With the an/a, ratios known, then l/e can be determined from equation (131), induced angle in the wake from equation (124), wing bending moment 

ratio from equation (ll), ncp from equation (12), and spanwise loading from 

equations (1) or (5). 

an/al coefficients for minimized induced drag and wing bending moment, 

and antisymmetric loading. - The added constraint of minimized wing bending 

moment is taken into account identically as done in the derivation of equation 

(77). For the cruciform wing solution the Inm 's of equation (77) are replaced 

by Pnm's. Or the equations are simply, mtf, added to the left side of equation 

(132). 

An antisymmetric loading solution can be obtained similarly as that leading to the equation (81), except the Inm's are replaced by Pnm's. With an/a2 

determined from equation (81), the drag efficiency factor is found from equation (80) in which Inm is substituted by Prim. 

38 
-- 

Cruciform wing with y = 90". - In a cruciform wing when Y = 90" there is --- 

no induced normal velocity at one vorticity sheet due to the other. The 

solution thus simplifies to the case of a single planar wing banked at an 

angle v/2. For the constraint condition of lift, the loading is elliptical 

along the span, n = s/se. The CL is given in equation (129), then from equation (131), e-1 = 2. For constraint conditions of lift and bending moment 1 

the loading characteristics are the same as those of equations (30) through 

(39), except that since by equation (129), C,(Y) = CL cos (v/2), then CL is 

replaced by 2%CL in equations (35) and (38), then 

Cb = $ (f' - kt), em1 = 2 (1 + kt2) (133) 

When y = 90", the influence coefficients Pn and Pnm should be zero if 

there is no normal velocity induced from one vorticity sheet onto the other. 

Examination of equations (C2) and (C15) shows that when y = 90", the Pn and. 

P nrn are zero for odd integer values of n and m, that is, for symmetric spanwise loading. and thus 

the two wings 

For antisymmetrical loading, Pn and P,, have values 

are not aerodynamically independent of each other. 

Flow F 

The cruc 

ield at a y Banked Plane, Induced by a Flat Vorticity Sheet 

iform wing solution is also the solution for determining the 

flowed field about a banked plane, as 

shown at the right. At the polar coordinate point (n,y) the normal velocity 

to the y plane is given by the second 

-, 

/ 

term in equation (124). Thus . / / 

WhY) - y 

V n=l 

n Pn a 

n (134) 

The lateral velocity along the y plane is analytically similar to equation 

(121), except that it is the cosine component, thus 

dr dv = -=COS A 

39 
where by the sine formula, cos A = (s,siny)/r. In dimensionless parameters, 

with equation (l), and integrating 

;= -sin y ? n a 1 COS~~COS n+, d01 

n=l ri7r cos~l-~cosy)2 + rl2sirGy 035) 

Since 2 cos$, cos n$l = cos (n + 1)+1 + cos (n - 1)$1, then equation (135) 

reduces to 

v(rl,y) = _ sin y 

V 

y 

2 n=l n (Jn+l + Jn-l) an 036) 

where the Jn values are given in equations (Cll) through (C13). The Pn 

values of equation (134) are given in equations (C2) through (C8). Equations 

(134) and (136) are exact solution equations for predicting the normal and 

tangential induced velocities on a y-banked plane, induced by a flat vorticity 

sheet of arbitrary loading distribution, i.e., arbitrary an (see eq. 1). 

V-Wing 

The V-wing induced angle equation is similar to that of the cruciform wing, 

except the spanwise integration for velocity normal to the surface is made 

only over the upper wing panels. Then equation (122) becomes 

I 

T/2 

"w(+) = ; 

G'($l) d+l+ 1 COS~COS~~-~) G'&) d+l (137) 

0 

cos@l- COS$ Tr cos I$ plcosy) 2 +n%in+ 

By adding and subtracting the first integral w 

that the pertinent geometric parameter for the 

related to y by 

ith lim its IT/~ to IT and noting 

V-wing is the dihedral angle 

r=& 

then equation (137) becomes 

038) 

40 

I 

' 

Tr 

G'(d),) d$, 

0 

cos41 

- 1 (1 - cos 2r) 

- cosc$ Tr 

I v2 

(cos+,+,,) cos+, G'(+, > d91 

(cOs+l-n>L (cOs+l-ncos 2rJ2 + n2 sin22rJ 039) 
With the G series of equation (1) 

a&d = n z ‘I (% + Q,) n a,, (140) 

where 

I 

IT 

Q, = - $ sin2r (cos@,+,) cos$, cos n$, d9, 

(cos~1-~)L(cos+1-+os2r)2 +n2sin22rJ (141) 

T/2 

Using equation (140), the induced drag equation, derived similarly to 

that of equation (69), is 

CDi = p ( ? n a 2 + 2 y an y n* Q,, an*) 

n=l n n=l n*= 1 

where 

(142) 

l-r 

Q nn* 1, =- Q,* sin 4 sin n$ d$ (143) 

0 

where Q,, is given in equation (141) in which n is replaced by n*, and n = cos+. 

The lift coefficient is 

CL = $! al cos r (144) 

where CL, A, and CD are based on S = 2secav, A = (~s,)~/S, CL = L/qS, CD = 

i i 

Di/qS. For symmetric spanwise loading only odd integer n and n* apply, then 

equation (142) with (144) becomes 

- - & Cl + 2 Q,, + n E 3n($)2 + 2 n 1 3(n Q,, + Q,,,) $ + 

odd odd 

a 

2 Y "Y 

n = 3 q n*= 3 

(145) 

odd odd 

Recurrence formula for Qn and Qnn*. - To formulate a recurrence formula 

for the induced velocity influence coefficient it is simpler to work with Q, 

41 
in the form 

Q, = q,., - rn 

where 

(146) 

I 

IT 

qn = + 

T/2 

I 

IT 

1 

r =- 

n TT 

T/2 

(cos 2r coso,-0) cos n$, d@, 

(cos~l-~cos 2r)2 +q2sin22r (147) 

cos n$, dGL 

COS$l - n (148) 

The recurrence formula for q, and rn are derived in the same manner as detailed 

in appendices B and C. Then qn+2 = 

I 

-+ cos 2r - 2n, n=O + 4~ cos 2r q,+,-2(1+2n2)qn 

cos 2r + p n, n=l 

4 (yy cos F + + sin y), n > 1 

+ 4~ ~0s 2r 9n-1-9n-2 

1 

(‘49) 

Noting that qSn = q, and rWn = rn, then 

92 = -$ cos 2r - 77 + 4n cos 2r q, - (1 + 2n2) q. 

(150) 

43 = cos 2r + $ n + 40 cos 2r q2 - (3 + 4~~) q, + 411 cos 2r q. 

(152) 

The induced drag influence coefficient in terms of q and r, is equation 

(146) with n = n*, inserted into (143), then 

Q nn* = qnn* - rnn* (153) 

42 
where 

9 

1 7T =- 

nn* n 

I 

q,* sin+ sin n,+ d+ 

0 

IT 

r 

1 =- 

nn* TT 

I 

rn* sin+ sin n+ d+ 

0 

Then using the recurrence formulas of equations (149) and (151) 

qn,n*+2 = -$ cos 2r , n = 1, n*= 0 

1 

-- 2 ,n=2,n*=O 

i cos 2r , n = 1, n*= 1 

1 = Tr , n 2, n*= 1 

2 cos 2r n*r 

71 n*2-1 2 cos - ’ n=l,n*>l 

1 . n*r 7 sin -p 7rn n = 2, n*>l 

Noting that q, 

,-n 

-k = qnnk and rn,-n* = rnn*, then 

(154) 

055) 

+ 2cos2r(q n+l,n*+l+qn-l,n*+l > 

- 4q nn*-qn+2,n*-qn-2,n* 

+ 2cos2r(qn+l ,n*-l+qn-l ,n*-1 ) 

- q n,n*-2 

(156) 

-I 

9 n2 = i -5 cos 2r, n = 1 1 +2 cos 2r hn+l 3 ,+q,-, 

, 

,) - 2q,&~,+2 

, 

0 

-- ; ,n=2 

1 

- F&-2,0 (157) 

n = 1, n*= 0 

I 

+r n+l,n* +r n-l,n*-rn,n*-l (158) 

n = 1, n*>l 

[ 

+, 1 1 1 

ml = 

n=l + 2 rn+l,O + 2 rn-l,O (15% 

an/al coefficients for minimized induced drag. - In equation (145), since 

a, represents lift, this equation has the constraint condition of lift, 

that is, induced drag per unit lift. The partial derivative of equation (145) 

43 
with respect to am/al is identically similar to that of equation (73) and (74). 

In equation (74) Imn is replaced by Q,,, then 

mQ, m + Q,, + m $ + n p 3 (mQnm + “Q,,) $ = 0; m = 3,5,7. . .N (160) 

odd 

A simultaneous solution of these equations leads to evaluation of the an/al 

ratios. With the an/a, values known, then the spanwise loading is determined 

from equation (5). This is the loading distribution for minimum induced drag 

per unit lift of the V-wing. Other aerodynamic characteristics are l/e from 

equation (145), induced angle normal to the wake from equation (140), wing 

bending moment ratio, Cb cosr, from equation (ll), and ncp from equation (12). 

The solution for the constraint conditions of lift plus bending moment is 

obtained from equation (160) which is modified by adding the term mtf,, similar 

to that shown in equation (77). 

V-wing in ground effect. - When the apex of the V-wing touches ground the 

vorticity sheet system including ground images becomes that of the cruciform 

wing except the sense of the ground image vorticity is reversed. This can be 

taken into account by reversing the sign on the integrals in equation (122) 

when integrating along the lower panels of the sheet. Thus 

(cos 2r cosol - Q) G’( Q1) d$l 1 

rl cos 2r)L + q2 sin’2r -T 

(cos 2r COST, - 0) G’(@,) d@l 

(cosgq - I ( cost)1 -rl cos 2r)L + $ sin22r TT 0 

1 =- I G'&) d@, 1 I 7 

IT 

cos 2r cod1 

jcos@q - 

- d G’(%) d+l 

COSi#Jl - rj - T Q cos 2r)l + T-I~ sin22r 

+ 1. 

7r 71 

0 

0 I r/2 IT 

(cos 2r cod, - 17) G'(~Q) d+] 2 

Q cos 2r)L + qL sin22r - T 

I 

G'($) d@L 

(CO@1 - COS$l - n 

n/2 

(161) 

With G'($l) = l? n an cos n+l, comparing with equation (124), the first 

n=l 

integral of equation (161), has the term sin n+/sin$, the second integral, 

44 
comparing with equation (125), has the term -P,, and the third and fourth 

integrals, comparing with equations (146) through (148), has the term 2Q,. Then 

equation (161) becomes 

aw(0) = ? (w+2Qn-Pn)nan 

n = 1 

(162) 

The solution for the V-wing in ground effect follows the same procedure as 

that for the V-wing, but with the substitution of 29, - Pn for Q,, and 2Qnnk - 

Pnn* for Qnn* in equations (141), (142), (145), and (160). The influence 

coefficients of Pn and Pnn*, in which y = 2r, are given in appendix C. 

Planar-Wing Winglet Configuration 

An aft view of this configuration consists of a flat wing span with winglets at the tips, as shown below. 

The dihedral cant angle, y, of the winglet is the angle between the extended 

wing ar,d the winglet. Theoretical aerodynamic characteristics of the planarwinglet have been developed in reference 9. The minimization method objective 

is to determine the loading distribution along the span extent of the wing and 

up the winglet such that induced drag for a given lift, and induced drag for 

a given root bending moment, are minimum. 

Planar-wing wing!et aerodynamic chqracteristics from reference 9. - From - = _ _ - - i = -. -- 

equation (69) of reference 9, the induced drag coefficient of a planar-wing 

winglet configuration with symmetric loading is 

ITA 

'Di = it- ( ! n an2 + ! ,an ! n* 4 Dnn* a,,*) 

n=l n = n*= 1 

(163) 

odd odd 

45 
where 

I 

T/2 

D =2 nn* Ln*w sin+ sin n$ d$ 

0 

where for $. I @I s % 

I 

$0 

I 

7i 

L =- 1 

n*w HIT L- cos n*$l d$l + k 

0 r-+0 

and for 0 s $I g $. 

m-90 

1 

I 

IT 

L* =- nw ~II L- cos n*$l d$l + & 

$0 I 

(164) 

L, cos n*$l d$, (165) 

Lo cos n*$l d$l (166) 

where (noting that no = COSQJ~) =-Go 

L, = - [( (COS~,~q)) (cos~~+cos~+2~()) (1 - cosy) 

cos~~-cos~)2 -I- 2(COS$$qo) ( cos~~~o)(l - cosy>J (cos@q-COS$) (167) 

Lo = - 

(l-2 cos2y) cos$, + ;o;~s$;,2nfl (1-cosy) cosy 

[(cos~~-cos~) cosy - 2110 ( - )J2 + (cos$l+cos+)2 sin2y - 

1 (168) 

COS@l - cos$l 

For the winglet span position of $. = 51~/32, or no = .881921, values of 

D nn* had been computed for y = 90", and 75", and presented in table II of 

reference 9. This table of Dnn* has been reproduced as table 13 in the present 

report. 

From equations (55) and (59) of reference (9), the lift coefficient and wing 

root bending moment coefficient are 

TAa 

CL = + [l - f (1 - cosy) (T, + 

N 

c Tn a,)1 a 

n=3 1 

odd 

(169) 

46 
n+l -- 

C cos~)rl~ T, + COSY ho 

odd 

where 

T, = +. - no sin o. 

T = sin(n-l)Q _ si;(;+;)+Q 

n n-l 

('70) 

(171) 

Equations (163) and (169) are combined to form e-l = m ACD /C 2, 

iL 

then 

1-1-i D,,t y n(k)2t c (- D,,+$ D )a"' y !.!! N4 N4 an* 

n=3 al n=3 ' T In al nasal n*=3* 1 - Dnn* - 

al 

1 - odd odd odd odd -- 

e 

[1 - ; (1-cosy) (T,+ ! Tn 

(172) 

2 

n=3 

odd 

The ratio of equatidn (170) to (169) leads to 

n+l 

5-i ('-cosyhoT1+ ! 

2 

n=3 

[',;'- 1 -; (1-cosy)noTn] $ 

Cb = 

odd 

lT[l- p (1 

N 

-COSY> (T1+,z3 T,, %I al 

(173) 

odd 

These induced drag, lift, and bending moment coefficients and aspect ratio 

are based on the extended span of the planar-wing span plus winglet spans, 

and on surface area of the planar-wing area plus the winglet areas. 

an/a, coefficients for minimized induced drag for a given lift. - The 

partial derivative of equation (172) with respect to am/al is 

47 
We> = 1 N 

3a,/a, 

[1 -f (1-cosy) (T,+ ! Tn ,"" 

2m~+$Dml+m~Dlmt c 

-y)l 2 al n=3 

n=3 odd 

odd 

(m 4 Dnm + n tDmn) %-+ $ ( 

al 

(T, + ! Tn 

n=3 

odd 

> 

1 - cosy > Tm [’ - ; (1 - cosy) 

(174) 

where m = 3, 5, 7, . . .N. The minimization solution is with equation (174) 

equal to zero. That is, the quantity within the braces in equation (174) must 

equal zero. Then 

am 2m - + p Dml 4 

al + m T Dim + n=3 ! (m $ Dnm + n $ D,,) $ + & (1 - COSY> 

odd 

Tm [l - : (1 - cosy) (T, + ! Tn :)] In= 0, m=3,5,7,. . .N. (175) 

n=3 

odd 

Equation (175) involves the solution of (N-l)/2 nonlinear simultaneous equations. 

It is nonlinear because of e in the last term. A method of solution is to 

iterate e with estimates for e computed from equation (172), then equation 

(175) can be solved as linear simultaneous equations. With these new values 

of an/al, e is recomputed from equation (172), and the linear solution repeated. 

A first estimate for l/e is that for elliptic loading, then an/al = 0 for n > 1. 

Then from equation (172) for elliptic loading 

4 

1 - ' +TDll -- (176) 

e C’ - p (1 -cosy) T112 

48 
an/al coefficients for minimized induced drag and wing root bending 

moment. - For this minimization process, equation (76) must be satisfied, thus 

=0 (177) 

where the first term is given in equation (174). The partial derivative of 

equation (173) with respect to am/al, inserted into equation (177) leads to 

a N 

2m m + !m. D 4 

al r ml + t-n T Dim + nz3 (m $ D,.,,,, + n z D,,) 5 + & (1 - 

al 

cosr> T,,, 

odd 

m+l 

Cl -f(lN 

cosy) (T, + c Tn a an), + 8t 

2 

n=3 1 i - 

[(,;)- 4 - ; (1 - cosy) T,,, 

odd 

COS$~] [1 - i (1 - cosy) (T, + ! Tn 1" 

n=3 

-p + I (1 - cosy) T,,, ; - ; 

odd 

n+l 

(1 - cosy) T, COS$~ + y $;)_2, - ; (1 - 

n cosy) T, COS$~] = 0 

n=3 

odd 

m = 3, 5, 7, . . . N (178) 

where t is a constant similar to that developed in equation (20). The solution 

of equation (178) with respect to an/al is the same as that of equation (175), 

but with the addition of the t term to the linear solution as e is iterated. 

Example numerical solution with 40 = 51~/32. - With the Dnn* for y = go", 

and T,, of tables 13 and 14, inserted into equation (175), this equation sitnplifies to the numerical form, for m = 3, 5, and 7 

49 
.001665 + 2.821910 - a3 - .166305 - a5 - .01'6025 - a7 + .117625 k = 

al al al 

0 

-.002125 - .166305 - a3 + 4.908620 - a5 + .099925 - a7 + 

al al al 

.126340 k = 0 1 (179) 

.011935 - 

a3 a5 a7 .016025 - + .099925 - + 7.184340 - + .076970 ; = 0 

al al al J 

where 

a3 a5 a7 E = .952165 - .117625 - - .126340 - - .076970 - 

al al al 

Equation (172) becomes 

a3 1.03335 + .00333 - - 

a5 a7 a3 

al 

.00425 - + .02387 -+2.82191 (-)2 + 

al al al 

2 + 7.18434 (-)2 a7.33261 a3 a5 

- -- - 

al al al 

.03205 a3 a7 

-- + 

al al 

.19985 ag 21 

al al 

and equation (173) becomes 

a3 Cb = $ (.30020 + .11853 - - 

a5 a7 

al 

.13513 - - .03109 --> 

al al 

(180) 

081) 

(182) 

For elliptic loading, equations (176) or (181), and (182) become 

& = 1.139785, Cbc = .100357 (183) 

C 

A one equation solution is obtained from the first equation of equation (179), 

starting with the e value of equation (183) and iterating until e converges. 

Similarly for a two equation solution, but starting with the 1 - eq. solution 

of e, etc. The converged three through one equation solutions are listed as 

follows: 

50 
1 a3 a5 a7 

e a 1 a 1 a 1 'b 

3 eq. solution 1.127466 -.047592 -.028843 -.012991 .,098846 

2 eq. solution 1.128776 -.047542 -.029110 .098826 (184) 

1 eq. solution 1.133285 -.045823 .097987 

The three equation result is an accurate solution for the wing with a winglet 

at 90 = 5~/32, or o. = .881921, with y = 90 degrees. The loading is given by 

inserting these an/al values into equation (85). This loading is along the 

span extent of the wing and up the winglet to the winglet tip. The spanwise 

loading for minimum induced drag for a given lift is inboard from elliptic 

loading, particularly the loading is substantially less over the winglet as 

indicated by the negative values of an/al. Compared with elliptic loading 

values the minimized solution has a -1.081 percent smaller e-1, and a -1.506 

percent smaller Cb. The e-1 value of 1.127466 is with CL, CD , and A based on 

i 

the combined areas and spans of the planar wing and winglet. When based only 

on the planar wing area and span, this e-1 value is multiplied by no2, then 

e-l = .876926. This means that the addition of two 13.39 percent wing semispan winglets at y = 90 degrees, to a planar wing, can reduce e-1 from one or 

larger to .877. 

The equations for the minimized induced drag and bending moment solution 

are given in equation (178). These equations are the same as those of equation 

(175) but with an addition of a t term. For +. = 5m/32, y = 90 degrees, the t 

terms to add to equation (179) are 

a5 t (.592669 - .123477 - - .051120 

al 

a3 a7 t (-.362955 + .123477 - + .025891 -) 

al al 

a3 t (-.025992 + .051120 - - .025891 

al 

(185) 

For t = 1 in equation (185) and adding to equation (179), gives 

51 
--. ..-...- 

a3 a.5 a7 .594334 + 2.82191 7 - .289782 - - .067145 - + -117625 5 = 0 

al al al 1 

-.365080 - 

a3 a5 a7 .042828 - + 4.90862 - + -125816 - + .126340 k = 0 

al al al 

(186) 

a3 a5 a7 -.014057 + .035095 - + .074034 - + 7.18434 - + .076970 ; = 0 

al al al J 

where E and l/e are given in equations (180) and (181). In the iteration of e 

it is simpler to do a one equation solution of equation (186) for the first 

estimate of e. The three equation iterative solution of equation (186) gives 

a3 a5 -= 

al 

-.25932, - 

al 

= .03994, a7 

- = 

al 

-.01069 

(187) 

1= e 1.2887, Cb = .086015 I 

where Cb is determined from equation (182). The spanwise loading is ghven in 

equation (85) which with these an/al coefficients shows that this loading is 

strongly inboard of elliptic loading. 

Combining equations (183) and (187), the e and Cb ratios are 

e C 

- = 1.13065, 

e 

'b 

- = .85709 

'bc 

(188) 

From equations (49 ) and (50), for the same lift and wing root bending moment 

S 

e 'bc -= - = 1.16674 

S (18% 

ec 'b 

Di -= 

D. 

1c 

($)' > = .83058 

e 

(190) 

where subscript c denotes a configuration with elliptic loading, and se is the 

semispan of the planar wing plus the span length or height of the winglet. 

Equations (189) and (190) means that for +. = 5~/32, y = 90 degrees, if the 

wing with the inboard loading of equation (187), has a span length 16.67 percent longer than the wing of elliptic loading, then the induced drag will be 

17 percent less, for the same lift and the same wing root bending moment. 

52 
RESULTS AND DISCUSSION 

In the present theory, minimization solutions are obtained by expressing 

the spanwise loading in a Fourier series, determining the induced angle normal 

,to the wake, formulating the induced drag as a series of products of Fourier 

loading coefficients and induced drag influence coefficients, imposing constraint conditions, then minimizing the constraint conditioned induced drag with 

respect to the Fourier loading coefficients. This results in a set of unknown 

Fourier loading coefficients with an equal number of equations. The solution 

of these simultaneous equations leads to values of the Fourier loading coefficients and thus the loading for minimum induced drag. 

Planar Wings 

Constraint that either lift or rolling moment is specified. - Example 

solutions are shown at the beginning of the planar wing chapter. For the 

constraint that lift is specified the solution is 

4 

---=;sin+p CD 

cL2 =x) e=l, cbzG, rl 4 

'L'av i 

(191) 

where sin@ = (1 - n2)', i.e., elliptic, and fl is given in equations (9), 

(lOa), or (36). 

For the constraint condition that rolling moment is specified, the solution 

for a planar wing is 

CRC F sin 24, CD 

32CQ2 

=p,e,=l,n 3Tr - = 

'RCav i 7rA cp = 16 (192) 

where sin 24 = 2n(l - n2)', and C, is wing rolling moment coefficient. 

Constraints of lift and of wing bending moment about nb. - The minimization .------- ----- 

solution in terms of infinite series is presented in equations (21) through 

(26). The solution is in terms of a constant, t, which governs the quantity 

of the bending moment constraint. When t is zero the solution simplifies to 

that of equation (191). An upper limit for t is given in equation (29) at 

53 
which point the loading becomes negative at the wing tips when t > tl. Series 

summation methods are developed in appendix A so that equations (21) through 

(26) can also be represented in terms of closed functions. In terms of closed 

functions the planar wing solution of minimized induced drag for a given lift 

and a given wing bending moment about span station nb, is presented in equations 

(30) through (39). For the constraint condition of wing root bending moment, 

Qb = 0, then equations (30) through (38) simplify to equations (40) through 

(44). For ob = 0, equation (41) can be written as t = 6 - 9nq cp/2, which 

substituted into equations (40) and (44), results in a loading function and 

induced drag function which correlate exactly with those presented in reference 

2 which is a wing-root bending moment method. Equations (30) through (39) 

represent an independent solution as determined by the use of the present 

minimization theory, and developed originally without knowledge of the previous 

work of references 1 through 4. In terms of qb, equations (30) through (39) 

are unique in the literature. 

Numerical values of k, kl, fl, and k. for various values of qb are 

presented in tables 1 and 2. With these constants the loading distribution, 

spanwise center of pressure, induced angle in the wake, wing bending moment 

parameter, and induced drag efficiency factor, are given in equations (30), 

(32), (34), (35) and (38), respectively, for arbitrary values of t. From 

equation (30) it can be seen that the loading distribution is elliptic loading 

times the factor, 1 - kot. With positive values of t, and since in table 2 

k. is negative inboard and positive outboard on the span, then this optimum 

loading will be inboard of elliptic loading. The aerodynamic characteristics 

for specified values of t are given in tables 4 through 8. Greater detail is 

given in the section 'Numerical values of loading characteristics for various 

ob and t'. A measure of the efficiency of wings with inboard loading is to 

compare induced drag with a wing with elliptic loading having the same lift 

and same wing bending moment. This is done in the section entitled 'Comparison 

with Elliptic Loading'. Aerodynamic characteristics ratioed to those of the 

wing with elliptic loading, for same nb, are presented in table 9. Corresponding spanwise loadings are given in table 8. In addition the data of tables 4 

through 7 contain some comparisons, that is, the Cb2/e ratios are the same as 

54 
Di/Dic, and Cbc/Cb = b/b,. 

Tables 10 and 11 contain data for the condition that wing bending moments 

are equal about the same yb = ybc span station. Since the elliptic loading 

span is smaller, then when yb = ybc, the dimensionless coordinates become, 

Ob ' "bc. The smaller 0b favors a higher bending moment which means there 

must be a larger inboard shift of loading which, in turn, increases induced drag. 

The induced drag ratios of table 10 are higher than those of table 9 where 

n'b = nbc- Corresponding spanwise loading coefficients are presented in table 

11. When b/be = 1, the loading is elliptic, that is 4(1-n2)%/r. The loading 

for the asterisk marked lines in table 10 is wingtip zero-slope loading presented at the top of table 8. 

Derivation of t for minimal induced drag and wing span. - Shown in the 

data of table 9 is a relative decrease of induced drag with increase in wing 

span, for wings with constant lift and constant wing bending moment. A design 

objective could be to get the smallest induced drag for the least increase in 

span. That is, to minimize the product of the span ratio and the induced drag 

ratio. The product of equations (52) and (57) gives, for same nb 

bDi 1 -=-= (1 

bcDic e .b- 

+ kt2 

bC 

> (1 - +t, 

1 

(193) 

where e is from equation (38). Equation (193) is minimized by taking the 

partial derivative with respect to t and equating to zero. This produces 

f 

1 

1 3 

.- = 

k l-kt=2+(,-3k)% 

fl f12 

1 

1 + kt2 

2 

= + 

2f12 

- e = 3 -[l gk - (1 

(194) 

(195) 

(196) 

55 
u. 

<,= e bc 

1 (b)-2. = (1 + kt2) (1 _ k t)" (197) 

Equations (194) through (197) are valid when the span-drag product of equation 

(193) is minimum, and when both wings have the same ran. 

For the constraint condition of wing-root bending moment, qb = 0. Then 

from equations (36) and (37), or table 1, k = 2/9, and fl = 4/3. Using 

equations (193) through (197) gives 

t = -418861 

b b, = 1.075049 

cL2 e=-= 

vAC .962475 

Di 1 (1% 

Di b D. 

- = 

D. -898987, b ; = -966456 

1c C ic J 

Using the above t, the spanwise loading distribution, and %p' awv and C b' are 

determined from equations (40), (41), (42), and (43), respectively. The values 

of equation (198) show that for a wing with the loading of equation (40), with 

just a 7.5 percent increase in span, a 10.1 percent decrease in induced drag 

can be realized compared to a wing with elliptic loading, with both wings at 

same lift and same wing root bending moment. 

For the constraint condition of wing bending moment about span station, 

Ob = .2, then k = -176851, fl = .784747, which are obtained from equations (36) 

and (37), or from table 1. Using equations (193) through (197) gives 

t = -928705 e = 

cL2 

___ = .867654 rAC 

Di 

ww 

; = 1.264693 

L 

Di b D. 

-= 

D. .720582, = .911315 

1c 

b ; 

C ic 

Using the above t and ob, the spanwise loading distribution, and qcp, aw, and 

Cb, are determined from equations (30), (32), (34), and (35), respectively. The 

values of equation (199) show that for a wing with the loading of equation (30), 

with a 26.47 percent increase in span, a 27.94 percent decrease in induced drag 

can be realized compared to a wing with elliptic loading, with both wings at 

56 
the same lift and wing bending moment about span station nb = .2 = nbC. Data 

from table 9 for ob = . 2 show that with just a 7.5 percent increase in span, 

a 11.8 percent decrease in induced drag can be realized. 

Nonplanar Wings 

Example applications of the minimization theory are made to the configurations described as, biplane, wing in ground effect, cruciform wing, V-wing, 

V-wing in ground effect, and planar-wing winglet. The biplane solution is 

obtained from equation (74). This equation is valid for a biplane with wings 

of equal span and same spanwise loading distribution; is independent of stagger; 

and is only dependent on height between wings or gap since In,.,* is a function 

of 5. Equation (74) is also valid for the case of a planar wing in ground 

effect, provided that In,,, is substituted by -I,,, and that 5 = 4h/b, where h 

is height from ground. For 5 = l/2, the solution for minimum induced drag for 

a given lift, for the biplane, the loading characteristics are those given in 

equations (84) and (88), which show an outboard shift of loading compared with 

elliptic. Similarly, for 5 = l/2, or h/b = l/8, for the wing in ground effect, 

the loading characteristics are those given in equations (97) and (98), which 

show an inboard shift of loading compared with elliptic. For 5 = l/2, the 

solution for minimum induced drag for a given lift and a given wing root bending 

moment, for the biplane, the loading characteristics are given in equations (90) 

through (96), while for the wing in ground effect, in equations (99) through 

(103). Compared to a wing with elliptic loading, the biplane with a 14.1 percent greater span, has 12.8 percent less induced drag, while the wing in ground 

effect with 28.6 percent greater span, has 13.6 percent less induced drag. 

The minimization theory application to a cruciform wing is similar to 

that for the biplane. The primary difference is that the induced drag influence 

coefficient, Pnn*, replaces In,.,* of the biplane. Minimized induced drag solutions are obtained from either equation (132) or equation (132) plus mtf,, which 

are analogous to equations (74) and (77), respectively. The Pnn* integrals are 

evaluated in appendix C. 

The analyses for a V-wing is the same as that for the cruciform wing except 

that the spanwise integrations for the induced velocity influence coefficient 

57 
is for half the wing span. The induced drag influence coefficient is denoted 

by Qnn*- Minimized induced drag solutions are obtained from either equation 

(160) or equation (160) plus tntf,,,, which equations are analogous to equations 

(74) and (77) with Inn*, or equation (132) with Pnn*. When the pointed end of 

the V-wing touches ground, the vortex sheet plus ground image vorticity resembles in part the cruciform wing (see eq. 161). The solution for the V-wing in 

ground effect (V-wing apex touching ground) follows the same procedure as that 

for the V-wing, but with the substitution of 24, - Pn for Q,, and 2Qnn* - Pnn* 

for Q,,* in equations (141), (142), (145), and (160). The Pn and P,,, coefficients, in which y = 2r, are given in appendix C. 

The planar-wing winglet configuration influence coefficients are derived 

in reference 9 and reproduced in the present report in equations (164) through 

(168) and in table 13. The minimization solution is obtained from equations 

(175) for a given lift, and (178) for a given lift and a given wing-root bending moment. The numerical solution for +. = 5~/32 (winglets starting at ,,. = 

k.88192) with winglets at right angle to the wing, is presented in equation 

(184) for the constraint of a given lift. This optimized loading is inboard 

relative to elliptic loading, particularly the loading is less than elliptical 

over the winglet. Comparing the same planar wing with and without added winglets, the induced drag is 12.3 percent less for the wing with winglets. The 

solution with constraints of lift and wing-root bending moment is presented in 

equation (187). The results show that if a wing-winglet with the optimized 

inboard loading has a span length 16.7 percent longer than that of the wingwinglet with elliptic loading, then the induced drag will be 17 percent less, 

for the same lift and wing-root bending moment. 

Other Applications of the Analysis 

An exact solution for the flow field about a flat vorticity sheet or 

planar wake is presented in equations (118) and (119). These equations are 

the vertical velocity or downwash, and lateral velocity or sidewash, induced 

by a flat vorticity sheet of arbitrary spanwise vorticity or loading distribution. The loading distribution is represented in equation (1) in which the 

an are Fourier loading coefficients which can have arbitrary values. Similarly, 

58 
equations (134) and (136) are presented for predicting the normal and tangential 

induced velocities on a y-banked plane, induced by a flat vorticity sheet of 

arbitrary loading distribution. 

With equations (118) and (119), the induced velocities can be predicted 

at any point in space. These equations can thus be used to set up the minimization solution for such configurations as multi-wings, wings in formation 

flying in arbitrary pattern, and banked wings. The minimization equation for 

the V-wing in ground effect is given in equation (162). The V-wing with apex 

at the boundary of an open wind tunnel has an identical solution to the 

cruciform wing presented in the section Cruciform Wing. The biplane solution 

is also the solution for wing flying near a wake, that is, over or under a wake 

which has the same loading distribution as that of the wing. The minimum induced 

drag solution for a wing under an arbitrarily loaded wake, is obtained by following the procedure in the biplane section. Let the wake loading be 

G,= '? 

n=l 

bn sin n+ 

then 

s,= 

y ?.jL!YLna+ y 

n = 1 sin 4 n n=l In n bn 

With equation (201), l/e of the wing becomes 

L y 

a 

e 

n (-J)2 + 2 

a bn* 

n=l al cL 

y" y 

n = 1 al n*= 1 

n*I - nn* b 1 

(200) 

(201) 

(202) 

The an/al coefficients for minimized induced drag are obtained by taking the 

derivative of l/e with respect to am/al, and setting equal to zero. This 

solution gives 

an cLw 1 -z-pal 

? n* bn* -1 

'L n n*= 1 bl nn* (203) 

where the lift coefficient ratio is that of the wake loading to that of the 

wing. The values of equation (203) in equation (1) give the loading distribution 

59 
of the wing with minimum induced drag, flying at a < distance above or below 

a wake which has the loading of equation (200). When the wake loading distribution is elliptic, then bn = 0 for n > 1, and equation (203) reduces to 

a 

n cLw In1 -=--- - 

al cL n 

(204) 

which loading is outboard relative to elliptic loading. 

The following two sections contain other applications in greater detail. 

Two-dimensional th 

same, differing only by 

substitute the longitud 

-us (w> for vs (y,d. 

Flow Field Solution of a Thin Wing Chordwise Vortex Sheet 

in wing theory and wake theory are in principle the 

coordinate definitions. In equations (114) and (1 

inal coordinate x for y, y for dr/dyl, c for b, and 

Then 

151, 

, j-c (x-x, > y dx, 

ws(x3z) = &y Jo (x-x;)2 + z; (205) 

u$x,z) = k 

f 

c 

Y dxl 

0 (x-x,)2 + 22 (206) 

where y is circulation per unit length along the wing chord, related to total 

circulation and local lift coefficient by 

I 1 

L. = 

cv 

$dii=> 

C (207) 

0 

In terms of dimensionless coordinates, chordwise trigonometric coordinates, and 

chordwise loading represented by a cotangent term plus Fourier series, define 

Z 25. = < = ; (1 - case), c = R 

C 

~d!JL 

2q - ;nc P 

= 2A, cot 5 + 2 : 

n=l 

An sin ng 

(208) 

(209) 

With equations (208) and (209), equation (205) becomes 

60 

..- 
wsk,fd 1 

[case,-(l-25)] ? 

=- n=l 

de1 

V 

-~_----- .-_I- .- 

IT [cose1-(1-2~)JL + 4s22 

i Jo 

!I (210) 

:s 

;j 

,I? These integrals compare with the integrals of equations (67) or (Bl), then 

f 

Ws(yd = A0 (IO + 11) + ; ?’ 

n=l An (In-1 - In+l) (211) 

where I 

n = In (1 - 25, 2n). By a similar derivation, noting correspondence 

with the Jn integral of equation (B12), equation (206) becomes 

u&n) 

V = 2 [A0 (Jo + J,) + ; : 

n=l 

A,, (Jnwl - Jn+l)ln (212) 

where Jn = Jn (1 - 25 , 2~). These I, and Jn induced velocity influence coefficients are evaluated by substituting 

l-j = 1 - 25, and c = 20 (213) 

into equations (B7) through (Bll), and into equations (B13), (B14) and (B18) 

through (B20). With these rl and r, substitutions the p and r values of In and 

Jn becomes (from eq. Bll) 

p = 4(52 + nq, r = 4[(1 - <)2 + nq (214) 

The dimensionless coordinate 5 is measured from the leading edge of the chordwise vortex sheet, and R is measured from the sheet. 

Equations (211) and (212) represent an exact analytical solution for 

induced downwash and induced longitudinal velocity, due to arbitrary chordwise 

loading as represented in equation (209). The total longitudinal velocity is 

U u,k 9n> 

-=l+ V v (215) 

where V is freestream velocity. 

61 
Formation Flying of Wings with Wingtips Linked 

The objective is to link wings together and determine the spanwise loading 

such that induced drag is minimum with.the constraint conditions that the bending moments at the wing connections be zero. This is because the term linked 

here means hinged and cannot support a 

bending moment. A three wing symmetrical combination shown here, has wings -: 

I I I I I 

-nb 0 

spanning the distances -1 to -Qb3 'b ' 

'Qb 

to Qb' and ,,b to 1. The constraint 'cpb 

condition when no lift transfers through the link is that the center of pressure 

of the outer wing loading, n cpb' be at the midpoint of the ,,b to 1 span. 

The minimization of induced drag for a given lift and wing bending moment 

has already been developed in the present planar wing chapter. These results are 

valid for an arbitrary value of t. For this linked wing problem, t must be 

determined such that the constraint condition on qcpb is satisfied. The span 

position, qcpb, is the sum of qb and the ratio of bending moment about nb, to 

wing lift from nb to 1. Then 

2cb 

%pb = Qb + L /L (216) 

b 

where Lb/L is the fraction of lift of the outer wing compared to the total lift 

of all the wings. Using the relation an/al = -tf, from equation (20), this 

ratio is 

fl 

Lb _ 

I 'b 

Gh 

-- 

L 

I 

1 

Gdrl 

= $ 0-q - t n r 3hnfn) 

odd 

-l 

where 

hl = +b - sin $b cos @by $-, = cos-l qb 

(217) 

(218) 

(219) h _ Sin (n-l)@b sin (n+l)+b 

n n-l - n+l 

62 
The constraint condition is that 11 @, = (l + $i2* Then equation (216), with 

equation (24) for cb' and equation (217), leads to 

fl - (1 - 'I,,) hl 

t - k - (1 - ,-,b) m hf (220) 

n=C3"" 

odd 

where fl, f,, and k are in equations (9), (lo), and (37), respectively, or in 

tables 1 and 3. 

Equation (220) represents the minimized induced drag solution for three 

symmetrically linked wings with arbitrary outer wing span. With t determined 

for a given ob, the three-wing aerodynamic characteristics are evaluated from 

equations (30) through (38), and the lift ratio from equation (217). In the 

three-wing problem when lift is able or allowed to transfer through the links, 

then ncpb can be reduced until the net three-wing optimum loading is elliptical 

and net e is unity. 

For the case of qb = 0, the solution reduces to a linked two-wing configuration. When qb = 0, then k = 2/9, kl = l/6, fl = 4/3, h, = ~/2, h, = 0 for 

n > 1, and equation (220) becomes 

t= - + (3lT - 8) = -1.068583 (221) 

then the two-wing aerodynamic characteristics from equations (40) through (44), 

and (217) become 

cRc 

r = : [(4 - n)(l - r12)' + (3~ - 8)r,2 cash-l 

1 

%p = 2 

TA 

T "w = -(3v9) + p (31~8) 

= -.42478 + 3.35705117 

e = [l + i (3~-8)~]-l = .79761 

I 

Thus, two equal wings flying linked together with the optimal net spanwise 

loading in equation (222), with no rolling moments will have 0.6269 as much 

63 
induced drag as when the wings are flying separately with elliptic loading. 

CONCLUSIONS 

An analytical minimization theory has been developed which is applicable 

to complex aircraft nonplanar configurations, with simple or complex constraint 

conditions. The application of the theory to many configurations and flow 

conditions have proven to be not excessively complicated and show that the 

analyses remain mathematically manageable and viable. This is because the 

solution method is based on a Fourier series representation which, in effect, 

breaks up ultra complex expressions into a sequence of analytically workable 

terms. The solution remains analytically exact since a Fourier series can be 

carried out to convergence, or final terms can be summed. In the planar wing 

minimization solution, the spanwise loading distribution is determined for which 

the induced drag is minimum for a given lift and wing bending moment about a 

given span station. The planar wing solution aerodynamic characteristics are 

given in equations (30) through (38). These equations are unique. They also 

provide the solution for three wingtip-linked wings. Minimization solutions 

are made for nonplanar type configurations. These show the optimum spanwise 

loading for different constraint conditions. In many cases the reduction of 

induced drag can be substantial, particularly with either the constraint of 

wing bending moment, ground effect, or formation flying. Some of the solutions 

show marginal reductions in induced drag, relative to that due to elliptic 

loading, but do provide the analytical loading for the minimum. 

For two of the simplified cases or conditions, previous theoretical solutions had been obtained by different analyses, which compare exactly with the 

present results. These include the planar wing with constraints of lift and 

wing-root bending moment (nb = 0), and the flow field of a planar vorticity 

sheet with elliptic loading distribution. This minimization theory has proved 

to be a useful analytical tool for obtaining the exemplary minimization solutions 

in the present report. The method can be applied to many other problems of 

interest, such as, complex aircraft configurations, wings with arbitrary number 

of structural or performance constraint conditions, and optimized pattern for 

wings in flying formation. 

64 
Vought Corporation Hampton Technical Center 

Hampton, Virginia 23666 

January 22, 1979 

65 
APPENDIX A 

'MATHEMATICS OF TRIGONOMETRIC SUMMATIONS 

Algebraic Series 

Need to evaluate series summations of series with terms of the type 

Z” 

n (n + a) (n + b) . . . (n + m) (Al ) 

where n are odd numbered integers and m are arbitrary numbers. This family 

of infinite series begins with the series of tanh -lZ, thus 

1+z z+23+25+ 

co 

tanh -1Z = $ ln ~ = Z" 

1 - z 3 5 *-- =Z+nE3n 

odd 

then 

m Z n+a 

nZ3-= -Z a+1 + Za tanh -lZ 

odd 

Similarly 

n-l 

tan -1z=z++p.. . =ztn13 (-1) 2 z" 

n 

odd 

then 

n-l 

00 

n:3 

(-1) 2 Pa = _Za+l + Za tan -lz 

n 

odd 

(A21 

(A31 

The l/n(n+a) terms summation is obtained by dividing equations (A2) and (A3) by 

Z, then integrating from 0 to Z, thus 

66 
m Zn+a+b a+b+l a+b b 

n g3n(n+a)-= - I: z a+1 d-- 

a 

tanh -lZ - 5 

I 

, faZ2 dZ; 

odd 0 ' 

a # 0 # - (odd integers) (A4) 

co Zn+b 

n=C3n2 

= [-Zbtl + Zb tanhZ-lZ dZi , a = 0 

odd I 0 

m Zn+b 

n=C3n(n-l) 

= Zb+' 

- $ (1 + Z) In (1 + Z) + $ (1 - Z) In (1 - Z), 

odd 

a = -1 

n p 3 e = [- ‘i’p’: + $2 tan -1Z _ $ 1 +.L!$ d , 

odd 

a # 0 # - (odd integers) 

n-l 

; (-1); zntb 

11 = 3 

= [-Zbtl + Zb tanZ-lZ ,,j , a=0 

odd 

Zf’ In (1 + Z2), 

a = -1 

(A5) 

uw 

(A7) 

uw 

(A% 

Apply equations (A4) and (A7) for various values of a, leads to 

67 
co Z n+b Zb+J Zb-J 

n=C3$iTiT =-2+- 

Zb-J 

2 (1 + Z) In (1 + Z) + 2 

odd 

(J - Z) In (J - Z) 

h z 

n+b Zb-J Zb+l Zb-2 

n=3n(n+2Tz2 

-- -- 

3 2 (1 - 22 

odd 

03 

Z n+b Zb+l Zb 

n =C 3ii7j7 2 = - - 2 (J - Z2) tanh -lZ 

odd 

n-l 

) tanh - lZ 

n p 3 q = -$!- + Zb f-an -1Z _ $2 ln 

odd 

(1 + 22) (A13) 

n-l 

(A’01 

(A”) 

(AJ2) 

NJ41 

odd 

n-l 

; (-1) 2 zntb 2 Zb+l Zb 

n=3n(n-2)- = - - 2 (1 + Z2) tan -lZ 

odd 

(Al5) 

In equations (A6), (A9), (AlO), and (Al3), Jet b = 0 then taking derivatives 

with respect to Z leads to 

co Z n+b 

n=C37= 

Zb+l 

- - In (1 2 - z2) (AJ6) 

odd 

co Zn+b Zb+l 

n$3n+l =--- 

Zb-l 

2 -1n (1 2 - Z2) (Al7) 

odd 

68 
n-J 

co 

n=C3 

(-1) 2 Zn+b = 

Zb+J 

n-J- - - In (J + Z2) 2 

odd 

n p 3 t-1 IyJZntb = _ $A + $I!- Jn (J + 2.2) 

odd 

Extension of equation (A2) leads to the two summations 

m Zn+b Zb+J 

nE377=--- 3 

Zb-l + Zbe2 tanh -lZ 

odd 

(AW 

(AJ 9) 

(A201 

co Zn+b 

n=C3-= Zbt2 tanh -lZ (A21) 

odd 

The integrations of equations (A4), (A6), (A7), and (A9) result in the following 

summations where a and b have values such that no denominator vanishes: 

m Z n+a+b+c a+b+c+ J 

n i 3 n(n+a)(n+a+b) 

= 

Za+b+c 

[- ( ’ )(a+b+J) t a(a+b) tanh -lz - a+1 

odd 

& I &T dZ)dZf 

0 

03 Z n+b+c 

n 5 3 n(n-J )(n+b) = 

b+c+l Zb+c 

c*z -- 

Zb+c+l 

b tanh -lZ - 2(b+l) 

odd 

In (1 - Z2) + & 

(A22 > 

(A23) 

69 
03 n+c 

nZj$iiIiJF=[$- ' (1 + Z) In (J + Z) - g (J - Z) In (J - Z) - 

odd 

ZctJ In (1-Z2) dZf 

2 I Z 0 

n z 3 e = [- &t&S t 2% tan -lZ - & 

odd 

I Z atb 

1+z2 dZ - $ 

n-J 

n I 3 *f = [ fi Zbtctl _ $2 tan-l Z _ $!$l$ 

odd 

‘n (J + Z2) + & JfZ2 dZ; 

I 

b Z 

n i 3 k$!?&$? = CZb+l Jn z _ Zb-+l j tanZ-lZ dZ Zf' 

odd 

In (J+Z2) dzf 

Z 0 

Ser ies of the type given i 

by using these equations or by 

evaluate the example 

(A241 

(A25) 

(A26) 

(A27 1 

n equations (A22) through (A27) can be determined 

solution with partial fractions. Thus to 

70 
m zn+l 

n.2 3 n(n-2)(n+l) 

odd 

then in equation (A22), a = -2, b = 3, c = 0, hence the summation is 

22 

b- - f tanh -lZ t % If$++j z2 +&-z-$ dZi 

thus 

co zn+l 

n 5 3 n(n-2)(ntl) = 3 z2 - L-I-L (1 - $- - $-f-) ln (1 _ Z) _ 6 

odd 

I+ (1 + $ - g) In (J t Z) (A28 > 

By partial fractions, 

1 

n(n-2)(ntJ) = 3& - 

1 

3n(n+I) 

then the summation is l/3 of the difference between equations (AJ2) and (AlO), 

and equals that given in equation (A28). 

Trigonometric Series 

The trigonometric series can be obtained from the algebraic series by 

letting Z be a complex function. That is 

Z = ei4 = cos$ + i sin 0 

zn = ,in+ = cos ng+ i sin n$ I 

(A29) 

71 
thus the cosine series is the real part and the sine series the imaginary part 

of the algebraic series. The real and the imaginary part of the functions in 

the algebraic series are listed as follows where absolute values are taken in 

the log terms: 

In (J + Z) = In 2 cos !+i$ 

In (1 - Z) = In 2 sin $ + i (- 

In (1 + Z2) = In 2 cos$ + i 4 

In (J - Z2) = In 2 sin@ + i (- 

--i? 2714 + 44 

tanh -lZ = $ tanh ml cos$ + i 

(A30) 

tan + i $ tanh ml sin@ J 

In addition since dZ/Z = id+, then with equation (A30) 

I tanh -lZ dZ = 

- $$ + i[ $ (ln 2 + 1 - In+) - + F (22m-1 - 1) Bm 

Z m=l m(2m +l)! 

(A31) 

where B, are Bernoulli's numbers, J/6, l/30, J/42, l/30, 5/66, 691/2730, 7/6, . . 

I tan -lZ 

Z dZ = $(; - 4) [ln 2 + 1 - In (F - +)] - 4 ; (22m-J - 1) Bm 

m=l m(2m+l)! 

(~32) 

(; - Jm+’ t it+ 

72 
I ~dZ=-$? + i[$ Jn 2 - m i J w 02mtJ] (A33) 

I 

v dZ = % (T - +) 9 t i [I$ (ln 2 - J + In 4) - 

m 22m-l B, 

m~lm(2m+l)!- @ 

2m+J 1 (A34) 

Bernoulli's number is related to the Zeta function by the relation 

r(2m)! 

Bn = 22m-l T2m+l d2d 

where c(2m) is the Zeta function given by 

m J G(2m) = - k 2 1 k2m 

(A35) 

(A36) 

For m = 1, 2, 3, c(2m) = ~r2/6, r4/90, n6/945, respectively. With equations 

(A35) and (A36) 

m 22m-l Bm a3 

m 2 ,m(2m+l)! +2mtJ = ."E 

($$2mt1 (A37) 

By partial fractions, J/m(2m+l) = l/m - 2/(2m+l), then the m-summation of 

equation (A37) is 

2- 2 tanh-l & - k In [J - (& I21 

73 
, also 

F Jn[J - (k)2] = In sin C+ - In + 

k=l 

thus the summation term of equation (A34) simplifies to 

o. 22m-l Bm 2m+l 

m 2 Jm(2m+l)’ 4 ln sin o d + + 4 - 4 Jn @ 

=I$1 t-t 4 - + In sin I+ + 2lT; P - k tanhk = lT 

1%) (A3B) 

Similarly, the summation terms of equations (A3J), (A32), and (A33) are 

2 

= $Jn$-mln----- 

tan $ 

2~; (St k 

k=l 

tanh-1 $ - 4k tanh-1 &) (A39 

3 the summation term of equation (A32) is the same as that of equation (A39) 

but with F - I$ replacing 4, also 

m i 122m-1m[~~~J;Jl) Bm $2m+l = 5 k I J[ 2 - k(J + 2 ) In (1 + 2) + 

odd 

k(’ - 2) In (J - g)] (A40) 

Example summations of trigonometric series. - With equations (A21), (A29), 

and (A30) 

74 
m Zntb Zb+2 

nE3"-2= tanh -lZ = [cos (bt2) + + i sin (bt2) $1 

odd 

(i tanh-1 cos I+ + i $ 9 

TV 

then since the cosine series is the real part and the sine series the imaginery 

part 

O3 w = k ~0s (bt2) 4 tanh.-1 

nE3 - 

cos I$ - $$-,- sin (b+2) 9 (A41 > 

odd 

O3 w = k sin (bt2) @ tanh-1 cos 4 t 

n23 - 

cos (b-+2) 4 (~42) 

odd 

where an alternative function for tanh-1 cos + is -In tan (+/2). 

With equations (Al2), (A29), and (A30) 

Z n+b 

ni3m = i cos (btl) 4 t i sin (b+J) $I - i (cos b 9 + i sin b 4) 

odd 

(1 - cos 2 4 - i sin 2 +)(itanh-l cos +I + i q-p 

~0 cos n+b 

n 2 3 

1Lf" = =,a - g!k [sin (btl) 4 tanh-1 ~0s $ t * 

n n-2 

odd 

cos (b+J) ~$1 (A43) 

75 
- sin n+b) 2 3 7+-f= n n-2 sin(b+J)$ 2 sine + 2[cos (b+l) + tanh-l cos 0 - 

n 

odd 

sin (b+l) $1 (A44) 

With equation (A28) divided by Z4, and equations (A29) and (A30) 

co Z n-Q % Z + 

n $ 3n(n-2)(n+l) = - :- 6 

-” (Z-l- $ _ $) Jn (J - Z) _ ’ 

b -+ + 22 

6 

odd 

(z-l t + - i$-) In (1 + Z) 

= 1 cos $ -f sin $ (sin 2 $ti $ sin +)[ln 2 sin a 2 + i 3 

(3 - $$-)I - i& cos $ (~08~ $ - i $- sin $j[ln 2 cos St 

Qil + 1 ; sin if 

then letting $ = $/2, the trigonometric summations become 

~13 cos(2n-l)+ 

: 3n(n-2)(n+l) s!?$!k- Y-r 671+ 

sin3 4 - !IL 3 sin 3 0 + sin2 + cos $ In 2 

n 

odd 

sin (p - + In2 cos I$ 

a sin(2n-l)+ 

n : 3n(n-2)(n+l) =9- Tr 7-r 2+ 

sin2 + cos 4 

odd 

In 2 cos 4 - y In 2 sin I+ 

(A45) 

f cos 3 4 + sin 4 cos2 4 

(A46 > 

76 
I - 

Spanwise Loading Distribution 

This loading function is given in equation (21) and contains the summation 

F f, sin n 4 

n = 3 

odd 

(A47) 

Substituting equation (JO) into equation (A47) results in 

y f, sin n I$ = 

; c sin(n-2)9b sin nf$ 

n=3 n=3 n(n-2) 

odd odd 

2 COS+b Sin(n-l)+b Sin n $J + 2 cos+b sin(n+l)+b sin n +I 

n(n-1) n(n+J) I (A48) 

The first term within the brackets, by trigonometric identities can be written 

as 

sin (n - 2)$b sin n $I = i cos [n $ - (n - 2)$b] - i cos [n 4 + (n - 2)$,,] 

= ; cos I/J, cos (n - 1) $-+i sin Qt sin (n - 1) Q--i cos $- cos (n - l)$, 

sin (n -1) Qt (A@) 

where JI+ and I+ are defined as 

lb, = $-, ++, $-=$,-$ (A50) 

Apply equation (A49) to equations (A43) and (A44) with b = -1, results in 

77 
n p ;w = $ [Cos $, - cos I)- t : sin (I++ - I$-1) t 

odd 

sin Gt sin $- (tanh-l cos $- - tanh-l cos IJ,)] (A51 ) 

The second term in equation (A48) is of the form obtained from equation 

(Al J ). In trigonometric summation it becomes 

~0 cos(n+b) _ cos(b-J) _ cos(b+l) 

n $ 3 +- 2 ' 3 '-q[&cos(b-l)$+ n n-+2 

odd 

sin (b - 1) QI tanh-l cos $1 

m sin(n+ 

n 5 pm 

odd 

cos (b - 1) + tanh-1 cos $1 

The sine product is expanded as before, but with b = 1, then the second term 

I&= 2 sin(b-l)$ _ sin(b+J)+ 

3 - y [ fi sin (b - 1) $ - 

(~52) 

(A53) 

summation of equation (A48) becomes 

; E 3w = $ [-cos $, + cos I)- t + sin Gj (j.~, t +-) 

odd 

sin i (Q, - $-) - F sin (Q, - IQ-1) + sin $+ sin $- (tanh-1 cos +- + 

tanh-1 co-s $+)I (A54) 

The third term in equation (A48) is of the form of equation (A6), then 

78 
odd 

= ~0s (b + 1) ,+ _ sin $ cos (b + i) 4 + $ sin (b + 1) $ 

-cos (b + i) 4 cos $ In 2 cos $-+ sin (b + i) $ sin $ In 2 sin $ (A55) 

n i 3w = sin (b+ 1) 4 - F$-$- sin $ sin (b + k) 4 - $ cos (b + 1) I+ 

odd 

-sin (b + i) I$ cos 8 In 2 cos $ - cos (b + k) 0 sin $ In 2 sin $ (A561 

The sine product is expanded in terms of n - i, then with b = -i in equations 

(A55) and (A56) the third term of equation (A48) becomes 

-2 cos$, n i 3w = cos2'b [(& _ $- _ $+) sin !$ 

odd 

QQ+ cos - 

2 

(lT 

e+ Q- Q+ Q- 

- l/l - Q,) sin 

2 

cos 

2 

+ 2 cos 

2-- 

cos 

2 

In 

cos 2 

Q+ cos- 2 

2 sin sin ++ Q2 

sin 

2 

In 2 

sin 5 

I (A571 

Similarly the fourth term in equation (A48) is developed as 

2 cos+bn i 3 Z-I!!- = COs2’b ~-2 sin (q,, + q,,> sin i (++ - a-1 - 

odd 

($-Q- 

++ 

- $+) sin gcos 2+ (r - $- ++ Q- 

- $+) sin 2 cos 2 + 2 

e+ @- 

dJ- cos - 

cos 2~0s z-ln 

cos ; 

$+ 

- 2 sin 2 sin > In 

sin p 

sin $= 

1 (A581 

79 
where $+ and I+- are defined in equation (A50). 

The desired summation of equation (A48) is obtained by adding together 

equations (A51), (A54), (A57), and (A58). Thus 

F 

n=3 

fn sin n+ = - v (1 + 2 cos2 4,) sin@ + i ( cos $, + cos $)2 

odd 

tanh-l sin +b sin 4 

1 + cos @b COS I$ + ; (cos (jb - cos $)2 tanh-l 

sin +b sin $ 

l- cos @b cos $I (A59) 

where from equations (2) and (6), n = cos 4, nb = cos 4b. Using hyperbolic 

function identities, equation (A59) in terms of n and nb becomes 

Y fn sin n$ = 

n=3 

- ; (1 + 2 nb2) (1 - Qb2)k (1 - n+ + ; (11, + n)2cosh-1 

odd 

-j$$f + ; (t,b - 0)' cash-l m uw 

The condition of wing bending moment taken about the midspan is nb = 0, then 

equation (A60) reduces to 

y 

n=3 

fn sin n$ = - ; (1 - n2)' + 02 cash-$ 

odd 

(A61 1 

When either equation (A59) or (A60) is substituted into equation (21) the loading 

distribution becomes a closed function for any value of span station nb about 

which bending moment is taken. 

80 
Combining equations (26) and (10) results in 

The k-Factor Summation 

4cosl$b [- sin(n-2)9bsin(n-l)+b _ sin(n+l)+bsin(n+2)@b 

n(n-2)(n-1) n(n+2)(n+l) 

+ sin(n-l)+bsin(n+2)@b, + 4cos2+ 

n(n-l)(n+2) b 

c 2sin(;-ln)~bs~:l(n+l)+b 

( - N 1 

si;;if;;? Q2j 

1 (A@) 

The summations are obtained by using equations (A22) through (A24) with the trignometric relations given in equations (A29) through (A34). An example the sixth 

term in equation (A62) has the form of the summation example of equation (A45). 

By trigonometric identity 

sin (n - 2) Ob sin (n + 1) $b = ; cos 3+b - ; cos (2n - 1) +b (A63) 

then the desired summation is i cos 3+b times equation (A45) in which 9 = 0, 

minus k times equation (A45) in which #I = I#~. Thus 

Q) sin(n-2)+hsin(n+l)$h _ @b 

- 6 sin 3$b + k sins+b - $ 1 

n%3 n(n-2)(n+l) sin2+b cos fj~~ - 2- 

odd 

sin2$b cos$bln sin$b + & cos3$b ln cos$bb (A64) 

This mathematical procedure is used to develop the other summations of equation 

(462). The results are listed as follows: 

81 
n 1 ,w = $. ($b - + sin 4$b) - i sin2$b - i sin2f+bc0s2+b In 

odd 

tan ob 

n i ,w =-t (+b - $- sin 4$b) + $ sin2$b - $ sin2$bcos2$b ln 

odd 

tan$b +36+ 7 COS 18 6$/, --- COS 4 2+b 

(A65) 

VW 

-2 ; sin(n-2)@hsin(n+2)$h = 

n=3 n(n-2)(n+2) - i (1 - 4 cos2Gb) sin2$b + sin2#bc0s2+b In 

odd 

tan$b (A67) 

sin2$bcos $b ln sin$b + cos3$b ln cos$b (A68) 

m sin(n-l)$bsin(n+2)@h = @b 

nf3 n(n-l)(n+2) - 6 sin 39b - b sin3+b + i sin2$bcos$b($- 

odd 

In sin$b)+ & COS~$~ In CO.S$~ (A691 

-2 ; sin(n-l)+hsin(n+l)@h 

n=3 n(n-l)(n+l) = sin2$b In sin+b - cos2$b ln cos$b (A70) 

odd 

82 
n i ,e= - y + i Qb - z sin 2$b - k sin2$b ln sin$b - i 

odd 

cos2+b In COS+~ (A7J) 

n E ,e= $ - $ $b + z sin 21$~ - t sinZ+b - i sin2+b In sin$b -i 

odd 

cOS~I+~ In cosjb (A72) 

Then by equation (A62) the k factor is the sum of equations (A65) through 

(~67), PIUS 4 ~0~4~ times the sum of equations (A68), (A64), and (A69), plus 

4 cos2+b times the sum of equations (A70) through (A72). Thus with an extensive 

reduction the k factor simplifies to 

k = $ (sin6+b 

Since nb = COS+~, 

- 3 sin20b cos2+b - 6 cos4+b ln cos2+b) (A73) 

k can be written as 

k =$(l-6 

The functiona 

k factor in terms 

nb 2+3n4+2ns-6 b b nb4 In qb2) (A74) 

1 behavior of k as nb -f 1 can be assessed by expanding the 

of 1 - nb' Then 

k = % (1 - nb)4 [l - ; (1 - qb) + & (1 - ,,b)2 + 36 n ; ,w 1 

odd 

(A75) 

83 
It can be seen that k decreases rapdily as nb approaches unity and remains 

positive. The summation terms are quite small. When an analytical summation 

of this series is made, equation (A75) converts back to equation (A74). 

Spanwise Center of Pressure, ncp 

The qcp of the spanwise loading distribution of equation (21) is given in 

equation (22) which contains the summation 

n-l n-l 

-3 y El& fn = y 

n=3 - n=3 

[ ??$k$kh _ S~~-h-&t% _ 

odd odd 

2 cos$bsin(n-l)$b + 2 cos@nb;ln(n+l)+b] 

n(n-1) (A76) 

These summations are obtained by expanding the n2 - 4 term in partial fractions, 

as 

-3 

n'-4 = 4 n-2 + n+2 

3 (IL 1) (A77) 

With equation (A77) the individual summations of equation (A76) appear as those 

of equations (A25) through (A27), which can be converted to trigonometric series 

by application of equations (A29) through (A34). The first summation by using 

equation (A25) with a = -2, b = c = 0, is 

n-l 

-n E 3w = - Imag. (- &- + $ tan -1Z + & tan -1Z _ i 

odd 

tan -lZ 

Z dZ) = k (2#b + sin 2$b) - t sinjb - + cos2$b tanti-' sin$b (A78) 

The other summations are as follows: 

84 
n T i&= - b sin$b - i sin$b cos2$b + t sin$b cos3$b - 

odd 

$ COS21#lb cos 2$b tanh-l sin+b (AW 

n-l 

B (-1) 2 sin(n+Z)&= 

n = 3 no(n+2) - &- sinOb + d 5 sin+b ~0~~4~ - $ sin$b ~0~34~ - 

odd 

i COS~C$~ cos 21$~ tanh-l sin+b VW 

-- 

n-F 3+ = - %b + i$ sirqb + + sin$b cos2+b - i sin+b 

odd 

cos$, - ; cos2L$b tanh-1 sin+b (A81 > 

n-l 

y (-1) 2 sin(n-l)+h- @b 

n=3 n(n-2)(n-1) 

--_- 

2 "4 sin+b 

odd 

(A82 1 

n-l -- 

-y (-1) 2 sin(n-l)$b _ Ob 

n=3 n(n-l)(n+2) 

---- 

6 1: sin+b - % sin$b ~0~~4~ + f sin$b coqb + 

odd 

f COS~+~ tanh-1 sin$b (A83) 

85 
_ 

I 

_. _._ _ --.-.-........ ..___. -.. ._.._ __._ _. ___ 

,-r 3* = - $ + $ sin@b + a sin+b cos2Gb - 3 sineb 

odd 

cos$, + i COS~+~ tanh-l sin+b 

n-l 

(A84) 

odd 

(A55) 

The fn summation term of equation (A76) is 3/4 times the sum of equations 

(A78) through (A81), plus 1.5 COS$~ times the sum of equations (A82) through 

(A85). Then, defining kl as follows: 

kl = -3n T ,',:';" f = 

n 

- & sin$b + 6 sin3$b + $ COS~$~ tanh-l sin$b 

odd 

(A=4 

In terms of nb = COS+~, kl is 

kl = ; (1 - ; nb2) (l- nb2)J5 + + qb4 cash-1 1 

/'lb/ 

(A87) 

Induced Downwash Angle in the Wake 

The equation for induced angle is given in equation (23) which contains 

the summation 

86 
1 B -=G,=3 nfn sin n+ =v 1 H [- sin(n-2)+b sin n+ + sin(n+2)+h sin n$ + 

SW n = 3 n-2 n+2 

odd odd 

2cos9b sin(n-l)+b sin n$ _ 2cos9b sin(n+l)+b sin n$] 

n-l n+l VW 

where f, is defined in equation (10). By trigonometric identities the general 

sine product is formulated as 

sin (n + c)+~ sin n+ = i cos cob cos n$ - k sin c~+~ sin n$- - + cos c$,, 

cos n++ + i sin c+~ sin n$+ (A891 

where JI- and $+ are defined in equation (A50)., With c = -2 in equation (A89), 

the first summation of equation (A88) can be evaluated by using equations (A41) 

and (A42) in which b = 0, and @ equals +- or Q+. Then 

-y sin(n-2)0b sin n+ = 

n=3 n-2 sin 24 - $ (tanh- kosqJ_ - tanh-1 

odd 

COS$+) cos 24J 

Equation (A20) can be made into trigonmetr 

of equations (A29) and (A30). Thus 

Zn+b 

y -= 

n=3 n+2 - ; cos (b + l)$ - cos (b - 1) 

(Ago) 

c series by using the relations 

-7-r 

4719 sin (b - 2)+ + i 

odd 

-I 

cos (b - 2)$ tanh-l cos$+i [- $ sin (b + l)$ - sin (b - l)$ + & 

cos (b - 2)$ + k sin (b - 2)+ tanh-1 cos+] (A91 ) 

87 
where the summations of cos (n + b)$ and sin (n + b)$ are given by the real 

part and the imaginary part, respectively, of equation (A91). With c = 2 in 

equation (A89) and using equation (A91), the second summation of equation 

(A88) becomes 

Y sin(n+2)+b sin r-i@ _ 4 

n=3 n+2 - 3 sin3$b sin+ - i (1 + 

l&) sin 2 $+ $ 

odd 

(tanh-l cos$- - tanh-l cos$+) cos 24 (A921 

In similar procedures, working with equations (16A), (17A), and (89A), the 

third and fourth summations of equation (88A) are evaluated as 

y sin(n-l)+b sin n+ = 

n=3 n-l - a $b sin+ + t (1 + h) sin+ + + cos$ In 

odd 

sin++ 

sin$ 

-Y 

n=3 

sin(n+l)$b sin n@ = sin~b 

n+l cosGb sin4 

odd 

sin+ - 

sin++ 

t cost$ In v SW- 

(A931 

3 +b sin+ + i (1 + 

l+T) 

(A94 > 

The required summation is the sum of equations (A90) and (A92), plus 

2cos$1~ times the sum of equations (A93) and (A94), then all divided by sin+, 

thus 

88 
1 -v tit? sin+ n = 3 

nfn sin n+ = $ (2 + cos2gb) sirQb - 2$, cos+b 

odd 

tcos$ - cos$,,) = f, - ; (1 + ,,=!T) (,j - nb) (A95 1 

where fl is defined in equation (9). Shown in equation (A95) is a linear 

variation with spanwise coordinate TJ for r-~ r qb, and is a constant with 

q for q 5 qb. That is, the induced downwash in the wake is constant inboard 

of $3 then varies linearly with n at span stations outboard of nb. 

89 
APPENDIX B 

EVALUATION OF In AND Inn* INTEGRALS 

Integral of Equation (67) 

The In integral is 

I 

IT 

In = ; 

(cos+~-~) cos n@l d+l 

0 

(cos$q-np + c2 (BJ > 

This integral was originally evaluated, for arbitrary integer n, in reference 

14 in terms of a recurrence formula of a related integral. Here a simpler 

In recurrence formula will be developed. From trigonometric identities 

cos(n + 2)$1 = 2 cos 2$1 cos n$l - cos(n - 2)41 032) 

then integrating as in equation (Bl) 

7-r 

I 2 

n+2 = YY I ( 

cm 

0 

cos+q-T$ + 52 I n-2 033) 

In the integrant, cos 2+1 can be divided by the denominator, and noting that 

2 COS~~COS n+l = cos (n + 1)+1 + cos (n - l)@l, then II 

I 2 

n+2 = YY I{ ~(cos+~-~) cos n$l + 2~(cos~,-n)Ccos(n+l)~,~ cos(n-l)M _ 

0 (cosgq-rJ2 + 52 

(l+2n2+2~2)(cos~1-n)cos n$L 

(COS$p+ + 52 

> 

d+l - In-2 (B4) 

Then the recurrence formula for In is 

I n+2 = [ 

2, n=l 

I I 

4n, n = 0 

- 

0, nf 1 0, nf 0 I + 4n In+, - 2(l+2n2+ 2~~) In + 4n 

I _ I (B5) 

n-l n-2 

Examination of equation (Bl) shows 

I = 

-n In' In(-0) = In(q) for odd n, In(-Q) = -In(q) for even n 036) 

90 
If‘ - 

;: 

that is, In is symmetric with n for odd n and antisymmetric for even n. Then 

from equation (B5) 

! I2 =-2q + 4q I, - (1 + 2$ + 252) IO (B7) 

I3 = 2 + 4r, I2 - (3 + 402 + 452) I, + 417 IO w 

where based on the work in reference 14, IO and I, are 

IO = (l-2) p+-(l+n)r+ 

[(pZt& - 41% 

I, = 1 - 

(-q+02+c2) p-‘- (q+q2+52) r-l” 

[ ( p%+rk)2 - 41% 

=1+ IO- 

<2 ( p-%+r-%) 

[(p%+r%)2 _ 43% 

(B9) 

@JO) 

where 

p = (1 - n)2 + 52 

_ 1 r = (1 + u)~ + c2 = p + 4n 

(BJJ > 

Equations (B5) through (Bll) gives the evaluation of the In integral of 

equation (Bl) for all values of n, 5, and n. Another method is, to define 

IT 

Jn = + 

i 

cos n$l d$l 

(cosc#q-'112 + c2 (B12) 

0 

From reference 14 

Jo = 

p+ + r+ 

[(p+Q5)2 - 414 

+ 

3, = P _ r-4 

[(p%r%)2 _ 41% 

0313) 

0314) 

91 
Using a trigonometric identity, equation (Bl) can be written 

In = k 

I 

r [cos(n+l)+l+cos(n-l)$,-2ncos nOl] d$l 

0 

(COS$1-+ + 52 

then 

In = ; (Jn+, - 2rl Jn + Jn-,) 0315) 

Since 

= J Jn; -n 

Jn(-‘1) = -J,(q) for odd n; J,(-q) = J,(Q) for even n (BW 

that is, Jn is antisymmetric with n for odd n and symmetric for even n. 

Then 

IO = J, - rl Jo 

Equation (B2) can be used to establish a recurrence formula for Jn in a manner 

similar to that for In. Then 

J n+2 = I 4, n=O 

0, n # 0 3 

+ 4n Jn+J -2( lt2n2+2<2)Jnt40 Jn-, -Jnw2 (Em 

then 

J2 = 2 + 4rl J, - (1+ 2n2 + 2~~) Jo (B’g) 

J3 = 417 J2 - (3 + 4q2 + 4$) J, + 4q Jo 

Combining equation (B15) for n = 1, and equation (Bl9) gives 

(B20) 

I, = J + oJ1 - ($ + c2) Jo = 1 + nIo - C2 Jo WI> 

Inserting equations (B13) and (B14) into equations (817) and (B21) leads to 

the IO and I, values of equations (B9) and (BlO). The Jn recurrence formula 

of equation (B18), together with equations (Bll), (Bl3), and (B14) provide 

an evaluation of the J, type of integral (eq. B12) for arbitrary n, 5, and n. 

92 
With Jn known, a ,second method, by applying (Bl5), is available for evaluating In. 

Integral of Equation (70) 

The Inn* integral is 

I =--. ’ *I nn* m 

I 

,,,sin+ sin n+ d$ (B22) 

:O 

where I,.,* is that in equation (Bl) with n = n*. Examination of equation 

(B22), taking into account the symmetric and antisymmetric characteristics 

Of In* in equation (B6), shows 

IOn* = O; I-n,n* = -Inn*; ln,-n* = Inn*; 

I 

n odd, n* even 

= 0; I = 

n even, n* odd 0 

which means that Inn* has a value only when n and n* are both 

n* are both even integers. 

odd, or n and 

A recurrence formula for I,,* can be made by making the 

equation (B22) on equation (B5). Then 

integrat ion of 

(B23) 

In,n*+2 = + r{ E: ,“: ; ;] - [; cos’: ;: ; ;] + 41,*+, cos+ - 

0 ‘I 

2(1 + 252 + 2 COSQ) In* + 4 In*-, cos+ - In*-2 sin+ sin n+ d+ (~24) 

Inserting the trigonometric identities 

2 sin n$ cos+ = sin (n + l)+ + sin (n - l)$ 

4 sin n+ cos2+ = 2 sin n$ + sin (n + 2)+ + sin (n - 2)+ 

into equation (B24), then the integrations can be made in terms of I-values with 

lower n* integers, which is a recurrence formula. The recurrence formula for 

I nn* is 

93 
I 1, n 1, rl* 1 

= = 

I n 

= 

= 2, n* 0 

= 

n,n*+2 O,n#l,n*#l I - [ 0, 1n # 2, n* # 0 1 + 21 n+l,n*+l + 

21 n-l,n*+l - 4(1 +c2) Inn* - In+2 n* - In-2 n* + 21n+1 n*-1 + 21n-1 n*-1 

, , 3 3 

- I n,n*-2 (~25) 

The Inn* terms for low values of n and n* are obtained from equation (B25) by 

taking into account the Inn* identity relations given in equation (B23). Thus 

letting n* be zero in equation (B25) results in 

I n2 = - i [ il i, 3 $3 + 21n+1 ,lf21n-1 ,1-2(1+52)1n()-~ ‘n+2,0+ ‘n-2,) 

'02 = '12 = In odd,2 = ' @W 

I22 = 

- ; + 2131+21 ,j-2(1+52)120-; I4o 

'42 = 2151+2131-2(1+~~)140-q 1 I 6o-i I2o I 

Letting n* be one in equation (B25) results in 

‘03 = ‘00 = ‘23 = In even,3 = o 

I13 = 1 + 2122 - 4 (1+52)1,1 - I31 + 2120 

I33 = 2142 + 2122 - (5+4c2)131 - I51 - Ill + 2140 + 2I2o 

J 

For higher values of n*, equation (B25) is used directly, without the bracketed 

terms since n* is greater than one. 

94 
With the recurrence formula of equation (B25) the In.,* terms can be 

evaluated to very large values of n*. However, as can be seen in equations 

(B26) and (B27), the recurrence formula starts with a low value of n*, 

namely n* = 0 and 1. Needed are evaluations of In0 and Inl. 

For n << 1, by binomial theory, the radicals of equations (B13) and (B14) 

can be expanded into a power series of n, then 

Jo = 

1 

lCl(1+C2)4 

[l + (; - r;2) (&)2 + ($ - 3r2 + C4) (&)4 + 

5 45 15 (16 - g-- c2 + 2 c2 - 59 (&-#I (B28) 

Jl = + c; - 2c2) (&-A3 t (; - ; 52 t 3r;4) ( 17 

1+52 

)5 t 

(k - Fc2+ 15 c4-4P) (&-$I (B29) 

The influence coefficients IO and I1 are obtained by combining equations (B28) 

and (B29) with equations (B17) and (B21). Thus 

13 

IO = 1 5, rlr 

r=l 

odd 

(B30) 

where 

(1+<2)3'2 ' 

-AL!-- (15 - 552 + r4) 

‘5 = tl+r;2j11/2 8 

(B31> 

-Id (3- <2) , 

<3 = (,+<2)7/2 2 

-Id (&7<2 ++ - <6) 

<7 = t1+c2j15/2 J 

and cg3 cl1 3 and q3 are determined from the simultaneous solution of three 

equations, at n = .8, .9, and .95 of 

ng5g + nlk, 1 + Gq3 = Ioh) - c’ 5,nr (B32) 

r=l 

odd 

95 
where I,(U) is given in equation (B9). The inverse of equation (B32) for 

T-, = .8, .9, and .95, is 

r9 29241 AI01 .8) _ 1785 .8y- 23104 AIO(.g) 

- 629 .gg + ___ 82944 3885 

'11 

13700 AI~( 08) 

= -- 357 .8S + 

61700 AIO('g) 46400 

629 .99 -- 777 (833) 

513 = 

8000 A10(-8) 40000 AIo( -9) 32000 .%) 

- 357 .8g -- 629 .99 + -- 777 AIo( .9P-- J 

where 

AIoh) = am - c’ 5 qr 

r=l r 

odd 

And for I, 

(B34) 

12 

11 = c 5,,lr 

r = 0 

even 

(B35) 

where 

co=l- Irl 

i/2' 

(l+r;2) (B36) 

c2 = -31 

2(1t<2)5'2 ' ‘6 =$+i-= 

(E - p <2 + ; <4) 

29241 A+ t-8) 

'8 

=- 23104 A+@) 

82944 

A+(.%) 

1785 .88 

-- 

629 .gt! + 3885 .95" 1 

(B37) 13700 ~1, t.8) 

510 =-357 .88 

61700 AI,(.g) 46400 

+ 629--l-v-- 777 

512 

8000 A11(s8) 

= - 357 .88 

40000 AI1 (-9) 

-- 629 .98 + 32000 777 

96 
where 

(B38) 

even 

For evaluating In0 and Inl, equations (B30) and (B35) are inserted into 

equation (B22). Since n = cos$, and sin4 sin n$ = 0 5 [cos(n-l)@ 

. - cos (n+l)$l, then 

7r 

I 1 13 

n0 = Z ' <r 

r=l 

i 

cosr+ [cos(n-l)$ - cos(n+l)+l d$ 

odd 0 

which integrates to 

I LL 

n0 = 4 c 

n= 

3 +a 

n= 

1, 2 4 1 ;,2 1 +& 

4 

;, 4 

$3 21 128' 

3 

16 ' 

27 

256' 

1 

32' 

1 

256' 

n= 

2 

4 

6 

8 

10 

I- 

+? 

..I 

33 n; 

256 ' 

165 

1024' 4 

55 

512' ' 

::6 ' 8 

$7 , 10 

i&y l2 

+ ti3- 429 

4 4096 ' 

143 

1024' 

429 

4096 ' 

13 

256 ' 

4:;6' 

1:24' 

7 

n= 

2 

4 

6 

8 

10 

12 

14 L 4096’ 

&, 4 

3 

32' 6 

' 8 ST' 

(B39) 

where the brackets indicate that a value exists at the cited n, but is zero 

for any other n integer. The ~r's for odd r are given in equations (B31) and 

(B33). 

Similarly, with equations (B35) and (B22) 

97 
7 n= 

64' 1 

7 

32' 3 

5 

F5 

1:8' 7 

1:8' ' 

+b 4 

21 n= 

256' ' 

45 

256' 3 

75 

512' 5 

35 

512' 7 

582' ' 

&, 17 

a 

tpt4 

33 "; 

512' 

297 

2048' 3 

275 

2048' 5 

77 

1024' 7 

27 

1024' ' 

&' 11 

. 

5 

32' ;= 

$9 3 

5 

32' 5 

1 

32' 7 

+ 

(B40) 

where 5,'s for even r are given in equations (B36) and (B37). 

; Numerical exam le of I * for 5 = l/2. - An application of the equations 

for evaluating Inn* leads to the following values: From equations (B31) and (B36) 

51 = -.357771 50 = .552786 

53 = -.286217 52 = -.429325 

55 = -.100748 

<4 

= -.228973 (B41) 

57 = .042497 56 = -.025645 

From equations (B9) and (BlO) 

Io(.8) = -.442322 Il(.8) = .199389 

Io(.9) = -.529048 I&9) = .100090 (B42) 

Io(.95) = -.569635 1,(.95) = .051389 

Using the C'S of equation (B41) 

98 
.- 

I 

c’ crnr i 5,rlr 

n r=l Q r=O 

odd even 

.8 -.456861 .8 .177508 

.9 -.569810 .9 .041175 

.95 -.633557 .95 -.040031 

(B43) 

By equations (B34) and (B38) the AIM and AI,(~) terms are given by the 

differences between the values of equations (B42) and (B43). Then with 

equations (B33) and (B37) 

59 = .075149 58 = .051299 

511 513 = = -.086553 .107226 510 512 = = -.105817 .191347 1 

(B44) 

With the cr values established in equations (B41) and (B44) the In0 

and In1 coefficients are determined from equations (B39) and (B40). These 

coefficients are listed in the first two columns of table 12. In table 12 

the I nn* for n* > 1 are determined from the recurrence formula of equations 

(B25) through (B27). 

99 
APPENDIX C 

EVALUATION OF Pn AND Pn,,* INTEGRALS 

Integral of Equation (125) 

The recurrence formula of equation (125) is obtained in a similar manner 

as done for In in appendix B. With equation (B2), equation (125) becomes 

P 2 

n+2 = YF 

cos 24, (cosycos@l-0) cos n$, d9, _ P 

cos+l-q cosy)2 + q2 sin2y n-2 (Cl) 

In the integrant, cos 2$1 can be divided by the denominator, and noting that 

2 COS$~ cos n$l = cos(n+7)$1+ cos(n-l)$l, equation (Cl) becomes 

(C2) 

which is the recurrence formula for Pn. Examination of equation (125) shows 

P = 

-n 'n' P,(-Q) = P,(o) for odd n, P,(-~1 = -P,(Q) for even n (c3) 

Then from equation (C2) 

P2 = -2q + 4r) cosy Pl - (1 + 2$) PO cc41 

P3 = 2 cosy + 4n cosy P2 - (3 + 402) Pl + 4r-j cosy PO (c5) 

where 

PO = cosy JI - rl Jo (Cc) 

Pl = cosy + n cosily J, - T-,~ cosy Jo (c7) 

where Jo and Jl are given in equations (813) and (B14), respectively, but with 

the parameters QCOSY for Q and nsiny for 5 in p and r of equation (Bll). Then 

p=l - 2 Tyzosy + $ 

r = 1 + 2 QCOSY + n2 1 

@3) 

100 
Equations (C6) and (C7) are derived from a parameter similarity with the Jo 

and J, integrals. Thus 

7r 

cosy 

po= Tr 

I 

cos$, d$ 

(cos$l-ncosv)21+ $sin2y - 1 

d$, 

o cos~l-~cos~)2 + $sitGy 

Kg) 

which comparing with the Jn integral of equation (B12) results in 

pO = COSY J, (rlcosy, nsiny) - T-I Jo (T-ICOSY, nsiny) 

that is, these J's are functions of ncosy and nsiny rather than the n and 5 

of equation (B12). Equation (C7) is derived in a similar fashion. 

In terms of the ncosy, nsiny parameters the Jn integral is IT 

Jn = f I ( 

cos n@l d@, 

cos+l-~cosy)2 + ~-2 sin2y 

0 

(cm 

Using the equation (B2) relation, a recurrence formula can be developed, thus 

Jn+2 = 1: : i i] + 40 cosy Jntl - 2(1 + 202) Jn + 4rl cosy Jn-, - Jnm2 

Since J-, = Jn, then 

J2 = 2 + 4r, cosy Jl - (1 + 2$) Jo (CW 

J3 = 47) cosy J2 - (3 + 4~-,~) J, + 4rl cosy Jo (C’3) 

where Jo and Jl are given in equations (B13) and (814) in which p and r are 

those in equation (C8). 

When n and 5 are replaced by ncosy and nsiny in equations (B18) through 

(B20), then equations (Cll) through (C13) are reproduced or checked. Equation 

(125) can be written as 

y [cos(n+l)$l + cos(n-l)$ll - ncos n$l 

> d$l 

(~0~~~-~cos~)2 + 02 sir+ 

101 
'n = ; (Jntl COSY - 2~ Jn + Jn-, cosr) (C14) 

Equation (C14) provides a second method (to eq. C2) for evaluating Pn. 

Integral of Equation (128) 

The recurrence formula for Pnn* is made by making the integration of 

equation (128) on equation (C2) in which n is replaced by n*. Then in a 

derivation similar to that leading to equation (B25), the recurrence formula 

is 

'n , n*+2 = 

'cosy, 

[ 

n = 1, n* = 1 

-1, n = 2, n* = 0 1 + 2COSY (pn+, 

, n*+l +P n-l, n*+l > - 

4pnn* - 't-j+2 n* 

, 

-P n 2 n* + 2 'Osy ('n+l, n*-1 + 'n-1 , n*-1 ) - 'n, n*-2 

- , 

(C15) 

where the brackets indicate that this term is zero for values of n and n* 

other than those listed. The relations of equation (B23) apply also to Pnn*. 

Then from equation (C15) 

P n2 = [- +, n=21 + 2 COSY Pn+, ,+Pn-, 1 1 - 2Pno - i Pnt2 o - i Pne2 o 9 ‘) , , 

‘02 = ‘12 = ‘n=odd,2 = ’ (Cl61 

1 P22 = - 7 + 2 cosy (P3, + Pl,) - 2P20 

- ; p40 

P n3 = Ccosw=l I+2 COSy(Pn+l 2+Pn-1 ,2)-5Pnl -Pn+2 1-Pn-2 ,+2 cOsy(Pntl o:Pn-l ,o) 

, , , , 

'03 = '23 = 'n=even,3 = ' (C17) 

p13 = cosy + 2 cosy P22 - 4Pll-P3, + 2 cosy P20 

Values of Pno and Pnl are needed to start this recurrence formula. From 

equations (128), (C6), and (C7) 

102 
i 

IT 

P 1 

no=; o (cosy Jl-Jo cos$) sin+ sin n$ d$ 

PO0 = P n=odd,O =0 (Cl81 

where the zero values result because the term inside the parenthesis is antisymmetric about $I = r/2, so when sin n$ is symmetric, the integral is zero. 

7r 

P nl = c; cosy, n=l] + & 

I 

(cos 2y J,-cosy Jocos.+) sin 2$ sin n$ d$ 

0 

PO1 = P =0 (CW 

n=even,O 

where as before the term 

grals can be evaluated by 

quadrature formula given 

r= 

inside the parenthesis is antisymmetric. These intethe expansion method used in appendix B, or by a 

in reference 15, which has 

(cm 

where M is an odd integer and $V = ~n/(M+l). The integrant of equation (C18) 

is symmetric about $I = n/2 when n is even, also it is zero at 4 = ~/2. Then 

with equation (C20), equation (C18) becomes 

n=e 

P n0 = 

where Jo and 

equation (C8 

M-l ven 2 - 

- : (cosy Jl - 

M+l p=l 

Jo COS+~) sin$ll sin n$V cc21 1 

J, of equations (B13) and (B14) are evaluated with the p and r of 

) at Q = ~0~4~. 

The integrant in equation (C19) is symmetric when n is odd, also it is 

zero at I$ = IT/~. Then with equation (C20),equation (C19) becomes 

n=odd M-l 

P = nl [i cosy, n=l] + jj& ; (cos 2y Jl - 

u=l 

COST Jo cos~~) sin 2$F sin W,, 

W2) 

where as before, $u = vT/(M+l), and Jo and Jl are evaluated with n = COS$~ 

in equation (C8). 

103 
REFERENCES 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

104 

Prandtl, Ludwig: Uber Tragfliigel Kleinsten Induzierten Widerstandes. 

Zeitschrift Flugtechnik und Motorluftschiffahrt, Vol. 24, Nov. 1933, 

pp. 305-306. 

Jones, Robert T.: The Spanwise Distribution of Lift for Minimum Induced 

Drag of Wings Having a Given Lift and a Given Bending Moment. NACA TN2249, Dec. 1950. 

Klein, Armin; and Viswanathan, Sathy P.: Minimum Induced Drag of Wings 

with Given Lift and Root-Bending Moment. Zeitschrift fuer Ange-wandte 

Mathematik und Physik, Vol. 24, Dec. 1973, pp. 886-892. 

Klein, Armin; and Viswanathan, Sathy P.: Approximate Solution for 

Minimum Induced Drag of Wings with Given Structural Weight. Journal of 

Aircraft, Vol. 12, No. 2, Feb 1975, pp. 124-126. 

Blackwell, Jr., James A. : Numerical Method to Calculate the Induced Drag 

or Optimum Loading for Arbitrary Non-Planar Aircraft. NASA SP-405, 

May 1976, pp. 49-70. 

Feitel, Winfried M. : Optimization and Design of Three-Dimensional Aerodynamic Configurations of Arbitrary Shape by a Vortex Lattice Method. 

NASA SP-405, May 1976, pp. 71-88. 

Heyson, Harry H .; Riebe, Gregory D .; and Fulton, Cynthia L.: Theoretical 

Parametric Study of the Relative Advantages of Winglets and Wing-Tip 

Extensions. NASA TM X-74003, Jan. 1977. 

Cone, Clarence D., Jr.: The Theory of Induced Lift and Minimum Induced 

Drag of Nonplanar Lifting Systems. NASA TR R-139, 1962. 

DeYoung, John: Nonplanar Wing Load-Line and Slender Wing Theory. NASA 

CR-2864, Aug. 1977. 

Glauert, Hermann: The Elements of Aerofoil and Airscrew Theory, 2nd 

ed., Cambridge University Press, New York, 1947, pp. 138-142, pp. 164-165. 

von Karman, Theodore;and Burgers, J. M.: General Aerodynamic Theory - 

Perfect Fluids, Vol. II of Aerodynamic Theory, div. E, W. F. Durand, 
ed., First Dover edition, Dover Publications, New York, 1973, pp. 146-151, 

pp. 171-177. 

12. DeYoung, J.: Wing Loading Theory Satisfying all Boundary Points. Ph.D. 

Dissertation, Dept. of Aerospace Engineering, The University of Texas at 

Arlington, Arlington, Texas, 1975, pp. 13-17. 

13. DeYoung, John: Spanwise Loading for Wings and Control Surfaces of Low 

Aspect Ratio. NACA TN-2011, Jan, 1950. 

14. DeYoung, John;and Barling, Jr., Walter H.x Prediction of Downwash Behind 

Swept-Wing Airplanes at Subsonic Speeds. NACA TN-3346, Jan. 1955, 

pp. 39-41. 

15. DeYoung, John;and Harper, Charles W.: Theoretical Symmetric Span Loading 

at Subsonic Speeds for Wings Having Arbitrary Planform. NACA Report 921, 

1948, p, 16. 

105 
TABLE 1. - CONSTANTS OF PLANAR WING 

MINIMIZATION SOLUTION 

I I 

kl f 1 

k 

fl 

0 219 

.05 .218893 . 

.lO .209570 . 

.15 .195126 . 

.20 .176851 . 

.25 .156042 . 

.30 .133952 . 

l/6 4/3 l/6 

65424 1.181253 .185306 

61760 1.039157 .201673 

55839 .907010 .215131 

47886 .784747 .225361 

38175 .672280 .232109 

27012 .569493 .235213 

.40 .090328 .101679 .392358 .230218 

.5 .053025 .074704 .251841 .210548 

.6 .025900 .048928 .145912 .177501 

:ll 

.009611 .026977 .071919 .133638 

.002196 .010978 .026398 .083197 

.9 .000157 .002159 .004719 .033267 

1.0 0 0 0 0 
0 -l/3 -. 239745 -.059810 .161042 .406024 .534748 .613552 213 

.05 -.325360 -.238882 -.060532 .159478 .403914 .532429 .611122 .664168 

.lO -.308364 -.236111 -.062646 .154815 .397606 .525492 .603852 .656692 

.15 -. 286239 -.230808 -.065990 .147144 .387163 .513998 .591800 .644294 

.2 -.261029 -.221316 -.070279 .136621 .372713 .498050 .575063 .627069 

.3 -.206572 -.183739 -.079716 .107996 .332454 .453424 .528138 .578723 

.4 -.152579 -.139061 -.083345 .071782 .278779 .393352 .464721 .513248 

.5 -.103773 -.096063 -.066443 .032844 .214672 .320444 .387268 .433013 

.6 -.063166 -.059116 -.044010 .001076 .144553 .238432 .299239 .341333 

.8 -.012386 -.011761 -.009496 -.003586 .020040 .071386 .113089 .144000 

1.0 0 0 0 0 0 0 0 0 

T 0 .2 

1- 

TABLE 2. - SPANWISE FUNCTION, k,, OF PLANAR WING 

MINIMIZATION SOLUTION 

.4 

-- 

I .6 .8 ji 

.9 .96 1 
TABLE 3. - FOURIER SERIES COEFFICIENTS, f, x 10 

OF THE SPANWISE LOADING OF EQUATION (21) 

'b 

\ 

n 

i 

7 

11" 

13 

15 

17 

19 

21 

23 

25 

27 

29 

31 

33 

35 

37 

39 

t; 

45 

47 

49 

51 

- 

A0 .05 .lO .15 .20 .40 .60 .80 

2.6667 2.6500 2.6005 2.5192 2.4080 1.7245 .8738 .2074 

-.3810 -.3710 -.3418 -.2951 -.2339 .0690 .2347 .1220 

.1270 .1199 .0996 .0684 .0303 -. 0966 -.0423 .0453 

-.0577 -.0515 -.0366 -.0150 .0090 .0444 -.0269 .0003 

.0311 .0266 .0147 -.0013 -.0166 -.0079 .0167 -.0105 

-.0186 -.0149 -.0053 .0063 .0151 -.0080 .0040 -.0047 

.0121 .0089 .OOll -.0072 -.0114 .0070 -.0077 .0021 

-.0083 -.0055 .0009 .0066 .0077 -.0056 .OOlO .0030 

.0059 .0035 -.0018 -.0055 -.0045 .0005 .0034 -.0012 

-.0044 -.0022 .0016 .0043 .0021 .0024 -.0019 -.0023 

.0033 .0014 -.0022 -..0031 -.0003 -.0028 -.OOll -.0008 

-.0026 -.0008 .0020 .0021 -.0007 .0015 .0016 .0005 

.0020 .0005 -.0018 -.0012 .0013 .OOOl 0 .0007 

-.0016 -.0002 .0016 .0006 -.0014 -.OOll -.OOlO .0003 

.0013 0 -.0013 -.OOOl .0013 .OOll .0005 -.0005 

-.OOll .OOOl .OOll -.0003 -.OOlO -.0005 .0005 -.0002 

.0009 -.0002 -.0009 .0005 .0007 -.0002 -.0005 .0002 

-.0008 .0002 .0007 -.0006 -.0003 .0006 -.OOOl .0003 

.0007 -.0002 -.0005 .0006 0 -.0005 .0004 0 

-.0006 .0003 .0003 -.0006 .0002 .0002 -.OOOl .0002 

.0005 -.0003 -.0002 .0005 -.0003 .0002 -.0003 -.OOOl 

-.0004 .0003 .OOOl -.0004 .0004 -.0004 .0002 .OOOl 

.0004 -.0003 0 .0003 -.0004 .0003 .OOOl .OOOl 

-.0003 .0003 -.OOOl -.0002 .0003 -.OOOl -.0002 0 

.0003 -.0002 .OOOl .OOOl -.0002 -.OOOl 0 -.OOOl 

- 

* 

* At n’b =l,f,=O. ' 
- 

:- 

TABLE 4. - AERODYNAMIC CHARACTERISTICS OF WING WITH 

WINGTIP ZERI)-SLOPE LOADING 

IF--- 

l 

I 'b 

ii--- 

I I I 

t=t, I 

eq. (29) ' 

I 

b 

IlA 

qy nW 

312 

1.50564 

1.52278 

1.55209 

1.59472 

1.65247 

1.72794 

1.94837 

2.30940 

2.92969 

4.11847 

6.94444 

18.11177 

m 

.31831 3 -1.71239 

.31871 

j 

2.77854 -1.71506 

.31987 2 58241 

.32176 2140776 

-1.72315 

-1.73687 

.32432 2.25145 -1 75652 

.32751 2.11092 -1178262 

.33127 1.98405 -1.81589 

.34033 1.76446 -1.90813 

.35119 1.58160 -2.04600 

.36358 1.42748 -2.25408 

.37726 1.29620 -2.58537 

.39206 1.18332 -3.18000 

.40782 1.08547 -4.60451 

.42441 1 -m 

I 

CbXIO j e 

i- 

.79577 I 213 

.67774 .66835 

.57293 .67296 

.48077 .68025 

.4uoo5 .68977 

.32979 .70121 

.26900 .71431 

.17218 .74466 

.10296 .77954 

-r 

Cb2 

e x102 

.94989 .84375 413 

.68727 .77779 1.38697 

.43785 .71342 1.44322 

.33979 .65224 1.50128 

.23202 .59496 1.56101 

.15510 .54193 1.62220 

.10130 .49323 1.68473 

.03981 .40838 1.81340 

.01360 .33359 1.94643 

.00380 .28158 2.08343 

.00077 .23511 2.22412 

.00009 .19722 2.36830 

. 00000 .16608 2.51588 

0 .14063 2.66667 

T cb2/e (cb2/p)c 'bc 

T 
TABLE 5. - AERODYNAMIC CHARACTERISTICS OF WING WITH 

(Cbc/Cb) = 1.1 LOADING 

'b 

aA 

'b2 cb2/e 

t 

Ty clw CbXIO e 

qcP e x1o2 0, 

eq. (46) Oin+, ll=l 

.Y .45077 6/11.385839347 1.72727 1.46843 .013681989 .7517696458 .959163798 .9919253921 .881093664 

.2 .40339 .39909 1.31656 .30272 .56771 .97203 .33157 .85023 

1; .386509488 .40358737 1.22011 1.15494 .370164160 .4119928384 .980383611 .1731308170 .8429833809 

:86 1.09270 .51216 .41378932 1.02885 1.07473 .4311334228 .0191010556 .99738325 .000371122 .823613206 

1.0 m .42441 1 -m 0 0 0 .82645 
TABLE 6. - AERODYNAMIC CHARACTERISTICS OF WING WITH 

(Cbc/Cb) = 1.2 LOADING 

'b t 

2CL aw C2 

ncP 

CbXIO e $x102 

Cb2/e 

eq. (46) Oi'li'lb 9=1 (c b 

.Y .82642 1 .3676853 2.33333 1.85873 -.80826 -.47787 .88419691 .3131887479 .9555354234 .7938484877 

.2 .73955 .37800 1.53036 -.27834 .52040 .91180 .29701 .76162 

.3 .70358 .38622 1.40353 -.15471 .37766 .93698 .15222 .74115 

4 .72395 .39317 1.28405 -.08568 .26019 .95480 .07090 .72732 

.6 .93896 .40491 1.13701 -.04293 .09676 .97768 .00958 .71030 

1:; 2.00328 m .424411503 1.05288 1 -m.20581 .01751 0 .99126 1 .00031 0 .7005769444 
'b eq.t(46) ncP 

.:5 1.34912 312 .318312969 

.lO 1.23963 .33931 

.15 1.16203 .34755 

.20 1.1093,7 .35479 

.25 1.07708 .36125 

.30 1.06287 .36712 

.40 1.08593 .37755 

:Z 1.40844 1.18738 .33677951 

:; 3.00492 1.87073 .410410299 

1:; 7.51495 m .417532441 

TABLE 7. - AERODYNAMIC CHARACTERISTICS OF WING WITH 

(Cbc/Cb) = 4/3 LOADING 

O<‘l<ll,, n=1 

'b2 

e x102 

Cb2/e 

(Cb2/e)c 

3 -1.71239 .79577 213 .94989 .84375 

2.59365 -1.43281 .70501 .71510 .69506 .78661 

2.23317 -1.21680 .62020 .75641 .50852 .74365 

2.05402 -1.04915 .54133 .79145 .37026 .71072 

1.87054 - .91751 .46836 .32126 .26710 .68492 

1.72410 - .31371 .40124 -84672 .19014 .66433 

1.60530 - .73207 .33989 .86857 .13301 .64762 

1.42607 - .62035 .23417 .90374 .06068 .62242 

1.29903 - .56610 .15031 .93044 .02428 .60455 

1.20551 - .56439 .08709 .95113 .00797 .59140 

1.13454 - .62858 .04292 .96746 .00190 .53142 

1.07932 - .80372 .01575 .98055 .00025 .57366 

1.03546 -1.32543 .00282 .99121 .00001 .56749 

1 -03 0 1 0 .56250 

?lA 

2c, aW T L CbXIO e 

I 
TABLE 8. - SPANWISE LOADING DISTRIBUTION, cec/CLcav, OF WING 

WITH SPECIFIED CONDITION OF LOADING* 

__-_ .---. - _-. _ ._. .- ._ -._. .-- ..3,. .._ - -_. 

1 Wingtip zero-slope loading 

-- 

nb 

- _--_i 

n 

\ t 

3/2 

1.52278 

1.59472 

1.94837 

2.92969 

6.94444 

-- _--- 

elliptic load. 

ing,either t=( 

or nb=l 

_ ..-.--- -_-I_-- 

.Y 

.2 

.4 

:86 

I I 

: 

0 .2 .4 .6 .8 .96 

1.69614 1.27164 

1.69605 1.27827 

1.68781 1.29773 

1.58552 1.35644 

1.46357 1.31740 

1.34940 1.24390 

1.24751 1.16694 

.77254 .29867 .10982 

.77346 .30140 .11038 

.79667 .30988 .11419 

.87613 .34900 .12965 

1.01538 .44042 .16731 

1.04396 .65763 .27986 

1.01859 .76394 

: 

.90986 

.37112 

1 .30325 

1 .65175 

; 

.50386 

.38276 

1.27324 

___---- 

1.90986 

1.75994 

1.64193 

1.48420 

1.38651 

1.32063 

_ - . . _ _ _ 

.02840 

.02869 

.02957 

.03371 

.04397 

.07653 

.35651 

(Cbc/Cb) = 4/3 loading 

0 

:: 

.4 

.6 

.8 

--- 

0 

.l 

:: 

.4 

.6 

.8 

-__- _ _-~ 

3/2 

1.23963 

1.10933 

1.08593 

1.40844 

3.00492 

-------“---- ---_- _ --.- --- 

1.69614 1.27164 

1.61265 1.25757 

1.55330 1.25792 

1.43590 1.27256 

1.35138 1.23928 

1.29160 1.20024 

-L- -~-_- -----_ 

.10982 

.19346 

.24836 

.31793 

.36362 

.43594 

.02840 

.08964 

.12908 

-17659 

.20625 

.23536 

(Cbc/Cb) = 1.2 loading 

--- --_ 

.--- 

(Cbc/Cb) = 1.1 loading 

-~- 

1.41065 

1.38029 

1.35839 

1.33611 

1.31602 

1.23529 

1.26335 

- -- - 

_.-- _ -- 

1.20501 

1.19990 

1.20003 

1.20290 

1.20535 

1.19325 

1.17905 

- .-~--_I 

. _ .---__ 

.92912 

.94751 

.96246 

.97608 

.98972 

1.01803 

1.02258 

6/11 

.45077 

.40339 

.38650 

.39488 

.51216 

1.09270 

-------- 

1.50474 

1.45022 

1.40731 

1.37489 

1.34995 

1.31443 

1.29047 

-_-- .-- 

.39311 .23720 

.42353 .25947 

.44349 .27381 

.45773 .28373 

.46879 .29108 

.48722 .30187 

.51170 .31245 

.59475 

.62702 

.64908 

.66578 

.67985 

.70739 

.74722 

-___ 

* Equation (63) and table 9 apply when comparing with elliptic loading with same 

nb values. 

113 
--.-. .- -.-- 

TABLE 9. - COMPARISON OF AERODYNAMIC CHARACTERISTICS DUE 

TO OPTIMIZED SPANWISE LOADING WITH THAT DUE TO 

ELLIPTIC LOADING OF EQUAL LIFT AND BENDING MOMENT 

FOR BENDING MOMENTS AT THE SAME nb = qbc NORMALIZED SPAN STATION 

'b 

= 

'bc 

0 

* 

.l 

* 

.2 

* 

.4 

* 

.6 

,14 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.25 

4/3 

1.60 

2.00 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.44322 

1.60 

2.00 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.35 

1.56101 

2.00 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.40 

1.81340 

2.00 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.60 

t 

eq. (59) 

-.31579 

0 

.28571 

.54545 

.78261 

1.00000 

1.20000 

3/2 

2.25000 

3 

-.26097 

0 

.23612 

.45077 

.64676 

.82642 

1.52278 

1.35945 

2.47926 

-.23354 

.:1130 

.40339 

.57873 

-73955 

1.15042 

1.59472 

2.21366 

-.22862 

0 

.20634 

.39488 

.56657 

.72395 

1.24106 

1.94337 

2.17135 

-.29651 

.:6827 

.51216 

.73434 

.93896 

2.11266 

qcP 

.44675 

.42441 

.40420 

.33583 

.36905 

-35363 

.33953 

.31831 

.26526 

.21221 

.43656 

.42441 

.40820 

.39347 

.38001 

.36768 

.31937 

.29676 

.25420 

.43907 

-42441 

.41115 

.39909 

.38309 

.37800 

.35221 

.32432 

.28516 

.43428 

.42441 

.41549 

.40737 

.39996 

.39317 

.37086 

.34033 

.33069 

.43057 

.42441 

.41834 

.41378 

.40915 

.40491 

.38054 

- 1 I --- _ .-.__-- --- 

a la w WC 

-~_ 

-.-_c_ 

.64149 

ll25256 

1:42750 

1.54516 

1.62037 

1.66400 

1.68750 

1.56250 

1.25000 

.80754 

1 

1.12958 

1.21358 

1.26434 

1.29082 

1.23982 

1.14541 

.89409 

.90496 

1.:5743 

1.08807 

1.09958 

1.09747 

1.04405 

.92396 

.68527 

1.00864 

1 

.93064 

.95450 

.92423 

.39170 

.75864 

.53657 

.46304 

1.06009 

1 

.94253 

.38821 

.33722 

.78959 

.51104 

-___- -. 

CbXIO 

rl=l 

---- -~. - 

1.74075 1.11688 

1 1.06103 

.43842 1.01051 

.01131 .96458 

-.31392 .92264 

-.56129 .83419 

-.74374 .34883 

-.96322 .79577 

-1.19367 .66315 

-1.10619 .53052 

1.62515 .37046 

1 .32693 

.52404 .73756 

.16024 .75176 

-.11841 .71907 

-.33185 .63911 

- .82729 .57298 

-.90828 .51633 

-.85840 .41347 

1.55533 .65735 

1 .62448 

.57575 .59474 

.25018 .56771 

-.00033 .54303 

-.19329 .52040 

-.54240 .46258 

-.72084 .40005 

-.70375 .31224 

1.48613 .32866 

1 .31223 

.62700 .29736 

.33934 .28384 

.11670 .27150 

-.05950 .26019 

- .43490 .22302 

-.58026 .17213 

-.56043 .15611 

1.47296 .12222 

1 .11611 

.63675 .11058 

.35631 .10556 

.13898 .10097 

-.02931 .09676 

-.52601 .07257 

_ .---. __-. -- 

___ 

e 

-.--- 

.97832 

1 

.93213 

.93798 

.88020 

.81318 

.75753 

2/3 

.47059 

.33333 

.98593 

1 

.98845 

.95916 

.91940 

.37479 

.67296 

.57985 

.43703 

.99045 

1 

.99217 

.97203 

.94407 

.91180 

.31034 

.68977 

.53460 

.99530 

1 

.99615 

.98611 

.97132 

.95430 

.37787 

.74466 

.70123 

.99773 

1 

.99314 

.99325 

.98621 

.97768 

.89638 

-- 

-- 

Di 

D. 1C 

- ----- 

1.13259 

1 

.92348 

.88109 

.85906 

.34877 

.84480 

.84375 

.83008 

.75000 

1.12385 

1 

.91763 

.36164 

.82243 

.79334 

.71342 

.67367 

.57204 

1.11872 

.;1419 

.85023 

.80094 

.76162 

.67712 

.59496 

.46764 

1.11326 

1 

.91053 

.33309 

.77807 

.72732 

.58119 

.40338 

.35652 

1.11056 

1 

.90872 

.33206 

.76672 

.71030 

.43578 
TABLE 9. - Continued. 

--. 

'b 

= 

'bc 

* 

.8 

* 

__- 

. 

~_ -- . _ . 

b/be 

= 

'bc"b 

t 

eq. (59) 

2.0834: 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.60 

2.3683( 

..--.-. 

2.92969 

-.63261 

.:7236 

1.09270 

1.56778 

2.00328 

4.50737 

6.94444 

-._ - _ . . --- . _ 

* Values for wingt i 

QcP 

a /a w WC CbXIO 

n=l 

.36358 

.42736 

.42441 

.42175 

.41932 

.41711 

-41508 

.40341 

.39206 

.32886 

1.08953 

1 

.92073 

.85029 

.78744 

.73117 

.48358 

.21097 

--- 

-.51929 .05573 .81813 .28158 

1.52995 .02211 .99912 1.10706 

1 .02101 1 1 

.59454 .02001 -99928 -90768 

.28288 .01910 .99738 .82861 

.04259 .01827 .99463 .76023 

-.14292 .01751 .99126 .70057 

-.66917 .01313 .95729 .40806 

-.56696 .00887 .90424 .19722 

p zero-: ;l( )pe loading, t of equation (29). 

e 'i - 

D. 

1c 

115 
TABLE 10. - COMPARISON OF AERODYNAMIC CHARACTERISTICS DUE TO OPTIMIZED SPANWISE 

LOADING WITH THAT DUE TO ELLIPTIC LOADING OF EQUAL LIFT AND BENDING 

MOMENT FOR BENDING MOMENTS AT THE SAME yb = ybc SPAN STATION 

nb 

r-l 

= bLbC 

'bc"b 

t ncp I+1 Cbx10 e 

eq. (60) 

Di 

D. 1c 

.l 

* 

.2 

* 

.4 

* 

.6 

* 

.8 

* 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.32306 

1.6 

2.0 

.95 

1 .oo 

1.05 

1.10 

1.15 

1.20 

1.29721 

1.6 

2.0 

.95 

1.00 

1.05 

1.10 

1.15 

1.20 

1.22396 

1.6 

2.0 

2.5 

.95 

1.00 

1.05 

1.10 

1.14423 

1.15 

1.20 

1.40 

513 

.95 

1.00 

1.05 

1.06860 

1.10 

1.15 

1.25 

- 

Same values as those in table 9 for nb = 0. 

-.33009 

.;9820 

.56887 

.81558 

1.04134 

1.52278 

2.32882 

3.08624 

-.37441 

.:3663 

.64065 

.91632 

1.16718 

1.59472 

2.56054 

3.32803 

-.60668 

0 

.53345 

1.00378 

1.41938 

1.78725 

1.94837 

3.56422 

4.19758 

4.34371 

-1.44623 

1.200278 

2.19802 

2.92969 

3.01594 

3.68188 

5.21515 

5.63377 

-7.85059 

0 

5.43663 

6.94444 

8.94496 

10.94770 

12.01966 

- 

.44708 

.42441 

.40394 

.38536 

.36842 

.35292 

.37987 

.26453 

.21253 

.44791 

.42441 

.40328 

.38420 

.36690 

.35116 

.32432 

.26370 

.21553 

.45059 

.42441 

.40139 

.38110 

.36316 

.34729 

.34033 

.27060 

.24327 

.23697 

.45445 

.42441 

.39944 

.37877 

.36358 

.36179 

.34796 

.31612 

.30742 

.46099 

.42441 

.39908 

.39206 

.38274 

.37341 

.36841 

- 

.72796 

1.:881( 

1.31495 

1.39695 

1.44592 

1.47525 

1.33594 

1.05177 

.78247 

1 

1.14664 

1.24194 

1.29987 

1.33051 

1.33796 

1.17554 

.90292 

.84428 

1 

1.09687 

1.15193 

1.17725 

1.18142 

1.17782 

.93689 

.66174 

.43269 

.87421 

1 

1.06621 

1.09150 

1.09029 

1.08889 

1.06752 

.89845 

.65593 

.87840 

1 

1.03720 

1.03627 

1.02159 

.97467 

.84307 

1.76210 

1 

.42334 

-.01429 

-.34668 

-.59876 

-. 98439 

-1.23617 

-1.12976 

1.82513 

1 

.37926 

-.08874 

-.44149 

- .70659 

-1.04383 

-1.33827 

-1.18815 

2.11139 

.;8483 

-.41177 

-.84580 

-1.15809 

-1.27372 

-1.68748 

-1.31633 

-.87735 

2.88793 

-.A0472 

-1.19123 

-1.72164 

-1.77684 

-2.14552 

-2.44520 

-1.89273 

6.34397 

1 

-2.06115 

-2.78482 

-3.62327 

-4.22658 

-3.99032 

- 

.88198 .97767 

.82693 1 

.77720 .98170 

.73206 .93649 

.69092 .87765 

.65327 .81483 

.57298 .67296 

.43856 .46804 

.31224 .33376 

.67717 .97581 

.62448 1 

.57711 .98035 

.53432 .93233 

.49552 .87071 

.46022 .80585 

.40005 .68977 

.26413 .46307 

.15611 .33798 

.35584 .96782 

.31223 1 

.27388 .97494 

.24008 .91658 

.21020 .84604 

.18376 .77608 

.17218 .74466 

.05603 .46566 

.01050 .38587 

0 .36978 

.14592 .94861 

.11611 1 

.09133 .96388 

.07081 .88879 

.05573 .81813 

.05396 .80933 

.04023 .74013 

.00863 .58671 

0 .54883 

.03472 .88079 

.02101 1 

.01150 .93905 

.00887 .90424 

.00537 .85055 

.00187 .79164 

0 .75915 

1.13334 

1 

.92393 

.88250 

.86155 

.85226 

.84889 

.83460 

.74903 

1.13550 

1 

.92521 

.88643 

.86842 

.86175 

.86154 

.84356 

.73969 

1.14487 

1 

.93034 

.90166 

.89375 

.89481 

.89641 

.83887 

.64789 

.43269 

1.16806 

.;4101 

.92986 

.93358 

.93428 

.93827 

.86960 

.65594 

1.25800 

1 

.96590 

.96847 

.97166 

.95516 

.84305 

- 

1 

* Values for wingtip zero-slope loading, t of equation (29 

116 
TABLE 11. - SPANWISE LOADING DISTRIBUTION*, cllc/CLcav, 

FOR CONDITIONS OF TABLE 10 

-._ -..-- _~. 

.2 .4 .6 .8 .9 .96 

(b/be) = 1.20 loading 

1.55424 1.24307 .85438 .44764 .25129 .13233 

1.56977 1.26267 .85617 .43161 .23237 .11722 

1.55757 1.34077 .88791 .38331 .16482 .06040 

1.51905 1.35603 1.01456 .35735 .06778 -.03628 

_- 

.l 

:: 

.6 

.8 

- 

1.49659 

1.48616 

1.46824 

1.45002 

1.41430 

- 

(b/be) = 1.10 loading 

! ~ 

(b/be) = 1.05 loading 

(b/be) = 1.15 loading 

.51621 

.50304 

.46166 

.43089 

.59634 

.31713 .18093 

.30171 .16865 

.24513 .12135 

.15590 .03476 

.12126 -.08487 

[ 

* The loadings for qb = 0, and for wingtip zero-slope loading with b/b, 

of table 10, are the same as those in table 8. 

117 
TABLE 12. - Inn*, INDUCED DRAG INFLUENCE COEFFICIENT FOR <=1/Z, FROM APPENDIX B 

7 n n* 

0 

1 

2 

3 

4 

6" 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 0 0 .000003 0 

17 0 0 0 .000005 

n* 

\ n 

8 

1: 

11 

12 

13 

14 

15 

0 1 2 3 4 5 

0 0 0 0 0 0 

0 .211037 0 -.023182 0 -.000442 

-.126495 0 .107717 0 -.019769 0 

0 -.069557 0 .058029 0 -.013710 

-.017041 

.:02017 

0 

0 -.039669 

-.002698 0 

0 .001243 

.001782 0 

-.:22936 

0 

.001958 

.032101 0 

0 .017975 

-.013384 0 

0 -.007837 

.000806 0 .001031 0 .001766 0 

0 .000244 0 .000516 0 .001378 

-.000008 0 .000026 0 .000243 0 

0 -.000049 0 -.000025 0 .000066 

-.000037 0 -.000023 

0 -.000013 0 

-.000005 0 .000006 

0 0 0 

0 

.000006 

.~00020 

-.000037 

0 

.000022 

0 

.000031 

0 

0 

-.000079 

0 

.000013 

0 

.000036 

6 7 8 9 

0 0 0 0 

0 -.000042 0 -.000382 

.000418 0 -.000639 0 

0 .000550 0 -.000640 

-. 008969 

.:10127 

0 

0 .000549 0 

-.005641 0 .000423 

0 -.003581 0 

.005505 0 -.002419 

-.004694 0 .002362 0 

0 -.003203 0 .000485 

.000900 0 -.002706 0 

0 .000371 0 -.002310 

-.000118 0 -.000040 

0 -.000337 0 

-.000190 0 -.000506 

0 -.000391 0 

0 

.000077 

0 

118 
TABLE 13. - n*+ D,,,* INDUCED DRAG INFLUENCE COEFFICIENTS OF PLANAR-WING 

WINGLET WITH 0, = 5d32, FOR EQUATIONS (172) AND (175), (DATA FROM TABLE 

II OF REF. 9) 

90 1 .03335 .00666 -.00167 -.01243 -.06192 -.10534 

3 -.00333 -.17809 -.16927 -.05646 -.02895 -.06658 

5 -.00258 -.16334 -.09138 .08085 .11343 .03307 

7 .03630 .02441 .11900 .18434 .11512 - .00789 

9 .03156 .12523 .17174 .08391 -.03334 -.07625 

11 -.02442 .06277 .04171 -.11282 -.16338 -.02769 

75 1 .02254 .01464 .00887 -.00752 -.04600 -.06860 

3 -.02833 -.11801 -.09743 -.03368 -.02510 -.04331 

5 -.02362 -.10743 -.05579 .05572 .06810 .01018 

7 .02184 -01044 .07264 .11569 .06939 -.01221 

9 .01822 .08039 .10960 .04817 -.02089 -.03748 

11 -.01661 .04548 .02751 -.07520 -.09798 -.00506 

119 

- __ 
TABLE 14. - Tn, WING ROOT BENDING MOMENT/LIFT 

INFLUENCE COEFFICIENTS OF EQUATION (171) 

51~132 .075139 .184765 .198455 .120903 .009690 -.066188 

d4 .285398 l/2 l/6 -l/6 -l/10 l/10 

120 
Minimization Theory of Induced Drag Subject to June 1979 

Constraint Conditions 6. Performing Orgamzation Code 

7. Author(s) 

John DeYoung 

9. Performing Organizatton Name and Address 

8. Performing Organization Report No. 

_ 10. Work Unit No. 

Vought Corporation Hampton Technical Center 

3221 N. Armistead Avenue 

Hampton, VA 23666 

I2 Sponsoring Asncy Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546 

11. Contract or Grant No. 

NASl-13500 

13. Type of Report and Period Covered 

Contractor Report 

14. Sponsoring Agency Code 

15. Supplementary Notes 

Langley Technical Monitor: Harry H. Heyson 

Topical Report 

16. Abstract 

Exact analytical solutions in terms of induced drag influence coefficients 

can be attained which define the spanwise loading with minimized induced drag, 

subject to specified constraint condtiions, for any nonplanar wing shape or 

number of wings. Closed exact solutions have been obtained for the planar wing 

with the constraints of lift plus wing bending moment about a given wing span 

station. Example applications of the theory are made to a biplane, a wing 

in ground effect, a cruciform wing, a V-wing, a planar-wing winglet, and linked 

wingtips in formation flying. For minimal induced drag the spanwise loading, 

relative to elliptic, is outboard for the biplane and is inboard for the wing 

in ground effect and for the planar-wing winglet. A spinoff of the biplane 

solution provides mathematically exact equations for downwash and sidewash 

about a planar vorticity sheet having an arbitrary loading distribution. 

1. Report No. 

NASA CR-3140 

4. Title and Subtitle 

2. Government Accession No. 3. Recipient’s Catalcq No. 

5. Report Date 

7. Key Words (Suggested by Authafr)I 

Induced Drag Minimization 

Optimum Spanwise Loading 

Loading Constraint Conditions 

NonDl anar WinEi 

Ground Effect-and Velocity Field 

9. %zurity Clawif. (of this report) 20. Security Clasif. ( 

18. Dirrribution Statement 

Unclassified - Unlimited 

Subject Category 02 

this page) 21. No. of Pages 22. Price’ 

Unclassified I Unclassified 124 $6.50 

*For sale by the National Tchnical Information Service, Springfield, Virginia 22151 

NASA-Langley, 1979