NATIONAL ADVISORY COMMITIEE 

FOR AERONAUTICS 

TECHNICAL NOTE 

No. 1194 

EFFECT OF FI!'ITTE SPAN ON rrm: ArnLOAD DISTRIBUTIONS FOR OSCIUATING \'T.INGS 

I - AERODYNf\l\lIC THEORY OF OSCILLATING i-IINGS OF FINI'l'E SPAN 

By Eric Re1ssner 

Massachusetts Institute of Technology 

T(CHNICAl LlBRARV 

AIRF. ,'[ARCH ANUF CTURING 00. 

9851·9951 SEPUlVED 81W 

lOS ANGElES 

CALIFO 

"lashing ton 

March 1947 

• 

https://ntrs.nasa.gov/search.jsp?R=19930082160 2019-08-28T14:02:09+00:00Z
NATIONAL ADVISORY CO~roTTEE FOR J..ERONAt,'TICS 

TECHNICAL rmTE NO. 1194 

EbEC'I' OF FHITTE SPAN ON THE AIRLOAD DISTRIBUTIONS FOR OSCILLATIl'~G InNGS 

I - AERODYNAMIC THEORY OF OSCILLATING' lrJIHG3 OF FINITE SPAN 

By Eric Reissner 

SUtr.iMARY 

A formula is derived for the pres8ure distrib-!.lt:ion on an oscillatir,g HftinG surface of fini tiC; span unc.er the acsumpUon of a ratio of· 

span to average chord (aspect ratio) that is not too small. The ranGe 

of validity of this i'ormula, so far as aspect-ratio l~.mi tat~.om: are concerne·i, is not less than the rt.:0..ge of vali·ii ty of lifting-line theory 

for tho non-oscalating 1.rinG. 

It is found that the effect of three-dimensionali t~r of the flow may 

be incorporatec. in the results of the two-rlimensional theory by adding a 

correction factor cr to the basic function C (1;:) of the t.lo,...dimensional 

i.heory. 

The correction term cr is a function that depends o~ 

:form: .,ing defiection function, and reduced fregD_ency k. 

naticn reCluires the solution of e.n inteGral e1uation .Ihich 

the integral eq,uation of lifting-line theory. 

vring plan 

Its determiis similar to 

The present report cbncludes .Ti th an explici t statement of ~he form 

which the results of the theory assume for the s11amlise varia"i.:,ion:. of 

lift, total moment, and hinge moments on a wing ilhich is oscillating in 

uendin3, torsion, and aileron and tab deflection. 

Methods for the numerical evaluation of' the results obtained as 

well as numerical applications to specific problems are given in part II 

of this report.O 

INTRODUCTION 

1~e present report deals with the linear aero~rnamic theory of 

cscillatj.ng airfoils of finite syan. It contains the outcome of attempts 
2 IifACA TN No. 1194 

to obtain simplifications and extensions of earlier results given in references 1 to 3. The guiding thought in the de-relopments is to deduce, 

frcm a rigorous formulation of' the problem of nearly plane lifting surfaces; simplified results in applicable form which depend on the assUInption of sufficiently large aspoct ratio. It is found that such results 

can be obtained and that their range of applicability, so far as aspectratio limitations are concerned, is at least as inclusive as that of 

lifting-line theory for the airfoil in uniform motion. 

In reference 1 wings of rectangular plan form were considered. and 

expressions were obtained for the spanwise distribution of lift and mo.,.... 

ment on the airfoil for arbitrary, small periodic displacements of the 

points of the wing surface. In reference 2 these results vler0 extended 

to wings of arbitrary plan form. In reference 3 expressions '",ere 0-0""" 

tained, on the basis of results in references 1 and 2, for the spanwise 

variation of aileron hinge moment and tab hinge moment. Moreover, it 

was established that the effect of finite span could be ~ncorporated 

into the results of the two-dimensional theory, as given in referenc.es 

4 and 5, by a modif~cationof the basic function C = F + iG in an 

explicitly specified manner. The term to be added to the function C 

in order to account for the three-dimunsionali ty of the problem 'lora,s 

found to 'be the same for lift, moment, and hinge moments but to deper..d 

on the nature of the motion being dealt with. ])etermination of these 

three~dimensional cor;l."ection terms rec;t1.ires the solution of an integral 

equation for the variation of the circulation along the span. The deriva.tion of this integral eq,uation is an important part of ,the work ~ 

In thj,s report the foregOing results are obtained in what appears 

at present to be thesimp;Lest possible wfiY. In particular, a consider-.-

able reduction in the necessary analysis is accoillplised by showing that 

the eff'ect of finite span manifests i teelf in the expression for the 

chordwise press~re distributj,on solely by a modif:Lcation of the function 

C. Vlith this resalt, use P18.y be made of the known formulas for lift and 

moments of the twq-dimensional theo~J in order tq establish tpe final 

results of the present theory without further integrations. 

SYMBOLS 

u velocity of flight 

x J y, Z Cartesian coordinates 

u+U J Y J v, components of fluid velocity 

t time 
NACA TN No. 1194 

p 

p 

¢ 

'1', a 

x2 (Y) 

Xt(Y) 

b 

bo 

s 

wa 

Za 

pressure 

density 

velocity potential 

discontinuities o~ u and v in the ~J-p1ane 

coordinate of leading edge 

coordinate o~ trailing edge. 

semichord 

semi chord o~ midspan. 

ratio of span to chord at mi.dspan 

velocity cQmponent w at the airfoil 

instantaneous de~lection o~ points o~ wing surface n:easured 

normal to xy-plane 

RaJ Rvl' Rr regions in xy-plane (~irfoil, !ake, !emaining) 

w circ~ar frequency 

r 

K 

y*, Tj* 

z, t 

barring of terms, defined by equation (12) 

circulation per unit o~ span 

ci:rculation ~unction defined by equatj.on (15) 

function defined by equations G9a,'b) and occurri~~~ as 

kernel of the integral equat~on (62) . 

dimensionless spanwise coordinates; y* = ylboJ Tj-l~ = Tj/bo 

dimensionless chordwise coordinates defined by equation . (40b) . 

3 

k 

dimensionless coordinate of midchord line; zm=(x2 + Xt) 12° 0 

reduced frequency; k = w b/u 

k 

o 

reduced frequency at midspan 
4 NACA TN No. 1194 

kzn 

Q function defined oy e~uation (47) 

ill , Az functions defined by e~uations (52b) fu~d (53b) 

J.I. function d.efined by equation C39) 

Q( 2 ~ two-dimensional circulation function given by equat:'on (60) 

I.. variable of integration~ 

s function defined by equation (72) 

C(k) function defined by Theodorsen 

CJ correction term defined by equation (77) 

a location of elastic ~Xi8 in units of semi chord b 

c location of aileron leac'ting edge in units of b 

e location of ''I.lle~:on hi:i.i[)e line in units of b 

d locat:ion of tab leading edge in units of b 

f location of tab hinge line in units of b 

l, m aileron and tab overhang in units of b' ,7. ;: e - c, m = f - d. 

h bending deflection of wings 

a, angle of attack of wings 

13, /' aileron and tab deflection angles 

L lift per unit of span 

Ma moment about elastic axis per unit of span 

M!3' My aileron and tab hi~o moments per unit of span 

F function defined by equation (86) 
NACA TN No. 1194 5 

FORl'-1£JLATION O:l!"' THE FROBLEI"l 

The wing considered is of arbitrarily given plan form and is placed 

in the path of a uniform incompressible air streCWli of veloc1ty U. Small 

crunte:!.', thickness, anele of attack, and deformations are assumed for the 

wing ,\-1hich, for this reason, may be represented by a plane surface of 

discontinuity of pressure and tangential velocity in the fluid, parallel 

to the direction of U. In addition to the surface of discontinuity 

representin.-'3 the wing i teelf, there is admitted the possibiE ty of a 

surface of discontinuity of tangential velocity (but not of pressure) 

which extends downstream from the trailing edge of the wing and. which is 

also taken to be parallel to U. 

Let x and ybe the coordinates in the plane of the surfaces of 

discontinuity, the direction of U determining the x-axis, and let z 

be the coordinate perpend.icular to x and y. Let u + U, v, and. ..T 

be the components of fluid velocity in the x, y, and z directions, 

respectively. Let p designate the pressure in tho fluiq. and p the 

densi ty. With the h:t,gher order terms in the velocities neglected, the 

equations of fluid motion and of continulty bocome 

(a) em em -+U·_= 

dt Ox 

(b) Ov Ov -- + U .~- ::: 

ot ox 

(c) dv~ + U OvT = 

dt ox 

1. ~p"\ 

p ox 

1 £P. I p oy 

1 ~P I 

p dz ..J 

, Exterior to the surfaces of discontinuity the flow is without 

vorticity so that in terms of a velocity potential ¢, 

(1) 
6 NACA TN No. 1194 

(a) u = 91 

dX 

(b) v = ~ (3) d" .) , 

(c) i{ = ~ 

dZ 

From equt:l.tions (3) and (1) an appropriate form of the 13ern'lulli equation 

follol-is for the pressure P J 

L .l- Ou 

(a) 

(q) 

I = 

0 = 

:£60 T 91 

-, ==::.-+U_, 

p at, ox 

( 4) 

(Lu) z=o = ( 6 2f \ dX }z=o 

(5 ) 

(6v)z=0 = ( d¢) 

6 dy z=o 

stand for the discontinuity distributions of the velocity components u 

and v over the surfaces of discontinuity in the xy-plane. According 

.to the Biot-Savart theorem tho velocity component w in the interior of 

the fluid is then given by the integral 

11, t)(x.- ~ ) + o( ~, 11, t)(y - 11) e . . -- d sd11 r( !:)2 ( )2 2\:3 /2 x -s + Y - 11 + z l l 'I· J 

(6) 

From equations (5) it follovs that y and 0 are related to each 

other in the form 

dY CO _ .. - = 

Oy dX ( 7) 
NACA TN No. 1194 

From equaticns (4) 'and. (5)- it follows that the 'discontinuity of 

pressure 6p . is given b;y , 

(8) 

The problem of the oscillating lifting surface may now be formulated. Designate the coo~inates of the leading and trailing edge of 

the wing by Xt(Y) and Xt (y), respectively, with Xt{O) = -bo and 

Xt(O) = bO' so that bC) represents the semi chord at midspan. Let sbo 

stand for the length of the SenU,s:pa.n. Distinguish the follovling three 

regions in the xy-planej (l) the airfoil region Ra, (2) the "(-Take 

region Rw which is the' strip of width 2sbo extepding dOvIDstream from 

the trailing edge, (3) the r~main1ng region Rr • Then there is the 

condition of continuous flow outside the airfoil and wake regions, in Rr , 

and the condition of continuous pressure in the wake region Rw, 

6p = 0 (10) 

At the airfoil the condition of tangenti~l flow is tc be satisfied. If 

Za(x, Y, t) stands for the iristantaneous deflection of points C'f thewing 

surface measured normal to the xy-plane, this condition in the region 

Ra is 

( ) oZa o~a wa = w x, y, 0, t = - + U -, -' o t <1 x 

(11) 

By introduciIlG equa1:.10nS (9) to (11) in the integral representation 

for v1, equation (6): and taking account of equations (7) and (8), there 

may be obtained the general i:p.tegral equation of 'lifting-surface ,theory 

for the distribution of the velocity discontinuities I'a and 0a over 

the airfoU region. \-lith the solution of this integra;L equation, 
8 NACA TN No. 1194 

equation (8) flirnishesthe choi"dwise pressure distribution /:), Pa. Appro·-

priate integratior.; of 6 Pa results in expressions for spanwise lift, 

moment, and aileron and tab hinge-moment distributions. 

Further development'ia here based on the assum;ptioll of simply 

harmonic motion, 

'- iillt Z (x, y, t) c Z (x, y)e ,a . a (12) 

As a consequence of the linearity of the theorY tit follOlfS also that 

velocl ties and pressure become products of eW ) ,and of amplitude functions independent of time which are designated by bars. 

The first step in the deduction of the f:i..nal form of the integral 

equation o~ the problem consists in the determination of the values of 

'Y and 0 in the wake region, by means of equations (8), (rn, (9), and 

(10). :J!;qua,tion (8) takes on the form 

Xt x 

iill ( Ira l') 

0 = dx' - dx' +,/ Y"T \ 

:r~, Xt 

If the circulation r is defined by 

:-:t 

r = r Ya dx' j 

xl 

and the circulation function ("2 by 

1 . ill - ). - Xt 

("2 = - r e U 

b o 

) + UYw (13) 

(14) 

(15) 

1 t follows from equation (8) and its differentiated form that 'Yw is 

given by 
NACA TN No. 1194 9 

(16) 

\.-lhen "lw of equation (16) is substituted in equation (7), there is obtained for ow' 

x 

Ow = /' 

) 

-co 

,,_ '. m 

Of dO -1:-· x 

)..: dx I = b o - e U 

u,y dy (n) 

Equations (17), (16), and (11) are nOvT introduced in the integral 

representatj.on for l.>T, equation (6), in which the coordinate z is 

mad_6 to approach zero. As '-las shown in reference 1 J the two liIni t proc--

esses of integration and of letting z approach zero may be interchanged 

if the Cauchy principal value of the integrals is taken at the paints 

t = x, -" = y for '-Il~ich the integrand becomes infinite. There results 

m-()( ') dQ( ) -i U D T) x - ~ + an- Y - T) t 

r- ' - :3 /8 d., d" I (x - ~ )2 + (y - 1'])2 I 

L ~ 

(18) 

E~uation (18) holds for all points x, y inside Ra' Its left-hand 

side is the given function 

Wa = im 

and, according to equations (7) and (9), 

fa by means of the formula, 

° is expressed in terms of 

a 
10 NACA TN No. 119~ 

_ [ Xa7a "0" = - dx' 

a a"y . 1 

(20) 

In addition, there is imposed the Kutta-Joukowsky condition stating 

that rand 5" are finite alopg tbe tr~iling edge of the wing. The 

amplitude of the chordwiae pressure distribution ~ Pa in terms of the 

function ra , for which equation (~8) is to be solved, is given by 

dx' + U ra (21) 

The general problem is the solution of equaUons (18) to (20) for 

arb~trary shape (plan form) of the airfoil region Ra. Results equivalent to this solution are kn01'1n for the following two special cases: (1) 

the two-dimensional problem to which the general problem reduces when 

the region Ra is the strip I x I ~ b = bo and wa of equation (19) 

is independent of y (see for instance ref~rence 5), and (2) the problem 

of the wing with circular plan form (referooces 6 and 7). 

The purpose of the remainder of this report is to obtain approximate 

solutions of the problem which are applicable subject only to the restriction that the airfoil region Ra is of sufflciently large aspect 

ratio, a restriction that is roughly equivalent to requiring that the 

ratio s of span to chord at midspan is suffiCiently large. The nature 

of these results is, presumably, that of the dominant terms of an asymptotic development of the exact solution in terms of the parameter s. 

HOvlever, no investigation of this aspect of the problem is made in the 

present report. 

Concerning the range of validity of the results as far as aspectratio limitations are concerned the statement may be made, in view of 

the nature of the analySis, that the results obtained in this report for 

OSCillating wings are certainly applicable for all wings for which 

lifting-line theory is considered applicable in the case of uniform motion. There is some reason to believe that the results for oscillating 

wings may have a somewhat wider range of validity than lifting-line 

theory, as the chordwise waves which occur in the vibration problem may 

be responsible for a reduction of the effective chord length while at 

the same time having no influence on the e~fective span length. 
NACA TN No. 1194 11 

DERIVATION OF SIMPLIFIED INTEGRAL EQUATION OF LIFTING-SURFACE THEORY 

Equation (18) may be v~itten in the form 

(22) 

where 

(23) 

(b) 

(l) (l) - -, 

-i- ~ -1 U 0 (1)) (x ~ e) + n (1) ) (y - 1)) 

e U [ (x _ ( ) e + (y _ ~) 2 r I a

d

;.i>l 

and where, from equation (20) J 

_d _Y_a_( e_'_, -..,..11_) d ~ I 

d 1) 

The decisive step in the present solution of the problem as expressed by equations (22) and (23) is the reduction of the twodimensional integral equation to 2 one-dimensional integral equations 

which can be solved successively. For this reduction it is assumed. that 

the airfoil region is sufficiently elongated in span direction, that the 

rate of taper is moderate, and that the velocities do not vary· appreciably along the span over distances which are a fraction of the mean 

chord. With these assumptions it may be po~tulated that at every spanwise section the contribution to the normal velocity wa in~uced by the 

tangential velocitiee at the airfoil is approxi~tely as if every 
12 NACA TN No. 1194 

spanwise section \OTere part of a. wing· wi tho~t taper and as if the flOi.,r 

'vere two-di!IlSnsional. Equ$~ion (23a) then becomes 

A corresponding approximation -12 for the integral I2 is ob···-

tained by a less. direct procedure. Hri te 

where 

-i ~ ~ 

r- e U ~ [2 '_l!l.2lx-=~ d ~ d1) 

I ( )2 ( )2 13 /a l x -- t + y - 1) J 

(26) 

( 2'7) 

Write· next 

(28) 
NACA TN No. 1194 13 

In the integral extended fram x to Xt(~), that is, over part of the 

airfoil region, the same approximation can be made as in the expression 

for I 1 • If this is done, :l.ts contribution to I vanishes and 

3 

sbo CI' --i ~~ -, n f e U 0 (~)(y - T)) 

I3 :::: .f '"' (y _ T))2 ]3/2 d~ dt) 

I 2 l (x -;) + - sb x 

0 

. Wi th a new variable of integrat.1.on ). = ~ - x 

ill CD --i -'- ). ( r e 11, (y - 11) d'. 1. . 1 r,2 ( )2J-3/2 J

. o.T) 

.~ Vo l '" + y - ~ . 

In 9rder to approximate the remaining integral 14 this integral is 

separated into t'VlO pa:rts, one part which equals the value of 14 in the 

t,.,o-dimensional theory and the other part being the difference between 

the bTo-dimeneionaJ., and the three-dimensional value of 14 , 

(30) 

...,1 ?:. ~ e U (x - ~ ~d~ dT] 

(x - ~) + (y - ~) [ 

2 2J3/2 

• ill ~ CD -1 -- !, 

• u = 2 0 (y) J e t d t (x - ) 

~ Xt(Y) 

(31) 

The factorO (y) l1J.a0T be ,vritten in the alternate form 
14 NACA TN No. 1194 

o (y) .L 

, j~O -, Iy- ,1)1 = - o (1)) d1) (32) 2 Y - 11' 

-sb 0 

- if account is taken of the fact that 0 (±sbO ) = O. 

If equations (31) and (32) are combined, the value of 14(.2) may 

be written 

ill .t " , --1 -. ~ -, U' 0 (1)) IY-1)1 ded 

e ---!'T) 

(x- t) (Y-1)) 

The next step is transformatlon of 14 by integration by parts 

with respect to 1). Before this step is carried o~t it is noted that it 

is consistent with the preceding approximations to replace in equation 

(27) the limit of integration xt (1)) by xt (y) . Then 

sbo 

-i ~ ~ [ t 

e 'U (x - ~) l.J 

. sb r 0 

x L! (x-

-sbo 

. ill e -]. "'7'" !, 

e tJ 

'-sb 

o 

(x - ~ ) 

0' (1))(Y - 1)) d1) ;~ ~ 

0 2 J (x - ~)2 + {y - 1))2..i 

as the integrated portion of the inner integral vanishes because 

( 34) 
NACA TN No. 1194 

By combining equations (34) and (33) there fo1lol-1s 

r y - 1') 

x l -,===::;:::.:=:======== /(x - t )2 + (y -'1)2 

fi' ,CIJ) e -i U ; 

x:- e 

15 

Equation (35) for /:), 14 may aga.in be separated into tw'O integrals, :;'n 

the sa.me manner as 13 of equation (28). If this sepa:cation is made, 

the integral between the lirni ts x and xt (y) may again be neglected 

with the result that 

e -i ~ e fi'.ill 

x - e 

and wi th >.. = ~ - x 

6.1 . 4 

ill sbo 

:: e - i fj x r 0' ( 1') ) 

.jb -So 

(36} 
16 NACA TN No. 1194 

Now} by combining equaMons (36) and '(31), there follows: 

(Xl 

x[! 

o 

.., 

= I 

4 

1 ( Y-T) 

~ j 2 ~2 + (y - T]) 

From combination of equations (37)} (29)} and (25) there follows 

finally as an approximate value of the wake integral 1 2 } 

where 

CJ.) 

e -i U ~ d ~ 

(x - ~ ) 

• CJ.) sbo 

__ e -1. U X Jr. dO K [ ~ (y - T]) J dT] 

4rc. dT) U 

-sb 

o 

, txl 

K(x) = ~bof u. 

o 

-H. 

e x ----------~ dA 

2 2 3/8 

[A + x ] 

-iA 

_e A- ( -j:::~=2=X=+=X=2:"': 

(38) 

,I : I ) d,A (39a) 
· .\ .' ;. 

NACA TN No. 1194 17 

For the appltcations it is convenient to transform K(x) by an integratien by parts of the fir~t integral as follows: 

00 

.r 

o 

n 

+ i ,.J 

o 

The right ride assumes a denni te value ~nl;y if 

c(x) 

and then 

00 

(. e -i~ d}~ 1 

j _._------ = -- + (t.. 2 +x2 )3/B 2 

X 

0 

i 

-2 

= -x 

!XI 

n 

/ 

./ 

~ 0 

-no 

e 

X 

2 (-,,/-t..2 A.+ ~2 -l 

By introd.ucing this formula in equation. (39a) there follows: 

mbo 1 

K(x) ::; -- -- - i 

U x 

Ixl) dA. 

) dt.. 

(39b) 

Combination of equations (38), (24),and (22) results in the approximate integral equation of lifting-surface theory, which is the basic result of the pl'esent work. Because the establishment of this eCluation 

depends on the condition of a ratio of vring span to average wing chord 

that is not too small (so that the vrlng plan form has the appearance of a 

strip), it might be considered to call this equation the integral equation of lifting-str~p theo17 to distinguish it from the general equation 

of l:i.f~i:ng--surface 'theory on the one hand, and from lifting-line theory 
18 NACA TN No e' 1194 

for the stationary sufficiently elongated ~ing on the other hana. That 

it is not possible to speak of a lifting-lirie theory for the oscillating 

wing follows from the fact that according to equation (38) and (22) the 

effect of spanwise variation of circulation is, hot Uniform across the 

chord but varies in a manner depending on the frequency w. 

In order to obtain a normalized form of the eqUations of the problem, the following dimens~onless variables may be introduced: 

( 40) 

(b) 

As the space coordinate z does no longer occur, from equation (18) on, . introduction of a dimensionless variable z at this stage will give no 

rise to doubts as to its meaning in the subsequent developments. Write 

as abbreviations 

( 41) 

(b) 

The quantity zm 1s a measure of the local sweep of the midchord line. 

TIle quantity b stands for the magnitude of the local chord 80 that 

b(o) = boo "lith these symbols equation (40b) becomes 

Also let 

wb k = --U' 

( 42) 

( 43) 

where k is the local reduced frequency and ko the reduced frequency at 

midspan. 
/ 

NACA TN No. 1194 ,& ..l.::;p 

..... -.. 

Upon introducing equations (40)" to (4-3) inequatiorts (38), (21+), and 

(22) there follows as the normalized form of the integral equation to be 

solved 

"-, i' 1-' l: Y

a (t 'j y*) 

-wa (z, Y':~) = 2 1f..J z - t d ~ 

-1 

iko -ikm - - e O(y*) 2 'It 

-ik . k- s_. 

+ _e __ ... z e -L"m! • d~_ ) K k r (y* -- 11')(-) J drj':< 

4 rc • dij ,'I- L 0 

(41~ ) 

-8 

Equation (44) has ftrst oeen established in reference 2. For the 

wing of rectangular plan form (k = ko' kili = 0) it reduces to an equation given in reference 1. ' 

'l'he problem from' here on is to solve equation (44) fo!: the funct:i.on 

"I a' to obtain an equation for the spanw"ise variation of n) and to 

express the chordl·!ise pressure distribution in terms)f wa' 

In terms of t~e :variable' ~ as defined by equation (42) the function n of equations' (14) and (15) is to be written as' 

n (y*) 

-1 

and equatio~ (21) for 6Pa becomes 

, So _a 

--= 

!3 U 

z 

ik ,/ I>a (\z' + )' a ( z, y<') 

..... 1 

( 46) 

To be established is the manner in which the presence of the term 

containing dO /dT/* in equation (44) modifies the result of the twodimensional theory for 6Pa' 
20 NACA TN No. 119!t 

CALCULATION OF PRESSURE DISTRIBUTION· 

The next Eitep in the analysis is the d~termination of 'fa from 

equation (44) in which the following· abbreviation is .inserted: 

s 

Q = ~ r ~~,K[ko(Y* - ~*)] d~* (47) 

-8 

Use is made of the following pair of formulas of integral-equation the~ry: .' 

finite 

-1 ( 48) 

f( z) 

It may be noted that a derivation of the results of the two--dimens:i.onal 

.-.theory in this lllB!lJ.'ler has been given in reference 8. 

By applying equations (48) to equaMon (44) there follows, 
NACA TN No. 1194 21 

00 

-ikA. . ,. e ,\ d'~ --- QA. ,-- ~ - A. / z - ~ 

Q -i~ --e 

2 1t 

The double integral in equ,ation (49) is reduced to a single integral as 

follm·rs: 

00 

=/ 

1 

= 

~ 

z - S 

1 

-ikA. I i /~ d ~ e L.1 / l-~ (S-A.)(z-~) -1 

-ikA. r 1 

e i ----

L z - A. 

1 

f //f:i(-,~). + 

-1 

-ikA.,. / ----- e .! I).. + 1 ---- ! 1£ \ 1 - 1 __ = f l _ Z - A. L \ I ),,-1 

By introducing equation (50) in equation (49)) 

" 

I dA. 

j 

1 \ -, 

/

) d ~ J' d.A. 

z - t 

( 50) 
22 NACA TN No. 1194 

- r =._gjl~Z (_drf ~ we dt 

a. 1t 1 + Z j j ~~ z - ~ 

-1 

Q -ikJn .-- e ( 51) 

2 1t 

-1 

According to equations (46) and (45) it is necessary. to integrate 

equation (51) between the limits -1 and z, 1.fith z' as the variable 

of integration. B.r interchanging the order of integration with respect 

to z', t, and A, respectiv~ly, the following integrals occur (see 

reference 8): . 

Z 

I/~ ::: t d:' z' = ~ + sin1 z+ I ~ : 1 A, (z, tl (52a) 

z 

--- = - + sin-1 z + . - 11.2 (z, )..) /

r,JS-z' dz' 1t /Bl 

., 1 + Z, A - Zl 2 ).. + 1 

-1 

The functions A are defined by n 

(53a) 

__ ! Zn 1 - zt "'" J 1 - t

a j 1 - Z2 (52b) 11.1 (z, t ) t J 2 1 - z - j 1 _ tal _ Z2 

A ( z, )..) = 2 tan -1 ( I (1 - z)().. + 1) \ -1C ( 53b ) 

2 (1 + z)().. - 1) ) 
NACA TN No. 1194 23 

The following formulas will also be needed: 

( 53c) 

z 

/' 

-1 

'Hi th equations (52a) and (53a) there follows from equation (51)) 

"fa dz' o ;?o /[/ ~ : 1 G + 

-1 

..,...1 

sin 

-::0 n e -iIo. ./[) t : ~ ( ~ +Sin1 z ) dB J e -ikA d, 

1 

1 

- ~ e -iIo. I· [ ) i : i sin z + -1 ) 

-1 

Equation (54) lead~ immediately to the integral equation fer the 

spamTise variation of r... Let z::: 1 and, there fOllows) in view of 

equation (45) and in view of the fact that Al (l, ~) = 0 and 

Aa (l) >.) = -1{) 

( 54) 
24 NACA TN No. 1191.~ 

( )8+ l' \) -ik~ ---1 e d"- "--1 . 

(55) 

Equation (55) can be s1mplified by DlBllnS of tho following relations 

of the theory of Bessel functions: 

(a) r ( fi+1_1\ ~ \ j~ ) 

1 

1 

(b) /) ~1 + h -ikh f' 

/ 

---e d!.= ~. 1 - r, 

-1 

nence 

1 

- b i (k+km) t /1 + ~ (2 = -2 - e / t wa d~ bo ~. 1 -

-.1 

-ik 

e 

ik 
NACA TIi No. 1194 25 

and by canceling t't·lO of the 

Define now a function 1.l. by the relatj.on 

and write as an abbreviation 

(60) 

Equation (58) then assumes the form 

- b - o (y*) + ~(k) --. 2Q(y*) 

bo . (61) 

Equation (6~) is the integral 6

y

uation for the spanwis6 variation of 

c'~i'l;"uJatioh. The function 0(2 is the distribution of (2 according· 

to the two-dimensional theory. The Q-term represents the influence of 

finite span. "lith equation (47) for Q there may be written 
26 NACA TN No. 1194 

s 

(2 (y*) + ~(k) b· f ~ K[ k (y* - 1)*)] d1)* = n(a) (y*) (62) boo Q.'r}i:- 0 

-a 

The next· step in the analysis is to subtract oqustion (55) multiplied by ! ( 1! + sin -1 z) from equation (54!): 

1t 2 

z J Y

a dZ'-; (~ + a1n-1 Z ) ~o .-1(k+km) Q 

-1 

iko - -iIsn --Oe 

1t 

<Xl 

J 

1 

1 

(

1t • -1 2" + s~n z 

- ~ .-111m J A, (z, t ).-1k t d t 

-1 

(63) 

The second and third integral in the right-hand sia.e of equation (63) 

are now integrated by parts, 
NACA TN No. 1194 

x { [ .!!. + sin -1 z + A':) (z, A.) 1 

2 r. . J 

CI} 

-ikA. 

e 

--- -ik 

ik - -ikJn J' 0 A a._ e -ileA. - -..2. (2 e --- dA. 

1C • dA. -ik 

1 

CI} 

! 

J 

1 

( A 

e

-ikt Q -iltzn - -2' e 

1C 1 -ik 

-ik~ 

e d~ ) 

-ik 

According to equations (520) and (53b), 

1\.1 (z, ± 1) = 0 Aa (z, 1) = 0 1 

! 

( ( 1C • -1 \' A z, 00 ) = - " - + Sln z ) I 

a ·2 / J 

The third of these relations is verified by means of the identity 

12 -1 1 - -Z 1\ • _.1 2 tan -- = -- - Sln z 

1 + z 2 

27 

( 64) 

(65) 

By means of equation (65), and by noting that b Ib = k Ik, it is 

o 0 

seen that the integrated terruB in equation (64) cancel and 
28 NACA TN No. 1194 

(66) 

Now by combining equations (66) and (51), as in equation (46), the!B 

follo,.,s for the pressure distributions on the airfoil 

- 6!U a = - ~ f '( A : ~ J~::t -J 

ex:> 

l

' cj§-Z/)"+l 1 OAs)e-ik).,d)., i-:-;- ).,-1 z-).,-~ . 1 

Equation (67) simplifies considerably when equations (52c) and 

(53c) are taken into account. There follows 

( 67) 
NACA TN No. 1194 29 

1 " 

6

Pf1 =g!CP!? }1 + t 1 - ik Al ) d~ pU 1f • 1 + z 1 - t h vFa 

z -

.,-1 

CO 

ik . ~ .f -ik}" o n -lkm 1 ,- z e +- e -- d}" 

1t 1 + Z j'}..2 -1 

1 

(68) 

Equation (68) contains the noteworthy result that the term with 0 and 

also the term with Q lead to chcrdwise pressure distributions which 

are both proportional to j (1 - z)/(l + z). Note that according to 

equation (44) these two pressure terms are caused by downward velocities 

, co 

across the wing of the form .J e -ik~ (z - U-· d t 

1 

and -ikz 

e The 

fact that these two different velocity distributions lead 'to the same 

simple pressure distribution is here arrived at by an analysis which 

does not reveal the inner reason for this occurrence. A modification of 

the analysis so as to clarify this point appears to be 'Worth whiJ.e, par'-

ticularly as the fact itself accounts for the relatively simple form in 

"'hieh the aerod.Ynamic span effect modifies the expressions for the air 

forces and moments of the two-dimensional theory. 

In order to obtain the final form of the expression for ~ Pa, use 

the follo'inng known formulas of the theory of Bessel functions: 

(a) .f 1 

1 

(b) I .f' 

'-1 
30 NACA TN No. 119'4 

- and express the quantity Q in terms of n by means of equation (61), 

-(2) - - n - n Q = ...u 

2J..l(1;:) k/ko 

(70) 

Then write 

1 

:~a =;; ! (J ~ : ~ ) ~~ t z = t - ikA 1) Wa d t + JF:: s 

(71) 

where 

-ll1n (kO ( 2) - 1,. ' .. ,. ')' S=e -H O+-,J~ 2 0 1( 0, 

) ] 

)J (72) 

-(2) Introduce the value of 0 from equation (60) and the value of J..l' 

fr~m equation (59). Then 
NACA TN No. ·1194 

Introduce in conformity with reference 4 the function 

Equation (73) then becomes 

31 

)

1 + ( - t l=T wa d ;.> 

)J 

S = -; [' Ii: ~ Wadb [c - ).+ ( n~) ., ~ ) (c + J

o

i 

_J~J) ] (15) 
32 NACI,\ TN No. 1194 

By introducing equation (75) in equation (71) there follows: 

6Pa 2 -=-

pU 11; 

.. n.~ ( /11 -+ Zz J1;1 1 ) j l=-i z _ t - ik Al wa d t 

-1 

X ~ 

J1 + z 

C + ) 1 

~ r --- / . JL:Ll wa d t 

'. 1 - t 

-1 

(76) 

Equation (76) is the general result to be established. A corresponding formula for the two-dimensional theory is g1 ven in reference 8. 

Equation (76) shows that the aerodynamic span effect in the present 

theory manifests itself. solely by a mod.i:fication of the bastc function 

C(k). An additive correction term occurs, which is given in the form 

(77) 

and which is seen to depend on the!:atio of three-dimensional to two--

dimensional circulation function O. In order to evaluate the correction term for a given'fing def;Lection function Za, it is then 

necessa17 only to solve the integral equation (62) for the spanwise 

variation of circulation. 
NACA TN No, 1194 33 

LIFT AND MOMENTS FOR BENDING, TORSIONAL, AILERON} AND TAB DEFLECTION 

For the applications (see reference 10) the following deflection 

functions are of particular interest: 

Bending deflection: Za = h(y) = h~h(Y*) 

Torsional deflection: Za'= ~(y)(x - ab) = ~(x - ab)fa(y*) 

Aileron deflection: 

Tab deflection: 

Za = ~ (y ) (x - e b) 

Z = l(Y)(x -'fb) a 

for cb < x < b '" ..... 

for db:::: x :s b 

Lift ar..d. moment functions according to the two-dimensional theory may be 

found in reference 5. Modified so as to include the,c1erodynamic span 

effect these functions can be written in the following form} as was 

shown in reference 3: 

+11: { ~k + ak

2 

+ [1 + (~- a) ik J [c + (Ja II a 

\,"'- 2 2 _ ..1J 
34 

( ! + a) ik rC + cr 11 ~ 2 L hJjb 

NACA TN No. 1194 

+ • (~k Aae -:2 A,., - ( ; + a) [1 + (; - a ) 1k J [c + a" J} Ci 

+f ~ [B." + ikE." - k2 B", J '-.2 13 12 11 

(80) 
NACA TN No. li94 

1 

-I- -

2~ 

The terms A, B, and D are defined'in reference 5. The terms E are 

of the form 

El = T10 - IT21 E1(d) = T1o(d) - mT21 (d) ") 

E2 = Tll - 2lT1O E2 ( ct) = Tll (d) . .., 2mTlO( d) ~ (82) I 

E3 = T12 - 2lT2o Ei d ) = T12 (d) - 2mT2 o(d) J 

wi th the terms T also defined in reference 5. 

by 

The terms OJ(j = h, a, ~,r) as defined by equation (77) are given 
36 , NACA TN No. 1194 

(83) 

The functions OJ(2) of the tw<? .... dimensional theory as, defined by equation (60) are given by 

n (~,) 

h 

i.r-m 

b 4iCe [ ( 1 ) J = bo "';"kH;';";1~(-2-) (-k-) . 1 + 2" - a ik _ ex: 

-

(81\- ) 

The functions n j of the three-dimensional theory a:r-e, according 

to equations(62) and (39b), the solutions of the integral equation, 
NACA TN ~~. 1194 

s 

-ik ,r oJ 

-s 

The f'imc ti on J.L 

I-L(k) = 

The function F 

37 

is defined by equation (59) as 

is, according to equation (39b), 

00 

F(x) ./ (86) 

o 

This function occurred previously in reference 9, where a different 

theory of' the problem of the oscillating wing of finite span was put 

fo~{ard. A discussion of the theory of reference 9 is Biven in reference 1. Tables 1 and 2 contain values of the tl-TO functions I-L and F 

for a significant range of the variables k and x. 

It is apparent that the main task in obtaining three-dimensional 

corrections to the results of the t~o-dimensional theory consists in 

solving the integral equation for r;;. The second part of this report 

(reference 10), i{hich deale with applications of the theory, contains a 

practical method for doing this. 

~-1a.asachusetts InstHute of Technology, 

Cambridge, Mass., December 3, 1945. 

, 
NACA TN No. 1194 

REFERENCES 

1. Reissner, Eric: On the General Theory of Thin Airfoils for Nonunifonn 

Motion. NACA TN No. 91t6, 1944. 

2. Reissner, Eric: The Equations of Lifting-Btrip Theory for Tapered 

Hings. Report SB-76-S-1, Res. Lab., Curtiss--Wright Corp., 1943. 

3. Hunter, Maxwell "",., Jr.: 

in Flutter Analysis. 

Technology, 1944. 

Calculation of the Aerodynamic Span Effect 

M. S. Thesis, Massachusetts Inst. of 

4., Theodorsen, Theodore: General Theory of Aerodynamic Instability and 

the Mechanism of Flutter. NACA Rep. No. 496, 1935. 

5. Theodorsen, Theodore, and Garrlck, I. E.: Nonstationary Flow about a 

Wing-Aileron-Tab Comb:tnation including Aerodynamic Balance. NACA 

Rep. No. 736, 1942. 

6. Schade, Th.: Theory of the Oscillating Circular Arc Airfoil on the 

Basis of the Potential Theory. pt. I. Luftf'ahrtforschungi vol. 

17, nos. 11-12, Dec. 10, 1940, pp. 387-400. (Translation) 

7. Kriene s, H., and Schad,e, T.: Theorie der schwingenden Krei sformigen 

Tragflache auf potentialtheoretischer GrundlaGe. Pt. II. 

Luftf'a~rtf'orschung, vol. 19, 1942, pp. 282-291. 

8. Schwarz, L.: 'Berechnung der Druckverteilung einer harmonisch sich 

verformendsn Tragflache in ebener Stromurlg. Luftfahrtforschung, 

vol. 17, Dec. 10, 191+0, pp. 379-386. 

9. Cicala, P.: Comparison of Theory with Experiment in the Phenomenon 

of Wing Flutter. NACA TM No, 887, 1939. 

10. Reissner, Erlc, and Stevens, John E.: Effect of Finite Span on the 

Airload Distributions for Oscillating Wings. II - Methods of 

Calculation and Examples of Application. NACA TN No. 1195, 1947. 

lAvailable for reference or loan in the Office of Aeronautical 

Intelligence, NACA, Washington, D. C. Also pts. I and II (references 6 

and 7) will be published as NACA TM No. 1098. 
NACA TN ~10. 1194 39 

---~ -- - - - -- - - - - - -- --- - -- - ------ - - --- -- --- -- ---

TABLE I.- VALUES OF THE FUNCTION F OF EQUATION (86) 

F(x) X X - li'(x) 

0.00 00 - 1.57li 2.2 0.113 - 0.425i 

.05 2.750 - 1.468i 2.4 .097 - .3951 

.10 2.109 - 1.3751 2.6 .oB3 - .3691 

.20 1.490 - 1.2481 2.B .072 - .3451 ~30 1.155 - 1.1461 3.0 .063 - .321-11 

.40 .935 - 1.0601 3.2 .055 - .3051 

.50 .778 - .987i 3.4 .048 - .28-91 

.60 .65B -:- . 922j . 3.6 .043 - .2741 

.70 .561~ - . B65l 3.8 .039 - .2601 

.80 .489 - .813i 4.0 .035 - .2481 

.90 .427 - .7681 4.2 .031 - .2371 

1.00 .376 - .7261 4.4 .028 - .2261 

1.10 .333 - .6891 4.6 ~026 - .2171 

1.20 .297 - .6541 4.8 .024 -':- .2081 

1.30 .265 - .6241 5.0 .022 - .2001 

1.40 .238 - .5931 5.2 .020 - .1921 

1.50 .214 - .5671 5.4 .018 - .1851 

1.60 .194 - .54-01 5.6 .017 - .1791 

1.70 .;1.'76 - . 517i 5.8 .016 - .1721 

1. So .160 - .1~96i 6.0 .015 - .1671 

1.90 .146 - . tr'"(5i ex> 1/U~x2 ) 1/;L 2.00 .134 - .4581 

TABLE II.- VALUES OF THE FUNCTION I-L OF EQUATION (59) 

k I-L(k) k l:(k) 

0.000 0~5000 - O.OOOOi 0.650 0.2139 - 0.06651 

.020 .4810 - .04231 .700 .2062 - •. 06101 .01~0 .4607 - .06651 .750 .1991 - .0557t 

,060 ,4410 - .08291 .800 .1924 - .05071 

.0Bo .4226 ... .0942i .900 ,1801 - .0413i 

.1VO .4051 - .10i81 ),,000 .1688 - .03291 

.125 ,3857 - .10821 1.250 .1436 - .01591 

.150 .3690 - .11201 1.500 .121B - .0042i 

.175 .3522 .. - .1130i 1.750 .1027 + .00311 

,200 .3393 - .11391 2.000 .0864 + .00661 

.225 .3268 - ,11321 2.100 .0807 + .00701 

.250 .3154 - .1116i 2.150 '.0780 .+ .00721 

.275 .3049 - .10991 2.200 ~0754 + .0072i 

.300 .2955 - .10761 2.250 .0730 + .0072i 

.350 .2787 .- .10231 2.300 .0706 + .. 0070i 

.400 .2644 - .09641 2.350 .0684 + .0069i 

.450 .2519 - .09031 2.400 .0663 + .00661 

.500 .2408 - .084(.21 2.480 .0632 + .0061;i. 

.550 .2288 - ,07751 2.51~ .0610 + .0057i 

.600 .2220 - .07221 2.600 .0591 + .00521